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Full text of "Lectures On The Theory Of Functions Of Real Variables Vol II"

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OSMANIA UNIVERSITY LIBRARY 

Call No. 3/7- Sy /V? / Accession No. // S ^ 2 



Author 

Title 



This book should be ret,urnd orvor befoe the date last marked below. 



LECTURES 

ON 

THE THEORY OF FUNCTIONS OF 
HEAL VARIABLES 

VOLUME II 



BY 

JAMES PIERl'ONT, LL.D. 

PKOFESSOK OF MATHEMATICS IN YALE UNIVERSITY 



GINN AND COMPANY 

BOSTON NEW YORK CHICACiO LONDON 



COPYRIGHT, 1912, BY 
JAMES P1ERPONT 

ALL RIGHTS RESERVED 

PRINTED IN THK UNITED STATUS OF AMERICA 
926.1 



Cbe fltbcnicure grc 

GINN AND COMPANY PRO- 
PRIETORS BOSTON U.S A. 



TO 

ANDREW W. PHILLIPS 
THESE LECTURES 

ARE INSCRIBED 
WITH AFFECTION AND ESTEEM 



PREFACE 

THE present volume has been written in the same spirit that 
animated the first. The author has not intended to write a 
treatise or a manual ; he has aimed rather to reproduce his uni- 
versity lectures with necessary modifications, hoping that the 
freedom in the choice of subjects and in the manner of presenta- 
tion allowable in a lecture room may prove helpful and stimulating 
to a larger audience. 

A distinctive feature of these Lectures is an attempt to develop 
the theory of functions with reference to a general domain of 
definition. The first functions to be considered were simple 
combinations of the elementary functions. Kiemann in his great 
paper of 1854, " Ueber die Darstellbarkeit einer Function durch 
eine trigonometrische Reihe," was the first to consider seriously 
functions whose singularities ceased to be intuitional. The re- 
searches of later mathematicians have brought to light a collection 
of such functions, whose existence so long unsuspected lias revolu- 
tionized the older notion of a function and made imperative the 
creation of finer tools of research. But while minute attention 
was paid to the singular character of these functions, practically 
none was accorded to the domain over which a function may be 
defined. After the epoch-making discoveries inaugurated in 1874 
by G. Cantor in the theory of point sets, it was no longer neces- 
sary to consider a function of one variable as defined in an in- 
terval, a function of two variables as defined over a field bounded 
by one or more simple curves, etc. The first to make use of this 
new freedom was C. Jordan in his classic paper of 1892. He 
has had, however, but few imitators. In the present Lectures the 
author has endeavored to develop this broader view of Jordan, 
persuaded that in so doing he is merely carrying a step farther 
the ideas of Dirichlet and Riemann. 

Often such an endeavor leads to nothing new, a mere statement 
for any n of what is true for n = 1, or 2. A similar condition 

v 



vi PREFACE 

prevails in the theory of determinants. One may prefer to treat 
only two and three rowed determinants, but he surely has no 
ground of complaint if another prefers to state his theorems and 
demonstrations for general n. On the other hand, the general 
case may present unexpected and serious problems. For example, 
Jordan has introduced the notion of functions of a single variable 
having limited variation. How is this notion to be extended to 
two or more variables ? An answer is far from simple. One was 
given by the author in Volume I ; its serviceableness has since 
been shown by B. Camp. Another has been essayed by Lebesgue. 
The reader must be warned, however, against expecting to find 
the development always extended to the general case. This, 
in the first place, would be quite impracticable without greatly 
increasing the size of the present work. Secondly, it would often 
be quite beyond the author's ability. 

Another feature of the present work to which the author would 
call attention is the novel theory of integration developed in 
Chapter XVI of Volume 1 and Chapters I and II of Volume II. 
It rests on the notion of a cell and the division of space, or in fact 
any set, into unmixed partial sets. The definition of improper 
multiple integrals leads to results more general in some respects 
than yet obtained with Riemann integrals. 

Still another feature is a new presentation of the theory of 
measure. The demonstrations which the author has seen leave 
much to be desired in the way of completeness, not to say rigor. 
In attempting to find a general and rigorous treatment, he was 
at last led to adopt the form given in Chapter XI. 

The author also claims as original the theory of Lebesgue 
integrals developed in Chapter XII. Lebesgue himself considers 
functions such that the points e at which a <f(x) < 6, for all a, b 
form a measurable set. His integral he defines as 



where l m <f(x)<l m+l i n e m whose measure is e f m9 and each 
lm+i Z m = 0, as n = oo. The author has chosen a definition which 
occurred to him many years ago, and which to him seems far 
more natural. In Volume I it is shown that if the metric field 21 



PREFACE vii 

be divided into a finite number of metric sets S v S a of norm <2, 
then 

f / = M ax Zm& , f / = Min 2 M& 

/5!l *^2( 

where m t , Jf t are the minimum and maximum of/ in 8 t . What 
then is more natural than to ask what will happen if the cells 
&i ^2"* are infiftit 6 instead of tinite in number? From this 
apparently trivial question results a theory of ^-integrals which 
contains the Lebesgue integrals as a special case, and which, 
furthermore, has the great advantage that riot only is the relation 
of the new integrals to the ordinary or Riemannian integrals 
perfectly obvious, but also the form of reasoning employed in 
Riemaim's theory may be taken over to develop the properties 
of the new integrals. 

Finally the author would call attention to the treatment of 
the area of a curved surface given at the end of this volume. 
Though the above are the main features of novelty, it is hoped 
that the experienced reader will discover some minor points, not 
lacking in originality, but not of sufficient importance to em- 
phasize here. 

It is now the author's pleasant duty to acknowledge the in- 
valuable assistance derived from his colleague and former pupil, 
Dr. W. A. Wilson. He has read the entire manuscript and 
proof with great care, corrected many errors and oversights in 
the demonstrations, besides contributing the substance of 372, 
373, 401-406, 414-424. 

Unstinted praise is also due to the house of Ginn and Com- 
pany, who have met the author's wishes with unvarying liberality, 
and have given the utmost care to the press work. 

JAMES PIERPONT 
NEW HAVEN; December, 1911 



CONTENTS 



CHAPTER I 
POINT SETS AND PROPER INTEGRALS 



ARTICLES 

1-10. Miscellaneous Theorems . 

11-15. Iterable Fields .... 

16-25. Union and Divisor of Point Sets 



PAGE 

1 
14 

22 



CHAPTER II 
IMPROPER MULTIPLE INTEGRALS 



26-28. Classical Definition 
29 Definition of de la Valle'e- Poussin 


. 30 
31 


30. Author's Definition ......... 
31-01. General Theory 


. 32 
. 32 
59 


70 78. Iterated Integrals 


. 63 



CHAPTER III 
SERIES 

79-80. Preliminary Definitions and Theorems 

81. Geometric, General Harmonic, Alternating, and Telescopic Series 

82. Dini's Series 

83. Abel's Series 

84. Trigonometric Series 

85. Power Series 

86. Cauchy's Theorem on the Interval of Convergence . 

87-91. Tests of Convergence. Examples 

92. Standard Series of Comparison ...... 

93-98. Further Tests of Convergence ........ 

99. The Binomial Series 

100. The Hypergeornetric Series 

101-108. Pringsheim's Theory 

109-113 Arithmetic Operations on Series 

114-115. Two-way Series 



77 

81 

80 

.87 

88 

89 

90 

91 

101 

104 

110 

112 

113 

125 

133 



CONTENTS 



CHAPTER IV 



MULTIPLE SERIES 



ABTIOLIS 

116-126, General Theory 
126-188. Iterated Series . 



PAGE 

187 
148 



CHAPTER V 

SERIES Of FUNCTIONS 

184-145. General Theory. Uniform Convergence . 

146. The Moore-Osgood Theorem . 

J47-149. Continuity of a Series .... 

150-152. Term wise Integration .... 

158-156. Termwise Differentiation .... 



156 
170 
178 
177 
181 



CHAPTER VI 
POWER SERIES 

157-168. Termwise Differentiation and Integration 187 

169. Development of log (1 + #), arcsin x. arctan x, e*, sin ac, coax . 188 

160. Equality of two Power Series 191 

161-162. Development of a Power Series whose Terms are Power Series . 192 

163. Multiplication and Division of Power Series 196 

164^166, Undetermined Coefficients 197 

166-167. Development of a Series whose Terms are Power Series . . . 200 

168. Inversion of a Power Series 203 

169-171. Taylor's Development 206 

172. Forms of the Remainder 208 

173. Development of (1 4- x)n 210 

174. Development of log (1-fx), etc 212 

176-181. Criticism of Current Errors 214 

182. Pringsheiin's Necessary and Sufficient Condition .... 220 

183. Circular Functions ".222 

184. Hyperbolic Functions 228 

186-192. Hypergeometric Function 229 

193. Bessel Functions 238 



CHAPTER VII 
INFINITE PRODUCTS 



195-202. General Theory 

203-206. Arithmetical Operations 

207-212. Uniform Convergence 

213-218. Circular Functions . 



242 
250 
254 
257 



CONTENTS 



xi 



ARTIOLBS PAGE 

219. Bernouillian Numbers ......... 266 

220*228. B aud T Functions .......... 267 

CHAPTER VIII 

AGGREGATES 

229-230. Equivalence ........... 276 

231. Cardinal Numbers .......... 278 

232-241. Enumerable Sets .......... 280 

242. Some Space Transformations ........ 286 

243-260. The Cardinal c .......... 287 

251-261. Arithmetic Operations with Cardinals ...... 292 

262-264. Numbers of Liouville ......... 299 

CHAPTER IX 

ORDINAL NUMBERS 

266-267. Ordered Sets ........... 302 

268-270. EutacticSets ........... 304 

271-279. Sections ............ 807 

280-284. Ordinal Numbers .......... 310 

285-288. Limitary Numbers .......... 814 

289-300. Classes of Ordinals .......... 318 

CHAPTER X 
POINT SETS 

301-312. Pantaxis ........... 324 

813-320. Transfinite Derivatives ......... 380 

321-333. Complete Sets ........... 337 



CHAPTER XI 
MEASURE 



334-343. Upper Measure . 

344-368. Lower Measure 

369-870. Associate Sets . 

371-376. Separated Sets . 



343 
348 
365 
366 



CHAPTER XII 
LEBESGUE INTEGRALS 



377-402. General Theory 
403-400. Integrand Sets . 



371 
385 



Xll 



CONTENTS 



ARTICLES PAGE 

407-409. Measurable Functions . . 388 

410. Quasi and Semi Divisors 390 

411-413. Limit Functions 392 

414-424. Iterated Integrals 394 

IMPROPER IT-INTEGRALS 

425-428. Upper and Lower Integrals 402 

429-431. 7,-Integrals 405 

432-435. Iterated Integrals 409 

CHAPTER XIII 
FOURIER'S SERIES 

436-437. Preliminary Remarks 415 

438. Summation of Fourier's Series ........ 420 

439-442. Validity of Fourier's Development 424 

443-446. Limited Variation 429 

447-448. Other Criteria . 437 

449-456. Uniqueness of Fourier's Development 438 



CHAPTER XIV 
DISCONTINUOUS FUNCTIONS 

467-462. Properties of Continuous Functions . 

403-404. Pointwise and Total Discontinuity . 

405-473. Examples of Discontinuous Functions 

474-489. Functions of Class 1 

490-497. Semicontinuous Functions .... 



452 

467 
459 
468 
485 



CHAPTER XV 
DERIVATKS, EXTREMES, VARIATION 

498-518. Derivates 

519-525. Maxima and Minima 

526-534. Variation 

535-537. Non-intuitional Curves 

538-539. Pompeiu Curves 

540-542. Faber Curves 



493 

521 
531 
537 
542 
546 



CHAPTER XVI 

SUB- AND INFRA-UNIFORM CONVERGENCE 
Continuity 



643-650. 

551-556. Integrability , 

557-561 . Differentiability 



665 
662 
670 



CONTENTS 



xin 



CHAPTER XVII 
GEOMETRIC NOTIONS 

ARTICLES PA6X 

662-663. Properties of Intuitional Plane Curves 678 

664. Motion 679 

666. Curve as Intersection of Two Surfaces 679 

<3^66. Continuity of a Curve . 680 

667. Tangents 680 

668-672. Length 681 

673. Space-filling Curves 688 

674. Hilbert's Curve . 690 

676. Equations of a Curve 693 

676-580. Closed Curves 694 

581. Area 699 

682. Osgood's Curve 600 

583. Resume" 603 

584-585. Detached and Connected Sets 603 

586-591. Images 605 

692-597. Side Lights on Jordan Curves 610 

598-600. Brouwer's Proof of Jordan's Theorem . 614 

601. Dimensional Invariance 619 

602. Schonfliess* Theorem 621 

603-608. Area of Curved Surfaces 623 

Index 639 

List of Symbols 644 



FUNCTION THEORY OF REAL 
VARIABLES 

CHAPTER I 
POINT SETS AND PROPER INTEGRALS 

1. In this short chapter we wish to complete our treatment of 
proper multiple integrals and give a few theorems on point sets 
which we shall either need now or in the next chapter where we 
take up the important subject of improper multiple integrals. 

In Volume I, 702, we have said that a limited point set whose 
upper and lower contents are the same is measurable. It seems 
best to reserve this term for another nofcion which has come into 
great prominence of late. We shall therefore in the future call 
sets whose upper and lower contents are equal, metric sets. When 
a set 31 is metric, either symbol 



or 



expresses its content. In the following it will be often con- 
venient to denote the content of 21 by 



This notation will serve to keep in mind that 21 is metric, when 
we are reasoning with sets some of which are metric, and some 
are not. 

The frontier of a set as 21, may be denoted by 

Front . 

2. 1. In I, 713 we have introduced the very general notion of 
cell, division of space into cells, etc. The definition as there 

1 



2 POINT SETS AND PROPER INTEGRALS 

given requires each cell to be metric. For many purposes this 
is not necessary ; it suffices that the cells form an unmixed divi- 
sion of the given set 91. Such divisions we shall call unmixed di- 
visions of norm S. [I, 711.] Under these circumstances we have 
now theorems analogous to I, 714, 722, 723, viz : 

2. Let 33 contain the limited point set 21. Let A denote an un- 
mixed division of 33 of norm S. Let 2l fi denote those cells of 33 con - 
taining points of 21. Then 

lim H a = H. 

5=0 

The proof is entirely analogous to I, 714. 

8. Let 33 contain the limited point set 21. Let f(x^ # m ) be 
limited in 2(. Let A be an unmixed division of 33 of norm 8 info 
cells S v S 2 , . Let 2ft t , tn t be respectively the maximum and mini- 
mum of f in S t . Then 

lim S* = lira 22ftA = f /<% (1 

6=0 6=0 c/2l 

lim S = lim 2mA = f fdft. (2 

6=0 6=0 J% 

Let us prove 1) ; the relation 2) may be demonstrated in a similar 
manner. In the first place we show in a manner entirely analo- 
gous to I, 722, that 

(3 

The only modifications necessary are to replace S t , S[, S l/e , by their 
upper contents, and to make use of the fact that A is unmixed, to 
establish 5). 

To prove the other relation 

(4 

we shall modify the proof as follows. Let U be a cubical division 
of space of norm e < e . We may take e so small that 

(5 



PROPER INTEGRALS 3 

The cells of E containing points of 21 fall into two classes. 
1 the cells e M containing points of the cell S t but of no other cell 
of A ; 2 the cells e{ containing points of two or more cells of A. 
Thus we have _. 



where M^ M{, are the maxima of / in e itt , e{. Then as above we 
have __ 

4' o 

if e is taken sufficiently small. 
On the other hand, we have 



Now we may suppose S , e are taken so small that 



diflfer from 21 by as little as we choose. We have therefore for 
properly chosen S , , 



This with 6) gives 



which with 5) proves 4). 

4. Let f(xi - . XM) be limited in the limited field 21. Let A be 
an unmixed division oftyof norm 8, into cells S v 8 2 . Let 



where as usual ra t , Jf t are the minimum and maximum of f in 8 t . 

~ 
Max >S A , Cfd* = Min A^ A . 

J% 



The proof is entirely similar to I, 723, replacing the theorem 
there used by 2, 8. 

5. In connection with 4 and the theorem I, 696, 723 it may be 
well to caution the reader against an error which students are apt 
to make. The theorems I, 696, 1, 2 are not necessarily true if / 



4 POINT SETS AND PROPER INTEGRALS 

has both signs in 21. For example, consider a unit square 8 
whose center call O. Let us effect a division E of S into 100 
equal squares and let 21 be formed of the lower left-hand square 
and of 0. Let us define / as follows : 

/= 1 within * 

= - 100 at a 

For the division JE?, 

^=-l + Tta 

Hence, Min ^ < _ 

On the other hand, lim 



The theorems I, 723, and its analogue 4 are not necessarily true 
for unmixed divisions of space. The division A employed must 
be unmixed divisions of the field of integration 21. -That this is 
so, is shown by the example just given. 

6. In certain cases the field 21 may contain no points at all. 
In such a case we define 



7. From 4 we have at once : 

Let A be an unmixed division of 21 into cells 8j, 8 2 , Then 

S = Min 2S t , 
with respect to the class of all divisions A. 

8. We also have the following : 

Let D be an unmixed division of space. Let d v d v denote those 
cells containing points of 21. Then 



with respect to the class of the divisions D. 
For if we denote by S t the points of 81 in d t we have obviously 



Also by I, 696, 21 = Miu 



PROPER INTEGRALS 5 

with respect to the class of rectangular division of space JP= je t j. 
But the class E is a subclass of the class D. 

Thus 

Min 2^ < Min 2<? t <Min 22 t . 

A D JS 

Here the two end terms have the value 21. 



3. Let/^j # m ), ^(zj # m ) be limited in the limited field 
We have then the following theorems : 

1. Letf s g in 21 except possibly at the points of a discrete set 



T 7 
//=// 



For let | / |, \g\< M. Let D be a cubical division of norm d. 
Let MI, NI denote the maximum of/, g in the cell d^ Let A de- 
note the cells containing points of J)> while A may denote the 
other cells of 2l/>. 

Then, 2 M& = 2 W 4- 



Hence, 



and the term on the right =^ as d = 0. 

2. Letf > g in 21 except possibly at the points of a discrete set 
Then 



For let 91 = A + 



/>/> 

But in -4, />^, hence 



The theorem now follows at once. 



POINT SETS AND PROPER INTEGRALS 
fc..fr t 

, fc-.fr. 



For in any cell d, 

Max of = c Max/; Min </ = c Min/ 
when c > ; while 

Max <?/=<? Min/; Min- <?/=<? Max/ 
when <? < 0. 

4. If g is integrable in 21, 



For from 

Max/+Min#<Max 
we have 

fc + &*& + ^fc + fc' (2 

But g being integrable, 



Hence 2) gives 



which is the first half of 1). The other half follows from the 
relation 

Min / + Min g < Min (/ + #) < Min / + Max g. 

5. The integrands f, g being limited, 



For in any cell d, 

Min (/ + g} < Min / + Max g < Max (/ + g). 



PROPER INTEGRALS 
6. Letf*=g + h, \h\<H a constant, in 21. Then, 



For 

Then by 2 and 4 

-f ff +fff<ff<fff 

*s9f *^9f '91 *^ 9f 

or - r r r" 

--ff+J ^<J /<J y- 

4. Letf(x v # m ) 6e limited in limited 21. 2%ew, 



J)^ |/ 1 < Jlf, ^e have also, 



~3t 



0- 

(2 
(3 
(4 

(5 



Let us effect a cubical division of space of norm 8. 
To prove 1) let JV;=sMax|/| in the cell rf t . Then using the 
customary notation, 



Hence 



< 



Letting 8=0, this gives 



which is 1). 



POINT SETS AND PROPER INTEGRALS 
To prove 3), we use the relation 

-i/i </< 

Hence 



from which 3) follows on using 3, 3. 
The demonstration of 4) is similar. 
To prove 5), we observe that 



5. 1. Let f> be limited in the limited fields 39, g. Let 51 be 
the aggregate formed of the points in either 35 or . Then 



This is obvious since the sums 



may have terms in common. Such terms are therefore counted 
twice on the right of 1) and only once on the left, before passing 
to the limit. 

Remark. The relation 1) may not hold when /is not > 0. 

Example. Let 21 = (0, 1), 35 = rational points, and (= irra- 
tional points in 31- Let/= 1 in 35, and 1 in . Then 



and 1) does not now hold. 

2. Let 21 be an unmixed partial aggregate of the limited field 
Let S - 33 - 21. If 

ff=*f in 
=s in 6, 

then 



PROPER INTEGRALS 



But ~ f 

V = J/ by8 ' 1 ' 

and obviously 



3. The reader should note that the above theorem need not be 
true if 91 is not an unmixed part of $3. 

Example. Let 21 denote the rational points in the unit square 



Then 



4. jLetf 21 fte a part of the limited field 33- Letf>. 6e limited in 
91. i^ #=/ tw 21 awci = in g = S - 21- Then 



- (2 

For let J^, N L be the maxima of/, g in the cell d t . Then 



Passing to the limit we get 1). 

To prove 2) we note that in any cell containing a point of 31 



Min/> Min#. 

6. 1. Letf(xi a? w ) be limited in the limited field 21. Let 4B M 
be an unmixed part of 21 awcA Aa ) = 21 as u === 0. 



f/=li,nf/. 

*1 u=0 



10 POINT SETS AND PROPER INTEGRALS 

For let / <Jf in . Let <5 M = 21 - S u . Then 



But 

by 4, 1), 5). 



Hence passing to the limit u = in 2) we get 1). 

2. We note that 1 may be incorrect if the $d u are not unmixed. 
For let 51 be the unit square. Let SS U be the rational points in a 
concentric square whose side is 1 u. Let/= 1 for the rational 
points of 21 and = 2 for the other points. Then 



7. In I, 716 we have given a uniform convergence theorem 
when each 33 M <21. A similar theorem exists when each M .>21, 
viz. : 

Let $8 U < 95 U , if u< u f . Let 91 be a part of each $Q U . Let <B M = 
21 as u = 0. Then for each e> 0, there exists a pair u^ d Q such that 



For SS Uo < 21 + ^, UQ sufficiently small. 

Also for any division D of norm d < some d Q . 



But 

,/>< MO ,^ if 
Hence 



8. 1. Let 21 be a point set in m = r + a way space. Let us set 
certain coordinates as # r +i a? m= i n ea h point of 21. The 
resulting points $8 we call a projection of 21. The points of 21 



PROPER INTEGRALS 11 

belonging to a given point 6 of 33, we denote by ( 6 or more shortly 
by S. We write 

a = 33 e, 

and call 33* & components of a. 

We note that the fundamental relations of I, 733 



hold not only for the components y, ^, etc., as there given, but 
also for the general components 31, 53. 

In what follows we shall often give a proof for two dimensions 
for the sake of clearness, but in such cases the form of proof will 
admit an easy generalization. In such cases 33 will be taken as 
the ^-projection or component of a* 

2. If a = 33 & is limited and 33 is discrete, 91 is also discrete. 

For let 91 lie within a cube of edge ^ (7> 1 in m = r + s way 
space. Then for any d < some c# , 



Then a/> < C' D < e. 

3. That the converse of 2 is not necessarily true is shown by 
the two following examples, which we shall use later : 

Example 1. Let 91 denote the points #, y in the unit square 
determined thus : 
For 



Tfl 

= -i 7i=l, 2, 3, -, 7/1 odd and < 2 n , 



let 



Here a is discrete, while 48 = 1, where 33 denotes the projection 
of a on the #-axis. 

4. ISxample 2. Let a denote the points #, y in the unit square 
determined thus : 



12 POINT SETS AND PROPER INTEGRALS 

For 

2 * _, nt, n relatively prime, 
n 

let 1 

o<y<l 

Then, $ denoting the projection of 81 on the tf-axis, we have 
f = 0, # = 1. 

9. 1. Let 21= $ & 60 a limited point set. Then 

<. a 

For let/=l in 21. Let ^ = 1 at each point of 21 and at the 
other points of a cube A = B - containing 21, let g = 0. Then 



Byl ' 733 ' 

But by 5, 4, 



Thus 

21 

which gives 1), since 



2. Jw case 21 * metric we have 

i-Jt*. (2 

andf S w aw integrable function over $ 
This follows at once from 1). 



PROPER INTEGRALS 13 

3. In this connection we should note, however, that the converse 
of 2 is not always true, i.e. if is integrable, then 21 has content 
and 2, 2) holds. This is shown by the following : 

Example. In the unit square we define the points #, ^of 21 thus : 
For rational a?, 



For irrational #, 

Then = i for every x in 33. Hence 



/"-* 



But 91 = 0, -l. 

10. 1. Letf(x^ " # m ) be limited in the limited field 81 = 33 



(2 



,, 



Let us first prove 1). Let 91, S3, ( lie in the spaces 9t m , $ 9J 
r -f- g = m. Then any cubical division D divides these spaces into 
cubical cells rf,, d(, d" of volumes rf, c?', rf" respectively. Ob- 
viously d = d'd". J? also divides 33 and eacA S into unmixed cells 
S', 8". Let M. = Max/ in one of the cells d a while M'J = Max/ 
in the corresponding cell S". Then by 2, 4, 



since M MJ > 0. Hence 



Letting the norm of D converge to zero, we get 1). We get 
2) by similar reasoning or by using 3, 3 and 1). 



14 POINT SETS AND PROPER INTEGRALS 

2. To illustrate the necessity of making/ > in 1}, let us take 
31 to be the Pringsheim set of I, 740, 2, while / shall = 1 in 21. 
Then 



On the other hand 
Hence 
and the relation 1) does not hold here. 

Iterable Fields 

11. 1. There is a large class of limited point sets which do not 
have content and yet _ -._ 

21= f . (1 

%/jjg 

Any limited point set satisfying the relation 1) we call iterable^ 
or more specifically iterable with respect to $$. 

Example 1. Let 21 consist of the rational points in the unit 
square. Obviously __ - -^ 

21= | 6= | = 1, 

j/33 */( 

so that 21 is iterable both with respect to 48 and (. 

Example 2. Let 21 consist of the points x, y in the unit square 
defined thus : 

For rational x let < y < J. 

For irrational # let < y < 1. 
Here 21 = 1. 



Thus 21 is iterable with respect to S but not with respect to 53, 



ITERABLE FIELDS 15 

Example 8. Let 91 consist of the points in the unit square de- 

fined thus: ,. 11^ A ^ ^o 

For rational a? let < y < |. 
For irrational a? let \ < y < 1. 

Here 91 = 1, while 



Hence 91 is iterable with respect to & but not with respect to $8. 

Example 4. Let 21 consist of the sides of the unit square and 
the rational points within the square. 
Here 91 = 1, while 



and similar relations for &. Thus 21 is not iterable with respect 
to either 33 or (. 

Example 5. Let 21 be the Pringsheim set of I, 740, 2. 
Here 21 = 1, while 



Hence 21 is not iterable with respect to either 33 or @. 

2. Every limited metric point set is iterable with respect to any of 
its projections. 

This follows at once from the definition and 9, 2. 

12. 1. Although 21 is not iterable it may become so on remov- 
ing a properly chosen discrete set J). 

Example. In Example 4 of 11, the points on the sides of the 
unit square form a discrete set 5 ; on removing these, the deleted 
set 21* is iterable with respect to either S3 or . 

2. The reader is cautioned not to fall into the error of suppos- 
ing that if 2l x and 2^ are unmixed iterable sets, then 21 = 2l x + 21 2 
is also iterable. That this is not so is shown by the Example in 1. 

For let 2t x = 21*, 91 2 = > in that example. Then $) being dis- 
crete has content and is thus iterable. But 21 = 2lj + 21 2 is 
iterable with respect to either $Q or <. 



16 POINT SETS AND PROPER INTEGRALS 

13. 1. Let 21 be a limited point set lying in the m dimensional 
space 9t m . Let 48, & be components of 31 in 9t r , 9?,, r + s = m. 
A cubical division D of norm 8 divides 9t m into cells of volume 
d and W r and 9t, into cells of volume d^ c?,, where d = d r rf r Let 

b be any point of 48, lying in a cell d r . Let 2rf, denote the sum 

b 
of all the cells d# containing points of 31 whose projection is b. 

Let 2,d s denote the sum of all the cells containing points of 31 

dr 

whose projection falls in d r , not counting two d t cells twice. 
We have now the following theorem : 
is iterable with respect to 48, 



For 
Hence 



< 
8 * 



Let now 8 0. The first and third members = 31, using I, 699, 
since 31 i iterable. Thus, the second and third members have 
the same limit, and this gives 1). 

2. If 31 i* iterable with respect to 48, 

lim Zd^d, = I. 

=o 33 b 

This follows at once from 1). 

3. Let 31 =as g be a limited point set, iterable with respect to $&. 
Then any unmixed part Q of 31 is also iterable with respect to the 
SB-component ofQ*. 

For let b = a point of 48 ; 6' points of 31 not in @ ; C b = points 
of S 6 in g, (7^ = points of ( 6 in g'. Then for each >0 there 
exist a pair of points, b v 6 2 , distinct or coincident in any cell d f 
such that as 6 ranges over this cell, 



ITERABLE FIELDS 17 

Let $ denote, as in 13, l, the cells of 2cZ, which contain points of (', 

and F the cells containing points of both (, (' whose projections 
fall in d r . Then from 



ii t 

we have dr 

<E 6 <C*+Cl< Min G b + 0' + S< Max <? 6 4- " 
Multiplying by d r and summing over 33 we have, 



(i 

Passing to the limit, we have 



f tf + y + <*, (2 

*/ 



the limit of the last term vanishing since (g, (' are unmixed parts 
of 31. Here r/, 77" are as small as vye please on taking /9 sufficiently 
small. From 2) we now have 



4. Let 31 = 55 fo iterable with respect to SJ. i^ S be a part 
of $8 and A all those points of 31 whose projection falls on B. Then 
A is iterable with respect to B. 

For let D be a cubical division of space of norm d. Then 



= lim 



2d r d,\, (1 

r,* > 



where the sum on the right extends over those cells containing 
no point of A. Also 

(2 

where the second sum on the right extends over those cells d, 
containing no point of JB. 
Subtracting 1), 2) gives 

= lim { A D - 2rf r S } 4- lim 

dU <> M > d0 



18 POINT SETS AND PROPER INTEGRALS 

As each of the braces is > we have 



|S. 
JB 



14. We can now generalize the fundamental inequalities of I, 
733 as follows : 

Let f(x l XM) be limited in the limited field 21 = 33 S, iterable 
with respect to SQ. Then 



For let us choose the positive constants A, B such that 
f+A>0, /-^<0, in . 



Let us effect a cubical division of the space of 9t m of norm 8 into 
cells d. As in 13, this divides 9^, 91 ,, into cells which we denote, 
as well as their contents, by c? r , d s . Let b denote any point of 48. 
As usual let m, M denote the minimum and maximum of / in the 
cell d containing a point of 21. Let m' , M 1 be the corresponding 
extremes of /when we consider only those points of 21 in d whose 
projection is b. Let |/| < F in 21. 
Then for any 6, we have by I, 696, 



or __ 

4- 6)+ 2md s < \J\ (2 



snce 

In a similar manner 



? <M3, + 4(2d,-<). (3 

Thus for any 5 in S3, 2), 3) give 

- JB(2df,-6)+2wrf,< Cf<2Md s + 4(2<Z.-<). (4 

ft 6 .( 6 6 

Let /8 > be small at pleasure. There exist two points b v 6 2 dis- 
tinct or coincident in the cell rf r , for which 



ITERABLE FIELDS 19 

where | ff l |, | & | < and & v and (^ stand for S v S 6j , and finally 
where __ 

y-Minf/ f J" 
*/(j 

for all points 6 in d r . 

Let c = Min & in c? r , then 4) gives 



5 - c) + 2mrf, < y + ft < </+ /3 2 < SMi, -h ^(2^, - c) 

11 22 

where the indices 1, 2 indicate that in 2 we have replaced b by 






Multiplying by d r and summing over all the cells d r containing 
points of 53, the last relation gives 



~ c) 4- ^ 

1 SB 1 



5 - c). (5 

SB SB SB 2 SB 2 



2rf r 2rf 4 = 2l, by 13, 2. 

SB 2 



s= ) 6 = S, since 51 is iterable. 
^ 



Thus the first and last sums in 5) are evanescent with 8. On 
the other hand 



2<2,m - 2(2 4 w) I < 

d, 1 | SB efc 1 

= as 8=0, by 13, l, 

Thus 

(6 



(7 



Hence passing to the limit 8=0 in 5) we get 1), since 2 1 d r , 
2/8 2 d r have limits numerically <>8& which may be taken as small 
as we please as (3 is arbitrarily small. 



20 POINT SETS AND PROPER INTEGRALS 

2. If 91 is not iterable with respect to 33, let it be so on remov- 
ing the discrete set ). Let the resulting field A have the com- 
ponents B, 0. Then 1 gives 



snce 



3. The reader should guard against supposing 1) is correct if 
only 21 is iterable on removing a discrete set J). For consider 
the following : 

Example. Let the points of 51 = 9lj + T) lie in the unit square. 
Let 9lj consist of all the points lying on the irrational ordinates. 
Let !) lie on the rational ordinates such that, when 

tyy* 

x~~, m,n relatively prime, 
n 

-. 
n 

Let us define/ over 31 thus : 

/=! in 9l r 

/=0 in >. 

The relation 1) is false in this case. For 



/=!, 
.* 
while 



/ 
.* 



15. 1. Let /(tfj x m ) be limited in the limited point set 91. 
Let D denote the rectangular division of norm d. All the points 
of 9l/> except possibly those on its surface are inner points / of 91. 
[I, 702.] 

The limits H m Cf , u m Cf (1 

d=0 */ rf=0 %/W 

-7) J) 

exist and will be denoted by 

JTV , fV , (2 

^/a ^a 

and are called the inner, lower and upper integrals respectively. 



1TERABLE FIELDS 21 

To show that 1) exist we need only to show that for each c > 
there exists a d Q such that for any rectangular divisions 2>', D" of 
norms < d 

A: 



To this end, we denote by E the division formed by superimpos- 
ing D" on D 1 . Then E is a rectangular division of norm < d . 

Let %s ~ la* = 4', * - />" = ^" 

If d ft is sufficiently small, A , An 

VI , -A <^ ??, 

an arbitrarily small positive number. Then 
I//T 7* \ //*" 7*\l 

A= r - u ~r < + 



if i; is taken small enough. 
2. The integrals 



- 

, r/ 

/ 



heretofore considered may be called the outer, lower and upper in- 
tegrals, in contradistinction. 

3. Let f be limited in the limited metric field 31. Then the inner 
and outer lower (upper) integrals are equal. 

For y( D is an unmixed part of 31 such that 

Cont !, = , asd = 0. 

Then by 6, l, r r 

limj /=)/. 

d=0^/, ^ 

But the limit on the left is by definition 



4. JTAew 21 has no inner points, 



22 POINT SETS AND PROPER INTEGRALS 

For each fl/, = 0, ar^d hence each 



Point Sets 
16. Let 21 = 33 + be metric. Then 



For let D be a cubical division of space of norm d. The cells 
of 31^ fall into three classes : 1, cells containing only points of 33; 
these form $8 D . 2, cells containing points of (; these' form (/>. 
3, cells containing frontier points of 33, not already included in 1 
or 2. Call these fo. Then 

/> = / , + 6 1> +L. ( 2 

Let now d = 0. As 91 is metric, \ D = 0, since f^ is a part of 
Front 21 and this is discrete. Thus 2) gives 1). 

17. 1. Let 31, , 6 -- (1 

be point sets, limited or not, and finite or infinite in number. 
The aggregate formed of the points present in at least one of the 
sets 1) is called their union^ and may be denoted by 



or more shortly by 



g _ 



If 31 is a general symbol for the sets 1), the union of these sets 
may also be denoted by 



or even more briefly by ,> 

If no two of the sets 1) have a point in common, their union 
may be called their sum, and this may be denoted by 



The set formed of the points common to all the sets 1) we call 
their divisor and denote by 



POINT SETS 23 

or fe y Dv{%\, 

if 21 is a general symbol as before. 

2. Examples. 

Let 31 be the interval (0, 2); SS the interval (1, oo). Then 

, 8) = (0, oo), J(a, 8) = (1, 2). 



Let Sit = (0,1), *, = (!, 2) 

Then 91 2 -) = (0, oo), 



a 9 .-o=o. 

Let 

Then {jA)i gf ...^ = (o* 

Then 



3. Let a^a^aa^a,^- (i 

Let = D(a, a r a,, -) 

Let 2l = 2 

Then 2l = S + S 1 + S 2 + - 

Let us first exclude the = sign in 1). Then every element of 
21 which is not in ) is in some 9l n but not in 2l n+1 . It is therefore 
in S B+ i but not in ( n+3 , & n + 3 , The rest now follows easily. 

4. Some writers call the union of two sets 91, 8 their sum, 
whether a, $ have a point in common or not. We have not done 
this because the associative property of sums, viz. : 



does not hold in general for unions. 



24 POINT SETS AND PROPER INTEGRALS 

Example. Let SI = rectangle (123 4), 
SQ = (5 6 7 8), 



Then 

and (Z7(, #)-), (2 

are different. 

Thus if we write + for U, 1), 2) give 

* + (-)*(* +)-. 

18. 1. Let Slt^^^Sls be a set of limited complete point 
aggregates. Then 

-/>(*!. V )><> 
Moreover 33 is complete. 

Let a n be a point of 2l n , n = 1, 2, and 21 = & v a& a$ 
Any limiting point a of 31 is in every 2l n . For it is a limit- 
ing point of 

#im #m+H #m+2i " 

But all these points lie in 2l m , which is complete. Hence a lies in 
3l m , and therefore in every Sl^ 2l 2 i Hence a lies in 33, and 
>0. 

48 is complete. For let $ be one of its limiting points. Let 



As each 6 m is in each 8l n , and 3l rt is complete, /9 is in 2l n . Hence /3 
is in #. 



2. i< 91 6e a limited point set of the second species. Then 

', ",'", )><>, 



and is complete. 

For < n) is complete and > 0. Also w >. (<H>I) . 



19. f Slj, 31 2 ... ?ie tn ; /et 21= Z7{2l n S. X^ A n be the com- 
plement of 2l n with respect to 53, so that A n + 2l n = 53 i^ 
A and 81 arc complementary, so that A 4- 21 = 53* 



POINT SETS 25 

For each point 6 of 53 lies in some 2l n , or it lies in no 2l n , and 
hence in every A n . In the first case b lies in 31, in the second in 
A. Moreover it cannot lie in both A and 21- 

20. 1. Let 2l a <21 2 <21 8 (1 

i 
be an infinite sequence of point sets whose union call 21. This 

fact may be more briefly indicated by the notation 



Obviously when 21 is limited, 

n . (2 



That the inequality may hold as well as the equality in 2) is 
shown by the following examples. 



Example 1. Let 2l n = the segment f-, 1 J 
Then = (0*, 1). 



n 



Example 2. Let a n denote the points in the unit interval whose 
abscissae are given by 

x = , m < n = 1, 2, 3, m, n relatively prime. 
n 

Let . = a 1 + ... + o.. 

Here 21 = Z7{2U 

is the totality of rational numbers in (0*, 1*). 

A a _ _ 

91 = 1 and 21 B = 0, we see 

t > lim I B . 

2. Let i>l2>- (3 

Let SQ be their divisor. This we may denote briefly by 



Obviously when SB 1 is limited, 

< lim .. 



26 POINT SETS AND PROPER INTEGRALS 

Example 1. Let $ = the segment f 0, -V 
Then = Dv\% n \ =(0), the origin. 

Here fg = 0. lim B = lim - = 0, 

n 

and = -i. ^r 

S3 = lim 4Bn- 

Example 2. Let 2l n be as in 1, Example 2. Let b n = 31 3l n . 

Let g* ("\ 2^ 4- b 

Here ^ = the segment (j^ 2) and *g n = 2. 

Hence ^ < Um = 



3. i^ 33 j < $3 2 < 6e unmixed parts of 31. ie < n = S. 
= U { 33 n } . Then S = 21 - & discrete. 

For let 31 = 33 W + S n 5 th en Sn is an unmixed part of 31. Hence 



Passing to the limit n = oo, this gives 

lim g n = 0. 
Hence is discrete by 2. 

4. We may obviously apply the terms monotone increasing, 
monotone decreasing sequences, etc. [Cf. 1, 108, 211] to sequences 
of the type 1), 3). 

21. Let 6 = 31 + S. If 31, S are complete, 



For 8=Dist(3l, 

since 91, SB are complete and have no point in common. Let D be 
a cubical division of space of norm d. If d is taken sufficiently 
small 210, $8 D have no cells in common. Hence 



Letting d = we get 1). 



POINT SETS 27 

22. 1. If 21, 33 are complete, so are also 

g=(2l, 33), ) = !>* (21, 33). 

Let us first show that is complete. Let c be a limiting point 
of (. Let <?j, c 2 , be points of ( which = <?. Let us separate 
the c n into two classes, according as they belong to 21, or do not. 
One of these classes must embrace an infinite number of points 
which = c. As both 51 and 33 are complete, c lies in either 21 or 
33- Hence it lies in g. 

To show that J) is complete. Let d v d^ be points of ) which 
= d. As each d n is in both 21 and 33, their limiting point d is in 
21 and $Q, since these are complete. Hence d is in ). 

2. If 21, 93 are metric so are 

= (21,33) $D = Dt>(, 33). 

For the points of Front (5 lie either in Front 21 or in Front 33i 
while the points of Front ) < Front 21 and also < Front 33. But 
Front 21 and Front 33 are discrete since 21, 33 are metric. 

23. Let the complete set 21 have a complete part 33. Then how- 
ever small e > is taken, there exists a complete set in 21, having no 
point in common with 33 such that 



Moreover there exists no complete set S, having no point in common 

with 33 such that _ _ _ 

<>-. 

The second part of the theorem follows from 21. To prove 1) 
let D be a cubical division such that 

,= ! + ', * = + ", 0<',"<. (2 

Since 33 is complete, no point of 33 lies on the frontier of 33^ 
Let denote the points of 21 lying in cells containing no point of 
33. Since 21 is complete so is &, and 33> & have no point in common. 

Thus _ 

z>. (3 



28 POINT SETS AND PROPER INTEGRALS 

But the cells of (/> may be subdivided, forming a new division A, 
which does not change the cells of 8^, so that JBz> = 33*1 but so that 

S A = S + "', <'"<. (4 

Thus 2), 3> 4) give 



r l=t-#~ 

>!--- 

24. Ze 31, 8 i complete. Let 



Then _____ 

I + C-U + fc. (1 

Forlet 11 = 31 + A 

Then ^4 contains complete sets (7, such that 

>U-3l-6, (2 

but no complete set such that 

<7>U-I, (8 

by 23. On the other hand, 



Hence A contains complete sets (7, such that 

>->-, (4 

but no complete set such that 

<?>-$>. (5 

From 2), 3), and 4), 5) we have 1), since e is arbitrarily small. 

8- Let 



each 31 B being complete and tuck that 31 B > some constant k, 
Then 



POINT SETS 29 



For suppose Z = t-l)>0. 

Let 



Then by 23 there exists in 91, a complete set S r having no point 
in common with $) such that 



or as 2L > A, such that 

j v 

Let 6 a = JD V (91 2 , (, ), U 

Then by 24, 8,4-C^H + 

Thus 



Thus 21 3 contains the non-vanishing complete set S a having no 
point in common with 2). In this way we may continue. Thus 
Slj, 21 2 , contain a non-vanishing complete component not in D, 
which is absurd. 

Corollary. Let 21 = ( Slj < ?I 2 < ) be complete . ?%<w l n = t 
This follows easily from 23, 25. 



CHAPTER II 
IMPROPER MULTIPLE INTEGRALS 

26. Up to the present we have considered only proper multiple 
integrals. We take up now the case when the integrand f(x l '-x m ) 
is not limited. Such integrals are called improper. When m = 1, 
we get the integrals treated in Vol. I, Chapter 14. An important 
application of the theory we are now to develop is the inversion 
of the order of integration in iterated improper integrals. The 
treatment of this question given in Vol. I may be simplified and 
generalized by making use of the properties of improper multiple 
integrals. 

27. Let 91 be a limited point set in w-way space 9J m . At each 
point of 91 let f(x l # m ) have a definite value assigned to it. 
The points of infinite discontinuity of f which lie in 21 we shall 
denote by Q. In general $ is discrete, and this case is by far the 
most important. But it is not necessary. We shall call $ the 
singular points. 

Vfli 

Example. Let 51 be the unit square. At the point # = , 

r n 

y = -, these fractions being irreducible, let f=ns. At the other 
s 

points of 21 let /= 1. Here every point of 21 is a point of infinite 
discontinuity and hence $ = 21. 

Several types of definition of improper integrals have been 
proposed. We shall mention only three. 

28. Type I. Let us effect a division A of norm S of 9t m into 
cells, such that each cell is complete. Such divisions may be 
called complete. Let 2ls denote the cells containing points of 21, 
but no point of $, while 2t 6 ' may denote the cells containing a 
point of Q. Since A is complete, / is limited in 2ls. Hence / 
admits an upper and a lower proper integral in 21$. The limits, 
when they exist, ~ 

lim f /, lim f /, (1 

ao J% 8 =o s/a g 

30 



GENERAL THEORY 31 

for all possible complete divisions A of norm S, are called the 
lower and upper integrals of / in 21, and are denoted by 



(2 

*3l % 

or more shortly by 



When the limits 1) are finite, the corresponding integrals 2) 
are convergent. We also say/ admits a lower or an upper improper 
integral in 21. When the two integrals 2) are equal, we say that 
/ is integrable in 21 and denote their common value by 

ffd* or by ff. (3 

*/Sl */2l 

We call 3) the improper integral of f in 21 ; we also say that 
f admits an improper integral in 2( and that the integral 3) is 
convergent. 

The definition of an improper integral just given is an extension 
of that given in Vol. I, Chapter 14. It is the natural develop- 
ment of the idea of an improper integral which goes back to the 
beginnings of the calculus. 

It is convenient to speak of the symbols 2) as upper and lower 
integrals, even when the limits 1) do not exist. A similar remark 
applies to the symbol 3). 

Let us replace /by |/| in one of the symbols %), 3). The 
resulting symbol is called the adjoint of the integral in question. 
We write 

(4 



When the adjoint of one of the integrals 2), 3) is convergent, 
the first integral is said to be absolutely convergent. Thus if 4) is 
convergent, the second integral in 2) is absolutely convergent, etc. 

29. Type II. Let X, /A>0. We introduce a truncated func- 
tion/^ defined as follows : 

f^ ~f(?i - Zm) when - X </< p 
= X when/< X 

= fi when / > fji. 



32 IMPROPER MULTIPLE INTEGRALS 

We define now the lower integral as 



A similar definition holds for the upper integral. The other 
terms introduced in 28 apply here without change. 

This definition of an improper integral is due to de la ValUe 
Poussin. It has been employed by him and ft. Gr. D. Richardson 
with great success. 

30. Type IIL Let , /9>0. Let 2l a|8 denote the points of 21 

at which 

- 

We define now 

f/ = lim f f ; f/ = Urn f /. (1 

J* ,*-/./ *V a, 0=~ -Al a / V 

The other terms introduced in 28 apply here without change. 
This type of definition originated with the author and has been 
developed in his lectures. 

31. When the points of infinite discontinuity $ are discrete 
and the upper integrals are absolutely convergent, all three defini- 
tions lead to the same result, as we shall show. 

When this condition is not satisfied, the results may be quite 
different. 

Example. Let 21 be the unit square. Let 2l a , 2^ denote respec- 
tively the upper and lower halves. At the rational* points S3, 

x=* n \ y = -, in 2l r let/= m. At the other points of 2l p let 

H\j 8 

/==-! In2l 2 let/=0. 

1 Definition. Here $ = Sl r 

Hence 



2 Definition. Here 

' i, f/=+oo. 



* Here as in all following examples of this sort, fractions are supposed to be 
irreducible. 



GENERAL THEORY 33 

3 Definition. Here 51^ embraces all the points of 8 a , S and a 
finite number of points of $B for a > 2, ft arbitrarily large. Hence 

//--!, //--It 
i ^ 

and thus 



32. In the following we shall adopt the third type of definition, 
as it seems to lead to more general results when treating the im- 
portant subject of inversion of the order of integration in iterated 
integrals. 

We note that if /is limited in 91, 

lim jf = the proper integral J /. 

For a, y8 being sufficiently large, 9l aj8 == 91- 
Also, if 91 is discrete, 

fr- 

For 9l a8 is discrete, and hence 



Hence the limit of these integrals is 0. 

33. Let w=|Min/| , Jlf|Max/| in 91. 
Then 

lim I /= lim I /, m finite. 



lim I /=lim f /, M finite. 

a, 0=00*61^ =^ tt ,ir 

For these limits depend only on large values of , & and when 

m is finite. ftf ~ . ^ 

- > foralla>w. 



Similarly, when J!f is finite 

5l ft<8 ^j/ , forall/8>Jf. 



34 IMPROPER MULTIPLE INTEGRALS 

Thus in these cases we may simplify our notation by replacing 

2L, M i 3lm0 

by 2l_. , 21, , 

respectively. 

2. Thus we have: 

|/=limj /, when Min / is finite. 
J% p=*J%p 

J / = lim J / , when Max/ is finite. 

3. Sometimes we have to deal with several functions/, #, - 
In this case the notation 2l aj8 is ambiguous. To make it clear we 
let 3l/,a,|8 denote the points of 91 where 



Similarly, 2l g , at ^ denotes the points where 

a<_g<_ft, etc. 

34. I f is a monotone decreasing function of a for each ft. 

J%afl 

I f is a monotone increasing function of ft for each a. 
J ^ 

If Max / 1 finite 

Xf are monotone decreasing functions of a. 
-J 

If Min / is finite 

\ f are monotone increasing functions of ft. 
J&p 

Let us prove the first statement. Let a f > a. 
Let J!) be a cubical division of space of norm d. 
Then ft being fixed, 

X/=lim2 m&, (1 

_~0 9,0 

f /=lim2m;<, (2 

Jxa'p *= n a <p 

using the notation so often employed before. 



GENERAL THEORY 85 

But each cell d, of 3L0 lies among the cells dj of 31. /. Thus we 
can break up the sum 2), getting 



Here the second term on the right is summed over those cells 
not containing points of 2l a/3 . It is thus < 0. In the first term 
on the right m^ <m t . It is thus less than the sum in 1). Hence 



Thus r r 

I < I , '>. 

J^i-J^ 

In a similar manner we may prove the second statement ; let 
us turn to the third. 
We need only to show that 

( / is monotone decreasing. 
'a-'. 

Let '>. Then -* 

I == lim 2j|f t <* t . (3 

J%_ a d^ ^ a 



f ^limSJK'rf/. (4 

J%_ a , d =o %_ a> 

As before ^M(d( = 2Jf (d{ + I.M'Jd'J. (5 



But in the cells d t , MJ = M,. Hence the first term of 5) is 
ihe same as 2 in 3). The second term of 5) is < 0. The proof 
? ollows now as before. 

35. If Max / is finite and I fare limited, ( f is convergent and 

J&- a ! 



If Min / is finite and \ are limited, ( f is convergent and 
/2l0 */$! 

f/< f / 

/M- JVL 



36 IMPROPER MULTIPLE INTEGRALS 

For by 34 

/../ 4' 

are limited monotone functions. Their limits exist by I, 277, 8. 

36. If M = Max / is finite, and \ f is convergent, the correspond- 
ing upper integral is convergent and 



where f >; a in 2L a . 

Similarly, if w= Min/ is finite and I f is convergent, the corre- 
sponding lower integral is convergent and 



Let us prove the first half of the theorem. 

We have * -* 

I /= lim ( . 

*/2l a=J8J- a 

Now C C C 

U<J <L <^3l-a- 

*L2l ^_ a ^21-a 
We have now only to pass to the limit. 



37, ffyf w convergent, and 53 < 81, 

/ 

does not need to converge. Similarly 



does not need to converge, although \ f does. 

Example. Let 21 be the unit square ; let S3 denote the points 
for which x is rational. 



GENERAL THEORY 37 

e /as 1 when x is irrational 

= - when x is rational 

y 

Then r 

A/- 1 ; 

On the other hand, 



Hence x / 

I = lim I =lim Iogy8= -f-oo 
*/$ 0. ^p 

is divergent. 

38. 1. In the future it will be convenient to let ^J denote the 
points of 21 where /> 0, and 9Z the points where/ <. 0. We may 
call them the positive and negative components of 21. 



2. If\f converges, so do I f. 
If \ f converges, so do \ f. 



For let us effect a cubical division of space of norm d. Let 
ft f > . Let e denote those cells containing a point of ^ ; e r 
those cells containing a point of ^ but no point of ^ ; S those 
cells containing a point of 2l a/3 but none of typ,. 

Then . 



f = limjSJf. - e + SJtf"., *' + 2JK, - S{. 
/ d=o 



Obviously lf.>M. , 

Hence ~ -~ 

\ - f =li 

^a' / a <* 



38 IMPROPER MULTIPLE INTEGRALS 

We find similarly 

f - f -Um{2(JT t -Jf.> 

*/^/ */fp 5 <*=:<) 



Now 



for a sufficiently large a, and for any & ' > . 
Hence the same is true of the left side of 1> 

As corollaries we have : 

3. If the upper integral of f is convergent in 31, then 



If the lower integral off is convergent in 31, 



f < f < f etc. 

J p pJyfi "/v 

4. Iff> 0wrf I /i convergent, so is 
*/$( 

J/ ' < 

Moreover the second integral is < the first. 
This follows at once from 3, as 31 = ^. 

39. If J f and (f converge, so do J /. 

We show thatj /converges; a similar proof holds forj . To 
this end we have only to show that 



>0; , /3>0; 



; a< ' 



GENERAL THEORY 



3? 



Let D be a cubical division of space of norm d. Let ^ , %r 
denote cells containing at least one point of 2l a '0 f , 2l a '^ M at which 
/>0. Let n a ' > tla" denote cells containing only points of St a '/3' 
2l a -0" at which/< 0. We have 



9U"/3" 



Subtracting, 



(2 



Let Jf[ == Max / for points of S J? in d,. Then since / has one 
sign in 9t, 

ISJtfft-ZJtW, <|2JfJ(l t -2-af!dJ. (3 

"a' "a" "~ 9^a' ^a" 

Letting d = 0, 2) and 3) give 



Now if /3 is taken sufficiently large, the first term on the right is 

< e/2. On the other hand, since I / is convergent, so is I / by 

J$t ^W 

36. Hence for a sufficiently large, the last term on the right is 
<6/2. Thus 4) gives 1). 

40. Iff is intec/rable in 21, it is in any 53 < 21. 
Let us first show it is integrable in any 2t a/ g. 

Let 



Let D be a cubical division of space of norm d. 
Then A aft = lim 2o> t d t , o> t = Osc/ in d.. 

*-0 %afi 

Let ' > , ff > 0. Then 

X'^' ~ A^ = Km {2(^( 



40 IMPROPER MULTIPLE INTEGRALS 

Now any cell d t of 2l a/3 is a cell of 2t a ' /8 ', and in d^ &[ > ( t . 
Hence A a >p > A aft . Thus A^ is a monotone increasing function 
of a, . On the other hand 

lim A a p =s 0, 



by hypothesis. Hence A a p= and thus/ is integrable in 2l a/3 . 

Next let / be limited in , then |/|<some 7 in 48. Then 
< 3l y , y But / being integrable in 2l y , y , it is in $8 by I, 700, 3. 

Let us now consider the general case. Since/ is integrable in 21 



both converge by 38. Let now P, N be the points of $, 91 lying 
in. Then 



both converge. Hence by 39, 



both converge. But if $8 a ^ b denote the points of SS at which 
-a<f<b, 

f/= lim f / 

/S , &=>*/5B a6 

by definition. 
But as just seen, 



and /is integrable in $8. 

41. As a corollary of 40 we have : 

1 . Iff is integrable in 21, it admits a proper integral in any part 
o/2l in which f is limited. 

2. Iff is integrable in any part of 21 in which f is limited, and if 
either the lower or upper integral off in 21 is convergent, f is Integra- 
ble in 21. 



GENERAL THEORY 41 

For let 



7f ~ 

J/-limJ/ 

^a a^oo^SU 

exist. Since 



necessarily 



exists and 1), 2) are equal. 

42. 1. In studying the function/ it is sometimes convenient to 
introduce two auxiliary functions defined as follows : 

#=/ where/ >0, 

= where /<0. 

A=-/ where/<0, 

= where />0. 
Thus g, h are both > and 



We call them the associated non-negative functions. 

2. As usual let 3l a/3 denote the points of 21 at which ;</^$. 
Let Sip denote the points where g</3, and 2l a the points where h<a. 
Then 



f A = lim f A. (2 

^sr a^aw^a^ 

For 

by5 ' 4 ' 



Letting , y8 = oo, this last gives 1). 
A similar demonstration establishes 2). 

3. We cannot say always 



X^aslim f g ; fA = lim f 
a-ooi si 8oo^ 



_* 
as the following example shows. 



42 IMPROPER MULTIPLE INTEGRALS 

Let /= 1 at the irrational points in 21 = (0, 1), 



s w, for # = in 21. 
n 



Then / / 

= , J <? = 1. 

X 



Again let / = 1 for the irrational points in 21, 

= n for the rational points x = 

w 

Then 



/=!. 

!t .0 

43. 1. 

; (2 



3) JTA=- r/, r/.< - r/, (4 

' / ^ ^l ~ J W 

provided the integral on either side of the equations converges, or 
provided the integrals on the right aide of the inequalities converge. 

Let us prove 1); the others are similarly established. Effecting 
a cubical division of space of norm d, we have for a iixed /3, 



f ^lim 

^8 </-'> 

= limS^f t rf t = f /. (5 

*= Va J ^ 



Thus if either integral in 1) is convergent, the passage to the 
limit /3 = oo in 5), gives 1). 



2. If \ fis convergent, \ g converge. 

If I f is convergent^ I h converge. 
*/^( /5( 

This follows from 1 and from 88. 



GENERAL THEORY 43 



3. If I / is convergent, we cannot say that I / is always con- 

/$r Aft 

imilar remark holds for the lower integra 

/ = 1 at the rational points of 21 = (0, 1) 
Then 



vergent. A similar remark holds for the lower integral. 

For let 

J = A 

= at the irrational points. 

x 



4. That the inequality sign in 2) or 4) may be necessary is 
shown thus : 

Let -j 

/== for rational x in 21 = (0, 1) 

\Jx 

= for irrational x. 

Then r r 

J^ = , J/=2. 

44. 1. ff^fg- lim f h, (1 

j/ = lim f 9-fy ( 2 

provided, 1 the integral on the left exists, or 2 the integral and the 
limit on the right exist. 

For let us effect a cubical division of norm d. The cells con- 
taining points of 21 fall into two classes : 

a) those in which /is always <0, 

6) those in which /is >0 for at least one point. 

In the cells a), since /= g A, 

Max/ = Max (g - A) = Max g - Min A, (3 

as Max g = 0. In the cells 6) this relation also holds as Min A = 0. 
-Thus 3) gives - - 

f=] a-\ *- (^ 

*/9f 'M 



44 IMPROPER MULTIPLE INTEGRALS 

Let now a, /3 = oo. If the integral on the left of 1) is conver- 
gent, the integral on the right of 1) is convergent by 43, 2. Hence 
the limit on the right of 1) exists. Using now 42, 2, we get 1). 

Let us now look at the 2 hypothesis. By 42, 2, 

r r 

Thus passing to the limit in 4), we get 1). 
2. A relation of the type 

X'-X'-X* 

does not always hold as the following shows. 



ra 



Example. Let/ = n at the points x = 



92n 



ra 

, 



= 1 at the other points of SI = (0, 1). 

Then f/=-l f 17 = f h = 0. 

J% JK J.X 

45. 7/ ' I f is convergent, it is in any unmixed part 53 0/'2l 
J% 

Let us consider the upper integral first. By 48, 2, 

X' 

exists. Hence a fortiori, 

X' 

exists. Since 21 = 53 -f is an unmixed division, 
f h= f h+ f h. 

J%afi !#<# J&ap 

Hence I h < ) h. 

*^9T 



GENERAL THEORY 45 

As the limit of the right side exists, that of the left exists also. 
From this fact, and because 1) exists, 



exists by 44, 1. 

A similar demonstration holds for the lower integral over 



46. J/2lx, 21 2 " ^mform an unmixed division 0/21, then 

f/ = f/+ ... + f /, (1 

S& Jl thm 

provided the integral on the left exists or all the integrals on the 
right exist. 

For if 2l m$ a/3 denote the points of 2l a|3 in 2l m , we have 

- r + ... + r . (2 

-f J* 



Now if the integral on the left of 1) is convergent, the integrals 
on the right of 1) all converge by 45. Passing to the limit in 2) 
gives 1). On the other hypothesis, the integrals on the right of 
1) existing, a passage to the limit in 2) shows that 1) holds in 
this case also. 

47. If \ f and \ f converge, so does I I/I, and 
Jy Jw ^21 

f|/|< /*/-/"/ (1 

*/ *A A 



(2 

For let ^.3 denote the points of 21 where 

0<|/| 

Then since , /. , 

I/I = 



<f ff +Ch (3 

c/9f *s9I 

<ff-ff by 43,1. (4 

xflj *s 9ft 

Passing to the limit in 3), 4), we get 1), 2). 



4(> IMPROPER MULTIPLE INTEGRALS 

48. 1. If I \f\ converges^ both I /converge. 
J% j/a 

For as usual let ty denote the points of 21 where/>0. 
Then 



is convergent by 38, 3, since J |/| is convergent. 

*^*i 
Similarly, - 

)(-/)=- // 
An /< 

is convergent. The theorem follows now by 39. 

2. If I | /| converges, so do 
* 



For by 1, 



both converge. The theorem now follows by 43, '2. 
3. For 

ff (2 

V2f 

/>//^ ^ converge it is necessary and sufficient that 



i convergent. 

For if 3) converges, the integrals 2) both converge by 1. 
( hi the other hand if both the integrals 2) converge, 



converge by 38, 2. Hence 3) converges by 47. 
4. Iffis integrable in 31, so is \f\. 
For let Aft denote the points of 21 where < |/| < /3. Then 



and the limit on the right exists by 3. 



GKNKUAL THKORY 47 

But by 41, l,/ is integrable in A ft . Hence |/| is integrable in 
A ft by I, 720. Thus 



49. From the above it follows that if both integrals 



converge, they converge absolutely. Thus, in particular, if 



converges, it is absolutely convergent. 

We must, however, guard the reader against the error of sup- 
posing that only absolutely convergent upper and lower integrals 

exist. 

Example. At the rational points of 21 =(0, 1) let 



At the irrational points let 

/< = -- 

x 
Here 



(/=-oo. 

Jy 



Thus, / admits an upper, but not a lower integral. On the 
other hand the upper integral of / does not converge absolutely. 

For obviously 



50. We have just noted that if 

f /(*! 
J . 



is convergent, it is absolutely convergent. For m = 1, this result 
apparently stands in contradiction with the theory developed in 
Vol. I, where we often dealt with convergent integrals which do 
not converge absolutely. 



48 IMPROPER MULTIPLE INTEGRALS 

Let us consider, for example, 

sin- 



If we set x = -, we get 
u 



U 



which converges by I, 667, but is not absolutely convergent by 
I, 646. 

This apparent discrepancy at once disappears when we observe 
that according to the definition laid down in Vol. I, 

J = R lim I fdx, 

a=0 J a 

while in the present chapter 

7= lim I fdx. 

a, /3 = ao ^a/ 

Now it is easy to see that, taking a large at pleasure but fixed, 
I fdx ==oo as fi = QO, 

^a/3 

so that 7does not converge according to our present definition. 

In the theory of integration as ordinarily developed in works 
on the calculus a similar phenomenon occurs, viz. only absolutely 
convergent integrals exist when m > 1. 

51. 1. If J |/| is convergent, 
Jyn 

Iff < f I/I- (1 

\J% J% 

For 9l tt/3 denoting as usual the points of 21 where </<y8 
we have ' ~ 

i/. 



. 



Passing to the limit, we get 1). 



GENERAL THEORY 49 

2. If I |/ 1 is convergent, ( f are convergent for any JB<91. 

*^2l Ji33 

For j I/ 1 is convergent by 38, 4. 
*/<B 

Hence -- 

S* 

~sd 

converge by 48, 3. 

3. If, 1, f |/ 1 is convergent and Minf is finite, or if, 2, I f is 

J% 5/K 

convergent and Max / i finite, then 



zs convergent. 

This follows by 36 and 48, 3. 



52. ie/>0 in 21. Xe^ A0 integral 



converge. If 

then for any unmixed part 33 < 91, 

f/=f/+"', (2 

./$ -*/% 

<<*'< (3 



For let 21 = + S. Then ^=SS ft + ^ f is an unmixed division, 

Also 



f + by 1) 

^ 

f + f +. 

j^ e 



f><) IMPROPER MUI/riPLK INTEGRALS 

Hence 



u . (4 

j93 & *$0 :<$,fi 

From 2) 



-/* 

= -{/-/! by 4) 

l/( J/(B ' 

which establishes 3). 

53. If the, integral \ \ f \ 

8, then 



> 0, a- > 0, 
for any 33 < 31 situh tfi 



< (2 



Let us suppose first that/>0. If the theorem is not true, 
there exists, however small o->0 is taken, a 33 satisfying 3) such 
that 



Then there exists a cubical division of space such that those 
points of 31, call them (5, which lie in cells containing a point of 
3J, are such that (5<cr also. Moreover S is an unmixed part of 21. 
Then from 4) follows, as/>0, that 

f f>6 (5 

c/(" V 

also. 

Let us now take ft so that 



%/r = L + 
Then 



and 0<'<a 

by 52. But f 

1<#S,<* 



GENERAL THEORY 51 

Let now @<r < e, then 



which contradicts 5). 

Let us now make no restrictions on the sign off. We have 



But since 1) converges, the present case is reduced to the pre- 
ceding. 



54. 1 . Let I f I converge 
J<% ' 



Let as usual 3( a8 denote the points of 21 at which # </ < /3. Let 
A ab be such that each 2l a /3 lie* in some A ab in which latter f is limited. 
Let 3X/3 = A ab 2la0 and let a, b ~ ao with a, f$. Then 

lim IXp = 0. 

a, 0=n 

For if not, let 

Inn > a = /, I > 0. 

a, 0= x, 

Then for any < X < /, there exists a monotone sequence Ja n , /3 n | 
such that 

)a n j3 n > ^ f r w > some m. 



Let ^n=Min(a n , &), then |/| > ^ n in S) an ^ n , and /i n =oo. 
Hence r 

J 

^an 

But !) an p n being a part of 



by 38, 3. This contradicts 1). 

2. Definition. We say ^4 fli b is conjugate to 2l a /s with respect to/. 



55. 1. J.8 ?/*maZ Zef -a<f<j3in 3l aj3 . Ze* 
Z^^ ^4^ 6^ conjugate to 2l a/3 w;zYA reference to f; and A b conjugate 
to 2(0 with respect to |/|. 



52 IMPROPER MULTIPLE INTEGRALS 

If, 1, 



or if, 2, 

lim J 

=00 ^ 



77 T? 

lim I /= I / 

a, 6=00 *Z^4 a, 6 ^1^1 

For, if 2 holds, 1 holds also, since 



Thus case 2 is reduced to 1. Let then the 1 limit exist. 
We have 



as 4) in 44, l shows. Let now 

3X 3 = 
Then, 



L 9<\ 4 ff<L 9+ I 9- 

J *o# JA >> ^ *^>/S 



(3 



But 5DB = as , /3 = oo, by 54. Let us now pass to the limit 
, /8 = oo in 3). Since the limit of the last term is by 53, 54, we 
get -^ 

lim f # = lim J g. (4 

a, /3co ^St^ 0,6 = 00*^^0,6 

Similarly, * ~ 

lim I A = lim I A. (5 

a, /3-GO il^a/3 a, 6=-< *L A ab 

Passing to the limit in 2), we get, using 4), 5), 

f /= lim ( f g- f h } 

J * ,*- \ J *< *** f 

- lim 

a> ;>nao 

In a similar manner we may establish 1) for the lower integrals. 



GENERAL THEORY 53 

2. The following example is instructive as showing that when 
the conditions imposed in 1 are not fulfilled, the relation 1) may 
not hold. 

Example. Since / a ? 

I =+oo, 
/o x 

there exists, for any b n > 0, a < b n+1 < 6 n , such that if we set 



then n ^ n ^ ^ 

Cr 1 <Cr 2 < =00, 

as b n = 0. Let now 

/ = 1 for the rational points in 21 = (0, 1), 

= - for the irrational. 
x 

Then 



Let 

Let ^4 n denote the points of 21 in (6 n , 1) and the irrational points 
in (6 n +r *) 

Then J[>^-+. 

ii^n 

But obviously the set A H is conjugate to 210. On the other hand, 



while r 

lim I 

n-oo i^4n 



56. Jf ^Ae integral 
converges, then 



<r> 



(2 



for any unmixed part SB of 21 



54 IMPROPER MULTIPLE INTEGRALS 

Let us establish the theorem for the upper integral; similar 
reasoning may be used for the lower. Since 1) is convergent, 

fff (3 

JK 

and \ = lim f h (4 

a,0-r*/H aj8 

exist by 44, l. Since 3) exists, we have by 53, 

<' (5 



for any 53 < 21 such that 33 < some <r'. 

Since 4) exists, there exists a pair of values a, b such that 

4' 

since the integral on the right side of 4) is a monotone increasing 
function of a, b. 

Since 31 = 93 -f S is an unmixed division of 31, 

XC C 

h== J k + J<. ^ 

Since h > 0, and the limit 4) exists, the above shows that 
ft= lim I h , i>= lim I 7t 

a, /3=ao /SB a a, /3=^ ^afl 

exist and that 



Then a, 6 being the same as in 6), 

/i= f A + y, (8 

^Be> 
and we show that 

O^V<iy (9 

as in 52. Let now c > a, 6 ; then 



if we take g <-. 



GENERAL THEORY 55 

Thu8 ' <. (11 



by 44, 1. Thus 2) follows on using 5), 11) and taking <r <<r', a". 

57. If the integral I / converges and $Q U is an unmixed part of 
J.W 

2i such that $Q U = 31 as u = 0, then 

lim f /= f/. (1 

w=oc/ tt *? 

For if we set 31 = 33,, -f S M , the last set is an unmixed part of 91 
and S M = 0. Now 



Js M 
Passing to the limit, we get 1) on using 56. 

58. 1. Let ^ 

//, 1, Ae u/>/?t?r contents of 



== 8 a, =ao , 
^/, 2, A0 upper integrals off, g,f + g are convergent, then 



If\ holds, and if, 3, the lower integrals off, g,f-\-g are conver- 
gent, then 

(3 



Let us prove 2); the relation 3) is similarly established. Let 
Z> a . be a cubical division of space. Let S a denote the points of 
'Dap lying in cells of D^, containing no point of the sets 1). Let 



56 IMPROPER MULTIPLE INTEGRALS 

Then D aft may be chosen so that g a/3 = 0. 
Now 7? C 

J*fi a/ ""^aP 

since the fields are unmixed. By 56, the second integral on the 
right = as a, ft =00 . Hence 



lim f /= lim f f. 

a, //, a/3 a, /3=cc '<g a/8 



Similar reasoning applies to # and/-f 9- 
Again, 



Thus, letting a, /3=ao we get 2). 

2. TFAen ^A singular points of f, g are discrete, the condition 1 
holds. 

3. If g is integrable and the conditions 1, 2, 3 are satisfied, 



4. If f> g are integrable and condition 1 is satisfied^ f + g is i 
tegrable and 



5. 

provided the integral off in question converges or is definitely infinite. 
For 



Also _- _ 

lim 3X0 = lim ^ 

where 2l a/J refers to/. 

6. When condition 1 is not satisfied, the relations 2) or 3) 
may not hold. 



GENERAL THEORY 57 

Example. Let 91 consist of the rational points in (0, 1). 
Let 



at the point x = . Then 
n 



Now 



/=! + n , g = 1 n 

Then 
/ + g == 2 in 91. 



embrace only a finite number of points for a given a, {3. On the 
other hand, 

21,^ = 91 for/3>2. 

Thus the upper content of the last set in 1) does not == as 
, f3 = oo and condition 1 is not fulfilled. Also relation 2) does 
not hold in this case. For 



59. Ifc>Q, thenj^cf-ef^f, (1 

, (2 



provided the integral on either side is convergent. 
For 

f ifc>0 (3 



/ if<?<0. (4 

c/,a^ 

Let c > 0. Since 



n 
therefore ^ 



in this set. 



Hence any point of 9l f/ a/3 , is a point of 51 A ?. ? and conversely. 
Thus *-*=, s.fi whenc>0. 



58 IMPROPER MULTIPLE INTEGRALS 

Similarly ^ = a , . whenc<0 . 

6 C 

Thus 3), 4) give 



f < 0. 

^ * J 
c' c 

We now need only to pass to the limit a, /3 = oo 
60. Let one of the integrals 



converge. Iff = g, except at a discrete set !j) in 31, both integrals 
converge and are equal. A similar theorem holds for the lower 
integrals. 

For let us suppose the first integral in 1) converges. Let 



_ 

l/= }/+)/= )/. (2 

^81 ^y) ^ ^,1 

stt' 7 = Um Jar 9 = liln J ^ 

a a,p= 



then 
Now 



/, a/3 

Thus the second integral in 1) converges, and 2), 3) show that 
the integrals in 1) are equal. 

61. 1. Let f /, fff (1 

fa ^H 

converge. Letf>^g except possibly at a discrete set. Let 

$.* = /><,, #,,,) ; f = a/,.p-S>i ; 9^ = ^,^-^ 

If - 

f^ = 0, 9 ap =0, as , ^= oo, 

then - 

I /'> I y- ( 2 

/a y - /a f/ v 



RELATION BETWEEN THE INTEGRALS OF TYPES I, II, III 59 
For let @ aj8 be defined as in 58, l. Then 



Let , /9 = QO, we get 2) by the same style of reasoning as in 

58. 

2. If the integrals 1) converge, and their singular points are dis- 
crete, the relation 2) holds. 
This follows by 58, 2. 

8. If the conditions of 1 do not hold, the relation 2) may not 
be true. 

Example. Let 21 denote the rational points in (0*, 1*). Let 
f=n at 3 = - in 21. 



Then f>y in 21. 

But 



r r 

l/=0 I 

/a %/H* 



Relation between the Integrals of Type* /, //, /// 

62. Let us denote these integrals over the limited field 21 by 

<? , Vk , P 

respectively. The upper and lower integrals may be denoted by 
putting a dash above and below them. When no ambiguity arises, 
we may omit the subscript 21. The singular points of the inte- 
grand/, we denote as usual by $. 

63. Tf one of the integrals P is convergent, and Q is discrete, the 
corresponding O integral Converges, and both are equal. 

L^ _ _ 

PH = Pa a + Aty using the notation of 28, 

= <?,+ Fj. 
Now = as 8 = by 56. 



60 IMPROPER MULTIPLE INTEGRALS 

Hence 



5-0 



= Cki by definition. 

64. If O is convergent, we cannot say that P converges. A 
similar remark holds for the lower integrals. 

Example. For the rational points in 31 = (0, 1) let 

f (<r^ 
J \f) 

for the irrational points let 



Then 



On the other hand, _ - 

P H = lim j / 

a^-ft/K a/3 

does not exist. For however large y8 is taken and then fixed, 

I f^= oo as = 00. 
J **p 

65. If C is absolutely convergent and $ & discrete, then both P 
converge and are equal to the corresponding C integrals. 

For let D be any complete division of 21 of norm S. Then 

J*afi A aM A' aM 

using the notation of 28. Now since 

Ck |/| converges, C%' 8 1/| = as S = 0. 
But 

(2 



Again, D being fixed, if are sufficiently large, 
f f^C^f > , /S>/3 . 

^6 



RELATION BETWEEN THE INTEGRALS OF TYPES I, II, III 61 
Hence 1), 2) give 

f /= CK* + *' K| < 1 for any 8 < some S . 

^a|3 * 

On the other hand, if S is sufficiently small, 

tf=tfa s + e" |"|<! forS<S . 

Hence f /-ff + e'" |e"'|<. 

X 

Passing to the limit a, /3 = oo, we get 



66. .// P r si/ is absolutely convergent, the singular points Q are 
discrete. 

For suppose $ > 0- ^ et 33 denote the points of 21 where 
>yS. Then $8 > <J for any ^8. Hence 



as /8 == oo unless ^ = 0. 

67. Jf F^/ is absolutely convergent, so is O. 

For let D be a cubical division of space of norm d. 

Then 

|/ 1 < some /8 in 2l d . 

Hence Jj/|S/J/|, < r.|/|. 

Hence (7 is absolutely convergent. 

68. Letf>Q. If V%f is convergent, there exists for each >0, 
a a- > 



/or any _ 

. (2 



<]2 IMPROPER MULTIPLE INTEGRALS 

For 



for X sufficiently large. Let X be so taken, then 



Al8 ' X/A<XS<|, (4 

if a is taken sufficiently small in 2). 
From 3), 4) follows 1). 

69. If V<&f is absolutely convergent, both converge and are 
equal to the corresponding \ r integrals. 

For by 67, O is absolutely convergent. Hence C converge by 65. 

Thus ~ C e 

OW/= I f 4- a , la <- for some d. 
*/H ( / 8 

Also - r 

V^f = I ,/A^t -f P , \ P \ < - tor some X, JJL. 

Hence 

Now 
But 



and 7 < | if d is sufficiently small, and for any X, /A, by 68. 

o 

Taking a division of space having this norm, we then take X, 
so large that 

/AM=/ i Sid- 
Then Q 

i) = a p 7, 

and hence , , 

| 7? | < . 

From this and 1) the theorem now follows at once. 



ITERATED INTEGRALS 83 

Iterated Integrals 

70. 1. We consider now the relations which exist between the 

integrals 



and 

(2 

where 21 = 93 S lies in a space 9t m , m = p + q, and 83 is a projection 
of 21 in the space 9? p . 

It is sometimes convenient to denote the last q coordinates of a 
point x = (x l x p Xp+i x p+q ) by y l y q . Thus the coordinates 
#! x p refer to 93 and y l - y g to (. The section of 21 correspond- 
ing to the point x in S3 may be denoted by g x when it is desirable 
to indicate which of the sections S is meant. 

2. Let us set 



then the integral 2) is 



It is important to note at once that although the integrand / is 
defined for each point in 21, the integrand < in 4) may not be. 

Example. Let 21 consist of the points (#, y) in the unit square : 



n n 

Then 31 is discrete. At the point (#, y)'m 21, let 

~y 

Then f/ = by 32. 

c/21 

On the other hand ~ 

for each point of 93. Thus the integrals 2) are not defined. 



64 IMPROPER MULTIPLE INTEGRALS 

To provide for the case that <f> may not be defined for certain 
points of 5) we give the symbol 2) the following definition. 

f f/ = Urn f f/, (5 

&'' a^-oo^Jr 

where F = S when the integral 3) is convergent, or in the con- 
trary case F is such a part of S that 



and such that the integral in 6) is numerically as large as 6) will 
permit. 

Sometimes it is convenient to denote F more specifically by r oj8 . 

The points 93 aj8 are the points of S3 at which 6) holds. It will 
be noticed that each 93 a/3 in 5) contains all the points of 93 where 
the integral 3) is not convergent. Thus 



Hence when 93 is complete or metric, 

lim 93 a p=. (7 

a,/3~co 

Before going farther it will aid the reader to consider a few 
examples. 

71. ^Example 1. Let 31 be as in the example in 70, 2, while/ = ri* 



at x = -. We see that 
n 



J>=- 



On the other hand 93 aj 8 contains but a finite number of points 
for any a, ft. Thus 



Thus the two integrals 1), 2) exist and are equal. 

Example 2. The fact that the integrals in Ex. 1 vanish may 
lead the reader to depreciate the value of an example of this kind, 
This would be unfortunate, as it is easy to modify the function so 
that these integrals do not vanish. 



ITERATED INTEGRALS 65 

Let 21 denote all the points of the unit square. Let us denote 
the discrete point set used in Ex. 1 by >. We define /now as 
follows : /shall have in $) the values assigned to it at these points 
in Ex. 1. At the other points A = 21 J),/ shall have the value 1. 

Then ///*/* 

1=1 + 1=1=1. (3 

/ JA J JA ^ 

On the other hand 33 aj3 consists of the irrational points in 93 
a finite number of other points. Thus 



Hence again the two 3), 4) exist and are equal. 
Let us look at the results we get if we use integrals of types I 
and II. We will denote them by and V as in 62. 
We see at once that 

(7 a== ra=P = l. 

Let us now calculate the iterated integrals 

Cfetffc, (5 

and F$ F<. (6 

We observe that 

C(i = 1 for x irrational 

= + QO for x rational. 

Thus the integral 5) either is not defined at all since the field 
$85 does not exist, or if we interpret the definition as liberally as 
possible, its value is 0. In neither case is 



Let us now look at the integral 0). We see at once that 



does not exist, as V& = 1 for rational x, and = +00 for irrational 
x. On the other hand 



Hence in this case 



f>6 IMPROPER MULTIPLE INTEGRALS 

Example 3. Let 31 be the unit square. 

Let 

f=: n for x = n even 
n 

= n for # = ~ M odd. 

n 

At the other points of 31 let/= 1. 
Then 



Here every point of 31 is a point of infinite discontinuity and 
thus ^ = 31. 

Here Cfo is not defined, as 31 5 does not exist; or giving the 
definition its most liberal interpretation, 



The same remarks hold for C^Ots 

-O V&* 

On the other hand Tr 

V* = 4-00, 

while v^ 

does not exist, since rr m 

K ff = n tor x = 
* ft 

= 1 for irrational x. 
Moreover T , Tr fr rr , 

58^ = - r 8 r c = + Qo. 

Example 4. Let 31 denote the unit square. Let 
/=n 2 for z = -, neven, 0<y<i 

W 71 

= n a for ^ = ~ , n odd, < y < - . 

~~ 



At the other points of 31 let/= 1. 
Then * 

r/=i 

^ 



ITERATED INTEGRALS 67 

Let us look at the corresponding C and V integrals. 
We see at once that 



Again the integral O^Cg does not exist, or on a liberal interpre- 
tation it has the value 0. Also in this example 



do not exist or on a liberal interpretation, they = 0. 
Turning to the F' integrals we see that 



while V% Fjg does not exist finite or infinite. 

Example 5. Let our field of integration 21 consist of the unit 
square considered in Ex. 4, let us call it @, and another similar 
square gf> lying to its right. Let / be defined over ( as it was 
defined in Ex. 4, and let/= 1 in . 

Then r r 

f/= f f=2- 

*/2l J&J& 

Also __ v __ 9 

C 5l - K 2l = ^ 

Then 1 

33S"" A 

while V^V^ does not exist, 

and Tr Tr 77- fr 



72. 1. In the following sections we shall restrict ourselves as 
follows : 

1 21 shall be limited and iterable with respect to SB. 

2 S3 shall be complete or metric. 

3 The singular points $ of the integrand /shall be discrete. 

2. Let us effect a sequence of superposed cubical divisions of 

space 

A* JV- 

whose norms d n = 0. 



68 IMPROPER MULTIPLE INTEGRALS 

Let 2l n = SS n - S n denote the points of 21 lying in cells of D n 
which contain no point of $. We observe that we may always 
take without loss of generality 

=*. 

For let us adjoin to 31 a discrete set ) lying at some distance 
from 21 such that the projection of on 9? p is precisely $Q. 

Let 4 = 2l + ) = S-(7 , <?=<5+c , c = 0. 

We now set . - - 

<p = / in VI 

= in 3\ 
A nen * * 7* 7* 

+ 1 ^,= j ^ 

ife 



Similarly 

Hence 

8. The set S n being as in 2, we shall write 



73. Let B^ n denote the points of 33 at which c n > <r. Then if 21 
is iterable, with respect to S3, 

lim S a n = 0. (1 

n=* 

For since 21 is iterable, 

21 = I S by definition. 

*/33 

Hence S considered as a function of # is an integrable function 
in. 

Similarly 



x, _ 

-X s " 



and S n is an integrable function in $Q. 



ITERATED INTEGRALS 69 

We have now ~ 7? , - ~ ^> A 

6 = Sn+C n , C n >0 

as S n , c n are unmixed. Hence c n is an integrable function in SB* 
But 



_ r 

:cw I Src 

-9* = L(Sn' 
/S 



As the left side = as n = oo , 

lim ( c n = 0. (2 

*/9^ V 

But 



As the left side == 0, we have for a given cr 

lim B n = 0, 
which is 1). 

74. ie 21 = 93 fo iterable. Let the integral 

ff , />0 



convergent and limited in complete 33. S n denote the points 
of S3 a which 

< 2 



lim g n = SB. (3 

7i==' 

For let . . . A 

<r 1 ><7 2 > =0. 

Since fft n === as w === oo by 48, we may take i>j so large, and 
then a cubical division of 9t p of norm so small that those cells con- 



taining points of B fflVv have a content <^/2. Let the points of 
93 lying in these cells be called B v and let SQj = 93 J? r Then 
S v 53j form an unmixed division of 93 and 



j is complete since SB is. 



70 IMPROPER MULTIPLE INTEGRALS 

We may now reason on S8 t as we did on 93, replacing ij/Z by i?/2 2 . 
We get a complete set $ 2 .<_$Bi such that 



Continuing we get ^ > , 
Thus ^ - 



Let now b = Dt;{8 ll j. 

Then r, (4 



by 25. 

Let b n denote those points of b for which 2) does hold. Then 
b = 5b n j. For let ft be any point of b. Since 1) is convergent, 
there exists a o\ such that 

at ft, 

for any c such that c <<r t . Thus ft is a point of bi/ t and hence of 
jb n j. Thus b n = b as b is complete. But S n > b n . 
Hence 



which with 4), gives 3). 

75. Let 21 = 93 S ie iterable. Let the integral 



^j convergent arid limited in complete 93- 

TA0W. /* 7i 

lim j ( /=0. (1 

n=oo & ^ n V 

For let D be a cubical division of 9? p of norm d. 
Then 



/ r ^" 

(f I /=lim2rf 4 Min I / = lim 

1^ /C n (/^Q t ilCn r/~0 



Let df[ denote those cells of D containing a point of S n where (5 n 
is defined as in 74. 



ITERATED INTEGRALS 71 

Let d" denote the other cells containing points of 33. Then 



where - 

0< \f<M. 

J& 

Hence 



4 
Letting d = 0, we get 

f/< 

^SB Cn 

Letting now n = oo and using 3) of 74, we get 1), since e is 
small at pleasure. 



76. Let 21 = 33 &e iterable with respect to 33, w/m?A / is com- 
or metric. Let the singular points $ of f be discrete. Then 



if, f>-& , f/< (* f/< f / (1 

J ' ^ J& J -"*/ . ~ / 

if, /<^ , r/<r r/< c/ (2 

47 t/ " - */2l' J<%Jss. J Jyi J v 



any on o/ /if? members in 1) may ft^ infinite. Then all 
that follow are also infinite. A similar remark applies to 2). 

Let us first suppose : 

/> , 53 is complete , 1 1 / is convergent. 

We have by 14, ~ -~ -~. -* 

I /< I I /< I /. 
*. ""iZr</ "~ ^n 

Passing to the limit gives 

r/<limf f /. (3 

Jsr ~ J&J&n J 

and also ^ ^ =- 

lim I I / <. I / , finite or infinite. (4 

JBfe ^a 

Now e > being small at pleasure, there exists a # such that 



72 IMPROPER MULTIPLE INTEGRALS 

But for a fixed n ^ 

I is limited in 93. 

/ ( 

Hence for Gr sufficiently large, 

f f< f/ , at each point of S3, 6? < 6^. (6 

*cSw ^r 

Then 



where F n , y n are points ot F in ( n , c w . 
Hence 

J$(t*Ln 

Now S3^ ; may not be complete ; if not let B tt be completed 93#. 
As S3 is complete, 



n ~JBoJ^ ' 

We may therefore write 8), using 5) 

-+//V /+/ f^ff+S f 

J%J& /,/^ii "LBffJy* ^*/n JUff^Vn 

By 75, the last term 011 the right = as 71 = oo. Thus passing 
to the limit, 

n/<lim f f/, (9 

_-_/=/& V 

since > is small at pleasure. 

On the other hand, passing to the limit Q- = oo in 7), and then 

w=oo, we get 

lira f f < f f . (10 

^=00 /.Z(S --?/ V 

Thus 3), 10), 9), and 4) give 1). 

M$ now suppose that the middle term of 1) is divergent. 
We have as before 

f f<]mif f < f/. 
i/a < ; ^/r n-^^ ^ ^^ 

Hence the integral on the right of 1) is divergent. 



ITERATED INTEGRALS 73 

Let us now suppose 93 is metric. We effect a cubical division 
of 9tp of norm d, and denote by B d those cells containing only 
points of 93. Then B d is complete and 



Let A d denote those points of 91 whose projections fall on JS d . 
Then A d is iterable with respect to B d by 13, 3, and we have as 

i Y- f-Vi^i i~iTir<iri i M rp r>ocjo 



in the preceding case 



If the middle integral in 11) is divergent, j is divergent and 1) 

holds, also if the last integral in 11) is divergent, 1) holds. Sup- 
pose then that the two last integrals in 11) are convergent. 
Then by 57 

limf f=f f. 

d=0*//W< //( 



limf = f. 

d=^Aa ^21 



Thus passing to the limit d = in 11) we get 1). 

Let us now suppose f > (?, Q > 0. 

Then 



and we can apply 1) to the new function g. 
Thus 



ffl= f fj/< (V (12 

?3i <33 J& *'yi 

Now 



by 58, 6, since $ is discrete. Also by the same theorem, 

C g = Cf + a iim s y = f / + or, (14 

/S /^ y - /g 

denoting by S y the points of S where 



and setting 

T = lim < 

y==X 



74 IMPROPER MULTIPLE INTEGRALS 

Now for any n 



Hence #H = lim C f # = <?lim fg n , 

x /33 c/g^ n-oo 4/93 

or (15 



Now for a fixed /&, 7 may be taken so large that for all points 
of S, 



Hence _ 

6 > Urn 

yes oo 

Hence > 



f 

33 



Hence (16 



and thus F is integrable in S3. 

This result in 14) gives, on using 58, 3, 



c r<7= c f 

*Z ?Ze 5/ ?^s 
From 12), 13), and 17) follows 1). 

77. As corollaries of the last theorem we have, supposing 21 to 
be as in 76, 

1. Iff is integrable in 31 andf> 6r, then 

J 21 Jjd J< 



2. -?// > ff and | is divergent, then 
J% 

^ divergent. 



ITERATED INTEGRALS 75 

3. If f > Gr and one of the integrals I I f is convergent, then 

/<*/( 

's convergent. 

78. Let 21 = S3 S be iterable with respect to 33, which last is com- 
plete or metric. Let the singular points $ be discrete. If 

f/' (1 

r IT/, (2 

^93 ^(S 
>oA converge, they are equal. 

For let J>!, D 2 be a sequence of superimposed cubical divisions 
is in 72, 2. We may suppose as before that each 93 n = 33- 
Since 1) is convergent 



Since /is limited in 3t n , which latter is iterable, 



This shows that 

s an integrable function in $8, and hence in any part of S3- 
From 3), 4) we have 

I f - f f <^ n>n<r (6 

- |Ja J^J&n 2 

We wish now to show that 





When this is done, 6) and 7) prove the theorem. 
To establish 7) we begin by observing that 



iim r r. 

,/i-^^^r 



76 



IMPROPER MULTIPLE INTEGRALS 



Now for a fixed n, , y8 may be taken so that F shall embrace all 
the points of < for every point of SB- Let us set 



r = 



Then 






(8 



As 



lira f f = f f by I, 724. 
,0-v a 3~X *AB*X 



On the other hand, 



< 



i/i< 

1 - 7 ' - 

Thus 7) is established when we show that 



To this end we note that |/| is integrable in 31 by 48, 4. Hence 
by 77, i, 



Also by I, 734, 



From 10), 11) we have for n > j 
Ar ' Ar ' J<y\*/(s 





2 



( 12 



since the left side == 0. 
But as in 8) 



Passing to the limit 6?= oo gives 



n!f|= f f |/|+ f f I/I. 
- - J SB J ^ ' JsaJc* ' 



This in 12) gives 9). 



CHAPTER III 
SERIES 

Preliminary Definitions and Theorems 
79. Let }, a<p a% - be an infinite sequence of numbers. 
The symbol A = a 1 + <*t + <%+. (1 

is called an infinite series. Let 

yl n = a 1 + ^+ ..- + a n . (2 

If lim A H (3 

=oo 

is finite, we say the series 1) is convergent. If the limit 3) is infi- 
nite or does not exist, we say 1) is divergent. When 1) is conver- 
gent, the limit 3) is called the sum of the series. It is customary 
to represent a series and its sum by the same letter, when no con- 
fusion will arise. Whenever practicable we shall adopt the fol- 
lowing uniform notation. The terms of a series will be designated 
by small Roman letters, the series and its sum will be denoted by 
the corresponding capital letter. The sum of the first n terms of a 
series as A will be denoted by A n . The infinite series formed by 
removing the first n terms, as for example, 



will be denoted by A n , and will be called the remainder after n 
terms. 

The series formed by replacing each term of a series by its nu- 
merical value is called the adjoint series. We shall designate it 
by replacing the Roman letters by the corresponding Greek or 
German letters. Thus the adjoint of 1) would be denoted by 

A = j -f 2 4- 3 4- = Adj A (5 

Where n=K|. 

77 



78 SERIES 

If all the terms of of a series are >0, it is identical with its 
adjoint. 

A sum of p consecutive terms as 



----- n+p 

we denote by A n ^ p . 

B=a^ + a^ + a^-\ ---- , 

be the series obtained from A by omitting all its terms that vanish. 
Then A and B converge or diverge simultaneously, and when conver- 
gent they have the same sum. 



Thus if the limit on either side exists, the limit of the other side 
exists and both are equal. 

This shows that in an infinite series we may omit its zero terms 
without affecting its character or value. We shall suppose this 
done unless the contrary is stated. 

A series whose terms are all > we shall call a positive term 
series; similarly if its terms are all < 0, we call it a negative term 
series. If a n > 0, n>m we shall say the series is essentially a pos- 
itive term series. Similarly if a n < 0, n>m we call it an essen- 
tially negative term series. 

If A is an essentially positive term series and divergent, 
lim A n = -f <x> ; if it is an essentially negative term series and di- 
vergent, lim A n = QO. 

When lim A tl = oo, we sometimes say A is 00. 

80. 1. For A to converge, it is necessary and sufficient that 

e>0, m, |A n , p |<e, n>m, p = l, 2, ... (1 

For the necessary and sufficient condition that 

lim A n 

n=w 

exists is A A A , 

> 0, m, \ A v A n \ < e, v, n > m. (2 

But if v = n + p 



Thus 2) is identical with 1). 



PRELIMINARY DEFINITIONS AND THEOREMS 79 

2. The two series A, A a converge and diverge simultaneously. 
When convergent, __ 

A = A. + A.. (3 

For obviously if either series satisfies theorem 1, the other 
must, since the first terms of a series do not enter the relation 1). 

On the other hand, A __ A A 

-^+P == A t -{ A 9 ^ p . 

Letting p==<x> we get 3). 

3. If A is convergent, A n = 0. 

For lim A n = lira (A - A n ) 

= A lim A n = A A 
= 0. 

For A to converge it is necessary that a n == 0. 

For in 1) take p = 1 ; it becomes 

I +! I < c n > m 

We cannot infer conversely because a n = 0, therefore A is con- 
vergent. For as we shall see in 81, 2, 

l + i + + - 
is divergent, yet lirn a n = 0. 

4. The positive term series A is convergent if A n is limited. 
For then lim A n exists by I, 109. 

5. A series whose adjoint converges is convergent. 
For the adjoint A of A being convergent, 

>0, m, A n , p |<e, n>m, p =1, 2, 3 
But 



Thus 
and A is convergent. 

Definition. A series whose adjoint is convergent is called 
absolutely convergent. 



80 SERIES 

Series which do not converge absolutely may be called, when 
necessary to emphasize this fact, simply convergent. 

6. Let A = a l -f 2 + 
be absolutely convergent. 

Let B = a tj -f a l2 + ; 4 1 <* 2 < 

fo any series whose terms are taken from A, preserving their relative 
order. Then B is absolutely convergent and 

\B\<A. 
For | J BJ<B ra <A n <A, (1 

choosing n so large that A n contains every term in B m . Moreover 
for m > some m 1 , A n B m > some term of A. Thus passing to the 
limit in 1), the theorem is proved. 

7. Let A a l + a%+ The series B = ka l + ka% + 
converges or diverges simultaneously with A. When convergent, 



We have now only to pass to the limit. 

From this we see that a negative or an essentially negative term 
series can be converted into a positive or an essentially positive 
term series by multiplying its terms by k= 1. 

8. If A is simply convergent, the series B formed of its positive 
terms taken in the order they occur in A, and the series C formed of the 
negative terms, also taken in the order they occur in A, are both 
divergent. 

If B and are convergent, so are B, F. Now 
A n = B ni 4- r 2 , n = ^ + n r 

Hence A would be convergent, which is contrary to hypothesis. 
If only one of the series B, is convergent, the relation 



shows that A would be divergent, which is contrary to hypothesis. 



PRKLIMfXAKY DEFINITIONS AND THRORKMS 81 

9. The following theorem often affords a convenient means of 
estimating the remainder of an absolutely convergent series. 

Let A = a l -f a 2 + be an absolutely convergent series. Let 
_Z? = 6j -f > 2 -f be a positive term convergent series ivhose sum is 
known either exactly or approximately. Then if \ a n \ < 6 n , n > m 

\A n \<B n <. 



<B n <B. 

Letting p == a> gives the theorem. 

EXAMPLES 

81. 1. The geometric series is defined by 



The geometric series is absolutely convergent when \g\< 1 and di- 
vergent when |</|>1. When convergent, 

ft 1 (f> 

u- . 4 



. 

Hence -, 

a - 1 _ ff 

^^n ^ ^ 

1-^ 1-^ 

When |^|< 1, lim^ w = 0, and then 

1 



lim# w = : 

When \g\ >1, lim# w is not 0, and hence by 80, 3, & is not conver- 
gent. 

2. The series - - JL^ J^ J_ , /g 



82 SERIES 

is called the general harmonic series of exponent p. When /* = 1, 
it becomes j, 1 + ^ + ^ + { + ... (4 

the harmonic series. We show now that 

The general harmonic series is convergent when /i > 1 and is di- 
vergent for /*<!. 

Let /*>!. Then 

l + l^l+l ^J-^ . a< l 

2* 3<* 2* 2* 2^ 2'"- 1 ' * 

1, 1 + 1 + .1 <1 - 1 , 1 + 1 = 1 = ,2 

4,* ^ 5/ot ^ tj,x ^ 7/01 ^ 4/a ^ 4/m ^ 4^ ^ 4^ 4^ ^ ' 



< + + ... + = = ^3 etc. 
15" 8^ 8^ 8^ 8* <J/ ' 



Let 7i < 2". Then 



Thus lim H n exists, by I, 109, arid 

1 



Let ^<1. Then .. . 



Thus 3) is divergent for //,< 1, if it is for /i = 1. 
But we saw, I, 141, that 

lim J n = GO, 
hence 7 is divergent. 

It is sometimes useful to know that 



In fact, by I, 180, 



^"- =1. (6 

log n 



lira -^a- = lim "~ ""' = lim 



log n log n - log (n - 1) i og f n 



PRELIMINARY DEFINITIONS AND THEOREMS 83 

Since * n > log n > l^n - we have 

. (7 



n l r n 

Another useful relation is 



For log(l -f w) logm = log( 1 + ~ ) < ~. 

\ w/ m 



(8 



Let w = l, 2>--n. If we add the resulting inequalities we 
get 8). 

3. Alternating Series. This important class of series is defined 
as follows. Let a l > # 2 > # 3 > =0. 

Then A = a j a 2 + 3 4 -f- (9 

whose signs are alternately positive and negative, is such a series. 
The alternating series 9) is convergent and 



For let p> 3. We have 
A n>f = (- l)]a n+1 - a n+2 + -. ( 

-(-l)-P. 
If /? is even, 

^ = (n^l - fln+ 2 ) + - + (n4-p 

If jo is odd, 

P = ( a n+i - n+ 2 ) + +(a n+p 
Thus in both cases, 

^><*n+l-n+ 2 >0. (11 

Again, if p is even, 



* In I, 461, the symbol " lim " in the first relation should be replaced by lira. 



84 SERIES 

If p is odd, 

P = a n+ 1 (n+2 ^n+s) (#n+p-l ~ #+p) 

Thus in both cases, 

P < a n+l - (a n+2 - a n + 8 ) < n+1 . (12 

From 11), 12) we have 

< n+l ~ n+2 < | ^, p | < n fl ~ ( n+2 - "n+a)- 

Hence passing to the limit p = oo, 



moreover, _._ n 

^n M " 

Example 1. The series 

i-i + *-i+- 

being alternating, is convergent. The adjoint series is 



which being the harmonic series is divergent. Thus 13) is an 
example of a convergent series which is not absolutely convergent. 

Example 2. The series 



V2 - 1 V2 + 1 V3 - 1 V3 + 1 

is divergent, although its terms are alternately positive and nega- 
tive, and a n == 0. 

For 



2 77? - 1, 

If now A were convergent, 

lira A n = lim .A a 
by I, 103, 2. 



PRELIMINARY DEFINITIONS AND THEOREMS 85 

4. Telescopic Series. Such series are 

A = (! - a a ) 4- ( 2 - 8 ) + (03 - a 4 ) + 

We note that 

A n = O 1 -tf 2 )+ + (a n -a n41 ) 

= ! a nfl . (14 

Thus the terms of any A n cancelling out in pairs, A n reduces to 
only two terms and so shuts up like a telescope. 
The relation 14) gives us the theorem : 

A telescopic series is convergent when and only when lim a n exists. 

A = a l -f # 2 + denote any series. 
Then a n = A n -A n ^ , 4,-0. 

Hence A = (A l - A ) + (^ 2 - AJ + (A 3 - AJ + - 
This shows us that 
Any series can be written as a telescopic series. 

This fact, as we shall see, is of great value in studying the 
general theory of series. 

Example 1. 



1 2 2 3 3 4 



; l n. 

Thus A is a telescopic series and 

1 

n 
Example 2. Let a v # 2 , a 8 , > 0. Then 

A *^ a n 

A __ > n 

^A / , ~ ~ ' 



1 + !> (!+_,) 
is telescopic. Thus 






and A is convergent and < 1 . 



86 SERIES 



Examples. 4=X- - \ - - x^ 0, -1, -2, 

* i O + rc- 



1 # -f 
is telescopic. 

1 1-1 



x x + n x 



82. Dints Series. Let A = a l + # 2 + fo a divergent positive 
term series. Then 



is divergent. 
For 



2 



(^m-Hl "I" '" "H 



--m ~ m, p -m+p 

Letting m remain fixed and jt? = oo, we have D m >l, since 
m-hp ==oo. Hence D is divergent. 

Let 4=1 + 1 + 1+ ... Then A n = n. 

Hence j) = i + + i+... is divergent. 

Let 4=1 + | + J 

Then -. - 



is divergent, and hence, a fortiori, 



But A n _^ > log n. 

Hence -i -j 



21og2 31og3 
is divergent, as Abel first showed. 



PRELIMINARY DEFINITIONS AND THEOREMS 87 

83. 1. Abel's Series. 

An important class of series have the form ' 

B = a^j + a^ + a B t B + (1 



As Abel first showed how the convergence of certain types-'of 
these series could be established, they may be appropriately called 
in his honor. The reasoning depends on the simple identity 
(AbeFs identity), 



where as usual A n ^ m is the sum of the first m terms of the re- 
mainder series A n . From this identity we have at once the fol- 
lowing cases in which the series 1) converges. 

2. Let the series A = a^ + a% -f- and the series 2|f n+1 t n \ 
converge. Let the t n be limited. Then B = a l t 1 + a 2 t^ -f converges. 

For since A is convergent, there exists an m such that 

\A ntp \<e; n>m, jt? = 1, 2, 3 ... 
Hence 

i^nJ<e{|*n+l-W2| + |*n+2-^+3|+ - + | *n + p \ } - 

3. Let the series A = a^ -|- a 2 -f- converge. Let t v t^ t^-' be a 
limited monotone sequence. Then B is convergent. 

This is a corollary of 2. 



4. Let A = a l + # 2 -f- 5e 3M<?A ^Aa^ \A n \ < Gr, n = 1, 2, 
2 1 n+1 ^ n | converge and t n = 0. 2%e^ JS is convergent. 

For by hypothesis there exists an w such that 

' |^+l-*+2| + |*n+2-*iM.3|+ "+IWl< 

for any n > m. 

5. Let \A n \<& and t^ > 2 > f 3 > =0. TAen 5 i convergent. 
This is a special case of 4. 



88 SERIES 

6. As an application of 5 we see the alternating series 



is convergent. For as the A series we may take J. = 1 1 + 1 

1+ ... as \A n \<l. 

84. Trigonometric Series. 

Series of this type are 

= # + a l cos x + a 2 cos 2 x + # 8 cos 3 a? + (1 

$ = flj sin x + a 2 sin 2 x -f # 3 sin 3 x + (2 

As we see, they are special cases of Abel's series. Special cases 
of the series 1), 2) are 

F = , + cos x + cos 2 x -f- cos 3 x + (3 

2 = sin x 4- sin 2 # + sin 3 x + (4 

It is easy to find the sums F n , 2 n as follows. We have 

. . , 2ra-l 2m + l 

2 sin mx sin x cos -- # cos -- - x. 



Letting m = 1, 2, w and adding, we get 

2 sin x 2 n = cos \ x cos n #. (5 

^ 

Keeping # fixed and letting w = oo, we see 2 n oscillates between 
fixed limits when x^ 0, 2 TT, 

Thus 2 is divergent except when x= 0, TT, 
Similarly we find when x 3= 2 

r - s 





*1 v 

sin \x 

Hence for such values T n oscillates between fixed limits. For 
the values x = 2 mir the equation 3) shows that T w = + oo. 
From the theorems 4, 5 we have at once now 

If 2 1 a n+l a n | converges and a n = 0, and hence in particular if 
a i> a 2 '" ^-> t'h e series 1) converges for every x, and 2) converges 
for x^2 WITT. 



If in 3) we replace x by x 4- TT, it goes over into 

A = cos x -f cos 2 # cos 3 x -f (7 



PRELIMINARY DEFINITIONS AND THEOREMS 89 



Thus AH oscillates between fixed limits if x^ (2 m I)TT, 
when n ^ oo . Thus 

J/ 2 1 +! -|- a n converges and a n = 0, and hence in particular if 
a l > # 2 > - =0, the series a a l cos # -f- a 2 cos 2 x a% cos 3 x -f 
converge* for # = (2 m 1) TT. 

85. Power Series. 

An extremely important class of series are those of the type 

P = a + ^(z a) + a^(x a) 2 -f a 3 (>--a) 3 + (1 

called power series. Since P reduces to # if we set x = #, we see 
that every power series converges for at least one point. On the 
other hand, there are power series which converge at but one 
point, e.g. 

a + l](x- a ) + 2\(x-a)* + Xl(z-a)* + - (2 

For if x*f* a, lim n\ x a \ n = oo, and thus 2) is divergent. 



1 . If the power series P converges for z=b, it converges absolutely 

within n , . 

AGO > X=|a-i|. 

Jf jP diverges for x = b, it diverges without J9 A (a). 

Let us suppose first that P converges at b. Let # be a point in 
Z> A , and set | x a \ =f. Then the adjoint of P becomes for this 



u L X 

lim n X n = 0, 

since series P is convergent for x = b. 
Hence ^ n ^ i . 



and II is convergent. X 

If P diverges at x = 6, it must diverge for all V such that 
| J' a \ > X. For if not, P would converge at b by what we have 
just proved, and this contradicts the hypothesis. 



90 SERIES 

2. Thus we conclude that the set of points for which P con- 
verges form an interval (a p, a + p) about the point a, called 
the interval of convergence ; p is called its norm. We say P is 
developed about the point a. When a = 0, the series 1) takes on 
the simpler form + ^ + 0,2? + 

which for many purposes is just as general as 1). We shall 
therefore employ it to simplify our equations. 

We note that the geometric series is a simple case of a power 
series. 

86. Cauchy's Theorem on the Interval of Convergence. 
The norm p of the interval of convergence of the power series, 
P = a Q + a l x + a t2 x* + 

is given by i 

- = hmVa n ^n #nr 

p 

We show II diverges if >/>. For let 



p 

Then by I, 338, l, there exist an infinity of indices i^ t a for 
which 

Hence 
and thus , 

since /3>l. Hence y p n 

n 

is divergent and therefore H. 

We show now that II converges if < p. For let 



Then there exist only a finite number of indices for which 



Let m be the greatest of these indices. Then 
V n </3 n>m. 



TESTS OF CONVERGENCE FOR POSITIVE TERM SERIES 91 
Hence 

and 

Thus 



and IT is convergent. 

Example 1. 2 . 

1 4- 4- 4- -4- ... 

1! 2! 31 

Here vX^-L^O by I, 185, 4. 

vn! 

Hence /? = oo and the series converges absolutely for every x. 
Example 2. x a? tf> 



Here ^ = ^ = 1 by I, 185, 8. 

Vn 

Hence p= 1, and the series converges absolutely for | x |< 1. 

Tests of Convergence for Positive Term Series 
87. To determine whether a given positive term series 



is convergent or not, we may compare it with certain standard 
series whose convergence or divergence is known. Such com- 
parisons enable us also to establish criteria of convergence of 
great usefulness. 

We begin by noting the following theorem which sometimes 
proves useful. 

1. Let A, B be two series which differ only by a finite number of 
terms. Then they converge or diverge simultaneously. 

This follows at once from 80, 2. Hence if a series A whose 
convergence is under investigation has a certain property only 



02 SERIES 

after the wth term, we may replace A by A m , which has this 
property from the start. 

2. The fundamental theorem of comparison is the following : 

Let -4 = a x 4- </ 2 4- > B = &! 4- J 2 4- i &00 positive term series. 
Let r > denote a constant. If a n < rb n , A converges if B does and 
A < rB. If a n > rb n , A diverges if B does. 

For on the first hypothesis 

A n <rB n . 
On the second hypothesis 

A>rB n . 
The theorem follows on passing to the limit. 

3. From 2 we have at once : 

Let A = a l 4- # 2 + '" ^ == ^i "^ ^2 "^ "" ^ e ^ wo p s ^i ve term series. 
Let r, s be positive constants. If 

r<-~<i* ra= 1, 2, 

or if 

lim -~ 

exists and is * 0, A and B converge or diverge simultaneously. If 
B converges and == 0, A also converges. If B diverges and ^- =00, 
A also diverges. 

4. Let A = a x 4- ^ 2 "^" '"* B = b l -\- b 2 + i^ positive term series. 
If B is convergent and 

A converges. If B is divergent and 

a n+\ ^ g_n-f I 

j^, * i 

rt n ft n 

yl diverges. 

For on the first hypothesis 



TESTS OF CONVERGENCE FOR POSITIVE TERM SERIES 93 

We may, therefore, apply 3. On the second hypothesis, we 
have 



and we may again apply 3. 

Example 1. A = ^ + ^ + ^-f ~ 
is convergent. For 

tf||!SS fi.n + l < n 

and V is convergent. The series JL was considered in 81, 4, Ex. 1. 
^w 2 

Example 2. A== e~ x cos $ + e~ 2x cos 2 # + 
is absolutely convergent for x > 0. 

For la |< 

which is thus < the nth term in the convergent geometric series 

Example 3. A = ]P - log ^-t 

is convergent. 



Thus A is comparable with the convergent series ]~V 

^-< n 2 

88. We proceed now to deduce various tests for convergence 
and divergence. One of the simplest is the following, obtained 
by comparison with the hyperharmonic series. 

Let A = ! + a 2 -f- be a positive term series. It is convergent if 

lira a n n* < oo , ^ > 1, 
and divergent if 

lim na n > 0. 



94 SERIES 

For on the first hypothesis there exists, by I, 338, a constant 
G- > such that 

a< (? nssl g ... 

Thus each term of A is less than the corresponding term of the 

^^A 1 
convergent series Gr2^ 

On the second hypothesis there exists a constant c such that 

-, o 

#n>- n = 1, z, 

and each term of A is greater than the corresponding term of the 

divergent series c V - . 
** n 

Example 1. A = V m >0. 

^log m n 

Here ^ = --^- = 4-00, by I, 463. 

log m n 

Hence A is divergent. 

Example 2. A = V - . 

^ n log n 

Here 



. 
log n 

Thus the theorem does not apply. The series is divergent 
by 82. 

Example 3. 

=2Z n = 21og(l + + 

\ n n r 

where /A is a constant and | 6 n \ < G-. 

From I, 413, we have, setting r = 1 + a, 



Hence nl n = fJL , if /A ^= 0, 



TESTS OF CONVERGENCE FOR POSITIVE TERM SERIES 95 

and L is divergent. If /* > 0, L is an essentially positive term 
series. Hence = + <x>. If /A < 0, L~ oo. 

Let /i = 0. Then 



which is comparable with the convergent series 

T- r I 

^f n r 

Thus L is convergent in this case. 
Example 4- The harmonic series 

is divergent. For li m = !. 

Example 5. -< 

-4 = V : yS arbitrary. 
^ ft a loer ft 



Ail a 

w - = oo , a<l 



Here 



by I, 463, l. Hence A is divergent for a< 1. 
Example 6. 



Here -I 

na n = -^ = 1 by I, 185, Ex. 3. 

Example 7. 



log ft 

Here, if /* > 0, 



_ 

log 71 

(l\ n 
1 +- ] s . 
n) 

Hence A is divergent. 



96 SERIES 

89. D'Alemberfs Test. The positive term series A = a l + a% -f 
converges if there exists a constant r < 1 for which 



It diverges if 



Tins follows from 87, 4, taking for B the geometric series 
1+ r + 



Corollary. Let ?*l==Z. //' ?<1, .A converges. If Z>1, i 



Example 1. The Exponential Series. 

Let us find for what values of x the series 

is convergent. Applying D'Alembert's test to its adjoint, we find 

x n n ^ [ 



a n 



r n-\ 



Thus ^ converges absolutely for every x. 

Let us employ 80, 9 to estimate the remainder E n . Let x >0. 
The terms of J? are all > 0. Since 



(w+/>)! n\ n -h 1 n + 2 n+ p n!\n-f-V 

we have _ _ __ 

(2 



However large x may be, we may take n so large that x<n + 1. 
Then the series on the right of 2) is a convergent geometric series. 
Let #<0. Then however large \x\ is, JE n is alternating for 
some m. Hence by 81, 3 for w>m, 



TESTS OF CONVERGENCE FOR POSITIVE TERM SERIES 97 

Example 2. The Logarithmic Series. 
Let us find for what values of x the series 



is convergent. The adjoint gives 

a n+l\ = n . I 3,1^1 x ^ 
a n I n H" 1 

Thus i converges absolutely for any |#|<1, and diverges lor 
When x = 1, L becomes 

which is simply convergent by 81, 4. 
When x = 1, L becomes 

which is the divergent harmonic series. 
Examples. ^ = ~+ ~ + ~+ - 



As A is convergent when /*>! and divergent if /n.<l, we see 
that D'Alembert's test gives us no information when I = 1. It is, 
however, convergent for this case by 81, 2. 

Example 4* 

ao . f 

vp n. 

^r _. _ 

Here .. 

a a n-fl-f^ 
and D'Alembert's test does not apply. 

Example 5. 

A = 2n M ^. 
Here 



98 SERIES 

Thus A converges for |#|<1 and diverges for \x\ > 1. For 
| x | = 1 the test does not apply. For x = 1 we know by 81, 2 
that A is convergent for JJL < 1, and is divergent for p > 1. 

For x = 1, A is divergent for /i > 0, since a n does not = 0. A 
is an alternating series for JJL < 0, and is then convergent. 

90. Cauchy's Radical Test. Let A = a l + # 2 -f- be a positive 
term series. If there exists a constant r < 1 such that 

ya~ n <r n=l, 2, ^ 
A is convergent. If, on the other hand, 

v^>i 

A is divergent. 

For on the first hypothesis, 

a n <r n 

so that each term of A is <; the corresponding term in 
r -{. 7.2 _|_ r s _j_ ... a convergent geometric series. On the second 
hypothesis, this geometric series is divergent and a n > r n . 

Corollary. If lim Va n = /, and I < 1, A is convergent. Ifl>\, 
A is divergent. 

Example 1. The series 



*4 log n n 
is convergent. For 

n/ 1 



n 

logn 

Example 2. 



. f\ 

= 0. 



1 . 

"" 



is convergent. For 



Example 8. In the elliptic functions we have to consider series 
of the type 

0(t>) = 1 + 2 Sj*' cos 2 Trnv < q < 1. 



TESTS OF CONVERGENCE FOR POSITIVE TERM SERIES 99 

This series converges absolutely if 

? + ? 4 + 9 9 + - 
does. But here 

A/^ = V ? 2 :=2 n = 0. 

Thus 6(v) converges absolutely for every v. 
Example 4- Let < a <b < 1. The series 
.A = + 6 2 + <i 8 + 6* + ... 

is convergent. For if rt 

& n2m 

-^/a n = 2 V^ = 6. 

If rc = 2 m + 1, 2//J+1 

V0 n = va- m+1 = a. 

Thus for all n ni , 

Vn < * < 1- 

Let us apply D'Alembert's test. Here 



Thus the test gives us no information. 

91. Cauchy's Integral Test. 

Let <f>(x) be a positive monotone decreasing function in the interval 

(a, oo ). The series 



is convergent or divergent according as 

y^QO 

I (f> (x) dx 

*Ja 

is convergent or divergent. 

For in the interval (n, n -f 1), n>m> a, 



100 SERIES 

Hence n+l 



Letting n = w, TW + 1, w -|-jt?, and adding, we have 



, p+l 



Passing to the limit jt? = oo, we get 



which proves the theorem. 

Corollary. When 4> t8 convergent 



Example 1. We can establish at once the results of 81, 2. For, 
taking *(*) = !, 



is convergent or divergent according as /i>l, or /*<!, by I, 
036, 636. 

We also note that if 

~" " "*"""" 



then 



Example 2. The logarithmic series 
^ 1 



8 = 1,2, 



are convergent if JJL > 1; divergent if /* < 1. 
We take here ^ 



and apply I, 637, 638. 



TESTS OF CONVERGENCE FOR POSITIVE TERM SERIES 101 

92. 1. One way, as already remarked, to determine whether 
a given positive term series A = a^ -f a 2 -f is convergent or 
divergent is to compare it with some series whose convergence or 
divergence is known. We have found up to the present the 
following standard series S: 

The geometric series 



The general harmonic series 

1 
The logarithmic series 



1 + 1+.. . (2 

2" 3" V 



(3 



S * -, (4 

^W W/ 'W/^W 



We notice that none of these series could be used to determine 
by comparison the convergence or divergence of the series follow- 
ing it. 

In fact, let a n , b n denote respectively the nth terms in 1), 2). 
Then for#<l, /t>0, 

^^oo by I, 464, 



a n+l 



or using the infinitary notation of I, 461, 

t> n > a n . 

Thus the terms of 2) converge to infinitely slower than the 
terms of 1), so that it is useless to compare 2) with 1) for conver- 
gence. Let g > 1. Then 



a n >b n . 
This shows we cannot compare 2) with 1) for divergence. 



102 SERIES 

Again, if # n , b n denote the nth terms of 2), 3) respectively, we 
have, if /i > 1, 

^ = -^- = 00 by I, 463, 
a n log* n 

or 7 

*n > <*' 



> .. 

Thus the convergence or divergence of 3) cannot be found 
from 2) by comparison. In the same way we may proceed with 
the others. 

2. These considerations lead us to introduce the following 
notions. Let A = a l -f a 2 + , B = b 1 -f- J 2 -f be positive term 
series. Instead of considering the behavior of a n /b n , let us gen- 
eralize and consider the ratios A n : B n for divergent and A n : B n 
for convergent series. These ratios obviously afford us a measure 
of the rate at which A n and B n approach their limit. If now A^ 
B are divergent and . ^ 

^n ~ -Hf* 

we say A, B diverge equally fast ; if 



A diverges slower than jS, and B diverges faster than A. From 
I, 180, we have : 

Let A, B be divergent and 



According as I is 0, =0, oo, A diverges slower, equally fast, or 
faster than B. 

If A, B are convergent and 



we say A, B converge equally fast ; if A converges and 

B, <A n , 



TESTS OF CONVERGENCE FOR POSITIVE TERM SERIES 103 

B converges faster than A, and A converges slower than B. From 
I, 184, we have: 

Let A, B be convergent and 



According as I is 0, = 0, oo, A converges faster, equally fast, or slower 
than B. 

Returning now to the set of standard series >S T , we see that each 
converges (diverges) slower than any preceding series of the set. 
Such a set may therefore appropriately be called a scale of con- 
vergent (divergent) series. Thus if we have a decreasing positive 
term series, whose convergence or divergence is to be ascertained, 
we may compare it successively with the scale S, until we arrive 
at one which converges or diverges equally fast. In practice such 
series may always be found. It is easy, however, to show that there 
exist series which converge or diverge slower than any series 
in the scale S or indeed any other scale. 

Foplet A, B, a,... ( 2 

be any scale of positive term convergent or divergent series. 
Then, if convergent, 

1-1 > 5-1 >-!>...; 
if divergent, A n > B n > C n > ... 

Thus in both cases we are led to a sequence of functions of the 

type 



Thus to show the existence of a series H which converges (di- 
verges) slower than any series in X, we have only to prove the 
theorem : 

3. (Du Bois Reymond.^) In the interval (a, oo) let 



denote a set of positive increasing functions which =00 

Moreover^ let f 

J\ 



104 SERIES 

Then there exist positive increasing functions which == oo slower than 



Foras/ 1 >/ 2 there exists an a l >a such that /i>/ 2 4-l for 
x> a r Since / 2 >/ 3 , there exists an a 2 > a 1 such that / 2 >/ 8 + 2 
for x>a%. And in general there exists an a n >a n _ l such that 
f n >/n+i + n for x > a n . Let now 



n 



Then g is an increasing unlimited function in (, oo) which 
finally remains below any f m (x) + m 1, m arbitrary but fixed. 

Thus ff 



Hence 



< lim -*i Q 

/ w + W - 1 



93. From the logarithmic series we can derive a number of 
tests, for example, the following : 

1. (Bertrams Tests.) Let A = a l + a 2 -f 6e a positive term 
series. 

Let , 1 

log 



/-. x N a n nLn L_ift ^ ^ 7 ^ 

G.(^)= - V - LJ - = 1, 2, ... Z n=l. 

L 8 + \ n 

If for some s and m, 

(?,O)>A*>1 n>m, (1 

is convergent. If, hoivever, 

<?.(n)<l, (2 

i divergent. 

For multiplying 1) by /, +1 n, we get 



or t 

log - > ^ log ^ra = log Zyn. 
ajiltfi fr,_j?i 

Hence - 



or 



TESTS OF CONVERGENCE FOR POSITIVE TERM SERIES 105 

Thus A is convergent. 

The rest of the theorem follows similarly. 

2. For the positive term series .A=a 1 -f a 2 -f to converge it is 
necessary that, for % = oo, 

lim a n = 0, lim na n = 0, lim naj^n = 0, lim na^nl^n = 0, 
We have already noted the first two. Suppose now that 
lim %#{}% I 8 n > 0. 

Then by I, 338 there exists an m and a c > 0, such that 
na n l^n I 8 n > c , n > wi, 



or 



Hence A diverges. 
Example 1. * __ 



n a 



We saw, 88, Ex. 5, that J. is divergent for < 1. For = 1, 
^1 is convergent for /3 > 1 and divergent if y8 ^ 1, according to 
91, Ex. 2. 



Then if ^8 > 0, 



and A is convergent since V is. If /3 < 0, let 



ra a n* 
But log*' w < n*' by I, 463, 1 ; 

and A is convergent since ^\ - is. 

*-i n * 



106 SERIES 

Example 2. 



Here 1 

log 



Q r= 

1 



by 81, 6). 



Hence A is convergent for /x>0 and divergent for /i<0. No 
test for /i = 0. 

But for /i = 0, j 

lo g r- rr i 



= -00, 

since l^n > l$n. Thus A is divergent for p = 0. 

94. A very general criterion is due to Kummer, viz.: 

Let A == a l 4- a 2 -f - be a positive term series. Let k r k^ be a 

set of positive numbers chosen at pleasure. A is convergent, if for 

some constant k > 0. 



^4 is divergent if 

t 2 
is divergent and 

K n <0 w = l, 2, 

For on the first hypothesis 



TESTS OF CONVERGENCE FOR POSITIVE TERM SERIES 107 
Hence adding, 



1 ' r, v l l 

and A is convergent by 80, 4. 
On the second hypothesis, 

a n Ti ~~^n ' 
or ._! 

Hence -4 diverges since R is divergent. 

95. 1. From Rummer's test we may deduce D'Alembert's test 
at once. For take 

Then A = a^ 4- # 2 + '" converges if 
K n = ^-\ 

i.e. if 

a n - P 

Similarly A diverges if - n1 >.!. 

2. To derive Uaabe's test we take 

k n = n. 
Then A converges if 



i.e. if 



Similarly A diverges if 



108 SERIES 

96. 1. Let A = a 1 -f a 2 -f be a positive term series. Let 



A converges if there exists an s such that 

it diverges if -\ / \ ^ -t ^ 

y J X,( w ) < 1 for n > m. 

We have already proved the theorem for X (n). Let us show 
how to prove it for Xj(V). The other cases follow similarly. 
For the Kummer numbers k n we take 

k n = n log n. 
Then A converges if 

k n = n log n - n - - O -1- 1) log (n -f 1) > k > 0. 

As 



n+l 

j 
nj 



Thus A converges if X x (^) > S > 1 for n>m. 

In this way we see that A diverges if Xj(n) < 1, n > m. 

2. Cahen's Test. For the positive term series to converge it is 

necessary that , , \ ^ 

limn Uf J!._i)_il = 4-00. 

n=o ( \a n+1 J } 



TESTS OF CONVERGENCE FOR POSITIVE TERM SERIES 109 
For if this upper limit is not -f 00, 



for all n. Hence I n 

But the right side = 0. Hence Xj(w) < 1 for n > some ?w, and 
A is divergent by 1. 

Example. We note that Raabe's test does apply to the harmonic 
l '* es 1 4. i 4. i 4. (\ 

Here 



Hence 'P n = 0, and 

lim P n = 0. 
Hence the series 1) is divergent. 

97. Gauss' Test. Let A = ctj + ocg-f be a positive term series 
such that 



where s, a l b 1 do not depend on n. Then A is convergent if 
a l b > ] , and divergent if a b l < 1. 

Using the identity I, 91, 2), we have 

i . JL t 7 > 



Thus limX (w)=a 1 6j. Hence, if a 1 &j>l, ^L is conver- 
gent; if ! i 1 <l, it is divergent. If ^ ^ = 1, Raabe's test 
does not always apply. To dispose of this case we may apply 
the test corresponding to X 1 (n). Or more simply we may use 
Cahen's test which depends on \(n). We find at once 

lim P n == 2 J 2 b 1 < oo ; 
and A is divergent. 



110 SERIES 

98. Let A = a l + # 2 -f- be a positive term series such thtt 
-5a- =:! + - + * /x>l, /S n <oc. 

Then A is convergent if a > 1 and divergent if a < 1 . 
For 



and ^L converges if a > 1 and diverges if < 1 . If = 1, 

and A is divergent. 

EXAMPLES 

99. The Binomial /Series. Let us find for what values of x and 
LL the series 



converges. If ^ is a positive integer, f is a polynomial of degree /-t. 
For fji = 0, J?= 1. We now exclude these exceptional values of p.. 
Applying D'Alembert's test to its adjoint we nnd 

n -f 1 



= \x\. 



Thus B converges absolutely for \x\ < 1 and diverges for \x\ > 1 . 
Letx*=l. Then 

B=I+ +^'^"" 1 -i ^/ A - 1 '/ i - 2 
M 1.2 1-2-3 

Here D'Alembert's test applied to its adjoint gives 



As this gives us no information unless M< 1, let us apply 
Raabe's test. Here 



T 



, for sufficiently large- n 



TESTS OF CONYEU<;ENTK FOR POSITIVE TERM SERIES 111 



Thus B converges absolutely if M>0, and its adjoint diverges 
if /i<0. Thus B does not converge absolutely for /A<0. 

But in this case we note that the terras of B are alternately 
positive and negative. Also 



1 - 



so that the \a n \ form a decreasing sequence from a certain term. 
We investigate now when a n = 0. Now 



. ~ 

In I, 143, let a = /*, = 1 . We thus find that lim a n = only 
when /i> 1. Thus jB converges when ^> 1 and diverges 



when /&< ! 

a?=-l. Then 



i n 2 



~~' ^" 1-2 1-2-3 

Jf IJL > 0, the terms of B finally have one sign, and 

/ V 

Hence B converges absolutely. 

If /A < 0, let fj. = \. Then B becomes 



Here 



n 



1.2 ' 

iV 


1.2-3 
l-\ ^_ l 


^ / i 


\-l 



n 



Hence J5 diverges in this case. Summing up : 

The binomial series converges absolutely for |#|<1 and diverges 
for \x\>\. When x 1 it converges for p, > 1 and diverges for 
p < 1 ; it converges absolutely only for /i > 0. When x = 1, it 
converges absolutely for p, > awo? diverges for /x < 0. 



112 SERIES 

100. The Hyper geometric Series 



7 



2 7 7 



o 



1.2-3.7.7 + 1.7 + 2 
Let us find for what values of x this series converges. Passing 



to the adjoint series, we find 



x = x . 



(i 



Thus F converges absolutely for | x \ < 1 and diverges for | x \ > 1. 
Let x = 1. The terms finally have one sign, and 

a n +i w 2 -f n(l + 7) + 7 

' 



Applying Gauss', test we find _F converges when and only when 



Let x = 1. The terms finally alternate in sign. Let us find 
when a n = 0. We have 



= a P- (a + 1) Q 

' 



-f 



Now 



m 



( t+ 
\ m 



= m(l + *-} 



y + m = m 



Thus 



But by I, 91, 1), 
1 _ 

, . 1 



mm 2 



mm* 



m 



where o- m == 1, r m = 7 2 as m == oo. 



PRINGSHEIM'S THEORY 113 

Hence 



ff-7-1 ,. 



Hence 



and thus 

L = lira log | a n+2 \ = 2J l m 






Now for a n to == it is necessary that L n === oo. In 88, Ex. 3, 
we saw this takes place when and only when + /3 7 1<0. 
Let us find now when | a n+} \<\a n |. Now 1) gives 



*n+2 



n 



Thus when a -f /3 7 1 < 0, a n+2 \ < | a n+l |. Hence in this 
case J 7 is an alternating series. We have thus the important 

theorem : 

The Tiy per geometric series converges absolutely when \ x \ < 1 and 
diverges when \x > 1 . When x = 1, F converges only when a -f- j3 
7<0 and then absolutely. When x = 1, -f 7 converges only 
when tf-f/3 7~1<0, a9?cZ absolutely if a + /3 7 < 0. 

Pringsheinis Theory 

101. 1. In the 35th volume of the Mathematische Annalen 
(1890) Pringsheim has developed a simple and uniform theory oi 
convergence which embraces as special cases all earlier criteria, 
and makes clear their interrelations. We wish to give a brief 
sketch of this theory here, referring the reader to his papers for 
more details. 

Let M n denote a positive increasing function of n whose limit 
is -H QO for n = CXD . Such functions are, for example, p > 0, 

nf" , log* 1 /i , Ifn , l^nltfi Z. 



114 SKRIKS 

An, where A is any positive term divergent series. 
B n ~* where B is any positive term convergent series. 



It will be convenient to denote in general a convergent positive 
term series by the symbol 

<7= aB <? 1 + <? 2 + ... 

and a divergent positive term series by 

Z> = rf 1 + </ 2 + ... 
2. The series 



Is convergent, and conversely every positive term convergent series 
may be brought into this form. 

*-S 

= j. JL^JL 

M l M m +\ M\ 
and is convergent. 

Let now conversely C=c l -\- c^-\- be a given convergent 
positive term series. Let 



Then ^ -. 

^i ~"~~ -- ~~ ~^"~ 



8. TA series 

V = %(M n+l -M n ) (2 

?' divergent* and conversely every positive term divergent series may 
be brought into this form. 
For 



PRINGSHEIM'S THEORY 115 

Let now conversely D = d l -f <7 2 -f be a given positive term 
divergent series. Let M n 

lYL n JJ n _ j. 

Then d M M 

a n irj. n+l ^a n . 

102. Having now obtained a general form of all convergent 
and divergent series, we now obtain another general form of a 
convergent or divergent series, but which converges slower than 
1) or diverges slower than 101, 2). Let us consider first con- 
vergence. Let M' n < M n , then 



is convergent, and if M' n is properly chosen, not only is each 
term of 1) greater than the corresponding term of 101, 1), but 1) 
will converge slower than 101, 1). For example, for M' n let us 
take M*, < p < 1. Then denoting the resulting series by 
0' = 2^, we have 



-ji/f\-ii _ n i fe) 

~"r^7 " ' w~ c 

1 r J-U-n+l 

Thus O r converges slower than C. But the preceding also 
shows that O 1 and 



converge equally fast. In fact 2) states that 



Since M n is any positive increasing function of n whose limit 
is oo, we may replace M n in 3) by l r M n so that 



is convergent and a fortiori 

^l r M n+l -l r M n r = 12 , ... (4 

^ V*M^ ^ 

is convergent. 



116 SERIES 

Now by I, 413, for sufficiently large n, 
log M n+l - log M n = - log(l - &%=&) > **f*- 

\ 1VJ n+\ J 1YL n+l 

Replacing here M n by log M n , we get 

7 M 1 M ^ loRj^tiJ=J2j?^ ^ M n+i^L M n^ 

c '2 irx n + l *V u n ^ i" 7i/r " ^ TUT ~i 7i/r~ " ' 

log^n + l M n + l \^M n ^ 

and in general 



Thus the series 
V 



converges as is seen by comparing with 4). We are thus led to 
the theorem : 

The series ^M n ^-M n y^-^ . 

^ M n M n+l ' ^ ^i^ 



an infinite set of convergent series; each series converging 
slower than any preceding it. 

The last statement follows from I, 463, l, 2. 

Corollary 1 (Abel). Let D = d t + J 2 -f denote a positive term 
divergent series. Then 



z convergent. 

Follows from 3), setting M n+l = D n . 

Corollary 2. If we take ^f n = n we get the series 91, Ex. 2. 

Corollary 3. Being given a convergent positive term series 
(7 = (?j -f c? 2 -f- we can construct a series which converges slower 
than C. 



PRINGSHEIM'S THEORY 117 

For by 101, 2 W e may bring to the form 



Then any of the series 7) converges slower than C. 

103. 1. Let us consider now divergent series. Here our 
problem is simpler and we have at once the theorem : 

Tfie series M iw 



diverges slower than 

zC (2 



That 1) is divergent is seen thus : Consider the product 

, M m+ i - M m \ _ M m+l 

~~~' 



which obviously = oo. 
N W P n = ( 



n .. 

2 

Hence J9 n = oo and D is divergent. 

AS d = 1 =Q 

* Jf 

we see that 1) converges slower than 2). 



2. ^4w^ given positive term series D = d + d^ + can be put in 
the form I). 

For taking M l >0 at pleasure, we determine M v M z by the 
relations jur 



118 SERIES 

Then M n+l > M n and 



Moreover M n = oo. For 



> 1 + A, by I, 90, 1. 
But J9 B = oo. 

3. 2%e *m'e8 



an infinite set of divergent series, each series divergent slower 
than any preceding it. l Q M n = M n . 

For log M nn - log M n - log l + 



M n ' 

This proves the theorem for r = 0. Hence as in 102 we find, 
replacing repeatedly M n by log M n , 

1 M _ 7 M ^ M n+l M n ,o 



Corollary 2. If we take M n = n, we get the series 91, Ex. 2. 

Corollary 2 {Abel). Let D = d l + d 2 -f fo a divergent positive 
term series. Then , 



'$ divergent. 

We take here J!J, = .Z) n . 

Corollary 3. Being given a positive term divergent series D, we 
can construct a series which diverges slower than D. 
For by 101, 3 we may bring D to the form 



Then 1) diverges slower than D. 



PRINGSHEIM'S THEORY 119 

104. In Ex. 3 of I, 454, we have seen that M n +i is not always^ 
M n . In case it is we have 

1. The series 



is convergent. 

Follows from 102, 3). 

2. The series 



- M 



is convergent if p > 0; it is divergent if /*< 0. 

For #"* > i ^Ml - M. I /* > 0. 

Thus 



3. If M n+l ~ MM we have 

I M -I M ^ 

i r+l JXL n +i Lr+i^n 



For by 102, 5), 103, 3), 

M + M n 



i ]u- ___j M 
Lr+ i Mn+l l r+\ m * 

Now since M n+l ~ M n , we have also obviously 
. l m M n ~l m M n+l m=l, 2, ...r. 

105. Having obtained an unlimited set of series which converge 
or diverge more and more slowly, we show now how they may be 
employed to furnish tests of ever increasing strength. To ob- 
tain them we go back to the fundamental theorems of comparison 
of 87. In the first place, if J.= a a -f a 2 -f is a given positive 
term series, it converges if 



120 SERIES 

It diverges if 

^>#. (2 

d n ^ 

In the second place, A converges if 

^n C n 

and diverges if , 

a n d n ~~ 

The tests 1), 2) involve only a single term of the given series 
and the comparison series, while the tests 3), 4) involve two 
terms. With Du Bois Reymond such tests we may call respec- 
tively tests of the first and second kinds. And in general any 
relation between p terms 

of the given series and JP terms of a comparison series, 
0m tfn+ti '" <Vf P -ii or d n , d n+1 d w +p-i 

which serves as a criterion of convergence or divergence may be 
called a test of the p th kind. 

Let us return now to the tests 1), 2), 8), 4), and suppose we 
are testing A for convergence. If for a certain comparison 
series 

not always <.6r , n > m 

it might be due to the fact that c n = too fast. We would then 
take another comparison series O f = ^c' n which converges slower 
than C. As there always exist series which converge slower than 
any given positive term series, the test 1) must decide the con- 
vergence of A if a proper comparison series is found. To find 
such series we employ series which converge slower and slower. 
Similar remarks apply to the other tests. We show now how 
these considerations lead us most naturally to a set of tests which 
contain as special cases those already given. 

106. 1. General Criterion of the First Kind. The positive term 

series A = a + a% -f converges if 



n+l 



PRINGSHEIM'S THEORY 121 

It diverges if M n 



*I + -*! (2 

This follows at once from 105, 1), 2); and 101, 2; 103, 1. 

2. To get tests of greater power we have only to replace the 



senes 



M n 

just employed in 1), 2) by the series of 102 and 103, 3 which con- 
verge (diverge) slower. We thus get from 1 : 

The positive term series A converges if 

__ 7lf 7L/> __ If 7 JIT ... 7 TIT n+V-M 

llm "" rlim " a -' <00 - 



It diverges if M n LM n l r M n 

--- ~1tf. + "- jr. " an> 

Sonnet's Test. The positive term series A converges if 
lira nl^n l r ^7il\^n a n < oo , //. > 0. 

*7 



Follows from the preceding setting 7Jf n = n. 

3. J 7 ^ positive term series A converges or diverges according as 

'"">' 



"' " 

For in the first case 

-.5 

and in the second case 



< 1 , ,.>0, (3 



The theorem follows now by 104, 2. 

4. The positive term series A converges if 



log^i-* 1 -- lo gl 

lim _^2 >0 or Mm- 



122 SERIES 

It diverges if 

M n+ , - M n , M n+ , - M n 

P- <0 oriS g 
hm - TJJ - < u or nm 



r = 0, 1, 2, and as before l Q M n = 3f n . 
For taking the logarithm of botli sides of 3) we have for con- 

log M +i - M 



As /i is an arbitrarily small but fixed positive number, A con- 
verges if lim q > 0. Making use of 104, 3 we get the first part 
of the theorem. The rest follows similarly. 

Remark. If we take M n = n we get Cauchy's radical test 90 
and Bertram's tests 93. 



/T 
^l= log {'I = - log Va n 






it is necessary that n/ ^ -. 

Also if 



log - - - - -f log _ 



a n nl^i l r n __ a n nl^n / r _ 1 
? r+1 n / r+1 n 



log 

= 1 | 

it is necessary that j 



107. In 94 we have given Rummer's criterion for the conver- 
gence of a positive term series. The most remarkable feature 
about it is the fact that the constants k r & 2 which enter it are 
subject to no conditions whatever except that they shall be positive. 
On this account this test, whicli is of the second kind, has stood 
entirely apart from all other tests, until Pringsheim discovered its 
counterpart as a test of the first kind, viz. : 



PRINGSHEIM'S THEORY 123 

Pringsheims Criterion. Let p v p%-- be a set of positive numbers 
chosen at pleasure, and let P n = p l -f 4- p n > The positive term 

series A converges if 

log_!L 

15m _ ^L > 0. (1 

* n 

For A converges if 



Jim - -Ji2 - > , by 106, 4. (2 

-- M n 

But M n+l M n = d n is the general term of the divergent series 
/)= rfj + c^-f 

Thus 2) may be written 



log 
lim _ ^>0. (3 



Moreover A converges if 



that is, if lim5t>0 f 

n 

where as usual (7= <?j-f <? 3 -h is a convergent series. 
Hence J. converges if Cn 

lim^>0. (4 

^n 

But now the set of numbers p v p 2 gives rise to a series 
P=p l -f p^ 4- ... which must be either convergent or divergent. 
Thus 3), 4) show that in either case 1) holds. 

108. 1. Let us consider now still more briefly criteria of the 
second kind. Here the fundamental relations are 3), 4) of 105, 
which may be written : 

6' nM -^ -- c n > for convergence; (1 



4 .j - -- t/ n < for divergence. (2 



124 SERIES 

Or in less general form : 

The positive term series A converges if 



It diverges if 

0. (4 



Here as usual C=c l -\-c^-^~ is a convergent, and D=d l 
a divergent series. 

2. Although we have already given one demonstration of 
Rummer's theorem we wish to show here its place in Pringsheim's 
general theory, and also to exhibit it under a more general form. 
Let us replace c n , c nn in 1) by their values given in 101, 2. 
We get 



or snce 



n +2 - n 1 . n __ n+1 ~ n > Q 
M n + 1 <-! ^n 

or by 103, 2 a n , Q 

^n+l -- a n > v i 
^n+1 

where D = d 1 +d 2 -{' is a^y divergent positive term series. 
Since any set of positive numbers &j, & 2 , gives rise to a series 
&i 4- ^2 "+" '" which must be either convergent or divergent, we see 
from 1) that 5) holds when we replace the eTs by the Fs. We 
have therefore: 

The positive term series A converges if there exists a set of positive 
numbers k v k 2 such that 

(6 



a n+\ 

It diverges if 



where as usual d l 4- d% -f denotes a divergent series. 



ARITHMETIC OPERATIONS ON SERIES 125 

Since the k's are entirely arbitrary positive numbers, the rela- 
tion 6) also gives 

A converges if 



as is seen by writing 

if 
n ^F 

K n 

reducing, and then dropping the accent. 

3. From Rummer's theorem we may at once deduce a set of 
tests of increasing power, viz. : 

The positive term series A is convergent or divergent according as 



~M 7 TIT ... / /I/ fi 7VT 1 1\T ... / 7l/ 

LfJ -n+\ l \- LJ -n+\ t/ r- L ' J -n+\ "'n+l LY - L n l \- L - L n L r L1J -n 

is > or is j< 0. 

For & a , & 2 ... we have used here the terms of the divergent 
series of 103, 3. 

Arithmetic Operations on Series 
109. 1. Since an infinite series 

is not a true sum but the limit of a sum 

A = lim-A n , 

we now inquire in how far the properties of polynomials hold for 
the infinite polynomial 1). The associative property is expressed 
in the theorem :' 

Let A = a^ 4- # 2 4- be convergent. Let b^ = a l 4- 4- a m , 
^2 = a m l +i 4- 4- , , Then the series B = b l 4- & 2 + '" l8 C(m ~ 
vergent and A = . Moreover the number of terms which b n em- 
braces may increase indefinitely with n. 

For *-^ 

and limA mn = A by I, 103, 2. 



126 SERIES 

This theorem relates to grouping the terms of A in parentheses. 
The following relate to removing them. 

2. Let B = b l + 6 2 -f be convergent and let b l = a + + <*< m * 
* a = / Wl +i+ ' +, v y 1<D ^ = a i + a 2 + " ** Convergent, 
A = B. 2 /f f/i* terww n >0, 4 i* convergent. 8 ^ eacA 
in n m n _i <^p a constant, and a n = 0, A is convergent. 

On the first hypothesis we have only to apply 1, to show 
A = jB. On the second hypothesis 

> 0, ra, B n < e, w > w. 

Then -A.< e , s>m n . 

On the third hypothesis we may set 

A. = B r +b' r+l 

where b' r+1 denotes a part of the a-terms in b r+l . Since b r+1 con- 
tains at most p terms of A, b' r+l = 0. 

Hence 



.= m r , or = . 
Example 1. The series 

B = (1-1) + (1-1) + (1-1)+..- 
is convergent. The series obtained by removing the parentheses 

4 = 1-1 + 1-1+ .- 
is divergent. 

Example 2. 



^ vfi __ )= y __ ^ 

^ ^ 



As J^ is comparable with 5],, it is convergent. Hence A is 

^rr 

convergent by 3. 

110. 1. Let us consider now the commutative property. 

Here Riemann has established the following remarkable 
theorem : 



ARITHMETIC OPERATIONS ON SERIES 12? 

The terms of a simply convergent series A = a l + a a + can be 
arranged to form a series S, for which lim S n is any prescribed 
number, or 00. 

For let p , , 

/> = fl -f -f - - 



be the series formed respectively of the positive and negative 
terms of A, the relative order of the terms in A being preserved. 
To fix the ideas let / be a positive number ; the demonstration 
of the other cases is similar. Since B n == + oo, there exists an m l 
such that 

, > i. (i 

Let m l be the least index for which l)is true. Since (7 n = oo, 
there exists an /w 2 such that 

An, + O m , < I. (2 

Let z 2 be the least index for which 2) is true. Continuing, 
we take just enough terms, say m 3 terms of B, so that 



Then just enough terms, say m t terms of (7, so that 

S mt 4-O mt + B mi . mt +O m ^ mt <l, 
etc. In this way we form the series 

&'=B mi +C m , + m ,, m ,+ - 
whose sum is L For 



a, i < s > a ; 



r 1 Qr 

Hence 



'2. Let A = aj -f a 2 H- 6e absolutely convergent. Let the terms 
of A be arranged in a different order, giving the series B. Then B 
is absolutely convergent and A = B. 

For we may take m so large that 

A m < e. 



128 SERIES 

We may now take n so large that A n B n contains no term 
whose index is <. m. Thus the terms of A n n taken with 
positive sign are a part of A m and hence 

A n - B n | < A m < e n > m. 

Thus B is convergent and B = A. 

The same reasoning shows that B is convergent, hence B is 
absolutely convergent. 

3. If A = a l -f- a 2 -f -"enjoys the commutative property, it is 
absolutely convergent. 

For if only simply convergent we could arrange its terms so as 
to have any desired sum. But this contradicts the hypothesis. 



Addition and Subtraction 

111. Let A = a l + a% -f , B = b l -f- 6 2 -h 6e convergent. 
The series 

<?= (^^ + (^62) + ... 

are convergent and 0= AB. 

For obviously O n = A n B n . We have now only to pass to the 
limit. 

Example. We saw, 81, 3, Ex. 1, that 



is a simply convergent series. Grouping its terms by twos and 
by fours [109, l] we get 



W-3 4w-2 4w-l 

Let us now rearrange J., taking two positive terms to one nega- 
tive. We get 



ADDITION AND SUBTRACTION 129 

We note now that 



_!_V! 



+ 



tw-S 4w-2 4w-l 

, _J 



t n - 3 4n-2 

1 1 1 \ 

: n - 1 "*" r?r^~3 "" 2w/ ^ 

= by 109, 2. 
Thus B = | ^4. 

This example, due to Dirichlet, illustrates the non-commutative 
property of simply convergent series. We have shown the con- 
vergence of B by actually determining its sum. As an exercise let 
us proceed directly as follows : 

The series 1) may be written : 

8/i-3 ^Y-- ~ 



n\ n 
Comparing this with 



we see that it is convergent by 87, 3. Since 1) is convergent, 5 
is also by 109, 2. 

112. 1. Multiplication. We have already seen, 80, 7, that we 
may multiply a convergent series by any constant. Let us now 
consider the multiplication of two series. As customary let 



denote the infinite series whose terms are all possible products 
a, b K without repetition. Let us take two rectangular axes as in 
analytic geometry ; the points whose coordinates are x = *, y = K 
are called lattice points. Thus to each term aJt> K of 1), cor- 



130 SERIES 

responds a lattice point t, K and conversely. The reader will find 
it a great help here and later to keep this correspondence in mind. 

Let A = a l -f a a H , B = b l -f b z -f 6e absolutely convergent. 
Then (J = # A is absolutely convergent and A B = (7. 

Let w he taken large at pleasure ; we may take n so large that 
F n A m B m contains no term both of whose indices are <. m. 

Then T n - A m E m < ,B m + ,B m + ... + m B m 



+ /8 1 A m +/3 2 A ro +... + ft m \ m 

< A m E m + B m A m 

< e for m sufficiently large. 



Hence 



and is absolutely convergent. 

To show that (7= A B, we note that 

| C n - A m m < F,, - A m B m < e n > n,. 

2. We owe the following theorem to Mertens. 

If A converges absolutely and B converges (not necessarily abso- 
lutely*), then 



(7= a^j -h O^ + a J>i) -^ OA -f- #2*2 + 
is convergent and C = A - B. 

We set 0= 6*j 4- <' 2 + c z+ '" 



where ^ a l b l 



c- 2 = a^ -f a^ 

c z = a^g -I- a 2 i 2 -f- aa 



Adding these equations gives 

C n = a^ + a^B n ^ -f a 3 



ADDITION AND SUBTRACTION 131 

But - 

B m =B-B m m = l,2, ... 

Hence 



where 



The theorem is proved when we show d n = 0. To this end let 
us consider the two sets of remainders 

B l , B<i , ... B n ^ 

_ Wj 4- n 2 = n. 

DO ]5 

"wt+1 ' "n t +2 ' '" -^fii+n, 

Let * $ach one in the first set be | < | M^ and each in the second 
set < M Then since 



Now for each e > there exists an n such that 
also a i>, such that 



Thus 1) shows that , , , 

I <*;i I <. . 

3. When neither J. nor B converges absolutely, the series 
may not even converge. The following example due to Cauchy 
illustrates this. 

^ = JL_J_ + _L__L + ... 

Vl V2 V3 V4 

5 = J = -_L + J_-J_ + ... = A . 

VI V2 V3 V4 

*The symbols | < , | < | mean numerically <, numerically <. 



132 SERIES 



The series A being alternating is convergent by 81, 3. Its 
adjoint is divergent by 81, 2, since here /* = . Now 



ViVi WiV2 

WT V3 V2 V2 Vrt VT 

=J T 1 1 1 

A 

By I, 95, 



== G' a + <? 3 -f <? 4 4- 
and 



_ 

VI Vw - 1 V2 Vre - 2 V - 1 VI 



Vw ( >) < '' 



Hence 1_ > 2 



m) n n 

Hence is divergent since c n does not = 0, as it must if C 
were convergent, by 80, 3. 

4. In order to have the theorems on multiplication together, 
we state here one which we shall prove later. 

If all three series A, B, Q are convergent, then = A B. 

113. We have seen, 109, l, that we may group the terms of a 
convergent series A a l + a% -f- into a series B = b^ 4- ^ 2 + 
each term b n containing but a finite number of terms of A. It is 
easy to arrange the terms of A into a finite or even an infinite 
number of infinite series, jB', B n ', B' n For example, let 



B" = a 2 -f a p+2 -f- a 2p + 2 4- 



Then every term of vl lies in one of these p series B. To decom- 
pose A into an infinite number of series we may proceed thus : 
In B 1 put all terms a n whose index n is a prime number ; in B n 
put all terms whose index n is the product of two primes ; in 



TWO-WAY SERIES 183 

B (m) all terms whose index is the product of m primes. We ask 
now what is the relation between the original series A and the 
series JS', B ff 

If A = a l 4- # 2 4- is absolutely convergent, we may break it up 
into a finite or infinite number of series B* , fl , fn , Each of 
these series converges absolutely and 

That each J9 (m) converges absolutely was shown in 80, 6. Let 
us suppose first that there is only a finite number of these series, 
say p of them. Then 

A n = B' ni + B\ + - + B% M = n, + - + n r 

As n = oo, each n v 7i 2 --.=oo. Hence passing to the limit 
n = QO , the above relation gives 

Suppose now there are an infinite number of series B (m \ 

We take v so large that A B n , n>v, contains no term a n of 
index < w, and m so large that 

4 ^ 



-\ m 



M-5.|<A, 



Two-way Series 

114. 1. Up to the present the terms of our infinite series have 
extended to infinity only one way. It is, however, convenient 
sometimes to consider series which extend both ways. They are 
of the type 



which may be written 

a O+"l + a 2+ ---- 1" a ~l + a 2+ 

or 



134 SERIES 

Such series we called two-way series. The series is convergent 

if 

lira 2a B (2 

r,s=ao n=~r 

is finite. If the limit 2) does not exist, A is divergent. The ex- 
tension of the other terms employed in one-way series to the 
present case are too obvious to need any comment. Sometimes 
n = is excluded in 1) ; the fact may be indicated by a dash, 

<x 

thus 2'a w . 

00 

2. Let m be an integer ; then while n ranges over 
... _ 8 , -2, -1,0,1,2,3... 

v = n -h m will range over the same set with the difference that v 
will be m units ahead or behind n according as w^O. This 
shows that 



W,= oo H- a 



Similarly, | _ a 

,, ^ a - n 

ft= tX> Jl=zGQ 

3. Example 1. = nx+an2 



This series is fundamental in the elliptic functions. 
Example 2. 



l , , ^ ^ 

-- T / \ --- ) 

x ~ \x -f n nj 



The sum of this series as we shall see is TT cot TTX. 



TWO-WAY SERIES 135 

115. For a two-way series A to converge, it is necessary and 
sufficient that the series formed with the terms with negative indices 
and the series C formed with the terms with non-negative indices be 
convergent. If A is convergent, A = B + C. 

It is necessary. For A being convergent, 

\A-B r -C.\</2 , A-B,-C B ,\<e/2 
if s, s f > some <r and r > some p. Hence adding, 

\C 8 -C 8 ,\<e, 

which shows C is convergent. Similarly we may show that B is 
convergent. 

It is sufficient. For B, C being convergent, 

|j5- r |<e/2 , |<7-<7.|<e/2 
for r, s > some p. Hence 



TVma n= 

inus lim 2 n = B + O. 



Example 1. The series 



x T^ \x + n n 

is absolutely convergent if x = 0, 1, ^, 
For 



a n \ = 



1 



x -{- n n 



\x I 
~-H 



Hence s , 

^*a n and 2<a n 

00 

OP ^ 

are comparable with 5]-- 



Example 2. The series 

(#) = l^nx+an' x arbitrary (2 

00 

is convergent absolutely if a < 0. It diverges if a > 0. 



136 SERIES 



__ 

n > 0, Vtf n = e x e an = if a < 

= 00 if a>0; 
w = -w', /i'>0 \/ n = '*"*' = ifa<0 

== QO if a > 0. 
The case a = is obvious. 

Thus the series defines a one-valued function of x when a < 0. 
As an exercise in manipulation let us prove two of its properties. 

1 (*)(#) is an even function. 

For 

e(-z)=26r w * +a 2 . (3 

-x> 

If we compare this series with 2) we see that the terms corre- 
sponding to n = m and n m have simply changed places, as the 
reader will see if lie actually writes out a few terms of 2), 3). 
Of. 114, 2. 

2 O + 2 ma) = e~"**+ ma) (x). m = 1, 2, ... 
For we can write 2) in the form 

r 2 ^ (.r+Swtf) 2 

<H)(=e < 2<T 4 "" (4 

HPl 

18 8 -- 



which with 4) gives 3). 



CHAPTER IV 
MULTIPLE SERIES 

116. Let x = Zj, x m be a point in w-way space 9t m . If the 
coordinates of x are all integers or zero, x is called a lattice point, 
and any set of lattice points a lattice system. If no coordinate of 
any point in a lattice system is negative, we call it a non-negative 
lattice system, etc. Let f(x l # w ) be defined over a lattice 
system 1 = ^,...^. The set J/(^---OS * s called an m-tuple 
sequence. It is customary to set 

/('i- v) = ., .s.- 

Then the sequence is represented by 



The terms Um A , lira 



as 4 1 t m converges to an ideal point have therefore been denned 
and some of their elementary properties given in the discussion 
of I, 314-328 ; 336-338. 

Let r = .TJ *m V = Vi '" Mm be two points in 9t m . If 
y\ ^ x \ '" Hm ^L x m we shall write more shortly y > x. If x 
ranges over a set of points x r > x n > x'" we shall say that x is 
monotone decreasing. Similar terms apply as in I, 211. 

If now 
when y >_ x, we say /is a monotone increasing function. If 

<* '" ^ > 



we say /is a monotone decreasing function. 
Similar terms apply as in I, 211. 



138 MULTIPLE SERIES 

117. A very important class of multiple sequences is connected 
with multiple series as we now show. Let # tl ... tw be defined over 
a non-negative lattice system. The symbol 

or X# t ... lm , or A. v ^... Vm 

o 

denotes the sum of all the a's whose lattice points lie in the rec- 
tangular cell 0<x l <v l -0<x m <p m . 

Let us denote this cell by M v ^.. vm or by R v . The sum 1) may be 
effected in a variety of ways. To fix the ideas let m = 3. Then 



etc. In the first sum, we sum up the terms in each plane and 
then add these results. In the second sum, we sum the terms on 
parallel lines and then add the results. In the last sum, we sum 
the terms on the parallel lines lying in a given plane and add the 
results; we then sum over the different planes. 
Returning now to the general case, the symbol 

u4 = 2a tl ... lm i v ... * m =0, 1, QO, 

or A = 2a tl ... twi 

o 

is called an w-tuple infinite series. For m = 2 we can write it 
out more fully thus 



In general, we may suppose the terms of any w-tuple series dis- 
played in a similar array, the term a tl ... lm occupying the lattice 
point t = (6 1 -"t m ). This affords a geometric image of great 
service. The terms in the cell R v may be denoted by A v . 

If limA,, l ... Vm = limA v (2 



GENERAL THEORY 139 

is finite, A is convergent and the limit 2) is called the sum of the 
series -A. When no confusion will arise, we may denote the series 
and its sum by the same letter. If the limit 2) is infinite or does 
not exist, we say A is divergent. 

Thus every m-tuple series gives rise to an 7n-tuple sequence 
\A v ^... vm \. Obviously if all the terms of A are >0 and A is diver- 
gent, the limit 2) is 4- oo. In this case we say A is infinite. 

Let us replace certain terms of A by zeros, the resulting series 
may be called the deleted series. If we delete A by replacing all 
the terms of the cell R v ^... vm by zero, the resulting series is called 
the remainder and is denoted by J.^...^ or by A v . Similarly if 
the cell R v contains the cell R^ the terms lying in R v and not in 
R^ may be denoted by A^ . 

The series obtained from A by replacing each term of A by its 
numerical value is called the adjoint series. In a similar manner 
most of the terms employed for simple series may be carried over 
to 7w-tuple series. In the series 2a tl ... lm the indices i all began 
with 0. There is no necessity for this; they may each begin with 
any integer at pleasure. 

118. The Geometric Series. We have seen that 
= 1 + a + a 2 + | a \ < 1, 

= ! + &+ 6 2 + - |6|<1. 



1- b 
Hence 



1 



(1 - a)(l - b) o 

for all points a, b within the unit square. 
In general we see that 



is absolutely convergent for any point x within the unit cube 

< K| < 1 *= 1, 2, W, 
and 1 



140 MULTIPLE SERIES 

119. 1. It is important to show how any term of A = 2a tl ... lw can 
be expressed by means of the ^l, v .. l//t . 



Then /),, ..... ,_,-, = '*,,,, -. I-K,,. - 4-,-, ..... -i-i, --i- ( 2 

I^et D, lft ..... ,_,= A,, ,...,._, -A,K, ..... .,-,- (3 

Similarly 

A,, a ..... , . = ^,,v ..- A v , ..... _,-,, (4 



Finally D V} = D v ^- D v ^ . p (6 



If now we replace the Z>\s by their values in terms of the As, 
the relation 7) shows that a v ^... Vm may be expressed linearly in 
terms of a number of ^4 Mi ... Mm where each JJL,, = v r or v r 1. 

For in = 4 2 we find 



!2. From 1 it follows that we may take any sequence \A Ll ... l}n \ 

to form a multiple series 

A ?a 

^ -- 'tt tl ..., m . 

This fact has theoretic importance in studying the peculiarities 
that multiple series present. 

120. We have now the following theorems analogous to 80. 

1. For A to be convergent it is necexsary and sufficient that 

6>0, p, \A^ \< R P <R<R V . 

2. If A is convergent, so is A^ and 

A^ = A A^ = lim A^ v . 

Conversely if A^ is converyent, so is A. 



GENERAL THEORY 141 

3. .For A to converge it is necessary and sufficient that 

lim A v = 0. 

V <X> 

4. A series whose adjoint converges is convergent. 

5. Let A be absolutely convergent. Any deleted series K of A is 
absolutely convergent and \ B \ < A. 

6. If A = 2# ti ... tm is convergent, so is B = 2^ ti ..., w a 

B = A^4, A: a constant. 



121. 1. jFor ^4 to converge it is necessary that 

A^ 2 ... ,_! , A-,,, ... ,. w - 2 , A, < <V 2 ... ,, = 0, as v = oo . 
For by 120, 1 , . - , 

^ ! A 1 ...A m ~A Ml ... MMI | <6 

if X 1 ---A m , ^ ---fin > p. 

Thus by 119, 1) 

[I> Vl , z .... m - l \<e v>p> 

Hence passing to the limit p = oo , 

lim D,,,...^,^ e. 

r = QO 

As e is small at pleasure, this shows that ^...^.^ 0. In this 
way we may continue. 

2. Although ,. A 

5 hm^ i ..., m = 

n ... ^=* 

when vl converges, we must guard against the error of supposing 
that a v = when v = (v l i/ m ) converges to an ideal point, all of 
whose coordinates are not oo as they are in the limits employed 
iul. 

This is made clear by the following example due to Pringsheim. 



Then by 119, 8) . _ 1 , 1 

a rM ~T* * 
1 a r a 8 



142 MULTIPLE SERIES 

As 



lim A r ^ g = 

r, 5=00 

A is convergent. But 

lim I a r J = , lim I a rt \ = 
r=oo ' a* *=oo a r 

That is when the point (r, ) converges to the ideal point 
(oo, ), or to the ideal point (r, oo ), a T9 does not = 0. 

3. However, we do have the theorem : 



converge. Then for each e > there exists a \ such that tl ... t < 
for any i outside the rectangular cell R^. 

This follows at once from 120, l, since 



122. 1. Letf{x l # m ) be monotone. Then 

Xn) = 1 x l < a v x m < a m , a may be ideal. (1 



exists, finite or infinite. If f is limited, I is finite. If f is unlim- 
ited, I = -f GO when f is monotone increasing, and I = oo whenf is 
monotone decreasing. 

For, let/ be limited. Let J. = j < o^ < = a. 
Then 



is finite by I, 109. 

Let now B = /3 r /3 2 , = a be any other sequence. 

Let Km/08.) = ' 11^/08.) - 

, - 5 

Then there exists by I, 338 a partial sequence of B> say 
(7= 7 X , 7 2 such that 



also a partial sequence J9= Sj, S 2 such that 

lim/(S n ) = i. 



GENERAL THEORY 143 



But for each a n there exists a 7 >. a n ; 
hence 



and therefore Z >. I. (2 

Similarly, for each d n there exists an a <n > S n ; 

hence /a> </(<o 

and therefore 7 < / f3 

Thus 2), 3) give ]im/ ~ = L 

B 

Hence by I, 316, 2 the relation 1) holds. 

The rest of the theorem follows along the same lines. 

2. As a corollary we have 

The positive term series A = 'a i . is convergent if A v ... is 
limited. 

123. 1. Let A = 2# tl ... l = 2a t , B 26 tl ... lg S5 t 6e two non- 
negative term series. If they differ only by a finite number of 
terms, they converge or diverge simultaneously. 

This follows at once from 120, 2. 

2. Let A, B be two non-negative term series. Let r> denote 
a constant. If a L < rl\ , A converges if B is convergent and A j< rB. 
If a t > rb t , A diverges if B is divergent. 

For on the first hypothesis 

and on the second A 

3. Let A, B be two positive term series. Let r, s be positive 
constants. If 



or if 

limf-' 

1=00 6 t 

exists and is = 0, A and B converge or diverge simultaneously. If 
converges and -* == 0, A is convergent. If B diverges and ~ == oo, 

* ' *n 

A is divergent. 



144 



MULTIPLE SERIES 



4. The infinite non-negative term series 

2^,...,. and 2 log (1 -f ,,...,.) 
Converge or diverge simultaneously. 
This follows from 2. 

5. Let the power series 



converge at the point a = (a r - #,), A#? tV converges absolutely for 
all points x within the rectangular cell H whose center is the origin, 
and one of whose vertices is a; that is for \ x, \ < | a t | , t= 1, 2, - s, 

For since P converges at a, 

lim * Wimi ...af ...<= 0. 

M=oo 

Thus there exists an Tiff such that each term 



Hence 



<M 



Thus each term of P is numerically < than TUf times the cor- 
responding term in the convergent geometric series 



We apply now 2. 

We shall call R a rectangular cell of convergence. 

124. 1. Associated with any m-tuple series .<4 = 2a li ... l are 

an infinite number of simple series called associate simple series, 
as we now show. 

Let R>, , R^ , R x , ... 

be an infinite sequence of rectangular cells each lying in the 
following. Let 

^\ 1 *> ' " ' ' *1 

be the terms of A arranged in any order lying in J? AI . Let 



GENERAL THEORY 145 

he the terms of A arranged in order lying in R^ R^ and so on 
indefinitely. 

Then s^ = ai + a ^ + ... + ^ + rtfi+i + ... 

is an associate simple series of ^L. 

2. Conversely associated with any simple series 21 = 2a n are an 
infinity of associate m-tuple series. In fact we have only to arrange 
the terms of 21 over the non-negative lattice points, and call now 
the term a n which lies at the lattice point i l L m the term <7 4 ... lw . 

3. Let$\ be an associate series of A = S^ ti ... lw| . Zf 21 ^8 convergent, 
so is A and ^ __ s ^ 

For A Vl ... Vm = % n . 

Let now v = oo, then n = oo. But ?I W = 31, hence J.^ ... Vm = 31. 

4. If the associate series ?l ?'s absolutely convergent, so is A. 
Follows from 3. 

5 If A = Sa^ ... ^ m i a non-negative term convergent series, all its 
associate series 21 converge. 

For, any 2l,, NP lies among the terms of some A^ v . But for X 
sulliciently large ^ < X < M < ^ 
Hence 



6. Absolutely convergent series are commutative. 

For let 5 be the series resulting from rearranging the given 
series A. 

Then any associate 93 of B is simply a rearrangement of an 
associate series 21 of A. But 21 = 33, hence A = B. 

7. A simply convergent m-tuple series A can be rearranged, 
producing a divergent series. 

For let 21 be an associate of A. 21 is not absolutely convergent, 
since A is not. We can therefore rearrange 21, producing a series 
33 which is divergent. Thus for some 33 

lim SB W 

does not exist. Let 33' be the series formed of the positive, and 
33" the series formed of the negative, terms of 33 taken in order. 



146 MULTIPLE SERIES 

Then either 33^ = + 00 or JB(J = oo, or both. To fix the ideas 
suppose the former. Then we can arrange the terms of 33 to 
form a series & such that S n == + oo. Let now S be an associate 
series of 0. Then 

^v = ^v^t vtn ~ &n 

and thus 

lim G v = lim 6 n = + oo. 

Hence (7 is divergent. 

8. If the multiple series A is commutative, it is absolutely con- 
vergent. 

For if simply convergent, we can rearrange A so as to make the 
resulting series divergent, which contradicts the hypothesis. 

D. In 121, 2 we exhibited a convergent series to show that 
a ti mlm does not need to converge to if ^ i m converges to an ideal 
point some of whose coordinates are finite. As a counterpart we 
have the following : 

Let A be absolutely convergent. Then for each e > there exists 
a \, such that any finite set of terms B lying without R^ satisfy the 

relation \ T>\ ^ s-\ 

\B\<e\ (1 

and conversely. 

For let SI be an associate simple series of Adj A. Since 21 is 
convergent there exists an n, such that 

<. 

But if X is taken sufficiently large, each term of B lies in 2l w , 
which proves 1). 

Suppose now A were simply convergent. Then, as shown in 7, 
there exists an associate series ) which is infinite. 

Hence, however large n is taken, there exists a p such that 



Hence, however large X is taken, there exist terms B= ( $) n ^ p which 
do not satisfy 1). 

10. We have seen that associated with any m-tuple series 



GENERAL THEORY 147 

extended over a lattice system 9K in 9t m is a simple series in 9? r 
We can generalize as follows. Let 2ft = \i\ be associated with a 
lattice system 9ft = \j\ in 9t w such that to each L corresponds &j and 
conversely. 

If i~ j we set a, ,= a. 

J 4... im ./!> 

Then .A gives rise to an infinity of w-tuple series as 



We say JS is a conjugate n-tuple series. 
We have now the following : 

Let A be absolutely convergent. Then the series B is absolutely 
convergent and A = . 

For let A\ B' be associate simple series of A, B. Then A, B 1 
are absolutely convergent and hence A f = '. But A = A r , B = B f . 
Hence A B, and B is absolutely convergent. 

11. Let A = 2a tl ... lm be absolutely convergent. Let B = Sa^...^ 
be any p-tuple series formed of a part or all the terms of A. Then 
B is absolutely convergent and 



For let A, B 1 be associate simple series of A and B. Then B f 
converges absolutely and |jB ; |< Adj A. 

125. 1. Let A-So,...^. (1 



in the cell 
Then 



.. 
1 ..., m = f /*r 1 .-*r m . (2 

c//2..., 



Let 72 denote that part of 3t m whose points have non-negative 

coordinates. Let 

(3 



If e/is convergent, A = J. We cannot in general state the con- 
verse, for A is obtained from A v by a special passage to the limit, viz. 



148 MULTIPLE SERIES 

by employing a sequence of rectangular cells. If, however, 
a v >_ we may, and we have 

For the non- negative term series 1) to converge it is necessary and 
sufficient that the integral 3) converges. 

2. Let f(2\ :r w ) > he a monotone decreasing function of 
x in R, the aggregate of points all of whose coordinates are non- 

negative. Let *f x 

* ^....m =/Oi '"O. 

The series A ^ 

" '*% ITO 

is convergent or divergent with 

J= ( fd*\ " dx m . 
J n 

For let JKj, J? 2 , be a sequence of rectangular cubes each R n 
contained in R n +\ . 

Let R n ^R s -R n *>n. 

Then X, /i being taken at pleasure but > some v, there exist an 
I, m such that 



But the integral on the right can be made small at pleasure if J 
is convergent on taking I > m > some n. Hence A is convergent 
if t/is. Similarly the other half of the theorem follows. 

Iterated Summation of Multiple Series 
126. Consider the finite sum 

2a lt ... lm ^ = 0, 1, /ij " i m = 0, 1, n m . (1 

One way to effect the summation is to keep all the indices but 
one fixed, say all but ij, obtaining the sum 



Then taking the sum of these sums when only * 2 is allowed to 
vary obtaining the sum m m 



ITERATED SUMMATION OF MULTIPLE SERIES 149 



and so on arriving finally ;it 

m n 



(2 



whose value is that of 1). We call this process iterated summa- 
tion. We could have taken the indices L I i m in any order 
instead of the one just employed; in each case we would have 
arrived at the same result, due to the commutative property of 
finite sums. 

Let us see how this applies to the infinite series, 

^. = 2tf 4i ... lw , ^ ^ = 0, l,...oo. (3 

The corresponding process of iterated summation would lead us 
to a series .^ ^ ^ 



which is an m-tuple iterated series. Now by definition 

*,n v m 1 v \ 

21 = lim 2 lim 2 lim 2 (i ... lw (5 

= lim lim lim A v ^.,. Vrn , (6 

while A i- * /T 

A = lim <4 , ...^. (7 

Thus A is defined by a general limit while 21 is defined by an 
iterated limit. These two limits may be quite different. Again 
in 6) we have passed to the limit in a certain order. Changing 
this order in 6) would give us another iterated series of the type 
4) with a sum which may be quite different. However in a large 
class of series the summation may be effected by iteration and this 
is one of the most important ways to evaluate 3). 

The relation between iterated summation and iterated integra- 
tion will at once occur to the reader. 

127. 1. Before going farther let us note some peculiarities of 
iterated summation. For simplicity let us restrict ourselves to 
double series. Obviously similar anomalies will occur in m-tuple 
series. 



150 MULTIPLE SERIES 



+ a o2 + + a 10 + a n + 
be a double series. The m ih row forms a series 



, 

n=0 

and the n ih column, the series 



0= (7" = a mn 

n-Q n~0 w 

are the series formed by summing by roivs and columns, respec- 
tively. 

2. A double series may converge although every row and every 
column is divergent. 

This is illustrated by the series considered in 121, 2. For A 

is convergent while 2 M , 2a rj are divergent, since their terms are 
not evanescent. ' <s ~ 

8. A double series A may be divergent although the series R ob- 
tained by summing A by rows or the series G obtained by summing 
by columns is convergent. 

Forlet A^Q if r or 8 = 

if r, s > 0. 



r H- s 

Obviously by I, 318, lim A T8 does not exist and A = 2# r , is di- 
vergent. 

On the other hand, 

R = lim lim A r8 = 0, 

(7= lim lim A r8 = 1. 
Thus both R and (7 are convergent. 



ITERATED SUMMATION OF MULTIPLE SERIES 151 

4. In the last example R and converged but their sums were 
different. We now show : 

A double series may diverge although both R and C converge and 
have the same sum. 

For let A rtS = if r or s = 

= -^- T) ifr.'.X). 



Then by I, 319, Km A r8 does not exist and A is divergent. On 
the other hand, fl = H m l im 4, = 0, 






C=\\m lim.A r , = 0. 

=00 r=00 

Then R and S both converge and have the same sum. 

128. We consider now some of the cases in which iterated sum- 
mation is permissible. 

CD 

Let A = S/7, t ... be convergent. Let t' v ^, i r m be any permutation 
of the indices i r i%, i m . If all the m \-tuple series 

00 00 00 

2 2 ... S a.,... . 

t '=rO t '=() t ' .=30 * " 

23 m 

are convergent, A= l "\2 tl ... lw . 

l i~ l ~ 

This follows at once from I, 324. For simplicity the theorem 
is there stated only for two variables ; but obviously the demon- 
stration applies to any number of variables. 

129. 1. Let f(%i - # m ) be a limited monotone function. Let the 
point a = (# x a nl ) be finite or infinite. When f is limited, all the 
s-tuple iterated limits jj m . 



exist . When s = m, these limits equal 

lim/to-aw). (2 

#== 

limits we suppose z<a. 



152 MULTIPLE SERIES 

For if /is limited, Hm/ ^ ^^ (3 

.Tt a =rti a 

exists by 122, 1. Moreover 3) is a monotone function of the re- 
maining m 1 variables. 

Hence similarly Um lim y 

^i_ l =flt_ l xi a =ai s 

exists and is a monotone function of the remaining m 2 vari- 
ables, etc. The rest of the theorem follows as in I, 324. 

2. As a corollary we have 

Let A be a non-negative term m-tuple series. If A or any one of 
its m-tuple iterated series is convergent, A and all the ml iterated 
m-tuple series are convergent and have the same sum. If one of these 
series is divergent, they all are. 

fS. Let a be a non-negative term m-tuple series. Let s < m. All 
the s -tuple iterated series of A are convergent if A is, and if one of 
these iterated series is divergent, so is A. 

130. 1. Let A = Sa ti ... tm be absolutely convergent. Then all its 
s -tuple iterated series s = l, 2---m, converge absolutely and its 
m-tuple iterated series all = A. 

For as usual let a ti ... tw = |a tl ... tw |. Since A = Adj A is con- 
vergent, all the 8-tuple iterated series of A are convergent. 

QO oo 

Thus $! = S tl ... lm i g convergent since 2 tl . . i w = <r t . Moreover 

t 



I s l I < <r r Similarly 2 Sa tl ... lw = Ssj is convergent since 

1^=0 tt =0 l a 

2 2a lt ... lm =s 20-j is convergent; etc. Thus every s-tuple iter- 
, a =o i t =o l2 

ated series of A is absolutely convergent. The rest follows now 
by 128. 

2. Let A = 2a tt ... lm . If one of the m-tuple iterated series B 
formed from the adjoint A of A is convergent, A is absolutely con- 
vergent. 

Follows from 129, 2. 

3. The following example may serve to guard the reader against 
a possible error, 



ITERATED SUMMATION OF MULTIPLE SERIES 158 

Consider the series 

a 2 a* 



as ... !=s 

and R = e" + e 2(t + ^ -f 

This is a geometric series and converges absolutely for a < 0. 
Thus one of the double iterated series of A is absolutely conver- 
gent. We cannot, however, infer from this that A is convergent, 
for the theorem of 2 requires that one of the iterated series formed 
from the adjoint of A should converge. Now both those series 
are divergent. The series A is divergent, for \a n \ == oo , as 
r, s = oo . 

131. 1. Up to the present the series 

2., ...... (1 

have been extended only over non-negative lattice points. This 
restriction was imposed only for convenience ; we show now how 
it may be removed. Consider the signs of the coordinates of a 
point x (x v x m ). Since each coordinate can have two signs, 
there are 2 m combinations of signs. The set of points x whose 
coordinates belong to a given one of these combinations form a 
quadrant for m = 2, an octant for m = 3, and a 2 m -tant or polyant 
in 9? m . The polyant consisting of the points all of whose coordi- 
nates are > may be called the first or principal polyant. 

Let us suppose now that the indices i in 1) run over one or more 
polyants. Let JS A be a rectangular cell, the coordinates of each of 
its vertices being each numerically < X. Let A^ denote the terms 
of A lying in # A . Then I is the limit of A K for X = oo, if for each 
e > there exists a X such that 

\A,-A^\<e X>X . (2 



154 MULTIPLE SERIES 

If lim A k (3 

A= 

exists, we say A is convergent, otherwise A is divergent. In a 
similar manner the other terms employed in multiple series may 
be extended to the present case. The rectangular cell R Xo which 
figures in the above definition may without loss of generality be 
replaced by the cube 

K|<A, - km|<V 

Moreover the condition necessary and sufficient for the exist- 
ence of the limit 3) is that 

| A - Ap | < X, /JL > \ Q . 

132. The properties of series lying in the principal polyant 
may be readily extended to series lying in several polyants. For 
the convenience of the reader we bring the following together, 
omitting the proof when it follows along the same lines as before. 

1. For A to converge it is necessary and sufficient that 

lim A x = 0. 

A= 

2. A series whose adjoint converges is convergent. 

3. Any deleted series of an absolutely convergent series A is 
absolutely convergent and 

\B\< Adj A. 



4. If A = 2a h ... tn is convergent, so is B = TLka^... ln and A = kB. 

5. The non-negative term series A is convergent if A^ is limited, 



= oo. 



6. If the associate simple series 21 of an m- tuple series A converges, 
A is convergent. Moreover if 21 is absolutely convergent, so is A. 
Finally if A converges absolutely, so does 21. 

7. Absolutely convergent series are commutative and conversely. 

8. Let ./(#! # m ) >.0 be a monotone decreasing function of the 
distance of x from the origin. 

Let 



ITERATED SUMMATION OF MULTIPLE SERIES 155 

Then 4 v 

A = ^,.. lm 

converges or diverges with 



the integration extended over all space containing terms of A. 

133. 1. Let B, C, D - denote the series formed of the terms of A 
lying in the different poly ants. For A to converge it is sufficient 
although not necessary that B, C, converge. When they do, 

A = B+ C + -D+ - (1 

For if 7? A , CA denote the terms of B, which lie in a 
rectangular cell R^ 



Passing to the limit we get 1). 

That A may converge when B, (7, do not is shown by the 
following example. Let all the terms of A= 2a tl .. tm vanish ex- 
cept those lying next to the coordinate axes. Let these have the 
value +1 if i v i 2 - i m >0 and let two a's lying on opposite sides 
of the coordinate planes have the same numerical value but opposite 
signs. Obviously, A^ = 0, hence A is convergent. On the other 
hand, every B, is divergent. 

2. Thus when B, converge, the study of the given series 
A may be referred to series whose terms lie in a single polyant. 
But obviously the theory of such series is identical with that of 
the series lying in the first polyant. 

3. The preceding property enables us at once to extend the 
theorems of 129, 130 to series lying in more than one polyant. 
The iterated series will now be made up, in general of two-way 
simple series. 



CHAPTER V 
SERIES OF FUNCTIONS 

134. 1. Let i = (* r i. 2 ) run over an infinite lattice system ?. 
Let the one-valued functions 



be defined over a domain 21, finite or infinite. If the jt?-tuple series 



extended over the lattice system 8 is convergent, it defines a one- 
valued function F(x l # m ) over 21. We propose to study the 
properties of this function with reference to continuity, differen- 
tiation and integration. 

2. Here, as in so many parts of the theory of functions depend- 
ing on changing the order of an iterated limit, uniform convergence 
is fundamental. 

We shall therefore take this opportunity to develop some of its 
properties in an entirely general mariner so that they will apply 
not only to infinite series, but to infinite products, multiple inte- 
grals, etc. 

3. In accordance with the definition of I, 325 we say the series 
1) is uniformly convergent in 21 when F^ converges uniformly to its 
limit F. Or in other words when for each e>0 there exists a \ 
such that 



- , 

I -ff -T jUl |< 

for any x in 31. Here, as in 117, F^ denotes the terms of 1) lying 
in the Fectangular cell R^, etc. 

As an immediate consequence of this definition we have : 

Let 1) converge in 21. For it to converge uniformly in 21 it is 
necessary and sufficient that \ F>, | is uniformly evanescent in 21, or in 
other words that for each e > 0, there exists a X such that F^ \ > e for 
any x, in 21, and /A>\. 

156 



GENERAL THEORY 157 

135. 1. Let 

lim /(^ x m < ^ 't n ) = $(x l a; m ) 

tT 

in 21. Here 21, r may be finite or infinite. If there exists an 
77 >0 such that /==</> uniformly in V^a), a finite or infinite, we 
shall say f converges uniformly at a ; if there exists no rj < 0. we 
say / does not converge uniformly at a. 

2. Let now a range over 21. Let 93 denote the points of 21 at 
which no rj exists or those points, they may lie in 21 or not, in 
whose vicinity the minimum of 77 is 0. Let D denote a cubical 
division of space of norm d. Let 93^ denote as usual the cells of 
D containing points of 93. Let (/> denote the points of 21 not in 
93^. Then/=< uniformly in (# however small d is taken, but 
then fixed. The converse is obviously true. 

3. Iff converges uniformly in 21, and if moreover it converges at a 
finite number of other poi tits 33, it converges uniformly in 21 -f 33. 

For if / = (f> uniformly in 21, 

|/- 0| < e x in 21, t in F 6o *(T). 
Then also at each point b a of 93, 

I/ -<l<e x = b. inr 4j *(T). 

If now S < 8 , 8j, S 2 these relations hold for any x in 21 + 93 

and any t in F 5 *(r). 

4. Let f(x " x m , j n ) == </> (^ # m ) uniformly in 21- 
/ Je limited in 21 /or eac/i m PyO"). 7%ew $ is limited in 21. 



for any x in 21 and t in V s *(r). Let us therefore fix t. The 
relation 1) shows that <J> is limited in 21. 

f>. jfjf 2 |/ ti ... ^(^ XM) | converges uniformly in 21, so dWs 2/ tl ... v 
For any remainder of a series is numerically < than the corre- 
sponding remainder of the adjoint series. 

6. Let the s-tuple series 



158 SERIES OF FUNCTIONS 

converge uniformly in 21. Then for each e > there exists a X 
such that , p I ,+ 

for any R v > R^ > jR A . When 8=1, these rectangular cells re- 
duce to intervals, and thus we have in particular 

f n (x 1 x m ) | < e for any n > n'. 
When 8 > 1 we cannot infer from 1) that 

\f ll ...t.(ix l -x m )\<e , in 21, (2 

for any i lying outside the above mentioned cell J? A . 

A similar difference between simple and multiple series was 
mentioned in 121, 2. 

However if f t > in 21, the relation does hold. Cf. 121, 3. 



in 



136. 1. Let f \x 1 #,, t l n ) fo defined for each x in 21, #wd t 

. .Z^ ,. ,, . . w 

hm/= 0^ a: m ) ^n 81, 

t=r 

finite or infinite. The convergence is uniform if for any x in 21 

O tin ^*(r 
lim ir = 0. 



For taking e>0 at pleasure there exists an ?;>0 such that 

|l/r|<6 , *in r,*(T). 

But then if 8< ?;, 

i/-*i< 

for any ^ in Fi*(r) and any ^ in 21. 

Example. 

T . sin x sin y A , or ^A x 
hm T^ T2 = = ^ m 21 = (0, oo). 
y==![ l + #tan 2 y 

Is the convergence uniform ? 
Let 

*- = -; 

then w = 0, as y = 



GENERAL THEORY 159 



Then 



sin x cos u 



\+x cot 2 u 
sin x sin^u 



SHI x cosi4sin 2 t6 



. A 
u = 0. 



I x cos 2 
Hence the convergence is uniform in 21- 

2. As a corollary we have 

Weierstrass Test. For each point in SI, let \f^... lp \^.M^... lp 
The series ^Lf ll ... lp (x l #,) is uniformly convergent in 21 if 
is convergent. 

Example 1. 

' 

Here 



and F is uniformly convergent in 21 since 

y L 

^y On 

is convergent. ^ 

Example %. 

JF T (o;) = 2a 7l sin X n a? 

is uniformly convergent for ( 00, oo) if 

2 | a n | 
is convergent. 

137. 1. The power series P = '2a mi ... mp x'T l *'"!* converges 
uniformly in any rectangle R lying within its rectangle of con- 
vergence. 

For let b = (J r l p ) be that vertex of R lying in the principal . 
polyant. Then P is absolutely convergent at 6, i.e. 



is convergent. Let now # be any point of R. Tlion each term in 

2<v ...,,? -&"' 

is < than the corresponding term in 1). 



160 SERIES OF FUNCTIONS 



an 



2. If the power series P a^-\- a^x -\- a^x 1 -\- converges at a 
end point of its interval of convergence, it converges uniformly at 
this point. 

Suppose P converges at the end point x = R > 0. Then 



however large n is taken. But for < .> < H 



<e by Abel's identity, 83, i. 

Thus the convergence is uniform at x~R. In a similar 
manner we may treat x = R. 

3. Let ,/*(#! :r m ), M = 1, 2 be defined over a set 21. If each 
\f n < some constant c n in 21, / is limited in 31. If moreover the 
r n are all < some constant f, we say the f n (x) are uniformly 
limited in 21. In general if each function in a set of functions 
{/! defined over at point set 21 satisfy the relation 

|/ 1 < a fixed constant 6 Y , x in 21, 
we say the jf s are uniformly limited in 21. 

The series F= ^gji n is uniformly convergent in 21, if G = c/ l -h// 2 -f 
is uniformly convergent in 21, while 2 1 h n+l h n \ and \ h n \ are 
uniformly limited in Jl. 

Tliis follows at once from Abel's identity as in 83, 2. 

4. The series F=^.o n h n is uniformly convergent in 21, if in 21, 
2 | h n+l h n | is uniformly convergent, h n is uniformly evanescent, 
and the Gr n uniformly limited. 

Follows from Abel's identity, 83, l. 

5. The series F= ^g n h n i* uniformly convergent in 21 if 
Gr = g l -f g% -f is uniformly convergent in 21 while h v h% - are 
uniformly limited in 21 and \h n \ is a monotone sequence for each 
point of 21- 

FOP by 88,1, . 



liKXKltAL THEORY l(it 

(). The series F~ 7LgJ l n ^ s uniformly convergent in ?l if Gr l = <j r 
(jr., <j l 4- # 2 , are uniformly limited in 31 and if h v h^ not only 
form a monotone decreasing sequence for x in 91 hut also are uni- 
formly evanescent. 

For by 83, 1, , F , , ff 

I -^w, P \ < "'n+l tr ' 

Example. Let A = a l + a. z 4- be convergent. Let b v {)% = 
be a limited monotone sequence. Then 



converges uniformly in any interval 21 which does not contain a 
point of 



A 

For obviously the numbers 
A.= 



form a monotone sequence at each point of ?I. We now apply 5. 

7. As an application of these theorems we have, using the re- 
sults of 84, 

The series , , 

tf -f a t cosx-h 2 cos %x 4- 

converges uniformly in any complete interval not containing one of 
the points 2 mir provided 2 | a n ^ a n \ is convergent and a n == 0, 
and hence in particular if a l > a 2 > == 0. 



8. , , 

^ a x cos x 4- a 2 cos 2 # 

converges uniformly in any complete interval not containing one of 
the points (2 m I)TT provided 2 a n+1 4- a n \ is convergent and 
a n = 0, 6?wJ A^n^ in particular if a l > a, 2 > = 0. 

9. The series . , . . 

j sin # 4- a 2 sin 2 # 4- 3 sin 6 x 4- 

converges uniformly in any complete interval not containing one of 
the points 2 mn provided 2 | a n+l a,, | i convergent and a n = 0, 
^///(^ hence in particular if a l >^ a 2 >^ ==0. 



162 SERIES OF FUNCTIONS 

10. Theories ^ ^ x _ ^ g a; + a 3 sin Sx _ ... 

converges uniformly in any complete interval not containing one of 
the points (2m I)TT provided 2 | #n+i+ a w | convergent and 
a n = 0, and hence in particular if a^>_a^ _>= 0. 

138. 1. Let F-V/* f* .. *^ 

-* ~ ~/ii i.V 1 ! ^m7 

6e uniformly convergent in 21. ie .A, jB Je ^o constants and 



i uniformly convergent in ?I. 
For then 



But F being uniformly convergent, 

I^A,J<6. 

2. Let f^Zf^fr...^ />0 

converge uniformly in 31. 



i uniformly convergent in 21. Moreover if F is limited in 21, so 
is L. 



or / t > in 21, hence 



for any t outside some rectangular cell 72 A . 
Thus for such i 

4/c < log (1 +/ t ) < Bf, in 21. 

139. 1. Preserving the notation of 136, let g^ # 2 , g m be chosen 
such that if we set 

formly in 21, 

lira A = Km \f(g l "(/ m ,t l O- 

tT 1 = T 



GENERAL THEORY 103 

For if /= $ uniformly in 31, 

e>0, 8>0 |/_4>| < 

for any x in 21 and any t in Fa*(r), S independent of x. 
But then | A | < e t in F 6 *(V). 

2. As a corollary we have : 

Let #}, # 2 ) == # -^^ -^ = ^/ & e uniformly convergent at a. 

Then iU0-o. 

140. Example 1. 

r ,. r sin w sin 2 w ., (2for#=0, 

hm/ = lim - -=<f>(#j=J ' 

u =o w =o sin 2 !/ -h x cos 2 16 1 for x^ 0. 

The convergence is not uniform at # = 0. For 



/.__ _ _ 

\ + x cot 2 ^ 
Hence if we set x = v? 

lim/= 1, since w 2 cot 2 u^= 1. 

M = 

Thus on this assumption 

Urn |/-</| = 1 1-2| = 1. 

Example 2. F =\ - x + x(\ - x)+ x*(l - a:)-f 2^(1 - x) -h 

Here ^ . 

J 7 = E(l #) a: n . 

o 

Hence F is uniformly convergent in any ( r, r), < r < 1, by 
136, 2. 

We can see this directly. For 



Hence ^ is convergent for -l<z<l, and then 
except at x = 1 where F = 0. 

Thus | F n (x) | = | x | n , except at # = 1. 
But we can choose m so large that r m <e. 
Then | J^ m (a;) | < for any a? in (A r). 



164 SERIES OF FUNCTIONS 

We show now that F does not converge uniformly at z~ 

For let 1 

a n = 1 --- 
n 



and F does not converge uniformly at x = 1, by 139, 2. 
Example S. <*> ,2 



Here -, 

f ^ 

J n -| 



and JP is telescopic. Hence 



-rob 

<T ' ^ -y" 



1 + 

-0 , * = 0. 
Thus __ l 

Let us take ^ 

Then __ i 

and JP is not uniformly convergent at x=0. It is, however, in 
(00, GO) except at this point. For let us take x at pleasure 
such, however, that I x \ > S. Then 



n \ i , f 

1 -f 

We now apply 136, 1. 
Example 4. 



-f- n 



GENERAL THEORY 165 

Here , t i \ 

f n n -f 1 I 



and F is telescopic. Hence 

x (n + l)a 



-rb '"* = <-* 

The convergence is not uniform at x = 0. 

For set a n = ^. Then 

n + 1 



It is, however, uniformly convergent in 21 except at 0. For 

if | x | > 8, 

(n + l)a? 



< e for n > some m. 

141. Let us suppose that the series .F converges absolutely and 
uniformly in 21. Let us rearrange JP, obtaining the series Gr. 
Since F is absolutely convergent, so is Gr and F = Gr. We can- 
not, however, state that Gr is uniformly convergent in 21, as Bocher 
has shown. 

Example. -. __ 

x 

F 2n =* 0. 



Hence ^is uniformly convergent in 21 = (0, 1). 
Let 



X 

Then 



a? 



166 SERIES OF FUNCTIONS 

Let j 

n 
Then 



== -f 1 -- ) as n = QO. 
e\ ej 

Hence O- does not converge uniformly at x = 1. 

142. 1. Let f =* (f> uniformly in a finite set of aggregates 21^ 
2J 2 , - 2l p . Then f converges uniformly in their union (2lj, 2l p ). 

For by definition 

> 0, . > 0, |/ - (f> I < e a? in ., * in JV(r). (1 

Since there are only p aggregates, the minimum 8 of Sj, S p 
is > 0. Then 1) holds if we replace S, by 8. 

2. The preceding theorem may not be true when the number 
of aggregates 2lp 21 2 is infinite. For consider as an example 



which converges uniformly in 21 = (0, 1) except at x = 1. Let 



Then .F is uniformly convergent in each 21,, but is not in their 
union, which is 21- 

3. Letf^ <f>, g = ^ uniformly in 21. 

Then f g == <f> ty uniformly. 

If <, -\/r remain limited in 21, 

fg~<f)'\lr uniformly. (1 

Jf moreover | ^ | > some positive number in 21, 

L ~ 2L uniformly. (2 

* ' 

The demonstration follows along the lines of I, 49, 50, 51. 



GENERAL THEORY 167 

4. To show that 1), 2) may be false if <, ^r are not limited. 
Let - 



Then $ = $ = - and the convergence is uniform. 
But 9 



Let # = . Then A = 2 as t = 0, and fg does not 
uniformly. 

Again, let - 



the rest being as before. 

*=-, 



Then 1 



But setting x = 



A I = 



= QO as t = 



and - does not converge uniformly to 2. . 
9 * 

143. 1. As an extension of I, 317, 2 we have : 



uniformly in 31. Let 



Let y^rj in F*(r). 

lim/C^ - rc m , ^ y p ) = ^(^ 2? m ), uniformly. 

t=r 

The demonstration is entirely analogous to that of I, 292. 
*' Let linni^...^,^... iO^^i-O ^ t = l,2 

t= T 

uniformly in 91. i^f the point* 



168 SERIES OF FUNCTIONS 

form a limited set 23- Let F(u u p ) be continuous in a complete 
set containing 23. Then 

lira F^ . u^ = F^ ... p ) 

l=T 

un'formly in 21. 

For jP, being continuous in the complete set containing 23, is 
uniformly continuous. Hence for a given e > there exists a 
fixed cr > 0, such that 

| F(it) - jF(v) | < e u in F^O) , v in 25. 
But as U L = v t uniformly there exists a fixed S > such that 
| U L - v l I < e' , 2; in 81 , in F 6 *(r). 

Thus if e' is sufficiently small, W = (M I , w p ) lies in V ff (v) 
when x is in 21 and t in F" 6 *(r). 

144. 1. 



uniformly in 31. ]jm ^ _ 

f^T 

uniformly in 2t, ^y > </> f limited. 
This is a corollary of 143, 2. 

lim/O^ -. a: m , ^ - O = <#)(^ - a: m ) 

t=T 

uniformly in 21. -Z/^f </> J^ greater than some positive constant in 31. 



uniformly in $,, if $ remains limited in 21. 
Also a corollary of 143, 2. 

3. Let f == (f> and g === i/r uniformly, as t === r. 

Ze (/>, i/r be limited in 21, awc? c/> > so7/ie positive number. Then 

fp == 0^ uniformly in 21. (1 

For (2 



GENERAL THEORY 169 

But by 2), log/=Mog< uniformly in 21; and by 142, 3 
= i/r log 0, uniformly in 21. Hence 2) gives 1) by 1. 



145. 1. The definition of uniform convergence may be given a 
slightly different form which is sometimes useful. The function 

/Oi - * m , *i - O 

is a function of two sets of variables x and , one ranging in an $R m 
the other in an 9t n . 

Let iis set now w = (^ # m , ^ n ) and consider w as a point in 
m -f-jp way space. 

As # ranges over 21 and over PVOr), let w range over 3j fi . 
Then 



uniformly in 21 when and only when 

e>0, S>0 |/-0|< w in SBa, 8 fixed. 

By means of this second delinition \vc obtain at once the follow- 
ing theorem: 

2. Instead of the variables x x m , t l n Z^t tf introduce the 
variables ^ y m , Wj w n ^ ^^ * ' ranges over 33s, 



ranges over (5s, f/ae correspondence between 33s, CSs being uniform. 
Thenf^= $ uniformly in 21 wAe/i '/i^ 0/4^ ^/i^/i 

e > 0, 8 > \f <f>\<e , ^ m (3, 
3. Example. \ 



where /a A A 

; X 



Then </>(o;) = lim/(^, ri) = , in 21 = (0, oo). 

71=00 

Let us investigate whether the convergence is uniform at the 
point x in 2l 

First let x > 0. If < a < # < 6, wo have 

\f-*\<~ 




170 SERIES OF FUNCTIONS 

As the term on the right = as n^= <x> , we see/=0 uniformly 
in (a, ft). 

When, however, a = 0, or b = ao , this reasoning does not hold. 
In this case we set 



which gives ^ i og i//i . t 

nr/P 
As the point (x, ri) ranges over delined by 



the point (, w) ranges over a iield X defined by 

t > 1 , w > 1, 
and the correspondence between and Z i* uniform. Here 



The relation 2) shows that when x > 0, t ^ co as 7^ -^ oo ; also 
when x 0, = 1 for any n. Thus the convergence at x = is 
uniform when , 



The convergence is not uniform at x = when 3) is not satisfied. 
For take -j 

^ = -;-,- ' w= 1, 2, ... 

^A/a 

For these values of x ^L A 

|/- 0| =e- a -^ 

which does not = as n == QO . 
146. 1. (Moore, Osyood.) Let 



uniformly in 21. i# a be a limiting point of 21 



/or each t in Fi*(r). 

4> = Km 0(^ -. j-J , ^ = lim 

^=a /- T 

exist and are equal. Here a* r are finite or infinite. 



GENERAL THEORY 171 

We first show <I> exists. To this end we show that 







>0 , S>0 , 



n 



Now since f(x, f) converges uniformly, there exists an 77 >0 
such that for any x', x n in 31 

$(x') =/(*', t) + e' t in r,*(r) (2 

*(*'0 =/(*", *) + ''. |e'|,|e"|<*. (8 
On the other hand, since / = ty there exists a 8>0 such that 

/(*', = *(>+ e"' (4 

/(a;", = ^(0 + *' v I '" M *' T | < (5 



for any #', x" in FV*(a) ; t fixed. 

From 2), 3), 4), 5) we have at once 1). Having established 
the existence of <$>, we show now that <J> = Mf. For since f con- 
verges uniformly to <, we have 

|/(^0-<K*0|< > *m a , *iiiF,*(T). (6 

o 

Since /= i^-, we havo 

zin Fy*(a) , t fixed in F/(T). (7 



3 

Since c^> == <t>, 

1 ^(a;) - <I> <^ x in IV*O). (8 

3 

Thus 7), 8) hold simultaneously for 8 < 8', 8". 
Hence 



or lim 

f=T 

2. Thus under the conditions of 1) 

lim lim f = lim lim f ; 

a-=a <==T ^=T ai=a 

in other words, we may interchange the order of passing to the 
limit. 



172 SERIES OF FUNCTIONS 

3. The theorem in 1 obviously holds when we replace the un- 
restricted limits, by limits which are subjected to some condition ; 
e.g. the variables are to approach their limits along some curve. 

4. As a corollary we have : 

Let F = S/aO"! x m ) be uniformly convergent in SI, of which x = a 

is a limiting point. Let Iimf 8 = / a , and set L = 21 8 . Then 

ipft 

Urn F = L ; a finite or infinite, 

o>=a 

or in other words 

Urn 2/ a = 2 limf a . 

Example 1. 



converges uniformly in 21 = (0, oo) as we saw 136, 2, Ex. 1. Here 



and i = 2? n =5J = 1. 



Hence lim F(x) 1. 

<r==oo 

Also J21im/ n = 0; 

jr = () 

hence jK lim F(x)= 0. 

ar = 

Example 2. 



converges uniformly in any interval finite or infinite, excluding 
x = 0, where .F is not defined. For 



+ 

Hence lim F(x) = e. 



GENERAL THEORY 173 

Example 3. 



1 



for x = 



1 + a? 
= fora:=0. 

Here lira jP(z) = 1, 

o?=0 

while Slim /.(a;)= 20 = 0. 

35 = 

Thus here lim 2/.(*)*21i,n /.(*), 

<P = !T = 

But F does not converge uniformly at x = 0. On the other 
hand, it does converge uniformly at # = oc . 

Now lim.FO) = , lim/ n O)=0, 

# = ;/, vo 'J 

alld liui S 



as the theorem requires. 

Example 4. rr/- \ _ V f nx * ( n + 

- - - 






wliich converges about x = but not uniformly. 

However, r v^/-\ vv -c s \ r\ 

Inn 2/ n O?) = 2 Inn f n (x) = 0. 

#=0 ^^0 

Thus the uniform convergence is not a necessary condition. 
147. 1. Let lim f^ l --x m , ^ f n ) = <(>i ^m) uniformly at 

t=T 

x~a. Let f(x, t) be continuous at x=afor each t in I r 8 *(r}. 
Then </> is continuous at a. 

This is a corollary of the Moore-Osgood theorem. 
For by 146, 1 

lira lim /(a -f A, ) = lim lim /(a + A, t). 

A=0 <=T ^=T /i=o 

Hence 



im , , + A) = lim /(a, *) = <^(a 



174 SERIES OF FUNCTIONS 

A direct proof may be given as follows : 

f(x, t) = <KaO + e' | e' | < e, x in V^a) 

<K*0-<K*'0=/<y< -/(*", *)+*' 

But |/O", -/(*', | < e , if | z' - x" | < . 

2. Z0 ^= 2^ t ..., p (a: 1 # m ) ie uniformly convergent at x=a. 
Let each f Sl ... Sp be continuous at a. Then l?(z l # m ) is continuous 
at x = a. 

Follows at once from 1). 

3. In Ex. 3 of 140 we saw that 



is discontinuous at x = and does not converge uniformly there. 
In Ex. 4 of 140 we saw that 



^ 



(1 + ^ 2 )(1 + (w 

does not converge uniformly at x = and yet is continuous there. 
We have thus the result : The condition of uniform convergence in 
1, is sufficient but not necessary. 
Finally, let us note that 






is a series which is not uniformly convergent at x = 0, although 
F(x) is continuous at this point. 



4. Let each term of F ^f fv .-^ p (x l x m ) be continuous at x = a 
^ itself is discontinuous at a. Then F is not uniformly 
convergent. 

For if it were, F would be continuous at a, by 2. 

Remark. This theorem sometimes enables us to see at once 
that a given series is not uniformly convergent. Thus 140, 
Exs. 2, 3. 



GENERAL THEORY 175 

5. The power series P = 2a v ..., m #f i a? is continuous at any 
inner point of its rectangular cell of convergence. 

For we saw P converges uniformly at this point. 

6. The power series P = a -f a^ 4- a 2 a: 2 4- is a continuous 
function of x in its interval of convergence. 

For we saw P converges uniformly in this interval. In par- 
ticular we note that if P converges at an end point x = e of its 
interval of convergence, P is continuous at e. 

This fact enables us to prove the theorem on multiplication of 
two series which we stated 112, 4, viz. : 

148. Let 



converge. Ttien AB = C. 

For consider the auxiliary series 
F(x) = a + r* 



Since J., J?, (7 converge, ^, 6r, IT converge for 2:= 1, and hence 

absolutely for | a: | < 1. But for all | # | < 1, 



Thus L lim HT(aO = i lira F(x) i lim G- (a?), 

^=1 jr=l Jr=^l 

or (7= -A- A 

149. 1. We have seen that we cannot say that /= <f> uniformly 
although /and < are continuous. There is, however, an impor- 
tant case noted by Dini. 

Let f(x l ... x m , t n ) be a function of two sets of variables 
such that x ranges over 21, and t over a set having r as limiting 
point, r finite or ideal. Let 



Then we can set 



176 SERIES OF FUNCTIONS 



Suppose now | ty(x, t')\ <\ *fy(x, ) | for any t r in the rectangu- 
lar cell one of whose vertices is t and whose center is r. We say 
then that the convergence of f to <f> is steady or monotone at x. 
If for each x in 21, there exists a rectangular cell such that the 
above inequality holds, we say the convergence is monotone or 
steady in 31. 

The modification in this definition for the case that r is an ideal 
point is obvious. See I, 314, 315. 

2. We may now state Dims theorem. 

Let /(#! x m , t l w ) = ^(^i " r m) steadily in the limited com- 
plete field 31 as t = r; r finite or ideal. Let f and </> be continuous 
functions of x in 31. Then f converges uniformly to <p in 31. 

For let x be a given point in 31, and 



We may take t' so near T that | -^(x, t')\<~- 
Let x 1 be a point in V^(x). Then 

/(*', 

As /is continuous in 
Similarly, 

Tlms \^(x',t')\<e x' in 

Hence 



, .^ , . ,. / . 

, t) I < e for any x in 

and for any t in the rectangular cell determined by t 1 . 
As corollaries we have : 

3. Let Gr = S |/ tt ...i,(^i Zm) I converge in the limited complete 
domain 31. Let Gr and each f t be continuous in 31- Then Gr and 
a fortiori F= 2/l t ... ta converge uniformly in 31, furthermore f^... l8 = 
uniformly in 31. 

4. Let #=2 \f^... l8 (x l x^ | converge in the limited complete 
domain 31, having a as limiting point. Let Gr and each f, be con- 



GENERAL THEORY 177 

tinuous at a. Then Gf and a fortiori F = 2/ tl ... t , converge uniformly 
at a. 

5. Let Gr = 2 |/ tl ...i a (^i 3" m ) | converge in the limited complete 
domain 21, having a as limiting point. Let lim Gr and each lim/ t 
exit. Moreover, let lim # = 2 lim/i 

Then Q- is uniformly convergent at a. 

For if in 4 the function had values assigned them at x = a dif- 
ferent from their limits, we could redefine them so that they are 
continuous at a. 

150. 1 . Let lim f(x l x m , t l t n ) = < (x l # m ) uniformly in 

t^T 

the limited field 21. I>6tf </> i^ limited in 91. 
Km r/-= 



For let y=<-j->^r. 

Since /= <j> uniformly 1^1 < 

for any t in some V*(r) and for any x in 21. 

Thus f r 

J f~J <f> < 

Remark. Instead of supposing <J> to be limited we may suppose 
that/(^r, t) is limited in 21 for each t near r. 

2. As corollary we have 

Let lim/(o; 1 x m , ^ ^ n ) = ^>(^! ^ /n ) uniformly in the limited 

field 91. Let f be limited and integrable in W for each t in 
Then is integrable in 91 and 



lim f/= f(/>= film/. 

t~T *S% J% ^21 =T 



3. From 1, 2, we have at once: 

Let F='Sf ll ... lt (x l "'X m ) be uniformly convergent in the limited 
field 91. Let eachf ti ... ts be limited and integrable in 2L Then F is 

integrable and p _, / 

X I / tl ...i.. 
JK 



178 SERIES OF FUNCTIONS 

If thef tl ... Lt are not integrdble^ we have 

I yar ** 

Example. _ 



does not converge uniformly at x = 0. Of. 140, Ex. 3. 
Here ^ =1- l 

and p n for #=^0, 

~~ JO fora: = 0. 

Hence C l TU 1 

I Fax J , 

^o 



r^ =i -Tr 



dx 



Thus we can integrate -J 7 term wise although F does not converge 
uniformly in (0, 1). 

151. That uniform convergence of the series 



with integrable terms, in the interval 31 = (a < 6) is a sufficient 
condition for the validity of the relation 



X6 rb /6 

^d/:= I ^^4- fdx- 
J,, * l Ja ' '* 



is well illustrated grai)lii(;ally, as Osgood has shown,* 

Since 1) converges uniformly in 91 by hypothesis, we have 

F n (x^ = F(x}-F n (x) (2 

and 

iP n <|<6 n>m (3 

for any x in 21. 

* Bulletin Amer. Math. Soc. (2), vol. 3, p. 69. 



GENERAL THEORY 



179 



In the figure, the graph of F(x*) is drawn heavy. On either 
Bide of it are drawn the curves F e, F+e giving the shaded 
band which we call the e-band. 

From 2), 3) we see that the graph of 
each F n , n>m lies in the e-band. The 
figure thus shows at once that 



/' 

Ja 



Fdx 



and 



F n dx 



can differ at most by the area of the 
e-band, i.e. by at most 




152. 1. Let us consider a case where the convergence is not 
uniform, as 



Here 



nx 

~^' 



If we plot the curves y = F n (x)^ we observe that they flatten 
out more and more as n = oo, and approach the 2;-axis except 

near the origin, where 
they have peaks which 
increase indefinitely in 
height. The curves 
F n (x), n>m, and m suf- 
ficiently large, lie within 
an e-band about their 
limit F(x) in any inter- 
val which does not in- 
clude the origin. 

If the area of the 
region under the peaks 
could be made small at 
pleasure for m sufficiently large, we could obviously integrate 
termwise. But this area is here 




180 



SERIES OF FUNCTIONS 



r 71 j i c a d r * ~b if IT v-i i 

Jo ^-2 Jo &^ aL"^*^ 1 -^ 



as n === QO . 



Thus we cannot integrate the ^ series termwise. 

2. As another example in which the convergence is riot uniform 
let us consider 






Here 



The convergence of J 7 is uniform in 31 = (0, 1) except at x = 0. 
The peaks of the curves F n (x) all have the height e* 1 . 

Obviously the area of the 
region under the peaks can be 
made small at pleasure if m is 
taken sufficiently large. Thus 
in this case we can obviously 
integrate termwise, although 
the convergence is not uniform 
in 21. 

We may verify this analytically. For 

C x -n 7 C x nx 7 1 1 -f nx . A 
I F n dx =1 dx = ---- ^- = as n = oo . 
/o */o e nx n ne nx 

3. Finally let us consider 




+ 



Here 



n*x 



The convergence is not uniform at x = 0. 

The peaks of F n (x) are at the points x = w~ 2 , at which points 



GENERAL THEORY 181 

Their height thus increases indefinitely with n. But at the 
same time they become so slender that the area under them == 0. 
In fact 



Jf >.(*)& -J^ id log 



2n\_ Jo 2 n 

We can therefore integrate term wise in (0 < a). 

153. 1. Let Urn Gr(x, ^ t n ) = #(V) in 21 = (a, a -f 8), T 
or infinite. Let each Gr'^x, t) be continuous in 21 ; also let Gr r x (x, 



converge to a limit uniformly in 21 as t = r. 

Km (?i<>, =^(^) m 21, (1 

tT 

and g 1 (x) is continuous. 
For by 150, 2, 

lim ra f x dx= f'lim ^^. 

/= T *^a ^a /= T 

By 1, 538, r , 

I G-' z dx= G-(x, t)-G(a, t). 

c/a 

Also by hypothesis, Hm { & ^ () _ & f) ; = g ^ _ g ^ 

t=T 

Hence ~ x 

g(x)-g(a)=\ lim &' x (x, t)dx. (2 

/a t=r 

But by 147, 1, the integrand is continuous in 21. 
Hence by I, 537, the derivative of the right side of 2) is this in- 
tegrand. Differentiating 2), we get 1). 

2. Let F(x) = 2/ tl ... la (V) converge in 21 = (a, a 
fl(x) be continuous, also let 



uniformly convergent in 21. 
J ? '(a;) 
This is a corollary of 1. 



182 SERIES OF FUNCTIONS 

3. The more general case that the terms / t ,... t , are functions of 
several variables x r x m follows readily from 2. 

154. Example. 



Here F n * xa 

c x 

a function whose uniform convergence was studied, 145, a. We saw 

F(x)=z$ foranyz>0. 
Hence f'Cx)- 

Let Q, X , 

Then ^ , , _ JT,,^ 
I*>0, 



hence ^ ; (^)= 2/JOc), (2 

and we may differentiate the series termwise. 

If z=0, and = 1, X>0; (? n (0)= n A = oo as n = oo. 

In this case 2) does not hold, and we cannot differentiate the 
series termwise. 

For a;=0, and >1, (? n (0)=0, and now 2) holds; we may 
therefore differentiate the series termwise. But if we look at the 
uniform convergence of the series 1), we see this takes place only 

when 



ft 

155. l 



x 



converge in SI = (a, 6). Jor 0t;ery x in 21 Z# |//(X)| < <7p constant. 
Let Q- = 2^ 4 converge. Then F(x) has a derivative in 21 and 



or i^e Tway differentiate the given series termwise. 



GENERAL THEORY 183 

For simplicity let us take s = 1. Let the series on the right of 
1) be denoted by $0*0 ^ or eac ^ x * n ^ we suow that 

< e, | Arc | < 8. 



e>0, S>0, D 
AJP 



A A* 

where | n lies in F(z). 

Thus 



But (3 1 being convergent, 6r m < e/3 if 77^ is taken sufficiently large. 
Hence 



On the other hand, since -^ ==/n(^) and since there are only rw 

L\X 

terms in D m , we may take S so small that 

|A,|<e/3. 

Thus |J>|< for|Aa;|<S. 

2. Example 1. Let 



This series converges uniformly in 91 = (0 < 6), since 

Also 

Hence n 

As 2# n converges, we may differentiate 1) termwise. In 
general we have 

OP X 

valid in 21. 



184 SERIES OF FUNCTIONS 

3. Example 2. The ? functions. 
These are defined by 

flj (z) = 2 2 ( - 1 ) 0< n+J)i sin (2 tt + 1) TTZ 

o 

= 2 ql sin TTZ 2 9$ sin 3 THE -h 

iV 2 (a?) = 2 ! ? <+i> f cos (2 7i + 1) irx 
o 

= 2 * cos THE -J- 2 ^2 cos 3 THE -f- 



= 1 + 2 2g n2 cos 2 

= 1 -f 2 <? cos 2 TTX + 2 5 4 cos 4 TTX+ 



V (a;) = 1 + 2 2 ( - lj) n j n2 cos 2 

= 1 25' cos 2 THE -f- 2 j 4 cos 4 TTX 
Let us take , , < ^ 

Then these series converge uniformly at every point x. For 
let us consider as an example v r The series 



is convergent since the ratio of two successive terms is 



and this == 0. Now each term in >v t is numerically 



and hence < the corresponding term in T. 

Thus #j (x) is a continuous function of x for every x by 147, 2. 
The same is true of the other v's. These functions were discovered 
by Abel, and were used by him to express the elliptic functions. 

Let us consider now their derivatives. 

Making use of 155, 1 it is easy to show that we may differentiate 
these series termwise. Then 

f>{ O) = 2 TT! ( - l) n (2 n + 1) (f+W cos (2 n -f 1) TTX 
o 

= 27r(}i cos TTX 3 9* cos STTX -f ). 



GENERAL THEORY 185 

^ (V) = - 2 TrI (2 n + 1) ?< B+ S sin (2 w + 1) trx 



= 2?r(^ sin TTX -f 3j* sin 3 THE 4- ). 

00 

V 3 ' (.r) = 4 7r^,nq n2 sin 2 mrx 
i 

= 4 TT (g sin 2 7r# -f 2 j 4 sin 4 TTX + ) . 

oo 

iS' a; = 4 TT] 1 ) n n n * sin 



i 

= -f 4 TT (^ sin 2 THE 2 q* sin 4 TTO; -h ). 

To show the uniform convergence of these series, let us con- 
sider the first and compare it with 



The ratio of two successive terms of this series is 



2n+1 



_ 
2 ra + 1 | j , w8 2 n -f- 1 



which = 0. Thus S is convergent. The rest follows now as 
before. 

156, 1. Let 



uniformly for < | A | < T;, T finite or infinite. 
Let G' r (a, exist 

for each t near T. Then <f (a) exists and 



This is a corollary of 146, 1. Here 

G(a + h f) -&(<*, 

A 
takes the place of f(x, f). 

2. From 1 WQ have as corollary : 



186 SERIES OF FUNCTIONS 

converge for each x in 31 which has x = a as a proper limiting point. 
Letf((a) exist for each t = (i> v n ). Let 



I 

converge uniformly with rexpect to h. Then 



CHAPTER VI 
POWER SERIES 

157. On account of their importance in analysis we shall 
devote a separate chapter to power series. 

We have seen that without loss of generality we may employ 

the series , ,+ 

a 4- a^x + a^ + ... (1 

instead of the formally more general one 

# 4- a,i(x a) -f- 2 (^ <* ) 2 4- 

We have seen that if 1) converges for r = c it converges abso- 
lutely and uniformly in (7,7) where < 7 < | c \. Finally, 
we saw that if c is an end point of its interval of convergence, it 
is unilaterally continuous at this point. The series 1) is, of course, 
a continuous function of x at any point within its interval of 
convergence. 

158. 1. Let P(x) = # 4- a^x 4 a^x 2 4 converge in the interval 
9( = ( , a} which may not be complete. The series 

P n = 1 2 . ... na n + 2 3 . ... O 4- 1X +1 * 4- - 
obtained by differentiating each term of P n times is absolutely and 
uniformly convergent in S3 = ( & /3), /3< a, 



For since P converges absolutely for a; = /8, 



Let now x lie within 93. Then the adjoint series of P^x) is 

^ + 2^ + 
Now its ?7i th term 



187 



188 POWER SERIES 

But the series whose general term is the last term of the pre- 
ceding inequality is convergent. 

2. Let P = a 4- ^x -f- a z x 2 -h 

converge in the interval 21. Then 

Q= I Pdx = I a*dx-\- I a^xdx -f 

*^a ^ a ^a 

where a, # 7/0 iw 21. Moreover Q considered as a function of x con- 
verges uniformly in 21. 

For by 137, P is uniformly convergent in (a, x). We may 
therefore integrate termwise by 150, 3. To show that Q is uni- 
formly convergent in 21 we observe that P being uniformly con- 
vergent in 21 we may set 

P = P + P 

-* -* m i -* m 

w ieie 



Then 
where 



on taking <r sufficiently small. 



> / o , o- small at pleasure. 



<<72l< 



159. 1. Let us show how the theorems in 2 may be used to 
obtain the developments of some of the elementary functions in 
power series. 

The Logarithmic Series. We have 



1 x 
for any x in 21 = (- 1*, 1*). Thus 



1 x 
Hence 




lo !-)=- 



This gives also 

z) = z- + -... ; a: in 



GENERAL THEORY 189 

The series 1) is also valid for x = 1. For the series is conver- 
gent for x = 1, and log (1 -f- x) is continuous at x = 1. We now 
apply 147, 6. 

For x = 1, we get 



2. 2%0 Development of arcsin x. We have by the Binomial 

Series -i -i -i Q -i o c 

- 



V- 2-4 2-4. 

for x in 21 = (- 1*, 1*). Thus 

C* dx , 1 , 1-3 * , ,o 

i ----- = arcsin x = x H -- or* -f -^ - -ar-J- (:i 

/o y i _ ^ 2-8 2-4-5 

It is also valid for x = 1 . For the series on the right is conver- 
gent for x = 1. We can thus reason as in 1. 
For x = 1 we get 

7T , 1 1-3 1.3-5 

2 2-3 2-4-5 2-4-6-7 

3. 7/40 Arctan fteries. We have 



for x in 31 = ( - 1*, 1*). Thus 

C x dx C' r C x 

\ -- '- = arctan x = I dx \ x 2 dx + 

x 3 , a; 5 xo 

= ,-_ + _-... (3 

valid in 21. The series 3) is valid for x = 1 for the same reason as 
in 2. 

For x = 1 we get ^ 11 1 

4 ~S + 5~7 + '" 

4. The Development of e x . We have seen that 

*<* ) = 1+ ii + ii + iT + - 

converges for any #. Differentiating, we get 



190 POWER SERIES 

Hence E'(x) = E(x) (a) 

for any x. Let us consider now the function 



Wehave 



e >2x e* 

by (a). Thus by I, 400,/(V) is a constant. For # = 0,/(X) = 1. 
Hence r ^ ~3 

-"- 1+ fi + ^ + li + - 

valid for any x. 

5. Development of cos #, sin x. 

The series . 



converges for every x. Hence, differentiating, 

c> = -*+*'-- *L+. 

+ ^ 



Hence adding, C+ C" = 0. (b) 

Let us consider now the function 

/()= sin x+ C' cos #. 
Then Q , gin ^ _ ^^ gin ^ + g,,, cog f 



= ((7+ <7")co8a; 
= by (b). 
Thus /(a;) is a constant. But O 1, C" = 0, fora; = 0, hence 

/O)=o, 

or (7 sin x -f (7' cos a; = 0. (c) 

In a similar manner we may show that 
or #(X)= C^cosa; O 1 sin 2= 1. (d) 



GENERAL THEORY 191 

If we multiply (c) by sin # and (d) by cos x and add, we get 
(7= cos x. Similarly we get 0' = sin x. Thus finally 

, z 2 x 4 
oo.*. !-_ + _-... 



valid for any x. 

160. 1 . Let P = a m x m -f a m+l x m+l -f , a m = 0, converge in 
some interval 21 about the origin. Then there exists an interval 
SB < 21 in which P does not vanish except at x = 0. 



Obviously Q converges in 31. It is thus continuous at x = 0. 
Since Q = at x = it does not vanish in some interval 33 about 
a?*0by I, 351. 

In analogy to polynomials, we say P has a zero or root of order 
m at the origin. 

2. Let P = a 4- a^x -f a^ -f vanish at the points b v ?> 2 , ... = 0. 
2% m a^ Ae coefficients a n = 0. 2%^ points b n are supposed to be 
different from each other and from 0. 

For by hypothesis P(bn) = 0. But P being continuous at x = 0, 



Hence P(0)=0, 

and thus A 

a = 0. 

Hence P-x^. 

Thus P l vanishes also at the points b n . We can therefore 
reason on P l as on P and thus a l = 0. In this way we may 
continue. 

3. If P = o + 



192 POWER SERIES 

be equal for the points of an infinite sequence B whose limit is x = 0, 
tlien a n = b n , n = 0, 1, 2 

For P Q vanishes at the points B. 

Hence _ , __ OT -0 1 2... 

a n o n u , n u, i, A 

4. Obviously if the two series -P, $ are equal for all x in a 
little interval about the origin, the coefficients of like powers are 
equal; that is ^ = ^ ^ n = 0,1,2... 

161. 1. Let y = as + a l x 

converge in an interval 21. As x ranges over 21, let y range over 
an interval 33. Let 



converge in 33- Then 2 may be considered as a function of x de- 
fined in 21. We seek to develop z in a power series in x. 

To this end let us raise 1) to the 2, 3, 4 ... powers ; we get 

series 2 , , 2 , 

y = a 2o + a 2i x + V + 



which converge absolutely within 21. 
We note that a mn is a polynomial. 



in # , # n with coefficients which are positive integers. 
If we put 3) in 2), we get a double series 



+ V21' r + *2^22^ ;2 + ' ' ' ( 4 

+ ?> 3^31^ + VV 2 + ' ' ' 



If we sum by rows, we get a series whose sum is evidently 2, 
since each row of D is a term of z. Summing by columns we get 
a series we denote by 

= CQ + C^X -f CyX 2 -f - (5 



GENERAL THEORY 193 



c l = Vi + V 2 i + /> 3 a 31 -f- - (6 



We may now state the following theorem, which is a solution of 
our problem. 

Let the adjoint y-series, 



converge for = f to the value rj = r? . Let the adjoint z series 



converge for 7; = ?; . Then the z series 2) can be developed into a 
power series in x, viz. the series f>), which is valid for \ x \ < | () . 

For in the first place, the series 8) converges for 77 <_?? . We 
show now that the positive term series 



-f 



converges for < f . We observe that ) differs from Adj D, 
at most by its first term. To show the convergence of ) we 
have, raising 7) to successive powers, 



We note that ^l mn is the same function F m%n of , j, n as 
mn is of a , a n , i.^. 

^-m,n= -^nCfloi "* n)- 

As the coefficients of F m% n are positive integers, 

<*m,n = |m f | < 4, n (9 



194 POWER SERIES 

Putting these values of rj, rj 2 , ?; 3 in 8), we get 
A = (/3 4- /^io) ~t~ $i w i? + $\u< 2 ' 4- 

4- 

Summing by rows we get a convergent series whose sum is 
or 8). But this series converges for ( < since then ?/ < ?/ , 
and 8) converges by hypothesis for 77 = rj Q . Now by 9) each 
term of <J) is < than the corresponding term in A. Hence ) 
converges for f < f . 

2. As a corollary of 1 we have : 

Let 

y = /i 4- ^ -f a 2 2: 2 -f ... 

converge in 21, 



converge for all cc < ;y < 4 oo. 7%0w 2 <?aw 6e developed in a 
power series in x, 

z = <? 4- c^ 4- ^ 4- ... == C Y 
/'or aZZ a; within 91. 



3. j/0 ^Ae series 

y = a m a; m 4- m + i^ m+1 4- , m>_\ 
converge for some x > 0. If the series 

z = fy) + i>\y + *2# 2 + 

converges for some y > 0, i (%m i^ developed in a power series 

z= r? 4- ^^ 4- r 2 .r' 2 -f ... 
convergent for some s > 0. 

For we may take = | r j > so small that 

i? = mf w -l-m +1 f wlfl -f - 
has a value which falls within Ihr interval of convergence of 



4. Another corollary of 1 is the following : 

Let 

y= a 4- ! 4- ^ 2 4- 



GENERAL THEORY 195 

converge in 21 = ( A, A). Then y can be developed in a power 
series about any point c of 21, 



y= e? + c l (x c') -f- <? 2 0> < 
which is valid in an interval 93 whose center is c and lying within 21. 

162. 1. As an application of the theorem 161, 1 let us take 

" i+ ff + H + fi+~ 

__ X_ __ X 3 X 5 _ 

y ~n sT + 57 

As the reader already knows, 

z = e v , y = sin x, 
hence z considered as a function of x is 

z = 6 8ln *. 
We have 

z = l-f z + Q-x*- ^ 3 -f O-z 4 -f 

+ Ja^+ - i^ 4 4- 

+ i* 3 + o - T v^ 

+ ^+ 



Summing by columns, we get 

^ _ ^,8in x _ 1 i /v. i 1 , r 2 1 ^,4 1 ^.5 1 ^6 . . . 

Ze l-t-X-t-igX -gX ^X 23"$ 3T 

2. As a second application let us consider the power series 

z = 



convergent in the interval 21 = ( 72, 72). Let a; be a point in 31- 
Let us take 77 > so small that y = x + h lies within 21 for all 
| h | <. rj. 

Then . , , ,, 

s = -f- ^ (x 4- A) 



-h a 3 (^+3 2;% -f 3 zA 2 + A 8 ) 



190 POWER SERIES 

This may be regarded as a double series. By 161, 1 it may be 
summed by columns. Hence 

P(x 4- h) = a 4- ax 4- a^x 2 4- a^x 3 4- 

-f A(aj 4- 2 a 2 :r 4- 3 a^ 4- ) 

7.2 

**+...) (2 

21 a ! 

on using 158, l. 

This, as the reader will recognize, is Taylor's development of 
the series 1) about the point x. We thus have the theorem : 

A power series 1) may be developed in Taylor '# series 3) about 
any point x within its interval of convergence. It ix valid for all h 
such that x+ h lies within the interval of convergence ofV). 

163. 1. The addition, subtraction, and multiplication of power 
series may be effected at once by the principles of 111, 112. We 
have if P /^.L/T^J. 

X I*Q | tl<jC ~ 
W/ "~~ f) I Q y, -J- 

converge in a common interval 31 : 



P - Q = 'v\) + ("A + ^(A)* -f- 

These arc valid within 31, and the first two in 31. 

2. Let us now consider the division of P by M. Since 

^=P 1 ' 
R R 

the problem of dividing P by R is reduced to that of finding the 
reciprocal of a power series. 

I** P == a 4- a^ 4- a^ 4- , *= 

converge absolutely in R =( 7?, 7Z). Z/^f 

$ = a^-f a 2 a^4- 
/>^ numerically < \ a | w 33 = ( ^ /4 ) r < R. 



GENERAL THEORY 197 



TJieit \/P can be developed in a power series 



valid in 33. The fir 8t coefficient <? = . 



1 

" 



a l "i! 



for all # in S3* We have now only to apply 1<I1, 1. 
8. Suppose -n... 



To reduce this case to the former, we remark that 

P = :rQ 

where n . 

C = + ^m ^ + 

Then 1^1 j_ 

P .r- ' Q' 

But l/^ has been treated in 2. 



164. 1. Although the reasoning in 161 affords us a method of 
determining the coefficients in the development of the quotient of 
two power series, there is a more expeditious method applicable 
also to many other problems, called the method of undetermined 
coefficients. It rests on the hypothesis that/(#) can be developed 
in a power series in a certain interval about some point, let us say 
the origin. Having assured ourselves on this head, we set 

f(x) = a + ap + atf? + 

where the a's are undetermined coefficients. We seek enough 
relations between the a's to determine as many of them as we 
need. The spirit of the method will be readily grasped by the 
aid of the following examples. 

Let us first prove the following theorem, which will sometimes 
shorten our labor. 



198 POWER SERIES 

2 ' V f(x) = 0,0 + 0,^ + 0,^+ ...; -R<x<R, (1 

is an even function, the right-hand Me can contain only even powers 
of x; iff(x) is odd, only odd powers occur on the right. 

For if /is even, f( ^ ==/( _ x) . (2 

But 



/(- x) = a - ajtf + <* 2 z 2 - - (3 

Subtracting 3) from 1), we have by 2) 

= 2 (a^ + atf? -f a$? -f ) 
for all # near the origin. Hence by 160, 2 

i = 3 ==a 5 == ' ==:0 - 
The second part of the theorem is similarly proved. 

165. Example 1. /(*-) = tan *. 

Since sina . 

tan ic = - , 
cos x 

and ^ & a* 



we have 



Since cos x > in any interval SB = ( ^ + 8, ^ & J , 3 > 0, it 
follows that \Q\<1 in S3. 

Thus by 163, 2, tana: can be developed in a power series about 
the origin valid in S3. We thus set 

tan x = a<iX -f- ^x 3 + agx 6 -f (2 



GENERAL THEORY 199 

since tana; is an odd function. From 1), 2) we have, clearing 
fractions, 



Comparing coefficients on each side of this equation gives 
a 1 = l. 

a-, 1 1 



a -2! 3! 



9 2 ! 4 ! 
Thus 



17 

> 

62 



C ' ^ ' 1 6 ' 315 7 "" 2 nl?^ " V 



v^^W in ( , ). 



2 ' 2 

Example 2. */ ^ 1 

^ j (x) = cosec ^ = -T 



sin a; 
1 



3! 5! 
Since ^ , sin 2: 



^ 
we see that i m ^ 1 



when x is in 33=(~7r + S, TT - 8), 8>0. Thus xf(x) = 
can be developed in a power series in 33. As /(#) ^ s an 
function, xf(x) is even, hence its development contains only even 
powers of x. Thus we have 

-f 



200 POWER SERIES 

Hence 



Comparing like coefficients gives 

00=1. 



k- 

Thus I 1^1 ^ " 3J . -1 

I '}' I *->l> 1 ' 

"~" I . *-' I . - . **' I , .,*- 



166. Let J p (rc)=/i(A . )+/2 (,. )+ ... 

where / , N , ,, , -, 

f n (x) = a nQ -f a nl x + a n2 x~ + M = 1, 2 

Let the adjoint series 

*0 + "nl + 2 | 2 + * 

converge for | = R and have c/> n ii.s sums for this value of . 
Let cj> = 1 + ^ . h ... 

converge. Then jP converges uniformly in ?[ = ( /^ /^) and ^ 
may be developed as a power series, valid in 31, by summing by 
columns the double series 






GENERAL THEORY 201 

F converges uniformly in 31. For as \x\ <, 



We now apply 136, 2 as ^<f> n is convergent for = It. 
To prove the latter part of the theorem we observe that 

(< 10"f n ft 4- 12 A >2 4- ' 
4- , 20 4- <% ft 4- 22^ 2 + ' * ' 

4- ....... 

is convergent, since summing it by rows it has <l> as sum. Tlius 
tlie double series 1) converges absolutely for |#|<, by 1*28, 2. 
Thus the series 1) may be summed by columns by 130, l and has 
JF(jc) as sum, since 1) has .Fas sum on summing by rows. 

167. Example. 



This series we have seen converges in 21 = (0, J), 6 positive and 
arbitrarily large. 

Since it is impossible to develop the/ n (V) in a power series about 
the origin which will have a common interval of convergence, let 
us develop Fin a power scries about o? >0. 

We have 

1 1 1 



1 -f a n x 1 -f a n x Q -, a n (x XQ 
" 



t+a%l l+a% (l+%) 2 

where j __ ( l) K <y n< 



202 POWER SERIES 

Thus F give rise to the double series 

JD = A^ + A'^x - a? ) -f- A'^(x - z ) 2 

"J" -"-10 ~^~ ^llv 2 ' "" ^o) "^" -^-12 v^ ~ *^o) 



where * f __ ( l) n j 

"WK """" i -**-n K. 

The adjoint series to/ n (V) is, setting f = \x a? |, 

This is convergent if 

. a * <1 or if $<x y 
1 -f- a n a; 

that is, if Q ^ 

For any # such that # < a; < 2 ^ , g = x x Q . 

Then for such an x 

A 1 1 

(f) = 

' n -- ! 1 i /<.! X 



and the corresponding series 



is evidently convergent, since <f> n < - 

ft I 



We may thus sum D by columns ; we get 
F(x) = lB K (x-x Q Y 

K=0 

where 



The relation 1) is valid for < x < 2 z . 



GENERAL THEORY 203 

168. Inversion of a Power Series. 

Let the series , , 7 . . , . 2 , ^1 

v = b 4- o^ 4- V + - (1 

have b 1 ^= 0, and let it converge for t= t Q . If we set 

, , f - J o 

= 2rt , w =:__fi, 

Mo 
it goes over into a series of the form 

u = x a%x* agfi (2 

which converges for x = 1. Without loss of generality we may 
suppose that the original series 1) has the form 2) and converges 
for x= 1. We shall therefore take the given series to be 2). By 
I, 437, 2 the equation 2) defines uniquely a function x of u which 
is continuous about the point u = 0, and takes on the value x= 0, 
for u = 0. 

We show that this function x may be developed in a power 
series in w, valid in some interval about u = 0. 

To this end let us set 

x = u -f- c 2 u 2 4- c s ifi -f (3 

and try to determine the coefficient c, so that 3) satisfies 2) 
formally. Raising 3) to successive powers, we get 

x 2 = w 2 + 2 <? 2 tt 3 -f (V + 2 ^X + ( 2 6 4 + 2 <V3> 6 + - ' 

x 3 = n 3 -f 3 V 4 + (3 <? 2 2 -f 3 <? 3 > 5 + .- (4 

z 4 = M 4 + 4 <? 2 w 5 4- 

Putting these in 2) it becomes 

u = w + (6' 2 a 2 )w 2 -f (<? 3 2 a 2 tf 2 a 3 )w 3 



+ (<? 6 - 2 2 (c 4 + 6> 2 c- 3 ) - 3 a 3 (e' 2 2 + <? 8 ) - 4 a 4 <? a - a^u* (5 
-f ............... 

Equating coefficients of like powers of u on both sides of this 

equation gives 

2 a 2 

<? 3 = 2 a 2 c 2 -f a 3 

<?4 = ^(^ 3 2 + 2 * 8 )+ 8 a 3 ^ 2 -f- 4 (6 

(? 6 = 2 a 2 ( 4 + (? a (? 8 ) 4- 3 3 (c 2 2 4- c 8 ) 4- 4 <* 4 2 4- 5 . 



204 POWER SERIES 

This method enables us thus to determine the coefficient c in 
8) such that this series when put in 2) formally satisfies this 
relation. We shall call the series 3) where the coefficients c have 
the values given in 6), the inverse series belonging to 2). 

Suppose now the inverse series 3) converges for some w ^=0 ; 
can we say it satisfies 2) for values of u near the origin ? The 
answer is, Yes. For by 101, 3, we may sum by columns the 
double series which results by replacing in the right side of 2) 

a-, # 2 , 3?, 

by their values in 3), 4). Hut when we do this, the right side of 
2) goes over into the right side of 5), all of whose coefficients 
by 0) except the first. 

We have therefore only to show that the inverse series con- 
verges for some u = 0. To show this we make use of the fact that 
2) converges for A = 1. Then a n = 0, and hence 

| tt n | < some a n = 2, 3, (7 

On the other hand, the relations ft) show that 

^n=/n(" 2 ' r/ 3' '"<*) ( 8 

is a polynomial with integral positive coefficients. In 8) let us 
replace a 2 , a 3 -*- by a, getting 

7n=/n(i , ) 0* 

Obviously a n \ < y n . (10 

Let us now replace all the a\s in 2) by a ; we get the geometric 

SOL 1CS if O A / 4 4 

u = x w (U, a.t, ... (11 

(12 



1 - x 
The inverse series belonging to 11 ) is 

x = u + 7 2 2 + 7s w3 
where obviously the 7's are the functions 9). 

We show now that 11) is convergent about u = 0. For let us 
solve 12) ; we get 



GENERAL THEORY 205 

Let us set 1 2(2 a -f l)w -f ^ 2 = 1 v. For u near u = 0, 
v \ <l. Then by the Binomial Theorem 



Vl - v = 1 4- d^v + dtfp H ---- 

Replacing v by its value in w, this becomes a power series in u 
which holds for u near the origin, by 161, 3. Thus 14) shows that 
x can be developed in a power series about the origin. Thus 13) 
converges about u = 0. But then by 10) the inverse series 3) 
converges in some interval about u = 0. 

We may, therefore, state the theorem: 

Let u=b + V + V + V + -.&!* 0, (15 

con re rye about the point .r = 0. Then this relation defines r as a 
function of u which admits the development 



r = (M- />) P- 



about the point u b. The coefficients a may be obtained from 15) 
/>// the method of undetermined coefficient*. 

Example. We saw that 

u=iog (1 + ^) = ,.-+'-^- + -- -. (1 

If we set 

,u = /- 4- //a*- 2 -f <V ;>{ 4- rt 4 .* 4 4 ("2 

we liavc _ , __ | _ i 

If we invert 2), wo got 

x u 4- <? 2 ?/ 2 4- c,^ -f 
where e's arc given by (>) in 1<>8. Thus 

.-. e? 2 = |. 



120' 



206 POWER SERIES 

Thus we get 



But from 1) we have 

2 

l + z = e = l + + 

which agrees with 8). 



Taylor s Development 

169. 1. We have seen, I, 409, that if f(x) together with its 
first n derivatives are continuous in 31 = (a < 6), then 




where ^ ,7^-1 n^/i^i 

a <.a + h <_b , 0<a<l. 

Consider the infinite power series in h. 

*'=/<) + ^/'() + f'/'W + " (2 

We call it the Taylor's series belonging to f(x). The first n 
terms of 1) and 2) are the same. Let us set 



(3 

n \ 

We observe that R n is a function of n, A, a and an unknown 
variable 6 lying between and 1. 

Wehave /( + A)-r. + ^. 

From this we conclude at once : 



If 1, /O) and ite derivatives of every order are continuous in 
21 = (a, 6), <md 2 

lira fi n = lira ^/<">(a+0A) = , n = oo, (4 

n \ 

a<a + h<b < < 1. 



TAYLOR'S DEVELOPMENT 207 

Then , r 2 

/(a 4- A) -/() + Af(a) + |y/"(<0 + - (5 

The above theorem is called Taylor's theorem; and the equa- 
tion 5) is the development of f(x) in the interval 21 by Taylor's 
series. 

Another form of 5) f 
/ O) 

When the point is the origin, that is, when a = 0, 5) or 6) 
gives ^2 

/CO =/(0)W'(0) + !/'(()) + .- (7 

This is called Maclaurins development and the right side of 7) 
Maclaurirfs series. It is of course only a special case of Taylor's 
development. 

2. Let us note the content of Taylor's Theorem. It says : 
If 1 f(x) can be developed in this form in the interval 

51 = (a < 6) ; 
2 if f(x) and all its derivatives are known at the point 

x = a ; 

then the value of / and all its derivatives are known at every 
point x within 31. 

The remarkable feature about this result is that the 2 condi- 
tion requires a knowledge of the values of f(x) in an interval 
(a, a -f S) as small as we please. Since the values that a func- 
tion of a real variable takes on in a part of its interval as (a < <?), 
have no effect on the values that/(#) may have in the rest of the 
interval (c < 6), the condition 1 must impose a condition on f(x) 
which obtains throughout the whole interval 31. 

170. Let f(x) be developable in a power series about the point a, 
viz. let 



/ (n) () n i ,9 

a n = J y-- w = 0, 1, (2 

n ! 

i.e. the above series is Taylor's series. 



208 1'OWKIl SK1UKS 

For differentiating 1) n times, we get 
f^(x)= t i\a n + 7 llla n ^ 

Setting here x= a, we get 2). 

The above theorem says that if f(x) can be developed in a 
power series about x = a, this series can be no other than Taylor's 
series. 

171. 1. Tjet f (n) (x) exist and be numerically less than some con- 
stant M for all a < x < ft, and, for every n. Tien f(x) can be 
developed in Taylors series for all x in (, ft). 

For then I 7? I M '*" 

n ! 

But obviously v h n A 

7 Inn = 0. 

>,-*, n ! 

2. 'I'he application of the preceding theorem gives at once: 



(2 



which are valid for ererif r. 

Since * = ^^", tf>0, 

we have 

-, . IOP- a , 9 lop <2 </ 
' - 



, . 
(4 

valid for all x and a > 0. 

172. 1. To develop (1 -f- x)^ and log (1 + #) we need another 
expression of the remainder R n due to Cauchy. We shall con- 
duct our work so as to lead to a very general form for R n . 

From 169, 1 we have 



TAYLOR'S DEVELOPMENT 209 

We introduce the auxiliary function defined over (a, b). 



Then 
and 



M 1 ! 
Hence r> 



/ \ , > 

</() (2 

If we differentiate 1), we (hid the terms cancel in pairs, leaving 
^(^C*-^ V^CO- (8 

We apply now Cauchy's tlieorein, I, 448, introducing another 
arbitrary auxiliary function (7(^) which satisfies the conditions 
of that theorem. 



Then 



= / (l a<e<x 

" ' 



Using 2) and 3), we get, since .r = + //,, 



where < < 1 . 
2. Ifwenet 

we have a function which satisfies our conditions. Then 4) becomes 



n n = i .f<*\a + eh\ (5 

/# I I fJL 



a formula due to ISehlo milch and Roche. 
For /A = 1, this becomes 



n 

which is ( y au<Jii/8 formula. 



210 POWER SERIES 

For /i = ?i, we get from 5) 



n I 
which is Lagrange's formula already obtained. 

173. 1. We consider noiv the development of 

(1 -f x)* x^>l , p arbitrary. 
The corresponding Taylor's series is 



We considered this series in 99, where we saw that : 
T converges for | x \ < 1 and diverges for | x \ > 1. 
When x = 1, T converges only when /JL > 1 ; when x = 1, 
T converges only when p 5; 0. 

We wish to know when 






The cases when I 7 diverges are to be thrown out at once. Con- 
sider in succession the cases that T converges. We have to 
investigate when lim R n = 0. 

Case 1. 0<|aj|<l. It is convenient to use here Cauchy's 
form of the remainder. This gives 



1 2 n 



settinef ! , ^ 

S -M'^-1 ' M-M + 1 
*-- ' 



NowinTT n , 
hence lim W n = 0. 



TAYLOR'S DEVELOPMENT 211 

In u " \i + ex\<i + \x\, 

which is finite. Hence U n is < some constant M. 

To show that lim S n = 0, we make use of the fact that the series 
T converges for the values of x under consideration. Thus for 
every /* 



since the limit of the n th term of a convergent series is 0. In 
this formula replace ^ by p 1, then 



1 2 n 1 

Hence lim 

Thus li mj R 

Hence 1) is valid for | x \ < 1. 



Case 2. x = 1, /x> 1. We employ here Lagrange's form of 
the remainder, which gives 



J. 



Consider W n . Since increases without limit, p n becomes 
and remains negative. As 9 > 

lim W n = 0. 
For U n , we use I, 143. This shows at once that 

lim U n = 0. 
Hence 



and 1) is valid in this case, i.e. for x = 1, p > 1. 



212 POWER SERIES 

Case 8. x = 1, JJL 5 0. We use here for /i > the Schlomilch- 
Roche form of the remainder 172, 5). We set a = 0, h = 1 and get 



n 



Applying I, 143, we see that lim R n = 0. 
Hence 1) is valid here if /-t >0. 

When /i = equation 1) is evidently true, since both sides 
reduce to 1. 

Summing up, we have the theorem : 

The development of (1 -f x)* in Taylor s series is valid when 
| x < 1 for all JJL. When x == 4- 1 it is necessary that JJL > 1 ; 
when x~ \ it is necessary that 



2. We note the following formulas obtained from 1), setting 
= 1 and 1. 



174. 1. We develop now log (1 -h #) The corresponding 
Taylor's series is 



We saw, 89, Ex. 2, that I 7 converges when and only when 
| x | < 1 or x 1. 

< # < 1 . We use Lagrange's remainder, which gives here 



Thus 1 

!"'< 

Hence lim R n = 0. 



TAYLOR'S DEVELOPMENT 213 

Let 1 < x < 0. We use here Cauchy's remainder, which 
gives, setting x = |, < f < 1, 



if a. = 



n 1- 

W n = 



a - 

Evidently j. /S = 

Also -. 

7^7 ^ 1 

^n ^ ^ Z 



Finally 1 

liin TT n = since - - < 1. 

1 # 

We can thus sum up in the theorem : 

Taylor's development of log (1 -f- x) is valid when and only when 
\ x | < 1 or x = 1. That is, for such values of x 



2. We note the following special case : 

I - i + i - i + - = log 2. 

The series on the left we have already met with. 

175. We add for completeness the development of the follow- 
ing functions for which it can be shown that Km R n = 0. 



536 
which is valid for ( 1, 1). 

arctan x = x - ^ + ^ - ~+ (2 

o 5 7 

which is valid for ( 1*, 1 ). 

-- 1 a* 1 3 a* L3.5 a* , ,, 

log(a:+ VI + * 2 ) = * ~ 2 3" + 274 5 ~ ^TTe Y + '" ( 

which is valid for ( 1*, 1*). 



214 POWER SERIES 

176. We wish now to call attention to various false notions 
which are prevalent regarding the development of a function in 
Taylor's series. 

Criticism 1. It is commonly supposed, if the Taylor's series T 
belonging to a function /(#) is convergent, that then 

/<X> = T. 

That this is not always true we proceed to illustrate by various 
examples. 

Example 1. For f(x) take Cauchy's function, I, 335, 



For x*Q #00= ""** ; for x = G Y ( = 0. 

1 derivative. For x = 0, C'(x) = \ C(x). 

6 

For x = 0, C> (0) = lim 



/<=o 



h 



2 derivative. x^Q, C f '(x) = C(x) \ 4 - 4 

I X 2J 

# = 0, (7"(0) = lim ^-A~~" \ L = ii m jl ^ * 8 o. 
3 derivative, x ^ 0, (7"' (a?) = (7 (a?) j ~ - ^ + | 

a:=0, (7"'(0)= lim^-^=0. 

A 

J^i general we have : 

On 



h terms of lower degree 

x 3 " 

x = 0, (7 ( ^(0) = 0. 
Thus the corresponding Taylor's series is 

T= <7(0) + 



TAYLOR'S DEVELOPMENT 215 

That is, T is convergent for every x, but vanishes identically. 
It is thus obvious that C (x) cannot be developed about the origin 
in Taylor's series. 

Example 2. Because the Taylor's series about the origin be- 
longing to C(x) vanishes identically, the reader may be inclined 
to regard this example with suspicion, yet without reason. 

Let us consider therefore the following function, 

/O) = 0(x) + e* = G(x) + g(x). 

Then /(*) and its derivatives of every order are continuous. 
Since /(n)(a;) = O(H)( .^ + ff(n) ^ 

n = 1, 2 ... 

and <7 (n; (0)=0 

we have yr.j(0)=l. 

Hence Taylor's development for f(x) about the origin is 

y = 1 + T! + S + 3 l + - 

This series is convergent, but it does not converge to the right 
value since T _ x 

-L e 

177. 1. Example 3. The two preceding examples leave noth- 
ing to be desired from the standpoint of rigor and simplicity. 
They involve, however, a function, namely, C(x), which is not 
defined in the usual way; it is therefore interesting to have ex- 
amples of functions defined in one of the ordinary everyday 
ways, e.g. as infinite series. Such examples have been given by 
Pringsheim. 

The infinite series 



defines, as we saw, 155, 2, a function in the interval ?l = (0, 6), 
b >0 but otherwise arbitrary, which has derivatives in SI of every 
order, viz. : 

---'. (2 



216 POWER SERIES 

The Taylor's series about the origin for F(x) is 

^0) = J ^ (A) (0) ; X! = 1 for X= 0, 

A=0^- 

and by 2) 



X! v y 3 w! 
Hence 

(3 

As A >0 and Km A = ^ ^A+I<^A this series is an alternate series 
for any x in 21. Hence T converges in 21. 

2. Readers familiar with the elements of the theory of func- 
tions of a complex variable will know without any further reason- 
ing that our Taylor's series T given in 3) cannot equal the given 
function F in any interval 21, however small b is taken. In fact, 
F(x) is an analytic function for which the origin is an essentially 

singular point, since F has the poles -- - n= 1, 2, 3 , whose 
limiting point is 0. 

3. To show by elementary means that F(x) cannot be devel- 
oped about the origin in a Taylor's series is not so simple. We 
prove now, however, with Pringsheim : 

If we take a ^( e -^f=.68 -*, T(x) does not equal F(x) 
\e \J 

throughout any interval 21 = (0, 6), however small 6>0 is taken. 

We show 1 that if F(x) = T(x) throughout 21, this relation is 
true in 33 = (0, 26*). 

In fact let 0<# <6. 

By 161, 4 we can develop T about # , getting a relation 

ro^icu -*-*) (i 



valid for all x sufficiently near # . On the other hand, we saw in 
167 that 

F(x)^E K {x-x,Y (2 

o 

is also valid for Q<x<*2xQ. But by hypothesis, the two power 
series 1) and 2) are equal for points near X Q . Hence they are 



TAYLOR'S DEVELOPMENT 217 

equal for 0<x<2x Q . As we can take # as near b as we choose, 



By repeating the operation often enough, we can show that F = 
T in any interval (0, 5) where B > is arbitrarily large. 

To prove our theorem we have now only to show F 3=. T for 
some one x>0. 
Since 

F(X ^JA ___ i_Wl t ___ a 1 V-, 

^ Vl+z l + aav/ \2!l + <Ar 8!l + aW 
we have i 



-f :r 1 -f 
On the other hand 



Hence 



To find a value of x for which Gr>_- take # = #"*. For this 
value of # 



Observe that G- considered as a function of a is an increasing 
function. For //j_i\2 i 

=(- i) , = - 

\6-iy e 

Hence JP> 3T for >'*. 

178. Criticism 2. It is commonly thought if /(a?) and its 
derivatives of every order are continuous in an interval 21, that 
then the corresponding Taylor's series is convergent in 21. 

That this is not always so is shown by the following example, 
due to Pringsheim. 

It is easy to see that 



converges for every x>_Q, and has derivatives of every order for 
these values of z, viz. : 



218 POWER SERIES 

Taylor's series about the origin is 

T = l i ( - 1) A (X + <r 



Tlie series 3T is divergent for x > 0, as is easily seen. 

179. Criticism 3. It is commonly thought if f(x) and all its 
derivatives vanish for a certain value of x, say for x = a, that 
then /(a;) vanishes identically. One reasons thus: 

The development of/(V) about x= a is 



Asf and all its derivatives vanish at a, this gives 
f(x) = + - (x - a) + (x - a) 2 -f 
= whatever x is. 

There are two tacit assumptions which invalidate this conclusion. 

First, one assumes because f and all its derivatives exist and 
are finite at x = a, that therefore f(x) can be developed in 
Taylor's series. An example to the contrary is Cauchy's function 
C(x). We have seen that C(x) and all its derivatives are at 
x 0, yet 0(x) is not identically 0; in fact vanishes only once, 
viz. at x = 0. 

Secondly, suppose f(x) were developable in Taylor's series in a 
certain interval 21 = (a h, a 4- h). Then / is indeed through- 
out 21, but we cannot infer that it is therefore outside 21. In 
fact, from Dirichlet's definition of a function, the values that/ has 
in 21 nowise interferes with our giving / any other values we 
please outside of 21. 

180. 1. Criticism 4 Suppose f(x) can be developed in Taylor's 
series at a, so that 



for St=O<i). 



TAYLOR'S DEVELOPMENT 219 

Since Taylor's series T is a power series, it converges not only 
in 21, but also within 93 = (2 a #, a). It is commonly supposed 
that f(x) = T also in 93. A moment's reflection shows such an 
assumption is unjustified without further conditions on f(x). 

2. Example. We construct a function by the method considered 
in I, 333, viz. 



n= 1 + (1 4- #) n 

Then /(z) = cos z, in 21 = (0, 1) 

= l-f sin x, within 93 = (0, 1). 
We have therefore as a development in Taylor's series valid 

/w=1 _ +f i_ii + ... =3 , 

It is obviously not valid within 93, although T 7 con verges in 93. 

3. We have given in 1) an arithmetical expression for jf(#). 
Our example would have been just as conclusive if we had said : 

Let f(p) == cos x in 21, 

and = 1 -f- sin x within 33 

181. 1. Criticism 5. The following error is sometimes made. 
Suppose Taylor's development 



valid in 21 = (a < i) . 

It may happen that 7 is convergent in a larger interval 



One must not therefore suppose that 1) is also valid in 93. 
2. Example. 



and = 6* + sin (x - 6) in 93 = ( J, J9) . 

Then Taylor's development 

/ (a0 .l + iL + + + ... (1 

is valid for 21. The series T converging for every x converges in 
93 but 1) is not valid for 93. 



220 POWER SERIES 

182. Let f(x) have finite derivatives of every order in 

31 = (<). In order that f(x) can be developed in the Taylor's 

series 2 



valid in the interval 21 we saw that it is necessary and sufficient 
that 



Hut R n is not only a function of the independent variable A, but 
of the unknown variable 6 which lies within the interval (0, 1) 
and is a function of n and h. 

Pringsheim has shown how the above condition may be replaced 
by the following one in which 6 is an independent variable. 

For the relation 1) to be valid for all h such that 0<^A< H, it is 
necessary and sufficient that Cauchifsform of the remainder 



n - 



4- 



the h and being independent variables^ converge uniformly to zero 
for the rectangle D whose points (A, 0) satisfy 



1 It is sufficient. For then there exists for each e > an m 
such that 

I Rn(k 0) I < n^m 

for every point (A, 0) of D. 

Let us fix h ; then | R n < no matter how 6 varies with n. 

2 It is necessary. Let A be an arbitrary but fixed number in 
21 = (0, #*). 

We have only to show that, from the existence of 1), for A<C A , 
it follows that 

**(*,*)-<> 

uniformly in the rectangle D, defined by 



TAYLOR'S DEVELOPMENT 221 

The demonstration depends upon the fact that /2 n (A 0) is h 
times the w th term / n (, ) of the development of /'(#) about the 
point a + a. In fact let A = a + A. Then by 158 



/'(a + A) =/ 
whose n th term is 



., 

7i "-" JL . 

Let = 6h, then 



as stated. 

The image A , of D is the half of a square of side A ft , below the 
diagonal. 

To show that R n converges uniformly to in Z> we have only 

to show that * s T\ /\ I-A x 

/ n (, *)= uniformly in A . (2 

To this end we have from 1) for all t in 21 

f'(a+ t)=f'(a) + tf(a) + f r "(a)+- (3 

Its adjoint 

=!/'() +/"()! + - (4 



also converges in 21. 

By 161, 4 we can develop 4) about t = , which gives 



, 

w 1! 

But obviously <?(, A) is continuous in A , and evidently all its 
terms are also continuous there. Therefore by 149, 3, 

~ (a) = uniformly in A . (5 



, 
n 1 ! 

But if we show that 



it follows from 5) that 2) is true. Our theorem is then 
established. 



222 POWER SERIES 

To prove 6) we have from 1) 

/(>( + )=/ (n) (<0 + */<*+ (a) + ^/ n + 2) (<0+ - (7 

and from 4) 



The comparison of 7), 8) proves 6). 



Circular and Hyperbolic Functions 

183. 1. We have defined the circular functions as the length 
of certain lines; from this definition their elementary properties 
may be deduced as is shown in trigonometry. 

From this geometric definition we have obtained an arithmeti- 
cal expression for these functions. In particular 



cos*- !- + -+... (2 

valid for every x. 

As an interesting and instructive exercise in the use of series 
we propose now to develop some of the properties of these func- 
tions purely from their definition as infinite series. Let us call 
these series respectively S and O. 

QI rt /Y* 1 

Let us also define tan x = - , sec x = - , etc. 

cos x cos x 

2. To begin, we observe that both S and converge absolutely 
for every #, as we have seen. They therefore define continuous 
one-valued functions for every x. Let us designate them by the 
usual symbols s{ux ^ 



We could just as well denote them by any other symbols, as 



3. Since S =0 , (7=1 for^ 
we have sin()==0 cos0==L 



CIRCULAR AND HYPERBOLIC FUNCTIONS 223 

4. Since S involves only odd powers of #, and only even 
powers, 

sin x is an odd, cos x is an even function. 

5. Since S and are power series which converge for every #, 
they have derivatives of every order. In particular 



dC __#,^_^ 5 ,# 7 __ __ __ o 

Tx" 1 8T~5! 7! '" """ ' 

Hence dsinx dcoxx . ^ 

- = cos x , --- = sin x. Co 

dx ax 

6. To get the addition theorem, let an index as a?, y attached to 
, indicate the variable which occurs in the series. Then 



_ ,. , 

7! 5!2! 3!4! 6 1 



2 I I 3 ! 2 ! 4 ! 



Adding, 



_ , + 

7! 5!2! 314! 



1! 3! 5! 

= 'S'l-HT 

Thus for every a:, y 

sin (a; + ^) = sin x cos y + cos x sin y. 
In the same way we find the addition formula for cos a;. 



224 POWER SERIES 

7. We can get now the important relation 

sin 2 a? 4 cos 2 x = 1 (4 

directly from the addition theorem. Let us, however, find it by 
aid of the series. We have 



/!_ J^JL_ _!__!_ 1\ 
' \(T! + 4 ! 2 ! + 2 ! T! "*" 6!/ 



,- 

! 6 ! 2 ! 4 ! 4 ! 6 1 2 ! 8 ] 



Hence 



Now by I, 96, 
Thus 



== sin 2 x + cos a a; = 1. 



8. In 2 we saw sin 2;, cosa; were continuous for x\ 4) shows 
that they are limited and indeed that they lie between 1. 

For the left side of 4) is the sum of two positive numbers and 
thus neither can be greater than the right side. 

9. Let us study the graph of sin a:, cos a;, which we shall call 2 
and F, respectively. 

(t si 11 (r 

Since sin x = 0, - = cos x = 1, for x = 0, 2 cuts the a>axis at 
dx 

under tin angle of 45 degrees. 



CIRCULAR AND HYPERBOLIC FUNCTIONS 225 

Similarly we see y = 1 for x = 0. F crosses the #-axis there 
and is parallel to the a>axis. 



and each parenthesis is positive for < x 2 < 6, 

sinz>0 for < :r<V<3= 2.449 



we see _ 

eosa;>0 for 0<a:<V'2 = 1.414 ... 

Since -\ x * + x * *Yi 

L ~ 2! + 4i ~ T! V 7 - 

cosx< for x = 2. 

Since D x cos a; = sin x and sin ^ > for < x < Vfi, we see 
cos x is a decreasing function for these values of x. As it is con- 
tinuous and > for x = V2, but < for x = 2, cos # vanishes once 
and only once in (V2, 2). 

This root, uniquely determined, of cos x we denote by As a 
first approximation, we have 

V2<f<2. 

From 4) we have sin 2 -- = 1 . As we saw sin x > for x< Vti, 
we have 

sin|= + l. 

Thus sin x increases constantly from to 1 while cos x decreases 
from 1 to in the interval (0, )= L\> We thus know how sin x, 
cos# behave in I r 
From the addition theorem 

sin ( -" -}- x } sin ~ cos x 4- oos sin # = cos x. 

cos ( ~ + x ) = cos ^ cos x sin sii\# = sin x. 
\2 y 2 2 



226 POWER SERIES 

Knowing how sin x, cos# march in 1^ these formulae tell us 
how they inarch in j^ = f ^, TrV 
From the addition theorem, 

sin (TT + x) sin #, cos (TT + #) = cos x. 

Knowing how sin #, cos x march in (0, TT), these formulae inform 
us about their march in (0, 2 ?r). 
The addition theorem now gives 

sin (x + 2 TT) = sin #, cos (# + 2 TT) = cos x. 

Thus the functions sin x, cos # are periodic and have 2?r as period. 

The graph of sin x cos x for negative # is obtained now by 
recalling that sin x is odd and cos x is even. 

10. As a first approximation of ?r we found 

V2 < J < 2. 
By the aid of the development given 159, 3 

, 3? , O? 6 Z 7 , rx 

arctg a? = x - - g - + - y 4- 5) 

we can compute TT as accurately as we please. 

In fact, from the addition theorem we deduce readily 

S in=-lz , cosf=4=- 

4 V2 4 V2 

Hence tan | = 1. 

This in 5) gives Leibnitz' s formula, 



The convergence of this series is extremely slow. In fact by 
81, 3 we see that the error committed in stopping the summation 

at the n th term is not greater than - - . How much less the 
error is, is not stated. Thus to be sure of making an error less 
than - it would bv' accessary to take ^(10 m 4- 2) terms. 



CIRCULAR AND HYPERBOLIC FUNCTIONS 227 

11. To get a more rapid means of computation, we make use 
of the addition theorem. 
To start with, let 



a = arctgi . 
Then5)gives 1 11 11 11 



a rapidly converging series. 



f . 
^ 



The error J5/ T a committed in breaking off the summation at the 
h term is 1 ., 



2/i-l 5 2 "- 1 
By virtue of the formula for duplicating the argument 

t, 9 _ ^ ^ an a 

T^tan2a' 

wehave tan'2 =1 V 

Similarly * < ___ j 2 o 

Let 

/8 = 4-|. (7 

The addition theorem gives 

tan 8 = -^ = 



Then 6) gives . = _i__lj_ , 1_J 

ft 289 32S9 8 52S9 6 '" C 

also a very rapidly converging series. 

We find for the error -, ^ 



2n-l 2392"-'' 



The formula 7) in connection with 6) and 8) gives - . The 
error on breaking off the summation with the th term is 

+ 



228 POWER SERIES 

184. The Hyperbolic Functions. Closely related with the cir- 
cular functions are the hyperbolic functions. These are defined 
by the equations 

gZ g-X 

sinh x = (1 

^"*- (2 

sinh a: e x e~ x 

, = 

cosh x e x + <'~ x 

cosecha? = 



cosh a; sinh; 

Since 9 ? 



we have , 6 

. + ... (3 

x ~ + - (4 

-J. I 

valid for every #. From these equations we see at once : 
sinh ( x) = sinh x ; cosh ( x) = cosh a:. 
sinh = 0. cosh 0=1. 

(5 

(6 

Let us now look at the graph of these functions. Since sinh x, 
cosh x are continuous functions, their graph is a continuous curve. 
For x > 0, sinh a; > since each term in 3) is > 0. The relation 
4) shows that cosh x is positive for every x. 

If x 1 > x > 0, sinh x 1 > sinh a?, since each term in 3) is greater 
for x r than for a;. The same may be seen from 5). 



a . ! , , 

sum a;= 1 + 
dx 


/*2 /v-4 

^T + 4l + ' 


. . = cosh x. 


d , x , 

- cosh a: = r H 
aa; 1 ! 


X a ,3?, 

h 3! + 5! + 


= sinh x. 



THE HYPEKGEOMETRIC FUNCTION 229 

Evidently from 3), 4) 

lim sinh x = + oo , lim cosh x = 4- oo . 

X=+co Xss+ao 

At x = 0, cosh x has a minimum, and sinh x cuts the #-axis 
at 45. 

For x > 0, cosh x > sinh x since 

e x + e~ x >e x e~ x . 

The two curves approach each other asymptotically as a?== 4-oo . 

For the difference of their ordinates is e~ x which = as x = -f oo . 

The addition theorem is easily obtained from that of e x . In fact 

. , , e x e~ x 

sinh x cosh y = - - 



= (e* +v + e x ~ v e~ x+v e~ x ~ v 
Similarly ^ x+y ___ x _ y e _ x+y _ 



Hence 

sinh x cosh y + cosh x sinh y = (e x+v ' e~ (x + y) } = sinh (x + 

Similarly we find 

cosh (x + #) = cosh ar cosh # + sinh a; sinh y. 
In the same way we may show that 

cosh 2 x sinh 2 x = 1. 



Hyper geometric Function 

185. This function, although known to Wallis, Euler, and the 
earlier mathematicians, was first studied in detail by Gauss. It 
may be defined by the following power series in x: 



1 7 1 2 7.74-1 

2 /S -8+ l 



, : 

1...7.74--7 + 

The numbers a, y8, 7 are called parameters. We observe that 
a, /3 enter symmetrically, also when a = 1, y8 = 7 it reduces to 
the geometric series. Finally let us note that 7 cannot be zero or 
a negative integer, for then all the denominators after a certain 
term = 0. 



:0 POWER SERIES 

Tlie convergence of the series F was discussed in 100. The 
main result obtained there is that F converges absolutely for all 
| x | < 1, whatever values the parameters have, excepting of course 
7 a negative integer or zero. 

186. For special values of the parameters, F reduces to ele- 
mentary functions in the following cases : 

1. If a or is a negative integer -r- n< F is a polynomial of 
degree n. 

2. F(l, 1, 2; -*)==* log (l+:r). (1 

For vn -, v> ^ -, jc x l x 3 

J< (1, 1, 4 -#) = 1 - - + - - -+ -.. 

Also 2 



The relation 1) is now obvious. 
Similarly we have 

l, 1, 2; 3) = log O- 



, , -. 

2 x 1 .r 

3. !( - , & /8 ; x) = 1 - ^ + "-j^ 1 ^- - 

= (!-). 

4. zF(%, .|, |, s: 2 ) = arcsin a;. 

5. ^X-l' !' !' - .-c 2 ) = a re tan a;. 

6. limJ'fa, 1, 1, *")= e.*. (2 
=+ V / 



a + 1 + 2 1 .3.8/gV 



THE HYPERGEOMETRIC FUNCTION 231 

Let < G- < /3. Then 



is convergent since its argument is numerically < 1. Comparing 
3), 4) we see each term of 3) is numerically < the corresponding 
term of 4) for any | jr \ < Q- and any a > $. Thus the series 3) 
considered as a function of is uniformly convergent in the 
interval (/3 -f oo ) by 136, 2; and hereby x may have any value 
in (~ (?, (7). Applying now 146, 4 to 3) and letting = +00, 
we see 3) goes over into 2). 

7. lim xF( a, u< '.* ; ~ } = sin x. (5 

a^+oo \ 2 4 V V 

For 



Let x = (r > and = Cr. Tlien 



, , g - g - - - 

is convergent by 185. We may now reason as in 6. 
8. Similarly we may show : 

/ 1 2 \ 

lim F[ a, , - ; - ~ ) = cos x. 

a= + oo \ 2 4 V 



O 7-2 

, , "\ T^]= si nh ^. 
2 4 a 2 

i 2 

lim J 7 } , , -, - ) = cosh ^. 



24 

187. Contiguous Functions. Consider two F functions 
JF (a,&7; a?) , ^(', ^, y ; S). 

If differs from f by unity, these two functions are said to be 
contiguous. The same holds for 0, and also for 7. Thus to 
F(a/3yx) correspond 6 contiguous functions, 

F(al, 1, 71; af). 



232 POWER SERIES 

Between F and two of its contiguous functions exists a linear 
relation. As the number of such pairs of contiguous functions is 

6-5_ 15 

r^- 15 ' 

there are 15 such linear relations. Let us find one of them. 

Q n== <*+ '"* t2 y'' > rc +n ."' + f. 

Then the coefficient of x n in F(a/3yx*) is 
in jP( + 1, ft, 7, x) it is 
in F(a, ft, 7 1, x) it is 



In' 
. 

Thus the coefficient of x n in 

/3, 7, a) + ^( + 1, A 7, x) 



is 0. This being true for each n, we have 

(7 - - 1) J^O, ft, 7, a:) + ^ ( + 1, A % 



Again, the coefficient of x n in ^(, /9 1, 7, #) is (/5 
in ^(a + 1, /3, 7, #) it is 71(7 + n 1) ^ n . 
Hence using the above coefficients, we get 

( 7 _ a ^ ft)F(a, ft, 7, a:) + (1 - x) F (a + 1, ft, 7, a?) 

+ ( y 8- 7 )^(, ^3-1, 7, 2:) = 0. (2 

From these two we get others by elimination or by permuting 
the first two parameters, which last does not alter the value of 
the function F(a/3yx). 

Thus permuting a, ft in 1) gives 

(7 - - l)-P(a, /3, 7, *) + W* /8 + 1, 7, x) 

+ (1 - i)P(*> ft, 7 - 1, ) = 0. (3 



THE HYPEKGEOMETRIC FUNCTION 233 

Eliminating -F(ct, /3, 7 1, x) from 1), 3) gives 

08 - ) J* (, A 7. *) + ^( + 1, A % a;) 

-F(,/8+l,7,aO = 0. (4 
Permuting a, y9 in 2) gives 

( 7 - a - /3)^(, A % a?) + 0(1 - x)F(a, + 1, 7* x) 

+ ( a _ 7 )jp( a _ 1, & 7 , 3) = 0. (5 

From 3), 5) let us eliminate F (a, ft -f- 1^ 7i #)i getting 
( _ 1 - ( 7 _ _ 1)^)^(0, A 7, *) + (7 - )^( - 1, A 7, *) 

+ (1 - 7)(1 - x)F(a, ft 7 -!.)= 0- (6 
In 1) let us replace a by a 1 and 7 by 7 4- 1 ; we get 

(7 - a + 1)^(~ 1, A 7 + 1, ar) -f (a- l)JF(a, ^ 7 + 1, a) 

-7 J F(a-l,/9,7,a?)=0. (a) 

In 6) let us replace 7 by 7 + 1 ; we get 



(^A7^) = 0. (b) 

Subtracting (b) from (a), eliminates JP( 1, /3, 7 -f 1, #) and 

gives 

7(1 - x}F(afax) - 7^0 - 1, 0, % a?) 



4- (7 - /8>J T (* A 7 + 1,) = 0. (7 
From 6), 7) we can eliminate ^(oc 1, /8, 7, ^), getting 



f- (7 - )(7 - /3)^( /8, 7 + 1, ) 
-f 7(1 ~ 7)(1 - *)y(, /9, 7 - 1, a;) = 0. (8 
In this manner we may proceed, getting the remaining seven. 

188. Conjugate Functions. From the relations between con- 
tiguous functions we see that a linear relation exists between any 
three functions 

F(& /3, 7, *0 ^O', ', 7', x) F(a", /3", 7", x) 

whose corresponding parameters differ only by integers. Such 
functions are called conjugate. 



234 POWER SERIES 

For let j0, q, r be any three integers. Consider the functions 

F(aftyx), F(a + l, ft, /,x)>~F(a+p, ft, 7,2?), 
F(a + p, /3 + 1, 7, ), F<* + p, ft + 2, 7, ) - J F (a+ j>, ft + q, 7, *), 
,aO>^( a +^^ 



We have p -f q -h r 4- 1 functions, and any 3 consecutive ones 
are contiguous. There arc thus p + q -h r 1 linear relations 
between them. We can thus by elimination get a linear relation 
between any three of these functions. 

189. Derivatives. We have 



2 M 7- 7 



L 2 w + 1 7 7 -h 1 7 -h /* 

-- 2X + 1) T72TTr M v r-7+i-" '" 



Hence 
F" (, /8, 7, z) = ^ 1" ( + 1, /3 + 1, 7 + 1, *) 

= " ' " + ~j+'f- + - F ^ + 2, /3 + 2, 7 + 2, *) 

and so on for the higher derivatives. We see they are conjugate 
functions. 

190. Differential Equation for F. Since F, F 1 ', F n are conju- 
gate functions, a linear relation exists between them. It is found 
to be 



x(x - 1)*"' 4- {(a + ft + V)x - 7} F' 4- aftF= 0. (1 

To prove the relation let us find the coefficient of x n on the left 
side of 1). We set 

p == CC '^L +1 ' + 'H-l __ y8 + l ' ^-Mij-l 

n " 1 -2 ... n 7 7 -hi"- 7-hV- 1 



THE IIYPERGKOMETR1C FUNCTION 285 

The coefficient of x* in x*F u is 



in xF" it is 

n(>+ 

" "' 

in ( + /8 + 1)2^' it is 

M( + 
in 7^' it is 



in rtS/ 1 it is 



Adding all these gives the coefficient of x n in the left side of 1). 
We find it is 0. 



191. Expression of F^ajB^x) as an Integral. 
We show that for | x \ < 1, 



3, 7-)-JF(/87tf)= -iO -M)v-^-i(l-a^) dw (1 

/o 

where B{p>> y) is the Beta function of I, 692, 



/o 
For by the Binomial Theorem 

xi x -i , , tt 4- 1 > o , -f 1 + 2 

(1 xu)~ a =I + -xul -- T^ x u H -- . o 
1 1 2! 1 ^ o 

for | xu | < 1. Hence 

,7= fV-H! -M)y-*- 1 (l-^)~ a dtt 

A) 






236 POWER SERIES 

Now from I, 692, 10) 

Hence 



7 7 + 

etc. Putting these values in 2) we get 1). 

192. Value of F (a, /3, 7, x) for z = 1. 

We saw that the F series converges absolutely for x = 1 if 
4- ft 7 < 0. The value of F when x = 1 is particularly in- 
teresting. As it is now a function of a, /3, 7 only, w,e may denote 
it by .F(, /?, 7). The relation between this function and the T 
function may be established, as Gauss showed, by means of 187, 8) 

V17 * 

7 ^ 7 _ !+( + + 1- 
+ (7 - ) (7 - 



Assuming that a + _ 7 < , (2 

we see that the first and second terms are convergent for x = 1 ; 
but we cannot say this in general for the third, as it is necessary 
for this that a -f- /3 (7 1) < 0. We cap, however, show that 

& 7 - 1>V?) = 0, (3 



supposing 2) to hold. For if | x \ < 1, 

J^(, & y 1, a:) = + a^x + a^x* + (4 

Now by 100, this series also converges f or x = 1. Thus 

lim a n = 0. / (5 

n=oo 

From 4) we have 
(1 - x)F(a, & y - 1, a) = a + Oi -00)* + 0*2 - i)a? + 

Let the series on the right be denoted by Q-(x}. As 
# n+1 (1) = a n , we see Gr (1) is a convergent series, by 5), whose 
sum is 0. But then by 147, 6, G-(x) is continuous at x = 1. 

Hence L lim (}(x)=a (1) = 0, 



THE HYPERGEOMETRIC FUNCTION 237 

and this Establishes 3). Thus passing to the limit x = 1 in 1) 
gives 



- *<, A 7) = 



Replacing 7 by 7 + 1, this gives 



etc. Thus in general 
^<X ft, 7) = 



Gauss se^e now 



,-, , , _ 

^ X) ~ 



Hence the above relation becomes 



! n* 



w, 7 - a - w, 7 - / - 

N W 

W=oo 

For the series 



n) ' (6 



lira ^(a, ^ 7 + ) = 1. (7 

l + ... (8 



7 7 7 

converges absolutely when 2) holds. Hence 



1 . 6? 1 2 (?.+ 1 

is convergent. Now each term in 8) is numerically < the corre- 
sponding term in 9) for any 7 > Gr. Hence 8) converges uni- 
formly about the point 7 = -f oo. We may therefore apply 146, 4. 
As each term of 8) has the limit as 7 = + oo, the relation 7) 
is established. 



238 POWER SERIES 

We shall show in the next chapter that 

lim n (w, x) 

n=Go 

exists for all x different from a negative integer. Gauss denotes 
it by II (x) ; as we shall see, 

T (x) = II (x - 1) , for x > 0. 
Letting n == 20, 6) gives 

j. ( , A 7) = n(7 - 1)11(7 -- 0-D. 

< 1/3 ' 7; n (7- - 1)11(7 --1) 
We must of course suppose that 

7, 7 - , 7 - ft, 7 - - /3, 

are not negative integers or zero, as otherwise the corresponding 
n or F function are not defined. 

Bessel Functions 
193. 1. The infinite series 



converges for every #. For the ratio of two successive terms of 
the adjoint series is 1^12 



which ==s as * == <x> for any given x. 

The series 1) thus define functions of x which are everywhere 
continuous. They are called Bessel functions of order 

n = 0, 1, 2-.. 
In particular we have 



2 2 2 2 4 2 2 2 4 2 6 2 

1" '>*" / 7*" T 1 

U^ U./ . Ux C/ 



*^ i /^ 'J 

2 2 2 4 2 2 - 4 2 6 ~ 2 2 4 2 . 6 2 8 '" 
Since 1) is a power series, we may differentiate it termwise and 

+"-i C4 

' ( 



BKSSKL FUNCTIONS 239 

2. The following linear relation exists between three consecutive 
Bessel functions : 



(5 

/ *""* | jf. 1). ^ +n ~ l ( 

" ' l -2- '(n - 1 ) ! ,=i ) 2 + *- !(-!+)! 

* ,2*tn-l 

(T 



I leuce 



i y/^_ l^ 



2 1 n 1 



JC 

\\. We show next that 

2 ^(a;) = /_,(*) -J-^Cz) w>0. (8 

For subtracting 7) from <>) ^ives 



From 8) we get, on replacing e/ n _ f . 1 by its value as given by 5) : 

n>0. (9 



X 

From 5) we also get 

J n ^(x^ n>0. (10 



4. The Bessel function J n satish'es the following linear homo- 
geneous differential equation of the 2 order : 

n = Q. (11 



240 POWER SERIES 

This may be shown by direct differentiation of 1) or more sim- 
ply thus : Differentiating 9) gives 

Ttf _ n T n 77 i Tf 

J n=^ J n-- J " + J "-l' 

Equation 10) gives 

Tf n ~ 1 T T 

J n-\ = - J n-\ "* 

X 

Replacing here J n _^ by its value as given by 9), we get 



Putting this in 12) gives 11). 

5. e*^ ='uJ n (x) (13 

for any x, and for u 3= 0. 
For 



<T 2" = e* e 2 

"9 **" ~* 



(9-1/92 9 ! o/2 I ' 

A (A> Zj *j : 66 J 



Now for any a; and for any u 3* 0, the series in the braces are 
absolutely convergent. Their product may therefore be written 

in the form _&, /W _J N 

2 V2/2I2I "V 



+ 2 



BESSEL FUNCTIONS 241 

194. 1. Expression of ' J n (x) as an Integral. 



* f"co 8 ( 

n + l\Jo 



For 



2n + l\ 
V 2 ) 



Hence ^ __ -JN. 

cos (a; cos <) = j ^-5 , ^ cos2 * 

o (- 8 ) 
and thus 



^ _ 

cos (a: cos <) sin 2n </> = 2 ^ y ^ cos 2 * sin 2n <. 



o 



As this series converges uniformly in (0, TT) for any value of x, 
we may integrate termwise, getting 



jfcos O cos <) sin 2n </)J<^ = J? ^^ &* fcos 2 * <^> sin 2n 



(2)! 2 ' 2 



2+i\ b j 692 

' J ' 



We shall show in 225, 6, that 

1.3.5... 

~ 2- 

Thus the last series above 



2 



Thus 



CHAPTER VII 
INFINITE PRODUCTS 



195. 1. Let S# 4 ...tJ be an infinite sequence of numbers, the 
indices 1 = ^ i a ) ranging over a lattice system in 8- way 
space. The symbol p = ^ = ^ (1 

fi C 

is called an infinite product. The numbers a t are its factors. Let 
/^ denote the product of all the factors in the rectangular cell 

R ^ If lim P^ (2 

/UL 00 

is Unite or definitely infinite, we call it the value of P. It is 
customary to represent a product and its value by the same letter 
when no ambiguity will arise. 

When the limit 2) is finite and = or when one of the factors 
= 0, we say P is convergent, otherwise P is divergent. 

We shall denote by P^ the product obtained by setting all the 
factors a, 1, whose indices i lie in the cell JB M . We call this the 
co-product of P^. 

The products most often occurring in practice are of the type 

go 

P = a l . a 2 - 3 - ". = IIa n . (3 

The factor P M is here replaced by 



and the co-product P^ by 

-* m =: ^TO+1 ' ^m+2 * ^nif " '" 

Another type is +30 

P=na n . (4 

=:- 

The products 3), 4) are simple, the product 1) is 8-tuple. The 
products 3), 4) may be called one-way and two-way simple products 
when necessary to distinguish them. 

242 



GENERAL THEORY 243 

2 ' Let -P = l- A-*- f -4. - 

Obviously the product P = 0, as 

w 

Hence P=0, although no factor is zero. Such products are 
sailed 2m> products. Now we saw in 1, 77 that the product of a 
finite number of factors cannot vanish unless one of its factors 
vanishes. For this reason zero products hold an exceptional posi- 
tion and will not be considered in this work. We therefore have 
classed them among the divergent products. In the following 
theorems relative to convergence, we shall suppose, for simplicity, 
that there are no zero factors. 

196. 1. For P = Ha^.. lg to converge it is necessary that each JPV 
is convergent. If one of these P^ converges, P is convergent and 

The proof is obvious. 

2. If the simple product P = a l a 2 3 is convergent, its fac- 
tors finally remain positive. 

For, when P is convergent, | P n \ > some positive number, for 
n > some m. If now the factors after a m were not all positive, P n 
and P v could have opposite signs v > n, however large n is taken. 
Thus P n has no limit. 

197. 1. To investigate the convergence or divergence of an 
infinite product P = n ti ... t , when a t > 0, it is often convenient to 
consider the series 



called the associate logarithmic series. Its importance in this con- 
nection is due to the following theorem : 

The infinite product P with positive factors and the infinite series 
L converge or diverge simultaneously. When convergent, P = e L , 
L = log P. 

For logP M = ^, (1 

P, = e'<*. (2 



244 INFINITE PRODUCTS 

If P is convergent, jP M converges to a finite limit = 0. Hence 
Lp is convergent by 1). If L^ is convergent, P^ converges to a 
finite limit =0 by 2). 

2. Example 1. 

, 2,..- 



is convergent for every x. 

For, however large | x \ is taken and then fixed, we can take m 
so large that 



n>m. 
n 

Instead of P we may therefore consider P m . 

^ 

But by I, 413 

log(l + ?}=* + M n ^, \M n \<M. 
\ nj n n 2 

Hence L n =li t M n x> -\ 

w+i n* 

which is convergent. 

The product P occurs in the expression of sin x as an infinite 
product. 

Let us now consider the product 



>=1, 2, 

ni 

The associate logarithmic series It is a two-way simple series. 
We may break it into two parts Z/, 7/", the first extended over 
positive w, the second over negative n. We may now reason on 
these as we did on the series 3), and conclude that Q converges 
for every x. 

'6. Example 2. 



* 



n 
is convergent for any x different from 

0, -1, -2, -3, 



GENERAL THEORY 245 

For let p be taken so large that | x \ < p. We show that the 
co-product x -, v x 

(1+ I 

a p = n^ nj 

P+I l+ x 
n 

converges for this x. The corresponding logarithmic series is 



n 

As each of the series on the right converges, so does L. Hence 
Gi converges for this value of x. 

198. 1. When the associate logarithmic series 

i=21oga tl ... lt , a t > 
is convergent, 



Hm log = ^ by 

iti= 

and therefore v ^ 

lira ^...^ = 1. 

|t|=w 

For this reason it is often convenient to write the factors 
a tl ... lf of an infinite product JP in the form 1 -f- 6 tl ... v When P is 
written in the form 

we shall say it is written in its normal form. The series 



we shall call the associate normal series of P. 
2. The infinite product 



and its associate normal series 



converge or diverge simultaneously. 



246 INFINITE PRODUCTS 

For P and T v , /1 , 

// = 2 log (1 + a t ) 

converge or diverge simultaneously by 197. But A arid I/ con- 
verge or diverge simultaneously by 123, 4. 

3. If the simple product P == a 1 - a 2 - a 3 is convergent, <e w ==l. 

For by 196, 2 the factors a n finally become > 0, say for n > m. 
Hence by 197, l the series 

I log a n a n >Q 

tim 

is convergent. Hence log a n == 0. /. a n == 1. 

199. Let R^ < 7? A2 < X == oo ^^ a sequence of rectangular 

cells. TJien if P u convergent^ 



For -P is a telescopic series and 



200. 1. Let P^ 

We call ^= 

the adjoint of P, and write 

<P = Ad j P. 
2. P converges, if its adjoint is convergent. We show that 

e > 0, \ \P fA -P v \<e p,v>\. 
Since ^JJ is convergent, 



is also convergent by 199. Hence 

< $ - ^ < X < fJL < V. 

But P v P fJL is an integral rational function of the a's with 
positive coefficients. Hence 

IP P I < <B ^ (\ 

* v L - +V -K** V^ 



GENERAL THEORY 247 

8. When the adjoint of P converges, we say P is absolutely 
convergent. 

The reader will note that absolute convergence of infinite 
products is defined quite differently from that of infinite 
series. At first sight one would incline to define the adjoint of 

P =!!...,. 

tobe <P=n | ...,,!. 

With this definition the fundamental theorem 2 would be false. 
For let P=n ( -l); 

its adjoint would be, by this definition, 



Now $ n = 1. '$ is convergent. On the other hand, 
P n =( l) w and this has no limit, as n =^ oo. Hence P is 
divergent. 

4. Jn order that P = 11(1 -f- a tj ... tg ) converge absolutely r , z z 
necessary and sufficient that y 

converges absolutely. 

Follows at once from 198, 2. 
Example. ^ , - 

i V w 2 / 

converges absolutely for every #. 

For o i 

Vi? 2 ^ /2Vl 
^n 2 ^ ^n 2 
is convergent. 

201. 1. Making use of the reasoning similar to that employed 
in 124, we see that with each multiple product 

P=iK... u 

are associated an infinite number of simple products 

<^lTa tt , 

and conversely. 



248 INFINITE PRODUCTS 

We have now the following theorems : 

2. If an associate simple product Q is convergent, so is P, and 
P=Q. 

For since Q is convergent, we may assume that all the a's are 
> by 196, 2. Then 



= gSlogaij-i, by 124, 3, 

= P by 197, l. 

3. If the associate simple product Q is absolutely convergent, so 
is P. 

For let P-II(l + *,...,> 

<?=n(l+a n ). 
Since Q is absolutely convergent, 



is convergent. Hence 11(1 4- a tl ... t ) is convergent by 2. 

4. 7/e _P=H(1 -H^-i ) ^ absolutely convergent. Then each 
associate simple product Q= 11(1 -f # n ) ^ absolutely convergent and 

P=Q. 

For since P is absolutely convergent, 

2<V-, 
converges by 200, 4. But then by 124, 5 

2 n 
is convergent. Hence Q is absolutely convergent. 

5. If P Ha tl ... la is absolutely convergent, the factors a tli ... t9 >0 
if they lie outside of some rectangular cell 7? M . 

For since P converges absolutely, any one of its simple associ- 
ate products Q=Tla n converges. But then a n >0 for n>m, by 
198, 3. Thus a v .. t > if t, lies outside of some R^ 

6. From 5 it follows that in demonstrations regarding abso- 
lutely convergent products, we may take all the factors > 0, 
without loss of generality. 



GENERAL THEORY 249 

For P = P^P^ 

and all the factors of P^ are > 0, if /* is sufficiently large. This 
we shall feel at liberty to do, without further remark. 

7. A=II(l + a lt ..... ) a t >0 



and Z = 2 log(l + ,...) 

converge or diverge simultaneously. 

For if A is convergent, 

2<v.e, 

is convergent by 200, 4. But then L is convergent by 123, 4. 
The converse follows similarly. 

202. 1. As in 124, 10 we may form from a given m-tuple 
product 4-IIa,,...., 

as infinite number of conjugate w-tuple products 



where a t = 5 7 - if i and/ are corresponding lattice points in the two 
systems. 

We have now : 

2. If A is absolutely convergent, so is B, and A = B. 
For by 201, 6, without loss of generality, we may take all the 
factors > 0. 
Then 



= B. 



an absolutely convergent m-tuple product. 



be any p -tuple product formed of a part of or all the factors of A. 
Then B is absolutely convergent. 



250 INFINITE PRODUCTS 

For S log a, is convergent. 

Hence 2 log fy is. 

Arithmetical Operations 

203. Absolutely convergent products are commutative, and con- 
versely. 

For let A T-T 

4 = IIa tl ... lBI 

be absolutely convergent. Then its associate simple product 

21= lla n 

is absolutely convergent and A = 21, by 201, 4 . Let us now re- 
arrange the factors of A, getting the product . To it corre- 
sponds a simple associate series 93 and B = 93. But 21 = 93 since 
21 is absolutely convergent. Hence A = B. 

Conversely, let A be commutative. Then all the factors # 4 ... ljw 
finally become > 0. For if not, let 

jRj < /? 2 < =00 (1 

be a sequence of rectangular cells such that any point of 9i m lies 
in some cell. We may arrange the factors a t such that the partial 
products corresponding to 1), 

1 ' ^2 ' 3 "* 

have opposite signs alternately. Then A is not convergent, which 
is a contradiction. We may therefore assume all the a's > 0. 

Then A 2 log 

A = e l 

remains unaltered however the factors on the left are rearranged. 

Hence v , 

21og<v.. lw 

is commutative and therefore absolutely convergent by 124, 8. 
Hence the associate simple series 



is absolutely convergent by 124, 5. Hence 

2n 

is convergent and therefore A is absolutely convergent. 



ARITHMETICAL OPERATIONS 251 

204. 1. Let 

4. --I, 

be absolutely convergent. Then the s-tuple iterated product 



is absolutely convergent and A~ where ij i! 9 is a permutation of 
*r '2 ' * 

For by 202, 3 all the products of the type 

Ua t t IIa t t 

t t ..-l, I, ...I, 

i. !* l 

are absolutely convergent, and by I, 324 

n = nn. 

^-i i *-i i 

Similarly the products of the type 

n 

l -i l -a l - 

are absolutely convergent and hence 

n= n n n. 



In this way we continue till we reach A and B. 
2. We may obviously generalize 1 as follows : 

Let A = Ua t . 

4'" 1 * 

be absolutely convergent. Let us establish a 1 to 1 correspondence 
between the lattice system ? over which i = ( L I ... ,) ranges, and the 
lattice system 2ft oi^r which 



.9 = O 11,7 12 '".721^22 " JrlJrt "'Jrp) 

ranges. Then the p-tuple iterated product 
JS=II II . ... Ha, 7 f 

1 2 r J 1?l> ^ 

z absolutely convergent, and 

A = ^. 



252 INFINITE PRODUCTS 

3. An important special case of 2 is the following: 
Let A = Ua n , 71=1,2,-.. 

converge absolutely. Let us throw the a n into the rectangular array 

a n , 12 . 



converge absolutely, and 

4 = J^JBa J? r . 



4. 2%0 convergent infinite product 
P = (1 + ^X1 



is associative. 
For let 



^ 

< wi< ... =00. 



We have to show that 

0=(l+4i)( 
is convergent and P = $. 

This, however, is obvious. For 



But when w = <x> so does z/. 

Hence v ^ ,. 

lim Q n = hmP n . 

Remark. We note that m m+1 m m may = QO with n. 



ARITHMETICAL OPERATIONS 253 

205. Let A = Ua^..^ , = Ub ti ... Lt 

be convergent. Then 

(7=na t .6 t , D = U^ 

o, 
are convergent and 

C=A.B , D = ^. 

> 

Moreover if A, B are absolutely convergent, so are (?, D. 

Let us prove the theorem regarding (7 ; the rest follows simi- 

larly. We have A n 

J Of. = Ap Bp. 

Now by hypothesis A^ == A^ B^ == B as ft = oo. 
Hence ^ = ^.A 

To show that is absolutely convergent when A, B are, let us 

write a, = 1 -f- a t , 6 t = 1 -f b t and set | a t | = t , | b t | = y8 t . 
Since A, B converge absolutely, 

2 log (! + .) , 2 log (1 4- A) 
are convergent. Hence 

S {log (1 + <*) 4- log (1 + A) I = 2 lo ^ C 1 H- O C 1 + A) 

is absolutely convergent. Hence C is absolutely convergent 
by 201, 7. 

206. Example. The following infinite products occur in the 
theory of elliptic functions : 



They are absolutely convergent for all | q\ < 1. 
For the series 2 | q* n \ , 2 1 9 2 "" 1 1 

are convergent. We apply now 200, 4. 

As an exercise let us prove the important relation 

P *><} -I. 



254 INFINITE PRODUCTS 

For by 206, /> = IT ( 1 + ? 2n ) ( 1 + J 2 "' 1 ) ( 1 - 



Now all integers of the type i'n, are of the type 4n'2 or 4w. 
Hence by 204, 3, 

n (1 - r/ 2 ") = 1 1 ( 1 - ? 4 ") II (1 - ?*"- 2 ), 



x-| 

P = II ^ 



= 1 

Uniform Convergence 

207. Jft A0 limited or unlimited domain 31, ^ 

i = 21og/ ti ... ta (^....r m ) , / t >0 
he uniformly convergent and limited. Then 



is uniformly convergent in 21. 

For ^ / 

F A = e^A- 

Now L* = L uniformly. Hence by 144, l, F is uniformly con- 
vergent. 

208. If the adjoint of 



z' uniformly convergent in 21 (finite or infinite^ F is uniformly 
convergent. 

For if the adjoint product, 



is uniformly convergent, \ve have 

!$-$ | < 

for any .> in ?(. 



UNIFORM CONVERGENCE 255 

But as already noticed in 200, 2, 1) 

|JFV-/M<l**-*r|. 

Hence F is uniformly convergent. 
209. The product 



is uniformly convergent in the limited or unlimited domain $[, if 
* = 2<k 1 ..M.Ov--O , <k=l/J 

is limited and uniformly convergent in ?l. 
For by 138, 2 the series 



is uniformly convergent and limited in 21- Then by 207, the 
adjoint of F is uniformly convergent, and hence by 208, F is. 

210. Let ^ \ n ^ / \ 

F(x v - x m ) = n./ ti ... ta (#! ;r w ) 

be uniformly convergent at x = a. If each fi is continuous at a, F 
is also continuous at a. 

This is a corollary of 147, 1. 

211. 1. Let G = S | / tl ... i t (2i ^ w ) I converge in the limited 
complete domain 31 having a as a limiting point. Let Q- and each 
f t be continuous at a. Then 



is continuous at a. 

For by 149, 4, G- is uniformly convergent. Then by 209, F is 
uniformly convergent, and therefore by 210, F is continuous. 

2. Let Or =2 |/ tl ... t8 (^i # m )| converge in the limited complete 
domain ?l, having x = a as limiting point. Let 

lim/ t = a t , lim Q- == Sa t . 



256 INFINITE PRODUCTS 

For by 149, 6, # is uniformly convergent at x = a. It is also 
limited near x = a. Thus by 209, 



is uniformly convergent at a. To establish 1) we need now only 
to apply 146, l. 

212. 1. Let J r =n/ ll ... i /ar) , / t >0 (1 

converge in 2l=(, a-f-S). Then 

log F=L = Slog/.. (2 

If we can differentiate this series termwise in 21 we have 



Thus to each infinite product 1) of this kind corresponds an infi- 
nite series 3). Conditions for termwise differentiation of the series 
2) are given in 153, 155, 156. Other conditions will be given in 
Chapter XVI. 

2. JSxample. Let us consider the infinite product 

6(x) =2q*Q sin Trail (1 - 2 q 2tl cos 2 TTX + q* H ) (1 

which occurs in the elliptic functions. 

Let us set 

1 - u n =l - 2 2n cos 2 TTX + q* n . 

Then | u n \ < ;2 | q | 2 + | q | 4n . 

Thus if | q \ < 1, the product 1) is absolutely convergent for any x. 
It is uniformly convergent for any x and for | q \ < r< 1. 

If it is permissible to differentiate termwise the series obtained 
by taking the logarithm of both sides of 1), we get 

-. (2 
4n 



^<l-2q 2n cos 2 



If we denote the terms under the 2 sign in 2) by v n we have 



THE CIRCULAR FUNCTIONS 



257 



Now the series 2a w converges if | q \ < 1. For setting 6 W = | 
the series 2J n is convergent in this case. Moreover, 



. 

b n 

Thus we may differentiate term wise. 

Tfie Circular Functions 
213. 1. Sin # and cos a; as Infinite Products. 
From the addition theorem 

sin (mx + x) = sin (m -f- 1)# = sin TWO? cos x 4- cos w# sin x 
m = 1, 2, 3 we see that for an odd w 

sin nx = a sin n a: + a l sin 11 " 1 x + -f # n -i g i n # 
where the coefficients a are integers. If we set t ?= sin #, we get 

sin nx = jP n (0 = a/ -f a^^- 1 + + a n -i^ (1 

Now JF n being a polynomial of degree w, it has n roots. They are 



A . 7T . . 2 7T 

0, sm , sm , 
n n 



-- 
2 n 



corresponding to the values of x which make sin nx = 0. Thus 
-F.(l) - V (' - si n j|) (( + sin J) ... 



Dividing through by 



_ 
n 



ir>2: 



sn 



1 7T 



and denoting the new constant factor by , 1), 2) give 



sin nx = a sin 



1- 



2 



sn 



1 



in2: 



sn 



1 



INFINITE PRODUCTS 



To find a we observe that this equation gives 



si Ji nx 
sin x 



1 



sin 2 a;" 

. 2 7T 

sin 2 
n 



Letting x= we now get a = n. Thus putting this value of 

x 
in ,T), and replacing x by -, we have finally 



sin x = n sin - P(.r, 

72 



where 



/*7T 



-1 2 "- 1 
- i, ^, ... o 



We note now that as n == oo, 



Similarly 



sin 
. x n . 

n sin = x = x. 

n x 

n 



snr - 
n 



It seems likely therefore that if we pass to the limit n = oo in 

l\ we shall get n/ , ^ r 

71 & sin^ = r/ > (rr) (5 

ivhere 



r Fhe correctness of 5) is easily shown. 
Let us set 



L(x, n) = log P(#, w) = 2 log 



L(x) = log P(x) = 2 log l - 



sin 2 - 



THE CIRCULAR FUNCTIONS 

We observe that 

lim P (x, n) = lim ** n > = e**> = P (x) 



259 



provided 



lim L (#, n) = L (x) . 



We have thus only to prove 7). Let us denote the sum of the 
first m terms in 6) by L m (x, n*) and the sum of the remaining 
l>yZ m (*,w). Then 



Since for 



we have 



7T 

2' 



- < sin x < x, 

2* 



- , 

n 4 n^ x* 

"" 



\. (8 



and hence for an m l so large that -< 1, we have, 



-log 1- 



snr- 

" 



. o 
Sill 2 



-log(l-- 



a* 



But the series 



is convergent. Hence for a sufficiently large m 



r > m. 



Now giving m this fixed value, obviously for all n > some v the 
first term on the right of 8) is < e/3, and thus 7) holds. 



260 INFINITE PRODUCTS 

2. In algebra we learn that every polynomial 

a + a v x + a^x* + + a n x n 
can be written as a product 

0*O - i)O-2>> (s-O, 
where 04, 2 are its roots. Now 

x x* , r 6 /Q 

81 n*=-- -+.,-.-. (9 

is the limit of a polynomial, viz. the first n terms of 9). It is 
natural to ask, Can we not express sin x as the limit of a product 
which vanishes at the zeros of sin x ? That this can be done we 
have just shown in 1. 

3. If we set x = 7r/2 in 5), it gives, 



Hence ^ 2r-2r = 2 2 4 4 (S (i ... n() 

2 U (2r-l)(2r+D 1 3 3 5 5~. 7 ' ^ 



a formula due to 

4. From 5) we can get another expression for sin #, viz. : 

siux^xU(l-~^\e^ r= 1, 2, ... (11 

For the right side is convergent by 197, 2. If now we group 
the factors in pairs, we have 



This shows that the products in 5) and 11) are equal. 

5. From 5) or 11) we have 

sin x = lim P n (x) = lim x ri' ?-J!T 

n=oo s-n 87T 

where the dash indicates that s = is excluded. 



THE CIRCULAR FUNCTIONS 261 

214. We now show that 



To this end we use the relation 

sin 2 x = 2 sin x cos x. 
Hence 

cos = -= 



from which 1) is immediate. 
From 1) we have, as in 213, 4, 

^ n = 0, 1, 2, ... (2 



215. From the expression of sin ar, cos # as infinite products, 
their periodicity is readily shown. Thus from 213, 12) 

sin x = lim P n (x). 



. 

'_ _. _ I as n = Q0> 

P n (^;) r-wir 

Hence lim p ^ (rr + ^ = _ Hm p^^^ 

sin (x + TT) = sin #. 

Hence . ^ , \ 

sin(a: -f 2 TT) = sin # 

and thus sin x admits the period 2 TT. 

216. 1. Infinite Series for tan #, cosec a?, etc. 

If O<#<TT, all the factors in the product 213, 5) are positive. 

Thus 



262 INFINITE PRODUCTS 

Similarly 214, 1) gives 

. (2 



To get formula) having a wider range we have only to square 
the products 213, 5) and 214, 1). We then get 

log sin 2 x = log a* + 2 log (l - ^Y, (3 

valid for any x such that sin x = ; and 

/ 4# 2 \ 2 

logeo s ^log^l- (28 _ i)%2 j, (4 

valid for any x such that cos#^0. 
If we differentiate 3), 4) we get 



cot * = 

x 



tan:c=2 T - (6 



lid as in 3), 4). 

Remark. The relations 5), 6) exhibit cot #, tana; as a series of 
rational functions whose poles are precisely the poles of the given 
functions. They are analogous to the representation in algebra 
of a fraction as the sum of partial fractions. 

2. To get developments of sec a?, cosec a?, we observe that 

cosec x = tan \ x + cot x. 
Hence 



COSeC Z=2 2/ TZ^ TTo 12+~- 

2 a? 



_ , 



valid for 



THE CIRCULAR FUNCTIONS 
3. To get sec #, we observe that 

cosec f ~ x J = sec x. 
\** 1 



263 



Now 



cosec x = - - 
x 



X 87T -f X ) 
1 1 



Hence 

2"~ y TT 



Let us regroup the terms of 8, forming the scries 
I 1 1 1 f 1 11 

+ - o +o + 



7T 7T 



2 w - 1 
- - , 



we see that 7 is convergent and = 8. Thus 



valid for all x such that cos x = 0. 

217. As an exercise let us show the periodicity of cot x from 
216, 5). We have 

n \ 

cot x = lim F n (x) = lim V 

5= n 



' w ^+( + i>. 

Letting n = oo we see that 

lim 1^(2; + TT) = lim .F n (a; 
cot (x -f- TT) = cot a?. 



nir 



and hence 



264 INFINITE PRODUCTS 

218. Development of log siii x, tan z, etc., in power series. 
From 216, 1) 



If we give to ^L?its limiting value 1 as x = 0, the relation 1) 
holds for | x \ < TT. 
Now for x < TT 



Thus 

, sin x x* , 1 a; 4 , 1 .r 

_l () o- - =-3 + 4 + 0-1 
./' 7T 2 2 7T 4 -> 7T 6 



r l 1 J A 1 ^ 
i x i l _~ i l _J L ... 

^ 32 ^2^2 3 4 7T 4 8 3 6 7T 6 

4- 

provided we sum this double series by rows. But since the series 
is a positive term series, we may sum by columns, by 129, 2. 
Doing this we get 



sn x 



whoro 



1.1,1,1 



? relation 2) i valid for \x\ < TT. 
In a similar manner we iind 



7T 



valid for \ x \ < - Here 



THE CIRCULAR FUNCTIONS 265 

The terms of Q- n are a part of ff n . Obviously 



Tliese coefficients put in 3) give 
-log w*x=(&-l)ff 



'TT" - ' "TT* " " ' ~7T 6 

valid for | x \ < -- If we differentiate 4) and 2), we get 

7T 2 7T 4 7T 6 

valid for \ x \ < - ; 

cot *= i - 2 # 2 ^ _ 2 # 4 ^ - 2 JT,^ - ... (6 

^ 7T 7T* 7T 

valid for < \x\ < TT. 

Comparing 5) with the development of tan x given 165, 3) 
gives 

111 _2 1 <) _2 O '2 

77- L JL J 7T I ^ 7T 15 ^i 7T 

2 ~~ 72 ~" 92 "* ^2 """ * " "" 7r "" H ' *> ! "~ J ' 9 ! 

JL w > 'J vi rf . w 

zr = l , 1 ,1, = Z^ = ?i^ = R.?i^ /-? 

4 I 4 2 4 3 4 1>0 30 ' 4 ! 3 4! ^ 



// = j_tj_ = jl_l. ?!^! = ** 

6 is^^^.s 6 !>4f) 42 ' ti! 5 ' 6! 

rr _ 1 , 1 , 1 , 7T 8 1 2' 7T 8 V 7T R 

"8 - a ' ,-.0 ' .>u I" 



1 s 2 8 H 8 J t50 30 8 ! 7 8 ! 

Let us set I72n _ t 2n 

TT" ^ """ D /"Q 

_/Z 2n :== ~ -*-*' > n 1* V 

Then 5) gives 



valid for |a;|< The coefficients B v B% are called 
nouillian numbers. From 7) we see 



266 INFINITE PRODUCTS 

From 6), 8) we get 

cotan*-^-!^^*'"-' (10 

valid for < | x \ < ?r. 

219. Recursion formula for the Bernouillian Numbers. 

If we set f() == ^ an Xt > 

we have by Taylor's development 



where ^2n-i) (0 ) _ 2(2 2 - I)g 2n _ 2 2 "(2 2 - 1) 

(2n-l)!~ 7T 2 " ~ (2)I 2 "- 1 

Now by I, 408, 



From 1), 2) we get 
-i(2 2 - 1) D /2n - 1\ 2 ln - 3 (^"- z - 1) R 

-O2n-l ~ I 2 J - I7~j - -"2n-3 

-n\ -i _... =( _ 1) .., (3 



We have already found jB r 5 3 , J^ 5 , ^ 7 ; it is now easy to find 
successively : 



Thus to calculate i? 9 , we have from 3) 

98 27(2 8 - 1) _1_ 

1-2 4 '30 1.2.3.4 3 42 



2 9 (2 10 ~~ 1) _ 98 27(2 8 - 1) _1_ 9- 8> 7 6 2 5 ( 2 6 ~ 

' 



^ 9.8.7 23(2* -1) .1, 9 . 2r2 2 _ n . 1 - l 

1.2-3 2 30 + C 1; 6" 
Thus 

* = 512 Ao23 51 - 9 + 168 - 2016 4- 9792} 

^ 5 . 7936 = ^ 
512 1023 66' 



THE B AND T FUNCTIONS 267 

The B and F Functions 

220. In Volume I we defined the B and F functions by means 
of integrals: ... x , 

BO, v) = I -^ (1 

v ' Jo (l + a: )+* v 

Xoo 
e-*^- 1 ^ (2 



which converge only when u, v > 0. Under this condition we saw 

- 



We propose to show that F(M) can be developed in the infinite 
product / 1 \ u 

i( 1 + -) 

# = ln^ _ n -L. (4 



J 

n 

This product converges, as we saw, 197, 3, for any u^Q, 1, 
2, From 201, 7 and 207 it is obvious that Gr converges abso- 
lutely and uniformly at any point u different from these singular 
points. Thus the expression 4) has a wider domain of definition 
than that of 2). Since Gr = F, as we said, for w>0, we shall ex- 
tend the definition of the F function in accordance with 4), for 
negative u. 

It frequently happens that a function f(x) can be represented 
by different analytic expressions whose domains of convergence 
are different. For example, we saw 218, 9), that tan x can be de- 
veloped in a power series 



valid f or | x \ < . On the other hand, 



x_ __ x a _ 

1~!~~3T 5l "" sin a; 
tan x = - 5 - z - ~ - 
x* x* cosx 



268 INFINITE PRODUCTS 

and ^ 

tan. = 2S * by 216, 6) 



are analytic expressions valid for every x for which the function 
tan x is defined. 

221. 1. Before showing that Q- and F have the same values for 
u > 0, let us develop some of the properties of the product # given 
in 220, 4). In the first place, we have, by 210: 

The function G-(u) is continuous, except at the points u = 0, 1, 
-2,.-. 

Since the factors of 4) are all positive for u > 0, we see that 

Q-(u) is positive for u > 0. 

2. In the vicinity of the point x = m, m = 0, 1, 

G-(u) = H(u ^ 
x -f- m 

where H(u) is continuous near this point, and does not vanish at 
this point. 
For 

'! + - 



m 



where //is tho infinite product Gr with one factor left out. As we 
may reason on /fas we did on Q-, we see H converges at the point 
x = m. Hence H^ at this point. But II also converges uni- 
formly about this point; hence /Tis continuous about it. 



r -...- M ^ 

= lim -------- ^ ------ ------ - - n u . (\ 

n=ao W (u + 1)(W -h 2) -..(M + W 1) 



To prove this relation, let us denote the product under the limit 
sign by P n . We have 



THE B AND T FUNCTIONS 269 

Also 



Thus P n = G n . But Q- n = 6r, hence P B , is convergent and (} = 
lira P n . 

223. Huler's Constant. This is defined by the convergent series 



It is easy to see at once that 



by 218, 7). By calculation it is found that 

C' = .577215 

224. Another expression of Gr is 



n) 
where is the Eulerian constant. 

For when a > 0, a u = e u Ioga . 
Hence 






Now 
and 



270 INFINITE PRODUCTS 

are convergent. Hence 



from which 1 ) follows at once, using 223. 
225. Further Properties of Q-. 

1. #O+l)=M#(tt). (1 

Let us use the product 



employed in 222. Then 

AO + i) = MMPn(M) - (2 

M -f-w 



= M as /i == QO 



u -h n 
we get 1) from 2) at once on passing to the limit. 

2. G(u+ n)=u(u+ 1) (u + n- 1)<7(V). (3 
This follows from 1) by repeated applications. 

3. Gr(n) = 1 2 n 1 = (n 1) ! (4 
where n is a positive integer. 

sin TTU 
For Q-CL u^ss uGKu} bv 1 

eG " , , by 224, 1). 
Hence -i .-C'W.C'M 



. e rt 

wy 



i i 



THE B AND T FUNCTIONS 271 

We now use 213, 5). 

Let us note that by virtue of 1, 2 the value of Q- is known for 
all u > 0, when it is known in the interval (0, 1). By virtue of 
5) Q- is known for u < when its value is known for u > 0. 
Moreover the relation 5) shows the value of G- is known in (, 1) 
when its value is known in (0, J). 

As a result of this we see & is known when its values in the 
interval (0, |) are known ; or indeed in any interval of length \. 

Gauss has given a table of log Gr(u) for 1<^<1.5 calculated 
to 20 decimal places. A four-place table is given in " A Short 
Table of Integrals " by B. 0. Peirce, for 1 < u < 2. 



5. <?() = VT^. (6 

For in 5) set u = ^. Then 



Hence 

We must take the plus sign here, since Gr > when u > 0, by 221. 



where n is a positive integer. 



, etc. 



Thas 2+l\_2n-l 2. -8 3 1 

mas 



226. Expressions for log (?(w), and if Derivatives. 
From 224, 1) we have for w > 0, 

i(0 = log ff O) = - O^ 
Differentiating, we get 



Iw- u + n} 
That this step is permissible follows from 155, 1. 



272 INFINITE PRODUCTS 

We may write 2) 

' = - (7+ V f i -- - - -1 - (3 

rf (n u+ n - 1 J 

That the relations 2), 3) hold for any u^ 0, - 1, - 2 follows 
by reasoning similar to that employed in 216. In general we have 



-^ , ,>1. (4 

In particular, 

Z/(l) = _ O. (5 



227. Development of log 6r(w) i/i # Power Series. If Taylor's 
development is valid about the point ?/ = 1, we have 

log #< = < = i(l) + =li L'd) 

or using 226, 5), and setting u = 1 4- a;, 

log 



We show now this relation is valid for Jj r < x < 1, by proving 
that 



converges to 0, as s == QO . 
For, if 0<x<l, then 



Also if - 



i i 

x-=o. 



The relation 1) is really valid for 1 <#<_ 1, but for our pur- 
pose it suffices to know that it holds in 31 = ( , 1). Legendre 



THE B AND T FUNCTIONS 27:3 

has shown how the series 1) may be made to converge more 
rapidly. We have for any x in 81 

log (1 4- x) = x - I (- 1)" 
2 n 

This on adding and subtracting from 1) gives 

log 0(1 + *) = -log(l +*)+ (1- # )*+( 

2 

Changing here x into x gives 

log &(i - *) = - log (i - *) - (i - c> + 

Subtracting this from the foregoing gives 
log 0(1 +x)- log 0(1 - x) 



From 225, 4 



log 0(1 + X) + log 0(1 - 2?) = log ~ 



sin TTX 

This with the preceding relation gives 
log 0(1 + *) 

-a-^-t^T^^*^-*!^-- 1 ^ (2 

valid in 31. 



This series converges rapidly for 0<#<|, and enables us to 
compute 0(u) in the interval l<w<|. The other values of 6r 
may be readily obtained as already observed. 

228. 1. We show now with Prinf/sheim* that 0(^) =F(wK for 
u>0. 

We have for <><_!, 

l\u 4- w)= 



Math, Anuttlen, vol. .'Jl, p. 466. 



274 INFINITE PRODUCTS 

Now for any x in the interval (0, w), 

x u <n u , x u >xn u ~ l 
since u > and u 1 < 0. 

Also for any # in the interval (n, oo ) 

x u <xn u ~ l , x u >n u . 
Hence 



u - l je- x x n dx+n u e' x x n ~ l dx<T(u 4- ft) 

/* xao 

< rc u I ^-^^-^^-f /i 1 *" 1 I e~ x x n dx. 

*/0 /n 



Thus 



n u 



< I e~ x x n ~ l dx-\ I e~ x x n dx -- I e~ x x n dx. 
*so n*/o n^o 



Let us call these integrals A, J5, respectively. 
We see at once that 

= n 1== j , 



n n 

Also, integrating by parts, 



[e~ x x n ~\ n . 1 C" - x n7 n n , n 
\ - _j_ _ I e * x n dx = -- h O. 

L n J n* 70 ?ie n 



Thus 



Hence 
where 
Now 



THE B AND T FUNCTIONS 275 

But 

"n>l + 7-7+ + 7 . 1N n , ; r , for any w 



(n + 1) (w + m) 
Let us take 






or <. 
w m 

Then 

ra m 



Since m may be taken large at pleasure, 

lim z/ n = QO 

an*d hence ,. A 

hm j n = 0. 

Thus , , 



But from T(u + 1) = wF(^) we have 



?i u ^(w 1) ! w n u (n 1) ! 

also, as n = QO . Thus the relation 1) holds for 1 < w< 2, and in 
fact for any w 

As 
we have 



v J ( + !)... ( + -!) 
Hence using 1), ^ (-!)! . T(. 

V. y X^ . -1 \ X . -IN x' 



Letting ^= oo , we get T(u)= Gr(u) for any w>0, making use 
of 1) and 222, 1). 

2. Having extended the definition of T(u) to negative values 
of w, we may now take the relation 



as a definition of the B function. This definition will be in 
accordance with 220, 1) for w, v > 0, and will define B for negative 
w, v when the right side of 2) has a value. 



CHAPTER VIII 
AGGREGATES 

Equivalence 

229. 1. Up to the present the aggregates we have dealt with 
have been point aggregates. We now consider aggregates in 
general. Any collection of well-determined objects, distinguish- 
able one from another, and thought of as a whole, may be called 
an aggregate or set. 

Thus the class of prime numbers, the class of integrable func- 
tions, the inhabitants of the United States, are aggregates. 

Some of the definitions given for point aggregates apply obvi- 
ously to aggregates in general, and we shall therefore not repeat 
them here, as it is only necessary to replace the term point by 
object or element. 

As in point sets, 31 = shall mean that 31 embraces no elements. 

Let 91, 33 be two aggregates such that each element a of 91 is 
associated with some one element b of 33, and conversely. We say 
that 21 is equivalent to 33 and write 

cyr ci(\ 
l ~ 3O- 

We also say 31 and 33 are in one to one correspondence or are in 
uniform correspondence. To indicate that a is associated with b 
in this correspondence we write 

a ~ b. 

2. If 21 ~ 33 and 33 ~ 6, then 91 2. 

For let a^b, b ~ c. Then we can set 91, ( in uniform corre- 
spondence by setting a ~ c. 

3. Let 91 = 33+S + >+ 

A = S + C+ D 4- 
// 33 - B, 6 - C, , then 91-^1. 

276 



EQUIVALENCE 277 

For we can associate the elements of 21 with those of A by 
keeping precisely the correspondence which exists between the 
elements of S3 and J9, of ( and (7, etc. 

Example 1. 21 = 1, 2, 3, ... 



If we set a n ~ ft, 91 and S3 will stand in 1, 1 correspondence. 

ExampleS. 21 = 1, 2, 3, 4, 

S3 = 2, 4, 6, 8, ... 

If we set n of 91 in correspondence with 2 n of S3, 91 and S3 will 
be in uniform correspondence. 

We note that S3 is a part of 91 ; we have thus this result : An 
infinite aggregate may be put in uniform correspondence with a 
partial aggregate of itself. 

This is obviously impossible if 9t is finite. 

Example 3. 21 = 1, 2, 3, 4, ... 

S3 = 10 1 , 10 2 , 10 3 , 10 4 , ... 

If we set n ~ 10 n , we establish a uniform correspondence be- 
tween 9t and S3. We note again that 91 ~ S3 although 91 > S3. 

Example 4- Let & = j(, where, using the triadic system, 

=-& 8 - . = 0.2 

denote the Cantor set of I, 272. Let us associate with f the point 

#=.2:^3 ... (1 

where x n = when n = 0, and = 1 when n = 2 and read 1) in 
the dyadic system. 

Then \x\ is the interval (0, 1). Thus we have established a 
uniform correspondence between and the points of a unit interval. 

In passing let us note that if < and x, x 1 are the correspond- 
ing points in {#}, then x <x f . 

This example also shows that we can set in uniform correspond- 
ence a discrete aggregate with the unit interval. 

We have only to prove that is discrete. To this end consider 
the set of intervals marked heavy in the figure of I, 272. Ob- 



278 AGGREGATES 

viously we can select enough of these deleted intervals so that 
their lower content is as near 1 as we choose. Thus 

Cont (7=1. 

As Cont C < 1, C is metric and its content is 1. Hence is 
discrete. 

230. 1. Let 91 = a -f- J., 33 = /? 4- J?, where a, b are elements 
0/91, 33 respectively. If^K^- 33, then A ~ B and conversely. 

For, since 21 ~ 33, each element a of 21 is associated with some 
one element b of 33, and the same holds for 33. If it so happens 
that a ~ /?, the uniform correspondence of A, B is obvious. If 
on the contrary a. ~ b' and /3 ~ a', the uniform correspondence be- 
tween A, B can be established by setting a 1 ~~ b r and having the 
other elements in A, B correspond as in 3l~ 33. 

2. We state as obvious the theorems: 
No part 33 of a finite set 91 can be ~ 31. 
No finite part 33 of an infinite set 31 can be ~ 31. 



Cardinal Numbers 

231. 1. We attach now to each aggregate 31 an attribute 
called its cardinal number, which is defined as follows : 

1 Equivalent aggregates have the same cardinal number. 

2 If 91 is ~ to a part of 33, but 33 is not ~ 31 or to any part 
of 31, the cardinal number of 91 is less than that of 33, or the 
cardinal number of 33 is greater than that of 31. The cardinal 
number of 91 may be denoted by the corresponding small letter 
a or by Card 31. 

The cardinal number of an aggregate is sometimes called its 
power or potency. 

If 91 is a finite set, let it consist of n objects or elements. 
Then its cardinal number shall be n. The cardinal number of 
a finite set is said to be finite, otherwise transfinite. It follows 
from the preceding definition that all transfinite cardinal num- 
bers are greater than any finite cardinal number. 



CARDINAL NUMBERS 279 

2. It is a property of any two finite cardinal numbers a, b that 

either t 

o = b , or a > b , or a < b. (1 

This property has not yet been established for transfmite car- 
dinal numbers. There is in fact a fourth alternative relative to 
31, 33, besides the three involved in 1). For until the contrary 
lias been shown, there is the possibility that : 

No part of 91 is ~ 53, and no part of 53 is ~ 21. 

The reader should thus guard against expressly or tacitly 
assuming that one of the three relations 1) must hold for any 
two cardinal numbers. 

3. We note here another difference. If 21, 53 are finite with- 
out common element, 

Card (21 + 53) > Card 21. (2 

Let now 21 denote the positive even and 53 the positive odd 
numbers. Obviously 

Card (21 + 53) = Card 21 = Card 53 
and the relation 2) does not hold for these transfinite numbers. 

4. We have, however, the following : 

Let 21 > 53, then 

Card 21 > Card S3. 

For obviously 53 is ~ to a part of 21, viz. 53 itself. 

5. This may be generalized as follows : 

Let 



If Card 53 < Card B , Card < Card <7, etc., 

then Card 21 < Card A. 

For from Card 53 < Card B follows that we can associate in 1, 
1 correspondence the elements of 53 with a part or whole of B. 
The same is true for , tf; , Z>; 

Thus we can associate the elements of 21 with a part or the 
whole of A. 



280 AGGREGATES 

Enumerable Sets 

232. 1. An aggregate which is equivalent to the system of 
positive integers $ or to a part of Q is enumerable. 

Thus all finite aggregates are enumerable. The cardinal num- 
ber attached to an infinite enumerable set is K , aleph zero. 
At times we shall also denote this cardinal by e, so that 

2. Every infinite aggregate 91 contains an infinite enumerable set -33. 
For let a 1 be an element of 2( and 

Then 21 x is infinite ; let a 2 be one of its elements and 

Then 21 2 is infinite, etc. 

Then ^ saa 

is a part of 91 and forms an infinite enumerable set. 

3. From this follows that 

K is the least transfinite cardinal number. 

233. The rational numbers are enumerable. 
For any rational number may be written 

(\ 

n 

where, as usual, m is relatively prime to n. 
The equation 

admits but a finite number of solutions for each value of 

p = 2, 3, 4, ... 

Each solution m, n of 2), these numbers being relatively prime, 
gives a rational number 1). Thus we get, e.g. 

p = 2 , 1. 

/> = 8 , 2, J. 

jt> = 4 , 3, J. 

j = 5 , 4, J , | |. 



ENUMERABLE SETS 281 

Let us now arrange these solutions in a sequence, putting those 
corresponding to p = q before those corresponding to p=sq + 1. 

We 



r i * r 2 * r a " v, rf 

which is obviously enumerable. 

234. Let the indices t x , * 2 , * p range over enumerable sets. Then 

is enumerable. 

For the equation __ 

where the z/s are positive integers, admits but a finite number 
of solutions for each n = p, _p + 1, p + 2, p4-3--- Thus the 
elements of ^ _ ,, , 

may be arranged in a sequence 



by giving to n successively the values p, p -f 1, and putting the 
elements b Vi ... Vp corresponding to n = q + 1 after those correspond- 
ing to n = q. 

Thus the set 48 is enumerable. Consider now 31. Since each 
index i m ranges over an enumerable set, each value of i m as i' m is 
associated with some positive integer as m f and conversely. We 
may now establish a 1, 1 correspondence between 21 and $Q by 
setting 

J >nX"-;~ ai ; i ;"-v 

Hence 21 is enumerable. 

235. 1. An enumerable set of enumerable aggregates form an 
enumerable aggregate. 

For let 21, 33, 6 be the original aggregates. Since they form 
an enumerable set, they can be arranged in the order 

2lj , 2( 2 ? 2ls i * (1 

But each 2l m is enumerable ; therefore its elements can be 
arranged in the order 



282 AGGREGATES 

Thus the a-elements in 1) form a set 

\a mn \ m, n,= 1, 2, 
which is enumerable by 234. 

2. The real algebraic numbers form an enumerable set. 

For each algebraic number is a root of a uniquely determined 
irreducible equation of the form 



the a's being rational numbers. Thus the totality of real algebraic 
numbers may be represented by 

\Pn, a,a 3 --- a n i 

where the index n runs over the positive integers and a^*** a n range 
over the rational numbers. 

3. Let 31, 33 be two enumerable sets. Then 

Card 91= Card = K . 

Card (H + ) = Ko. 

And in general if Slj , 91 2 are an enumerable set of enumerable 
aggregates, Card (Slj, 2I 2 , ) = K . 

This follows from 1. 

236. Every isolated aggregate 21, limited or not, forms an enumer- 
able set. 

For let us divide $R m into cubes of side 1. Obviously these form 
an enumerable set^Cp CIj""' About each point a of 21 in any C n 
as center we describe a cube of side <r, so small that it contains no 
other point of 21. This is possible since 21 is isolated. There are but 

a finite number of these cubes in C n of side <r = -, z>= 1, 2, 3, 

v 

for each v. Hence, by 235, l, 21 is enumerable. 

237. 1. Every aggregate of the first species 21, limited or not, is 
enumerable. 

For let 31 be of order n. Then 



ENUMERABLE SETS 283 

where 2l t denotes the isolated points of 21 and 2lp the proper limit- 
ing points of 21' 
Similarly, 



Thus, 

qr _. or i or/ i or" i ... i 9f(n) 

<t - <lt I ^J0,t I v*p,l I I vlp * 

But 2l (n) is finite and ?(<?> < 2J (7l) . 

Thus 21 being the sum of n -f- 1 enumerable sets, is enumerable. 

2. If W is enumerable, so is 31. 
For as in 1, 



and {,<'. 

238. 1. Every infinite aggregate 21 contains a part 33 
33-21. 

For let (S = (a 19 a 2 , a 3 ) be an infinite enumerable set in 21, 
so that 

21 = <g + g. 

Let g == a x + ^- 

To establish a uniform correspondence between J5?, (5 let us 
associate a n in ( with a n+1 in E. Thus <&~ E. 
We now set 



Obviously 21 ^ 33 since E~ @, and the elements of % are common 
to 21 and SB. 

2. -Z^2I~ 33 are infinite, each contains a part 2l x , 33i such that 



For by 1, 21 contains a part 2l x such that 2l~2l r Similarly, 
33 contains a part 33 a such that 33 ~ 33 r As 2l~S3, we have the 
theorem. 



284 AGGREGATES 

239. 1. A theorem of great importance in determining 
whether two aggregates are equivalent is the following. It is 
the converse of 238, 2. 

Let* l <%, !<. If K^ and ^ ~ , 
then 21 ^ 33. 

In the correspondence 2lj ~ 33, let 21 2 be the elements of $1 
associated with SS l . Then 

31 2 ~ ! - 21 
and hence 91 91 f 1 

But as 2lj > 2(3 , we would infer from 1) that also 

21 -2l r (2 

As 2lj ~ 53 by hypothesis, the truth of the theorem follows at 
once from 2). 

To establish 2) we proceed thus. In the correspondence 1), let 
21 3 be that part of 2I 2 which ~ 21 1 in 21. In tfc correspondence 
! ~ 21 3 , let 21 4 be that part of 21 3 which ~ 21 2 in 2l t . 

Continuing in this way, we get the indefinite sequence 

21 > Slj > 21 2 > 3 > - 
such that Qr or fty 



Letnow a^ + t^ , 3^ = 



Then a = ^ + ^ + ^ + ^ + ^ + _ (g 

and similarly 9r-^a.^u.^j.pra.^a. 

^X = A) 4- & 2 4- i> 3 -f V2-4 4" V2, 6 H- ' 

We note that we can also write 

a 1 = s> + <E 8 + e a + e 6 + < 4 + - (4 

Now from the manner in which the sets 21 3 , 21 4 were obtained, 
it follows that 

61 ~ 6, , C 8 ^C5- ( s 

Thus the sets in 4) correspond uniformly to the sets directly 
above them in 3), and this establishes 1). 



ENUMERABLE SETS 285 

2. In connection with the foregoing proof, which is due to 
Bernstein, the reader must guard against the following error. It 
does not in general follow from 

21^ + Sj , 2t 2 =2*3 + 3 , 2(~21 2 , ^-Sls 

that K n- 

(>! ~ 3 

which is the first relation in 5). 
Example. Let 21 = (1, 2, 3, 4, ). 

2l x = (2, 3, 4, 5 -.) , 21 2 = (3, 4, 5, 6 ) 

2( 3 =(5, 6, 7,8 ) 

Then , e i= l 6 8 =(3, 4). 

Now 21, 21 j, 21 2 , 21 3 are all enumerable sets ; hence 

a -a, , a! -a,. 

But obviously Sj is not equivalent to S 3 , since a set containing 
only one element cannot be put in 1 to 1 correspondence with a 
set consisting of two elements. 



240. 1. 

For by hypothesis a part of 48, viz. @, is ~2l. But a part of 21 
is ~$, viz. 35 itself. We apply now 239. 

2. Let a be any cardinal number. If 

a < ("arc! 33 < oc, 

then a = CardS. 

For let Card 21 = . Then from 

a < Card $ 
it follows that 21 ~ a part or the whole of 3d ; while from 

Card $ < 
it follows that 53 is ^ u part or the whole of 21. 

3. Any part 38 of an enumerable set 21 is enumerable. 
For if 59 is finite, it is enumerable. If infinite, 

Card > HO- 
On the other hand 

Card < Card = *. 



286 AGGREGATES 

4. Two infinite enumerable sets are equivalent. 

For both are equivalent to $S ^ ne se t of positive integers. 

241. 1. Let @ be any enumerable set in 91 ; set 21 = g + 93. 
33 i* infinite, 91-93. 

For S3 being infinite, contains an infinite enumerable set 
Let = g + Then 



+ g~g. Hence ~. 

2. We may state 1 thus : 

Card (21 -<)= Card 9t 
provided 91 @ is infinite. 

3. From 1 follows at once the theorem : 

31 ie <my infinite set and ( an enumerable set. Then 
Card (31 + <5) = Card 31. 



Transformations 

242. 1. Let 7 be a transformation of space such that to each 
point x corresponds a single point X T , and conversely. 

Moreover, let #, y be any two points of space. After the trans- 
formation they go over into X T ^ y T . If 



we call To, displacement. 

If the displacement is defined by 

^ = 0?! + ^ , x f m ^x m + a m 
it is called a translation. 

If the displacement is such that all the points of a line in space 
remain unchanged by T, it is called a rotation whose axis is the 

fixed line. 



THE CARDINAL C 287 

If 9? denotes the original space, and 9? r the transformed space 
after displacement, we have, obviously, 



!l\ = tx \ ' " V = tX m , t > 0. (1 

Then when a; ranges over the m-way space , y ranges over an 
m-w f dy space 9). If we set x ~ y as defined by 1), 



Als Dist (0, y) = t Dist (0, x). 

We call 1) a transformation of similitude. If t > 1, a figure in 
space is dilated ; if t <1, it is contracted. 

3. Let Q be any point in space. About it as center, let us de- 
scribe a sphere S of radius R. Let P be any other point. On the 
join of P, Q let us take a point P' such that 

Dist (P', (?) = 



Dist (P, (?) 

Then P 7 is called the inverse of P with respect to S. This trans- 
formation of space is called inversion. Q is the center of inversion. 

Obviously points without S go over into points within, and con- 
versely. As P = oo , P r = Q. 

The correspondence between the old and new spaces is uniform, 
except there is no point corresponding to Q. 



The Cardinal c 

243. 1. All or any part of space S may be put in uniform cor- 
respondence with a point set lying in a given cube 0. 

For let @ t denote the points within and on a unit sphere S about 
the origin, while @ e denotes the other points of space. By an in- 
version we can transform @ e into a figure @y lying in S. By a 
transformation of similitude we can contract @ M (>,- as much as we 
choose, getting <g[, @J. We may now displace these figures so 
as to bring them within G in such a way as to have no points in 
common, the contraction being made sufficiently great. The 



288 AGGREGATES 

correspondence between @ and the resulting aggregate is obviously 
uniform since all the transformations employed are. 

As a result of this and 240, 1 we see that the aggregate of all 
real numbers is ^ to those lying in the interval (0, 1); for example, 
the aggregate of all points of 5R m is ~ to the points in a unit cube, 
or a unit sphere, etc. 

244. 1. The points lying in the unit interval 31 = (0*, 1*) are 
not enumerable. 

For if they were, they could be arranged in a sequence 



Let us express the as as decimals in the normal form. Then 
a n = a nl a n2 n3 

Consider the decimal 

b = b^b^ 

also written in the normal form, where 

J l^ a M ' ^2^^2,2 ' ^3^^3,3 " 

Then b lies in 31 and is yet different from any number in 1). 

2. We have (0*, 1*) ~ (0, 1) , by 241, 3, 

-(a, 4) , by 243, 

where a, b are finite or infinite. 

Thus the cardinal number of any interval, finite or infinite, 
with or without its end points is the same. 

We denote it by c and call it the cardinal number of the recti- 
linear continuum, or of the real number system 9t. 

Since 9? contains the rational number system R, we have 

0o- 

3. The cardinal number of the irrational or of the transcendental 
numbers in any interval 21 is also c. 

For the non-irrational numbers in 51 are the rational which are 
enumerable ; and the non-transcendental numbers in 91 are the 
algebraic which are also enumerable. 



THE CARDINAL c 289 

4. The cardinal number of the Cantor set S of I, 272 is c. 

For each point a of & has the representation in the triadic 

system n 

J a = a 1 a 2 a a , a = 0, 2. 



But if we read these numbers in the dyadic system, replacing 
each a n = 2 by the value 1, we get all the points in the interval 
(0, 1). As there is a uniform correspondence between these two 
sets of points, the theorem is established. 

245. An enumerable set 91 is not perfect, and conversely a perfect 
set is not enumerable. 

For suppose the enumerable set 

91 = !, a 2 (1 

were perfect. In D^^a^) lies an infinite partial set 91 j of 31, 
since by hypothesis 91 is perfect. Let a mt be the point of lowest 
index in 9l x . Let us take r 2 <r l such that D rs (a m2 ) lies in 
D r * (a x ). In -Z> r f(a m2 ) lies an infinite partial set 9^ of 9l r Let 
a ms be the point of lowest index in 91 2 , etc. 
Consider now the sequence 

<*i i <V < a m s 

It converges to a point a by I, 127, 2. But lies in 91, since this 
is perfect. Thus a is some point of 1), say a = a.. But this 
leads to a contradiction. For a, lies in every D r * n (am n ); on the 
other hand, no point in this domain has an index as low as m n 
which == oo, as n == oo. Thus 91 cannot be perfect. 

Conversely, suppose the perfect set 91 were enumerable. This 
is impossible, for we have just seen that when 91 is enumerable it 
cannot be perfect. 

246. Let 91 be the union of an enumerable set of aggregate* 9l n 
each having the cardinal number c. Then Card 91 = c. 

For let 48 n denote the elements of 9l n not in 91 1 ,91 2 9l n _j. 



Let S n denote the interval (n 1, w*). Then the cardinal 
number of gj 4- S 2 + is c. 



290 AGGREGATES 

But Card n < Cardg n . 

Hence Card 21 < c , by 231, 6. (1 

On the other hand, 

Card 21 > Card 2l x = c. (2 

From 1), 2) we have the theorem, by 240, 2. 

247. 1. As already stated, the complex x = (x^ x%, # n ) de- 
notes a point in w-way space. Let x, # 2 , denote an infinite 
enumerable set. We may also say that the complex 

x= (#!, # 2 , in inf.) 
denotes a point in oo -way space 9?^. 

2. Let 21 denote a point set in 9t n , n finite or infinite. Then 

Card 21 < c. (1 

For let us first consider the unit cube (5 whose coordinates x m 
range over 33 = (0*, 1*). Let ) denote the diagonal of . Then 

c = Card ) < Card 6. (2 

On the other hand we show Card ( < c. 

For let us express each coordinate x m as a decimal in normal 
form. Then __ 

x l ' a ll a 12 a 13 a !4 " 



Let us now form the number 



obtained by reading the above table diagonally. Let?) denote the 
set of #'s so obtained as the #'s range over their values. Then 



For the point y, for example, in which a ln = 0, n = 1, 2, lies 
in 53 but not in 2) as otherwise x l = 0. Let us now set x ~ y. 
Then g ^ 2) and hence Card g ^ (g 

From 2), 3) we have Card g = c. 



THE CARDINAL c 291 

Let us now complete by adding its faces, obtaining the set C. 
By a transformation of similitude T we can bring O T within &. 

Hence Card > Card (7. 

On the other hand, is a part of (7, hence 

Card 6 < Card (7. 
Thus Card = c. The rest of the theorem follows now easily. 

248. Let 3 = S/i denote the aggregate of one-valued continuous 
functions over a unit cube in 9? n . 

Then 



Let C denote the rational points of , i.e. the points all of 
whose coordinates are rational. Then any / is known when its 
values over C are known. For if is an irrational point of , 
we can approach it over a sequence of rational points a a , a 2 === . 
But f being continuous, /(a) = lim/(a n ), and f is known at . 
On the other hand, being enumerable, we can arrange its points 

in a sequence n 

^ C=c l , c%, ... 

Let now 9?^ be a space of an infinite enumerable number of 
dimensions, and let y = (y x , y 2 ) denote any one of its points. 

Let f have the value t] 1 at <?j, the value ?; 2 at <? 2 an( i s n f r 
the points of 0. Then the complex 7?j, ?; 2 , - completely deter- 
mines / in (. But this complex also determines the point 
?7 = (rjj, 7/2 ..) in 9?^. We now associate/ with ?;. Thus 

Card $< Card SR = c. 

But obviously Card $ > c, for among the elements of $ there 
is an/ which takes on any given value in the interval (0, 1), at 
a given point of @. 

249. There exist aggregates whose cardinal number is greater 
than any given cardinal number. 

Let 33= \b\ be an aggregate whose cardinal number b is given. 
Let a be a symbol so related to S3 that it has arbitrarily either 
the value 1 or 2 corresponding to each b of $&. Let 21 denote the 



292 AGGREGATES 

aggregate formed of all possible #'s of this kind, and let a be its 
cardinal number. 

Let /3 be an arbitrary element of 33. Let us associate with /3 
that a which has the value 1 for b = and the value 2 for all 
other 6's. This establishes a correspondence between S3 and a 
part of 91. Hence 

a>b. 

Suppose a = b. Then there exists a correspondence which 
associates with each b some one a and conversely. This is 
impossible. 

For call a b that element of 31 which is associated with b. Then 
a b has the value 1 or 2 for each j3 of S3. There exists, however, 
in 21 an element a' which for each /9 of S3 has just the other 
determination than the one a b has. But a f is by hypothesis 
associated with some element of S3, say that 

a' = a b >. 

Then for b = b f , a' must have that one of the two values 1, 2 
which a b ' has. But it has not, hence the contradiction. 

250. The aggregate of limited integrable functions $ defined over 
31 = (0, 1) has a cardinal number f > c. 

For let f(x) = in 31 except at the points of the discrete 
Cantor set of I, 272, and 229, Ex. 4. At each point of let / 
have the value 1 or 2 at pleasure. The aggregate formed of 
all possible such functions has a cardinal number > c, as the 
reasoning of 249 shows. But each f is continuous except in S, 
which is discrete. Hence / is integrable. But rj > . Hence 

f>c. 



Arithmetic Operations with Cardinals 

251. Addition of Cardinals. Let 31, S3 be two aggregates with- 
out common element, whose cardinal numbers are a, b. We define 
the sum of a and b to be 

Card (31, )=a + b. 



ARITHMETIC OPERATIONS WITH CARDINALS 293 

We have now the following obvious relations : 

K 6 + n = K , n a positive integer. (1 

o+---4-K = , n terms. (2 

KO + **o + " == NO * ^ w infinite enumerable set of terms. (3 
//* the cardinal numbers of 31, 33, G are a, b, c, 



a-f b = b + a. 

The first relation states that addition is associative, the second 
that it is commutative. 

252. Multiplication. 

1. Let 21 = ja!, 33 = J&5 have the cardinal numbers a, b. The 
union of all the pairs (a, b) forms a set called the product oftyt and 
53. It is denoted by 31 33. We agree that (a, b) shall be the 
same as (5, a). Then 

. = . 

We define the product of a and b to be 

Card 91 . 33 = Card 93-2I = a.b = b.a. 

2. We have obviously the following formal relations as in finite 

cardinal numbers : /* \ / \\ 

a(u c) = (a u)c, 

a b = b - a, 



which express respectively the associative, commutative, and dis- 
tripulative properties of cardinal numbers. 

Example 1. Let 9l=Jaj, 53 = ji| denote the points on two 
indefinite right lines. Then 

2l.$ = K^)5- 
If we take a, b to be the coordinates of a point in a plane 9? 2 , 



* The reader should note that here, as in the immediately following articles, c is 
simply the cardinal number of ( which is any set, like 51, 53 



294 AGGREGATES 

Example 2. Let 21 = \a\ denote the family of circles 

Let 33 = j6j denote a set of segments of length b. We can 
interpret (#, 5) to be the points on a cylinder whose base is 1) 
and whose height is 5. Then 2( S3 is the aggregate of these 
cylinders. 

253 1 K = n OT tit = c. (1 

For let m , x 

9e = (rt 1 , a a , ... a n ), 

(5 = (^, ^ 2 in inf.) 
Then SR . g = (a O Cd Ca e ) - 



==^ + (52+ ... +C. 

The cardinal number of the set on the left is ?iN , while the 
cardinal number of the set on the right is K . 

2. ec = c. ' (2 

For let ( = \c\ denote the points on a right line, and @ = (1, 2, 
3,.-). 

Then $,= \(n>c)\ 

may be regarded as the points on a right line l n . Obviously, 

Card JU=c- 
Hence 

ec = Card (gg = c. 

254. Exponents. Before defining this notion let us recall a 
problem in the theory of combinations, treated in elementary 
algebra. 

Suppose that there are 7 compartments 

^1' ^2' "* ^Y> 

and that we have k classes of objects 

J^i* -Sji ... K k . 



ARITHMETIC OPERATIONS WITH CARDINALS 295 

Let us place an object from any one of these classes in C v an 
object from any one of these classes in (7 2 and so on, for each 
compartment. The result is a certain distribution of the objects 
from these k classes K, among the y compartments C. 

The number of distributions of objects from k classes among y 
compartments is &. 

For in O l we may put an object from any one of the k classes. 
Thus O l may be filled in k ways. Similarly (7 2 may be filled in 
k ways. Thus the compartments O l , <7 2 may be filled in & 2 ways. 
Similarly (7j, (7 2 , (7 3 may be filled in A 3 ways, etc. 

255. 1. The totality of distributions of objects from k classes 
K among the y compartments C form an aggregate which may be 
denoted by j^c 

We call it the distribution of K over C. The number of distri- 
bution of this kind may be called the cardinal number of the set, 
and we have then -' K c = fr. 



2. What we have here set forth for finite (7 and TTmay be ex- 
tended to any aggregates, 21 = \a\, 93 = \b\ whose cardinal num- 
bers we call a, b. Thus the totality of distributions of the a's 
among the 6's, or the distribution of^[ over 33, is denoted by 

a, 

and its cardinal number is taken to be the definition of the symbol 
a *' Thus > Card- 21* = a*. 

256. Example 1. Let 

x n + a^ 71 " 1 + ' 4- a n = (1 

have rational number coefficients. Each coefficient a 9 can range 
over the enumerable set of elements in the rational number 
system R = \r\, whose cardinal number is K . The n coefficients 
form a set 51 = (a t , a n ) = \a\. To the totality of equations 1) 
corresponds a distribution of the r's among the a's, or the set 

B 

whose cardinal number is 



296 AGGREGATES 

As Card *= Ho = e 

we have the relation : 

%w __ /-|% />n _ A 

NO NO > or e c 
/or any integer n. 

On the other hand, the equations 1) may be associated with 
the complex 

0*i, <Oi 

and the totality of equations 1) is associated with 

= JOi> -<*)} 

But Ki*a2)!={i! *! 

{(!, a 2 , a 3 )| = {(a x , a 2 )} \a s l , etc. 

Hence ff __ , , , , 5 > 

^ = Ji! \ a <n J^ni- 

us Card = e-e---e , n ^Ws as factor. 

But Card 6 = Card JZ 

since each of these sets is associated uniformly with the equations 
'* s e n = e e e , w ft'wes as factor. 



257. Example 2. Any point # in w-way space 9t m depends on 
m coordinates 2^, a; 2 , a; m , each of which may range over the set 
of real numbers SR, whose cardinal number is c. The m coordi- 
nates x l x m form a finite set 



Thus to 9? m = {a;j corresponds the distribution of the numbers in 
9t, among the m elements of X, or the set 

K* 

whose cardinal number is 

c w . 

As Card 9J X = c 

we have 

c m = c for any integer m. (1 
As in Example 1 we show 

c m = cC'"c , m times as factor. 



ARITHMETIC OPERATIONS WITH CARDINALS 297 

258. a b+c = a 6 -a c . (1 

To prove this we have only to show that 
9l 8+e and . 31 S 

can be put in 1-1 correspondence. But this is obvious. For 
the set on the left is the totality of all the distributions of the 
elements of 31 among the sets formed of 33 and S. On the other 
hand, the set on the right is formed of a combination of a distri- 
bution of the elements of 91 among the 33, and among the . Hut 
such a distribution may be regarded as the distribution first con- 
sidered. 



259. (a*y = aK (1 

We have only to show that we can put in 1-1 correspondence 
the elements of 

(31 V and a?'*. (2 

Let 31= jaj, 33= |6j, = \c\. We note that is a union of 
distributions of the a's among the 6's, and that the left side of 2) 
is formed of the distributions of these sets among the c's. These 
are obviously associated uniformly with the distributions of the 
a's among the elements of 33 &. 



260. 1 . c w == (mO n = nc = mC = c (1 

where m, n are positive integers. 

For each number in the interval S = (0, 1*) can be represented 
in normal form once and once only by 

afya z in the w-adic system, (2 

where the < a s < m. [I, 145] . 

Now the set of numbers 2) is the distribution of 90? = (0, 1, 2, 
m 1) over (g = (a x , # 2 , a 3 ), or 



whose cardinal number is 

9?? e . 

On the other hand, the cardinal number g is c. 



298 AGGREGATES 

Hence, m e = c. 

As n* = e, we have 1), using 1) in 257. 

2. The result obtained in 247, 2 may be stated: 

c' = c. (3 

3. ec = c. (4 
For obviously n* < e c < c e . 

But by 3), c e = c and by 1) n e = c. 

261. 1. 27*0 cardinal number t #/" a/ functions f (x l # m ) which 
take on but two values in the domain of definition 21, of cardinal num- 
ber o, is 2 . 

Moreover, 2 ^ > a. 

This follows at once from the reasoning of 249. 

2. Zrtf f Je the cardinal number of the class of all functions de- 
fined over a domain 21 whose cardinal number is c. Then 

For the class of functions which have but two values in 21 is by 
On the other hand, obviously 

But 

c< = (2')S by 260, 1) 

= 2ec, by 259, 1) 

= 2^ by 253, 2). 

Thus, c c = 2 c. 

That f > c 

follows from 250, since the class of functions there considered lies 
in the class here considered. 

3. The cardinal number \ of the class of limited integrable func- 
tions in the interval 21 is = f, the cardinal number of all limited 
functions defined over 21. 



NUMBERS OF LIOUVILLE 299 

For let D be a Cantor set in 21 [I, 272]. Being discrete, any 
limited function defined over J) is integrable. But by 229, Ex. 4, 
the points of 21 may be set in uniform correspondence with the 
points of D. 

4. The set of ail functions 



which are the sum of convergent series, and whose terms are continu- 
ous in 21, has the cardinal number c. 

For the set $ ^ continuous functions in 21 has the cardinal 
number c by 248. These functions are to be distributed among 
the enumerable set @ of terms in 2). Hence the set of these 
functions is ^ 

$ > 

whose cardinal number is 

c e = c. 

Remark. Not every integrable function can be represented by 
the series 2). 

For the class of integrable functions has a cardinal number > c, 
by 250. 

5. The cardinal number of all enumerable sets in an m-way space 
5R m is c. 

For it is obviously the cardinal number of the distribution of 
9t m over an enumerable set (, or 

Card ffi = c c = c. 



Number* of Liouville 

262. In I, 200 we have defined algebraic numbers as roots of 
equations of the type 



where the coefficients a are integers. All other numbers in 9? we 
said were transcendental. We did not take up the question 
whether there are any transcendental numbers, whether in fact, 
not all numbers in 9t are roots of equations of the type 1). 



300 AGGREGATES 

The first to actually show the existence of transcendental num- 
bers was Liouville. He showed how to form an infinity of such 
numbers. At present we have practical means of deciding 
whether a given number is algebraic or not. It was one of the 
signal achievements of Hermite to have shown that e = 2.71818 
is transcendental. 

Shortly after Lindemann, adapting Hermite's methods, proved 
that ?r = 3.14159 is also transcendental. Thereby that famous 
problem the Quadrature of the Circle was answered in the negative. 
The researches of Hermite and Lindemann enable us also to form 
an infinity of transcendental numbers. It is, however, not our pur- 
pose to give an account of these famous results. We shall limit 
our considerations to certain numbers which we call the numbers 
of Liouville. 

In passing let us note that the existence of transcendental num- 
bers follows at once from 235, 2 and 244, 2. 

For the cardinal number of the set of real algebraic number is 
e, and that of the set of all real numbers is c, and c > e. 

263. In algebra it is shown that any algebraic number a is a 
root of an irreducible equation, 

f(x) = a x m 4- ap"- 1 + 4- a m = (1 

whose coefficients are integers without common divisor. We say 
the order of a is m. 

We prove now the theorem 

Let 

P 
r n = , p n , q n relatively prime, 

<}n 

= a, an algebraic number of order m, as n = ao. Then 

\ a - r \>-^{ ' n>v - ( 2 

!/ 

For let a be a root of 1). We may take 8>0 so small that 
f(x)= in -Z> 5 *(0> and 8 so large that r n lies in D 5 (), for n>8. 



NUMBERS OF LIOUVILLE 301 

for n > s, since the numerator of the middle member is an integer, 
and hence >1. 

On the other hand, by the Law of the Mean [I, 397], 



where /8 lies in -Z>s(). Now /()=0 and /'(/9)< some M. 
Hence 



on using 3). But however large M is, there exists a v, such that 
q n > M) for any n>v. This in 4) gives 2). 

264. 1. The numbers 

= J?i-+JSL + -*IL+... (1 

10 1! 10 21 10 3! ^ 

w^ere a n < 10 n , anrf rc0 aH of them vanish after a certain index, are 
transcendental. 

For if L is algebraic, let its order be m. Then if L n denotes 
the sum of the first n terms of 1), there exists a v such that 



But 



(2 



i/ being taken sufficiently large. But 3) contradicts 2). 
The numbers 1) we call the numbers of Liouville. 

2. The set of Liouville numbers has the cardinal number c. 

For all real numbers in the interval (0*, 1) can be represented 

"-& + & + & + - ' ^' 

where not all the 6's vanish after a certain index. The numbers 



_ 
10 21 10 8! 

can obviously be put in uniform correspondence with the set {/8j. 
Thus Card \\\ =c. But \L\ > {X{, hence Card {} > c. On the 
other hand, the numbers \L\ form only a part of the numbers in 
(0*, 1). Hence Card \L\<t. 



CHAPTER IX 
ORDINAL NUMBERS 

Ordered Sets 

265. An aggregate 31 is ordered, when a, b being any two of 
its elements, either a precedes ft, or a succeeds ft, according to some 
law ; such that if a precedes 6, and b precedes e% then a shall pre- 
cede c. The fact that a precedes b may be indicated by 

a<b. 

Then a>b 

states that a succeeds b. 

Example 1. The aggregates 

1, 2, 3, ... 

2, 4, 6, ... 



3, -2, -1,0, 1, 2,3, .. 



are ordered. 



Example 2. The rational number system R can be ordered in 
an infinite variety of ways. For, being enumerable, they can be 

arranged in a sequence .. 

' i * r a ' r a ' " * r n ' 

Now interchange r T with r n . In this way we obtain an infinity 
of sequences. 

Example 3. The points of the circumference of a circle may be 
ordered in an infinite variety of ways. 

For example, let two of its points P l , P 2 make the angles a-f^, 
-f-# 2 with a given radius, the angle varying from to 2 TT. 
Let P l precede P 2 when l < # 2 . 

302 



ORDERED SETS 303 

Example 4- The positive integers $ may be ordered in an infi- 
nite variety of ways besides their natural order. For instance, we 
may write them in the order 

1, 3, 5, ... 2, 4, 6, ... 

so that the odd numbers precede the even. Or in the order 
1, 4, 7, 10, ... 2, 5, 8, 11, ... 3, 6, 9, 12, ... 

and so on. We may go farther and arrange them in an infinity 
of sets. Thus in the first set put all primes ; in the second set 
the products of two primes ; in the third set the products of 
three primes; etc., allowing repetitions of the factors. Let any 
number in set m precede all the numbers in set n >m. The num- 
bers in each set may be arranged in order of magnitude. 

Example 5. The points of the plane 9t 2 may be ordered in an 
infinite variety of ways. Let L y denote the right line parallel to 
the a>axis at a distance y from this axis, taking account of the sign 
of y. We order now the points of 3J 2 by stipulating that any 
point on L v , precedes the points on any L^ when y 1 <y", while 
points on any L v shall have the order they already possess on that 
line due to their position. 

266. Similar Sets. Let 31, 53 be ordered and equivalent. Let 
a ~ b, a ~ ft. If when a < a in 21, b < ft in 53, we say 21 is similar 

to 53, and write OT ~ 

zl 3o. 

Thus the two ordered and equivalent aggregates are similar 
when corresponding elements in the two sets occur in the same 
relative order. 



Example 1. Let 



1 9 

1, -<, 



In the correspondence 21 ~~ 53, let n be associated with a n . Then 

^. 

Example 2. Let M 100 

VI = 1, A o, 

53= a l 2 a m , b^ b^ J 3 



304 ORDINAL NUMBERS 

In the correspondence 21 ~ 33, let a r ~r for r = 1, 2, m; also 
let 6 n ~ m + n, 7i = 1, 2 - Then 21 ^ S3. 

Example 3. Let 

* 



Let the correspondence between 21 and S3 be the same as in 
Ex. 2. Then 21 is not similar to S3. For 1 is the first element in 
21 while its associated element a l is not first in S3. 

Example A. Let or 100 

r 21 = 1, A o, ... 

S3 = tfj, a a ij, 5 2 
Let a n ~ 2 n, b n ~ 2 n - 1. Then 21 - S3 but 21 is not a* S3- 



267. i^2l^S3, S3^S. Then %**<$,. 

For let a- i, a 1 ~V in 21 -S3. Let 6 - <?, J'^^' in S3 ~ g. Let 
us establish a correspondence 21 (5 by setting a ~ c, a f ~c f . Then 
if a <a f in 21, c< c r in S. Hence 21 ^ S. 

Eutactic Sets 

268. Let 21 be any ordered aggregate, and S3 a part of 21, the 
elements of S3 being kept in the same relative order as in 21. If 21 
and each S3 both have a first element, we say that 21 is well ordered, 
or eutactic. 

Example 1. 21 = 2, 3, f>00 is well ordered. For it has a first 
element 2. Moreover any part of 21 as 6, 15, 25, 496 also has a 
first element. 

Example 2. 21 = 12, 13, 14, in inf. is well ordered. For it 
has a first element 12, and any part S3 of 21 whose elements pre- 
serve the same relative order as in 21, has a first element, viz. 
the least number in 53. 

The condition that the elements of S3 should keep the same rel- 
ative order as in 21 is necessary. For B = 28, 26, 24, 22, 20, 
21, 23, 25, 27, ... is a partial aggregate having no first element. 
But the elements of B do not have the order they have in 21. 



EUTACTIC SETS 305 

Example 8. Let 21 = rational numbers in the interval (0, 1) 
arranged in their order of magnitude. Then 21 is ordered. It 
also has a first element, viz. 0. It is not well ordered however. 
For the partial set 33 consisting of the positive rational numbers of 
21 has no first element. 

Example 4- An ordered set which is not well ordered may some- 
times be made so by ordering its elements according to another 
law. 

Thus in Ex. 3, let us arrange 21 in a manner similar to 283. 
Obviously 21 is now well ordered. 

Example 5. 21 = a^ # 2 4j, J 2 is well ordered. For a is the 
first element of 21 ; and any part of 21 as 



has a first element. 

269. 1. Every partial set 33 of a well-ordered aggregate 21 is well 
ordered. 

For 33 has a first element, since it is a part of 21 which is well 
ordered. If & is a part of 33, it is also a part of 21, and hence has 
a first element. 

2. If a is not the last element of a well-ordered aggregate 21, there 
is an element ofty. immediately following a. 

For let 33 be the part of 21 formed of the elements after a. It 
has a first element b since 21 is well ordered. Suppose now 

a< c < b. 
Then b is not the first element of 33, as c < b is in 33. 

3. When convenient the element immediately succeeding a may 
be denoted by 

a + L 

Similarly we may denote the element immediately preceding a, 
when it exists, by 



306 ORDINAL NUMBERS 

For example, let 

21 = a^a% ^1^2 ' ' " 
Then a n + 1 = a n+1 , b m + 1 = b n+1 

<*n - 1 = n-l i *m - 1 = *m-r 

There is, however, no b l 1. 



270. 1. .7/31 i Wtf/Z ordered, it is impossible to pick out an in- 
finite sequence of the type 

a x > rt a > a 3 > ... (1 

For m 

1O = ' #3, # 2 ' j 

is a part of 21 whose elements occur in the same relative order as 
in 21, and $ has no first element. 

2. A sequence as 1) may be called a decreasing sequence, while 

a 1 < a 2 < a 3 ... 
may be called increasing. 

In every infinite well ordered aggregate there exist increasing 
sequences. 

3. Let 21, 48, (5, be a well ordered set. Let 21= \ a\ be well 
ordered in the a #, 53 = 5 6 i ^ we# ordered in the 6's, ec. 



U = , , CS - 
ordered with regard to the little letters a, b ... 

For U has a first element in the little letters, viz. the first ele- 
ment of 21. Moreover, any part of U, as 55, has a first element in 
the little letters. For if it has not, there exists in 33 an infinite 
decreasing sequence 

t> s>r> '- 

This, however, is impossible, as siu-li a sequence would deter- 
mine a similar sequence in U as 

3T > @ > 5R > -.. 
which is impossible as U is well ordered with regard to 21, 33 

4. Let 9l<<e< (1 

Let each element of 21 precede each element of 53, etc. 



SECTIONS 307 



Let each 21, $J, be well ordered. 
Let #=3l+S, 

nen 



CM- 



is a well ordered set* @ preserving the relative order of elements 
intact. 

For <3 has a first element, viz. the first element of 21. Any 
part S of @ has a first element. F"or, if not, there exists in @ 
an infinite decreasing sequence 

r > q> p > (2 

Now r lies in some set of 1) as 9t. Hence <?, JP, also lie in 
JR. But in 5R there is no sequence as 2). 

5. Let 21, 33, S, be an ordered set of well ordered aggre- 
gates, no two of which have an element in common. The reader 
must guard against assuming that 21 -f- .SB + & -f > keeping the 
relative order intact, is necessarily well ordered. 

For let us modify Ex. 5 in 265 by taking instead of all the 
points on each L v only a well ordered set which we denote by Sl y . 

Then the sum ^ __ y^ 

has a definite meaning. The elements of 21 we supposed arranged 
as in Ex. 5 of 265. 

Obviously 21 is not well ordered. 



Sections 

271. We now introduce a notion which in the theory of well- 
ordered sets plays a part analogous to Dedekind's partitions in 
the theory of the real number system 9{. Cf. I, 128. 

Let 21 be a well ordered set. The elements preceding a given 
element a of 21 form a partial set called the section of 21 generated 
by a. We may denote it by 

Sa, 

or by the corresponding small letter a. 



308 ORDINAL NUMBERS 

Example!. Let 91 = 1 2 3 

Then 

100 = 1, 2,- .-99 

is the section of 21 generated by the element 100. 

Example 2. Let 

21 = 6^, a 2 -.- b l , 6 2 -.. 
Then 

*S* 6 = a 1 a a --. ^626364 

is the section generated by b^. 

Sb l = a^ 
that generated by 6j, etc. 

272. 1. Every section of a well ordered aggregate is well ordered. 
For each section of 21 is a partial aggregate of 21, and hence 

well ordered by 269, 1. 

2. In the well ordered set 21, let a<b. Then Sa is a section 
ofSb. 

3. Let @ denote the aggregate of sections of an infinite well 
ordered set ty. If we order @ such that Sa < Sb in @ when a<b in 
21, @ is 0cK ordered. 

For the correspondence between 21 and @ is uniform and similar. 

273. Let 21, 58 be well ordered and 21^93. If a~~b, then 

Sa ^ Sb. 

For in 21 let a ff <a f >a. Let b'~a' and b n ^a rr . Since 
21 2- 93, we have 

6"<6'<6; 
hence the theorem. 

274. If 21 is well ordered, 21 is not similar to any one of its 
sections. 

For if 21 ^$a, to a in 21 corresponds an element a l <a in Sa. 
To a l in 21 corresponds an element a 2 in Sa> etc. In this way we 
obtain an infinite decreasing sequence 

a> a l 

which is impossible by 270, 1. 



SECTIONS 309 



275- Let 21, 93 be well ordered and 21 ^ 93. 2%ew to Sa in 21 
7&0 correspond two sections Sb, /SyS each 2* Sa. 

For let b < fr and a ^ 6, #a ^ $8. Then 

Sb ^ #/3, by 267. (1 

But 1) contradicts 274. 

276. Let 21, S3 be two well ordered aggregates. It is impossible 
to establish a uniform and similar correspondence between 21 and S3 
in more than one way. 

For say Sa ^ Sb in one correspondence, and Sa ^ S/3 in an- 
other, b, /3 being different elements of S3- Then 

#6 ^ tf/3, by 267. 
This contradicts 275. 

277. 1. We can now prove the following theorem, which is 
the converse of 273. 

Let 21, 93 be well ordered. If to each section of 21 corresponds one 
similar section of 93, and conversely, then 93 21. 

Let us first show that 21 ~ 93. Since to any Sa of 21 corre- 
sponds a similar section Sb in 93, let us set a ~ b. No other 
a 1 ~ 6, and no other b' ~ a, as then Sa 1 ^ Sb or Sb' ^ Sa, which 
contradicts 274. Let the first element of 21 correspond to the 
first of 93. Thus the correspondence we have set up between 21 
and 93 is uniform and 21 ~ 93. 

We show now that this correspondence is similar. For let 

a ~ b and a' ~ b r , a! < a. 

Then b r < b. For a f lies in Sa ^ Sb and b f a r lies in Sb. 
2. From 1 and 273 we have now the fundamental theorem : 

In order that two well-ordered sets 21, 93 be similar, it is necessary 
and sufficient that to each section of 21 corresponds a similar section 
of 93, and conversely. 

278. Let 21, 93 be well ordered. If to each section of 21 corre- 
sponds a similar section of 93> but not conversely, then 21 is similar to 
a section of 93. 



310 ORDINAL NUMBKKS 

Let us begin by ordering the sections of 21 and 93 as in 272, 3. 
Let B denote the aggregate of sections of 53 to which similar sec- 
tions of 21 do not correspond. Then B is well ordered and has a 
first section, say Sb. Let /3 < b. Then to 8ft in 53 corresponds 
by hypothesis a similar section Sa in 2t. On the other hand, to 
any section Sa 1 of 21 corresponds a similar section fib' of 53- Ob- 
viously b'<b. Thus to any section of 21 corresponds a similar 
section of Sb and conversely. Hence ?l^/S7> by 277. i. 

279. Let 21, 93 be well ordered. Either 21 is similar to 93 or one 
is similar to a section of the other. 

For either : 
1 To each section of 21 corresponds a similar section of 93 

and conversely ; 
or 2 To each section of one corresponds a similar section of 

the other but not conversely ; 

or 3 There is at least one section in both 21 and 93 to which no 
similar section corresponds in the other. 

If 1 holds, 21 ^93 by 277, l. If 2 holds, either 21 or 93 is similar 
to a section of the other. 

We conclude by showing 3 is impossible. 

For let A be the set of sections of 21 to which no similar section 
in 93 corresponds. Let B have the same meaning for 93. If we 
suppose 21, 93 ordered as in 272, 3, A will have a first section say 
8a, and B a first section fl/3. 

Let a < a. Then to Sa in 21 corresponds by hypothesis a sec- 
tion Sb of Sft as in 278. Similarly if b r < 0, to Sb' of 93 corre- 
sponds a section 8a' of 8a. But then 8a^8f3 by 277, l, and this 
contradicts the hypothesis. 

Ordinal Numbers 

280. 1. With each well ordered aggregate 21 we associate an 
ittribute called its ordinal number^ which we define as follows : 

1 If 2193, they have the same ordinal number. 
2 If 21 a section of 93, the ordinal number of 21 is less than 
that of 93. 



ORDINAL NUMBERS 311 

3 If a section of 21 is ^ 33, the ordinal number of 21 is greater 
than that of 33. 

The ordinal number of 21 may be denoted by 

Orel 21, 

or when no ambiguity can arise, by the corresponding small letter a. 
As any two well ordered aggregates 21, 55 fall under one and only 
one of the three preceding cases, any two ordinal numbers a, b 
satisfy one of the three following relations, and only one, viz. : 

a = b , a<b , a > b, 
and if a < b, it follows that b > a- 

Obviously they enjoy also the following properties. 

2 If 

J a = b , b = c , then a = c. 

For if c = Ord , the first two relations state that 



But then aafg ^ by 

Hence _ 

*^ a > b , b > c , then a > c. 

281. 1. Let 21 be a finite aggregate, embracing say n elements. 
Then we set Onia=n. 

Thus the ordinal number of a finite aggregate has exactly similar 
properties to those of Unite cardinal numbers. The ordinal num- 
ber of a finite aggregate is called finite, otherwise transfinite. 

The ordinal number belonging to the well ordered set formed 
of the positive integers c\ _ i o o 

O A, *j, o, 

we call a). 

2. The least transfinite ordinal number is to. 

For suppose a = Ord 21 < o>, is transfinite. Then 21 is ^ a 
section of $. But every section of 3 is finite, hence the 
contradiction. 



312 ORDINAL NUMBERS 

3. The cardinal number of a set 51 is independent of the order 
in which the elements of $ occur. This is not so in general for 
ordinal numbers. 

For example, let or 1 o Q 

4\ 1, 4, O, 

33=1, 3, 5, -.2,4, 6, .- 
Here Card 21 = Card S=. 

But Ord 21 < Ord , 

since 21 is similar to a section of S3, viz. the set of odd numbers, 
1, 3, 5, ... 

282. 1. Addition of Ordinals. Let 1, S3 be well ordered sets 
without common elements. Let & be the aggregate formed by 
placing the elements of S3 after those of 21, leaving the order in 53 
otherwise unchanged. Then the ordinal number of g is called the 
sum of the ordinal numbers of 21 and 53, or 

Ord g = Ord 2t + Ord 53, 
or c = a + 6. 

The extension of this definition to any set of well-ordered aggre- 
gates such that the result is well ordered is obvious. 

2. We note that A , t ^ A rt _j_f,^f, 

a -f > a, a -f b > I). 

For 21 is similar to a section of S, and 53 is equivalent to a part 
of 6. 

3. The addition of ordinal numbers is associative. 

This is an immediate consequence of the definition of addition. 

4. The addition of ordinal numbers is not always commutative. 
Thus if 



let = (ojo, ... ^6 2 ... 6 n ), Ord = c, 

) = (Jj ... b n a^ -.), Ord = b. 

Then v , 

c = a) -f- ^ , b = ft + o> 



ORDINAL NUMBERS 313 

But 21 2* a section of S, viz. : ^ >S Y 6 a , while D ^ SI. Hence 

0) < C , = b, 

or 

CD + n> to , w-j- ft > : =fc>' 

5. If a > b, ^e/i c -f a > c + b, awd a + c > b + c. 
For let 



Since a > b, we can take for 93 a section Sb of 31. Then c 4- & is 
the ordinal number of (S 4. 9f (\ 

and c + b is the ordinal number of 

S + Sb, (2 

preserving the relative order of the elements. 

But 2) is a section of 1), and hence c 4- a > c -}- b. 
The proof of the rest of the theorem is obvious. 

283. 1. The ordinal number immediately following a is a 4- 1. 

For let a = Ord 91. Let 93 be a set formed by adding after all 
the elements of 91 another element b. Then 

a + 1 = Ord 33 = b. 
Suppose now 

a<c<b , c=Ordg. (1 

Then is similar to a section of 53. But the greatest section 
of 93 is 8b = 91. Hence 

c < a, 
which contradicts 1). 

2. Let a > b. Then there is one and only one ordinal number b 

such that - , , 

a = u -h o. 

For let 



a = Ord 31 , b = OrdS8- 
We may take 93 to be a section Sb of 31. Let ) denote the set 
of elements of 91, coming after Sb. It is well ordered and has an 
ordinal number b. Then 

91 = 93 + , 

preserving the relative order, and hence 

a = b + b. 
There is no other number, as 282, 6 shows. 



314 ORDINAL NUMBERS 

284. 1. Multiplication of Ordinals. Let 31, 93 be well-ordered 
aggregates having a, b as ordinal numbers. Let us replace each 
element of 21 by an aggregate ^ 33. The resulting aggregate & 
we denote by $.91 

As 6 is a well-ordered set by 270, 3 it has an ordinal number c. 
We define now the product b a to be c, and write 

b a = c. 

We say c is the result of multiplying a % b, and call a, b factors. 
We write 

a . a = a 2 , a a a = a 3 , etc. 

2. Multiplication is associative. 

This is an immediate consequence of the definition. 

3. Multiplication is not always commutative. 
For example, let 



S3 = (1, 2, 3 .-. in inf.). 
Then .a = (W 8 -.., <W* - 

21 33 =(*!, <?i, & 2 6 2 

Hence Qrd (93 21) = 2 



4. If a < b, then ca < cb. 

For 6 81 is a section of g 33. 

Limitary Numbers 
285. 1. Let 



be an infinite increasing enumerable sequence of ordinal numbers. 
There exists a first ordinal number a greater than every a n . 

Let n =0rd2l n . 



LIMITARY NUMBERS 315 

Since n ^ 1 < n , 2l n _ 1 is similar to a section of 2l n . For simplicity 
we may take 2l n _! to be a section of 2l n . Let, therefore, 



Consider now w w ^ ~ 

Zl = a x -f ^ 2 + ^3 + 

keeping the relative order of the elements intact. Then 21 is well 
ordered and has an ordinal number a. 
As any 2l n is a section of 21, 



Moreover any number /3<a is also < some a m . For if 33 has 
the ordinal number /3, 33 must be similar to a section of 21. Hut 
there is no last section of 21. 

2. The number a we have just determined is called the limit of 
the sequence 1). We write 

a = Km n , or n = a. 

We also say that a corresponds to the sequence V). 
All numbers corresponding 1 to infinite enumerable increasing 
sequences of ordinal numbers are called limitary. 

3. // every a n in 1) is < /3, then a < /3. 

For if /3<, a is not the least ordinal number greater than 
every n . 

4. If /3<, /3 is 



286. jfri or^r that . . ,* 

1 <0 2 < (1 

/3i</ 3 2<'" (2 

define the same number \ it is necessary and sufficient that each 
number in either sequence is surpassed by a number in the other. 

For let . Q . & 

n = , fin = fl- 

it no /3 n is greater than a m , ft<a m < , by 285, 3, and = @. 

On the other hand, if each m < some /3 n , </S. Similarly 
/3<a. 



316 ORDINAL NUMBERS 

287. Cantors Principles of Q-enerating Ordinals. We have now 
two methods of generating ordinal numbers. First, by adding 1 
to any ordinal number a. In this way we get 

a, a+ 1, a+ 2, .- 

Secondly, by taking the limit of an infinite enumerable increas- 
ing sequence of ordinal numbers, as 

! < 2 < 8 < - 

Cantor calls these two methods the first and second principles 
of generating ordinal numbers. 

Starting with the ordinal number 1, we get by successive appli- 
cations of the first principle the numbers 

1, 2, 3, 4, ... 

The limit of this sequence is CD by 285, 1. Using the first prin- 
ciple alone, this number would not be attained ; to get it requires 
the application of the second principle. Making use of the first 
principle again, we obtain 

oi + l, w + 2, o) + 3, ... 

The second principle gives now the limitary number &> + ct> = &>2 
by 285, 1. From this we get, using the first principle, as before, 

w2 + l, <2 + 2, 0)2 + 3, . 
whose limit is o>3. In this way we may obtain the numbers 

o)w + n , m, n finite. 
The limit of any increasing sequence of these numbers as 

ft> , o>2 , o)3 , o)4, 
is o) o) = o) 2 , by 285, 1. 

From o) 2 we can get numbers of the type 

aPl + com + n l,m,n finite. 

Obviously we may proceed in this way indefinitely and obtain 
all numbers of the type 



where # , ^ - a n are finite ordinals. 



LIMITARY NUMBERS 317 

But here the process does not end. For the sequence 

0) < O) 2 < ft) 3 < 

has a limit which we denote by a> w . 
Continuing we obtain 

a) w&> , ft) w&)W , etc. 

288. It is interesting to see how we may obtain well ordered 
sets of points whose ordinal numbers are the numbers just con- 
sidered. 

In the unit interval 21 = (0, 1), let us take the points 

J . I . i ' if- a 

These form a well ordered set whose ordinal number is a>. 
The points 1) divided 21 into a set of intervals, 

i , a , a 8 - ( 2 

In m of these intervals, let us take a set similar to 1). This 
gives us a set whose ordinal number is com. 

In each interval 2), let us take a set similar to 1). This gives 
us a set whose ordinal number is o> 2 . The points of this set 
divide 21 into a set of &> 2 intervals. In each of these intervals, 
let us take a set of points similar to 1). This gives a set of 
points whose ordinal number is o> 3 , etc. 

Let us now put in 2l t a set of points SS 1 whose ordinal number 
is co. In 2I 2 let us put a se ^ 33 2 whose ordinal number is o> 2 , and 
so on, for the other intervals of 2). 

We thus get in 21 the well ordered set 



whose ordinal number is the limit of 

a) , a) 4- co 2 , ft) + P 4- a> 8 1 
This by 286 has the same limit as 

CD , a) 2 , a) 3 , or G> W . 

With this set we may now form a set whose ordinal number is 
fl> wW , etc. 



318 ORDINAL NUMBERS 

Classes of Ordinals 

289. Cantor has divided the ordinal numbers into classes. 

Class 1, denoted by Z, embraces all finite ordinal numbers. 

Class 2, denoted by Z 2 , embraces all transfinite ordinal numbers 
corresponding to well ordered enumerable sets ; that is, to sets 
whose cardinal number is N . For this reason we also write 



It will be shown in 293, 1 that Z^ is not enumerable. Moreover 

if we set . % ~ , ~ 

Kj = Card Z 2 , 

there is no cardinal number between K and Kj as will be shown in 
294. We are thus justified in saying that Class 8, denoted by 
Z z or ^(Kj), embraces all ordinal numbers corresponding to well 
ordered sets whose cardinal number is Kj, etc. 

Let /3 = Ord S3 be any ordinal number. Then all the numbers 
a < /8 correspond to sections of S3. These sections form a well 
ordered set by 272, 3. Therefore if we arrange the numbers 
a < y8 in an order such that ' precedes a when Sa r < $, they are 
well ordered. We shall call this the natural order. Then the 
first number in Z l is 1, the first number of Z% is ct). The first 
number in Z z is denoted by fl. 

290. As the numbers in Class 1 are the positive integers, they 
need no comment here. Let us therefore turn to Class 2. 

If a is in Z<i , so is a 4- 1 

For let a = Ord 21. Let S3 be the well ordered set obtained 
by placing an element b after all the elements of 21. Then 

+ 1 = Ord S3. 

But S3 is enumerable since 21 is. 
Hence a + 1 lies in Z 2 . 

291 - Let 1 

be an enumerable infinite set of numbers in Z^. Then a = lim n lies 
in Z 



CLASSES OF ORDINALS 319 

For using the notation employed in the proof of 285, 1, a is the 
ordinal number of 



But ?lj, 33j, 33 2 '" are eacn enumerable. 

Hence 21 is enumerable by 2.35, l, and a lies in Z^ 

292. We prove now the converse of 290 and 291. 

Kach number a in /^ 2 , except <y, is obtained by adding 1 to some 
number in Z%; or it is the limit of an infinite enumerable increasing/ 
set of numbers in Z v 

For, let a = Ord 21. Suppose first, that 21 has a last element, 
say a. Since 21 is enumerable, so is Sa. Hence 

ft = Ord Sa 
is in Z r Then = + !. 

Suppose secondly, that 21 has no last element. All the numbers 
ft < in Z% belong to sections of 21. Since 21 is enumerable, the 
numbers ft are enumerable. Let them be arranged in a sequence 

ft r flv /V" (1 

Since they have no greatest, let ft[ be the first number in it 
>/3 1 , let /3 2 be the first number in it >/3(, etc. We get thus the 
sequence /^ < # < 2 ' < (2 

whose limit is X, say. 

Then \ = <*. For A, is >any number in 1), which embraces all 
the numbers of Z% < a. Moreover it is the least number which 
enjoys this property. 

293. 1 . The numbers of Z 2 are not enumerable. 

For suppose they were. Let us arrange them in the sequence 

i* 2 > 3 " C 1 

Then, as in 292, there exists in this sequence the infinite enu- 

merable sequence . , . / . /0 

^ ! < J < 2 < (% 

such that there are numbers in 2) greater than any given number 
in 1). 



320 ORDINAL NUMBERS 

Let = '. Then ' lies in Z% by 291. On the other hand, by 
285, a f is > any number in 2), and therefore > any number in 
1). But 1) embraces all the numbers of Z 2 , by hypothesis. We 
are thus led to a contradiction. 

2. We set p , 

Kj = Oard Z 2 . 

294. There is no cardinal number between K and ttj- 

For let a=Card 21 be such a number. Then 21 is ~ an infinite 
partial aggregate of Z%, which without loss of generality may be 
taken to be a section of Z 2 . But every such section is enumer- 
able. Hence 21 is enumerable and =K , which is a contradiction. 

295. We have just seen that the numbers in Z 2 are not enumer- 
able. Let us order them so that each number is less than any 
succeeding number. We shall call this the natural order. 

1. The numbers of Z% when arranged in their natural order form 
a well ordered set. 

For Z% has a first element co. Moreover any partial set Z, the 
relative order being preserved, has a first element. For if it has 
not, there exists an infinite enumerable decreasing sequence 



This, however, is not possible. For /3, 7, form a part of Sec 
which is well ordered. 

There is thus one well ordered set having Kj as cardinal num- 

ber " Let 



Let now 21 be an enumerable well ordered set whose ordinal 
number is . The set 



the elements of 21 coming after Z 2 , has the cardinal number Kj by 
241, 3. It is well ordered by 270, 3. It has therefore an ordinal 
number which lies in Z 3 , viz. H -j- by 282, l. Thus Z% embraces 
an infinity of numbers. 

2. The least number in Z B is fl. 

For to any number < li corresponds a section 21 of Z y Hence 
a lies in Z. 



CLASSES OF ORDINALS 321 

296. 1. An aggregate formed of an Kj set of Kj sets is an ^ set. 
Consider the set 

A = a u ,\a l2 , 

a 21 ' a 22 ' 



< 



Here each row is an Kj set. As there are an Kj set of rows, A 
is an Kj set of Kj sets. To show that A is an ^ set, we associate 
each a ilc with some number in the first two number classes. 

In the first place the elements a lie where i K. < CD may be associ- 
ated with the numbers 1, 2, 3, < co. The elements a t<0 , a^ 
lying just inside the &> th square and which are characterized 
by the condition that i = 1, 2, o>; K = 1, 2 < co form an 
enumerable set and may therefore be associated with the ordinals 
o>, to -f 1, ... < ft>2. For the same reason the elements just inside 
the ft) + 1 st square may be associated with the ordinals ft)2, ft)2 -f- 1, 
... < ft)3. In this way we may continue. For when we have 
arrived at the a th row and column (edge of the <* th square) we 
have only used up an enumerable set of numbers in the sequence 

i, 2, ... w ... < n (i 

in our process of association. There are thus still an 8 X set left 
in 1) to continue the process of association. 

2. As a corollary of 1 we have : 
The ordinal numbers 

n 2 , n 3 , ft 4 , ... 

lie in Z% . 

297. 1. Let </3<7< ... (1 

be an increasing sequence of numbers in Z z having K x as cardinal 
number and such that any section of 1) has K as its cardinal. 
There exists a first ordinal number \ in Z% greater than any number 
in 1). 
For let 



322 ORDINAL NUMBERS 

Since a < /3 we may take 21 to be a section of S3. Similarly 
we may suppose S3 is a section of (, etc. 

Letnow = + *, <-+ 

Consider now ^ _ ^ ^ ^ 

keeping the relative order intact. Then is well ordered by 
270, 4. Let 



Since Card H = Kj, by 290, l, X lies in Z 3 . 
As any 21, S3, is a section of , 

<<< X. 

Moreover, any number ^ < X is also < some , /3, 7 F*or if 
3ft has ordinal number /*, 3W must be similar to a section of . 
But there is no last section in 8. 

2. We shall call sequences of the type 1), an Sj sequence. 
The number X whose existence we have just established, we shall 
call the limit of]). We shall also write 

< ft< 7 =X 
to indicate that a, /8, is an Kj sequence whose limit is X. 

298. 1. The preceding theorem gives us a third method of 
generating ordinal numbers. We call it the third principle. 

We have seen that the first and second principles suffice to gen- 
erate the numbers of the first two classes of ordinal numbers but 
do not suffice to generate even the first number, viz. fi in Z B . We 
prove no\v the following fundamental theorem : 

2. The three principles already described are necessary and suffi- 
cient to generate the numbers in Z B . 

For let a = Ord be any number of Z 3 . If 21 has a last element, 
reasoning similar to 292, l shows that 



If 21 has no last element, all the numbers of Z B <a form an K 
or Kj set. In the former case 

a = n + ft, 



CLASSES OF ORDINALS 823 

where /? lies in Z a . In the latter case, reasoning similar to 292, 1 
shows that we can pick out an Kj increasing sequence 



299. 1. The numbers of Z form a set whose cardinal number a 
is >K r 

The proof is entirely similar to 293, 1. Suppose, in fact, that 
a == Hj . Let us arrange the elements of Z in the Kj sequence 

19 Ojj (1 

As in 292, there exists in this sequence an Kj increasing sequence 

a[<a< ... = a'. (2 

Then ex.' lies in Z% by 297, 1. On the other hand a! is greater than 
any number in 2) and hence greater than any number in 1). 
But 1) embraces all the numbers in Z 3 by hypothesis. We are 
thus led to a contradiction. 

2. We set 2 = CardZ 8 . 

3. There is no cardinal number between Kj and K 2 . 

For let a = Card 21 be such a number. Then 21 is equivalent to 
a section of Z%. But every such section has the cardinal num- 
ber K r 

300. The reasoning of the preceding paragraphs may be at 
once generalized. The ordinal numbers of Z n corresponding to 
well ordered sets of cardinal number K n _ 2 form a well ordered set 
having a greater cardinal number a than S w _ 2 . Moreover there is 
no cardinal lying between K n _ 2 and a. We may therefore ap- 
propriately denote a by K n _ r The K n _ 2 sequence of ordinal 
numbers 



lying in Z n has a limit lying in Z n , and this fact embodies the 
n th principle for generating ordinal numbers. The first n prin- 
ciples are just adequate to generate the numbers of Z n . They do 
not suffice to generate even the first number in Z n+1 . 
Finally we note that an S n set of N n sets forms an K n set. 



CHAPTER X 
POINT SETS 

Pantaxis 

301. 1. (JBorel.^) Let each point of the limited or unlimited set 
91 lie at the center of a cube (E. Then there exists an enumerable set 
of non- overlapping cubes jcj such that each c lies within some (5, and 
each point of 21 lies in some c. If 21 is limited and complete^ there 
is a finite set \t\ having this property. 

For let jZ)j, i> 2 "* be a sequence of superposed cubical divisions 
of norms === 0. Any cell of D l which lies within some and 
which contains a point of 21 we call a black cell ; the other cells 
of D we call white. The black cells are not further subdivided. 
The division D 2 divides each white cell. Any of these subdivided 
cells which lies within some & and contains a point of 21 we call a 
black cell, the others are white. Continuing we get an enumer- 
able set of non-overlapping cubical cells Jcj. 

Each point a of 21 lies within some c. For a is the center of 
some S. But when n is taken sufficiently large, a lies in a cell of 
Z> n , which cell lies within g. 

Let now 2X be limited and complete. Each a lies within a cube c, 
or on the faces of a finite number of these c. With a we associ- 
ate the diagonal of the smallest of these cubes. Suppose 
MinS = in 21. As 21 is complete, there is a point a in 21 such 
that Min S = 0, in any F^(). This is not possible, since if ?; is 
taken sufficiently small, all the points of V^ lie in a finite number 
of the cubes c. 

Thus Min B > 0. As the c's do not overlap, there are but a 
finite number. 

2. In the foregoing theorem the points of 21 are not necessarily 
inner points of the cubes c. Let a be a point of 21 on the face of 
one of these c. Since a lies within some S, it is obvious that the 

324 



PANTAXIS 325 

cells of some Z) n , n sufficiently large, which surround a form a 
cube (?, lying within . Thus the points of 21 lie within an 
enumerable set of cells }<?{, each c lying within some (. The 
cells c of course will in general overlap. Obviously also, if 21 is 
complete, the points of 31 will lie within a finite number of 
these c's. 

302. If 21 is dense, 21' is perfect. 

For, in the first place, 31' is dense. In fact, let be a point of 
21'. Then in any D*() there are points of 31. Let a be such a 
point. Since 31 is dense, it is a limiting point of 31 and hence is a 
point of 21'. Thus in any ./)*() there are points of 31'. 

Secondly, 31' is complete, by I, 266. 

303. Let 33 be a complete partial set of the perfect aggregate 31. 
Then g = 21 - 33 is dense. 

For if contains the isolated point c, all the points of 31 in D r *(<?) 
lie in 33, if r is taken sufficiently small. But $8 being com- 
plete, c must then lie in 33- 

Remark. We take this occasion to note that a finite set is to be 
regarded as complete. 

304. 1. 7/31 does not embrace all 9t n , it has at least one frontier 
point in 9f n . 

For let a be a point of 31, and b a point of 9? n not in 31. The 
points on the join of a, b have coordinates 

^ = ^ + 0(^-0=^(0), 0<0<1, i = 1, 2, ...n. 

Let & be the maximum of those #'s such that x(d) belongs to 
31 if < 0'. Then x(0 r ) is a frontier point of 31. 

2. Let 31, 33 have no point in common. If Dist (21, 33) >0, we 
say 31, 33 are exterior to each other. 

305. 1. Let 31 = \a\ be a limited or unlimited point set in 9? m . 
We say 33 < 21 is pantactic in 31, when in each D^a) there is a 
point SB. 

We say 33 is apantactic in 21 when in each -Z)$(a) there is a point 
a of 21 such that some J?T,0*) contains no point of 33. 



326 POINT SETS 

Example 1. Let 21 be the unit interval (0, 1), and S3 the ra- 
tional points in 21. Then 93 is pantactic in 21. 

Example 2. Let 21 be the interval (0, 1), and $B the Cantor set 
of I, 272. Then 93 is apantactic in 21. 

2. If 93 < 21 is pantactic in 21, 21 contains no isolated points not 
in 93. 

For let a be a point of 21 not in 93. Then by definition, in any 
D 5 (a) there is a point of 93. Hence there are an infinity of points 
of 93 in this domain. Hence a is a limiting point of 21. 

306. Let 21 be complete. We say 93 < 21 is of the 1 category 
in 21, if 93 is the union of an enumerable set of apantactic sets 
in 21. 

If 93 is not of the 1 category, we say it is of the 2 category. 
Sets of the 1 category may be called Baire sets. 

Example. Let 21 be the unit interval, and 93 the rational 
points in it. Then 93 is of the 1 category. 

For 93 being enumerable, let 93 = \b n \. But each b n is a single 
point and is thus apantactic in 21. 

The same reasoning shows that if 93 is any enumerable set in 
21, then 93 is of the 1 category. 

307. 1. If $8 is of the 1 category in 21, 91 - 93 = B is > 0. 

For since 93 is of the 1 category in 21, it is the union of an 
enumerable set of apantactic sets {93 n J. Then by definition there 
exist points a r a 2 , in 21 such that 



where D(a^) contains no point of 93j, -#(#2) no P ^ of 93 2 > e ^ c 
Let b be the point determined by 1). Since 21 is complete by 
definition, b is a point of 21. As it is not in any 93 n , it is not 
in 93. Hence S contains at least one point. 

2. Let 21 be the union of an enumerable set of sets |2l n j, each 2l n 
being of the 1 category in 93. Then 21 is of the 1 category in 93. 

This is obvious, since the union of an enumerable set of enu- 
merable sets is enumerable. 



PANTAXIS 327 

3. Let 93 be of the 1 category in 21. Then B = 21 -$8 is of the 
2 category in 21. 

For otherwise 33 + B would be of the 1 category in 21. But 

a - ( + j?) = o, 

and this violates 1. 

4. It is now easy to give examples of sets of the 2 category. 
For instance, the irrational points in the interval (0, 1) form a 
set of the 2 category. 

308. Let 21 be a set of the 1 category in the cube }. Then 
A = Q 21 has the cardinal number c. 

If A has an inner point, -Z) 6 (a), for sufficiently small 6, lies in A. 
As Card D& = c, the theorem is proved. 

Suppose that A luis no inner point. Let 21 be the union of the 
apantactic sets 2lj < 21 2 < in Q. Let A n = Q 2l n . Let q n be 
the maximum of the sides of the cubes lying wholly in A n . Ob- 
viously q n = 0, since by hypothesis A has no inner points. Let Q 
be a cube lying in A. As q n = 0, there exists an n such that Q 
has at least two cubes lying in A ni ; call them $ , Q 1 . There ex- 
ists an n% > n such that $ , Q l each have two cubes in A n ^\ call 

them v Q O (1 O 

Vo, o ' ^0,1 ' Vi, o ' Vi, i 

or more shortly $ 4t t2 . 

Each of these gives rise similarly to two cubes in some A n& , 
which may be denoted by $ ll} t2 lg , where the indices as before have 
the values 0, 1. In this way we may continue getting the cubes 

0, , 0, 4 , Q^- 

Let a be a point lying in a sequence of these cubes. It obvi- 
ously does not lie in 21, if the indices are not, after a certain stage, 
all or all 1. This point a is characterized by the sequence 



which may be read as a number in the dyadic system. But these 
numbers have the cardinal number c. 

309. Let 21 be a complete apantactic set in a cube O. Then there 
exists an enumerable set of cubical celts 5qJ such that each point of 
21 lies on a face of one of these q, or is a limit point of their faces. 



328 POINT SETS 

For let D l > D% > be a sequence of superimposed divisions 
of Q, whose norms S n = 0. Let 



be the cells of D l containing no point of 21 within them. Let 

^21' ^22' ^23 '" (^ 

denote those cells of D 2 containing no point of 21 within them and 
not lying in a cell of 1). In this way we may get an infinite se- 
quence of cells 3D = \d mn \, where for each ra, the corresponding n 
is finite, and m = oo. Each point a of A lies in some d m ^ n . For 21 
being complete, Dist (a, 21) > 0. As the norms S n === 0, a must lie 
in some cell of D n , for a sufficiently large n. The truth of the 
theorem is now obvious. 

310. Let 33 be pantactic in 21. Then there exists an enumerable 
set S<. S3 which is pantactic in 21. 

For let Dj >D 2 > be a set of superimposed cubical divisions 
of norms c? n == 0. In any cell of D l containing within it a point 
of 21, there is at least one point of 93. If the point of 21 lies on 
the face of two or more cells, the foregoing statement will hold 
for at least one of the cells. Let us now take one of these points 
in each of these cells; this gives an enumerable set @j. The 
same holds for the cells of D 2 . Let us take a point in each of 
these cells, taking when possible points of (Sj. Let (5 2 denote the 
points of this set not in @j. Continuing in this way, let 



Then (5 is pantactic in 21, and is enumerable, since each @ n is. 

Corollary. In any set 21, finite or infinite, there exists an enumer- 
able set (5 which is pantactic in 21. 

For we have only to set 93 = 21 in the above theorem. 

311. 1. The points S where the continuous function f(x rr m ) 
takes on a given value g in the complete set 21, form a complete set. 

For let tfj, <? 2 be [joints of (5 \\hirh = c. We show c is a 
point of (5. For ,, f 

1 === 



PANTAXIS 329 

As /is continuous, *, \ ss \ 

/(*)=/ 00- 

Hence B /oo=* 

and c lies in &. 

2. Letf^x^ # m ) 6e continuous in the limited or unlimited set 21. 
J/* /te va/we of f is known in an enumerable pantactic set (5 in 21, 
which contains all the isolated points of 21, in case there be such, the 
value off is known at every point of 21. 

For let a be a limiting point of 21 not in (. Since (5 is pantactic 
in 21, there exists a sequence of points e l ^ e z in S which = a. 
Since / is continuous, /(e n )==/00 As/ is known at each e n , 
it is known at a. 

X. Let g= j/{ be the class of one-valued continuous functions 
defined over a limited point set 21. Then 

f = Card3 = C. 

For let 9?^ be a space of an infinite enumerable number of 
dimensions, and let , x 

y = (yii y^ ) 

denote one of its points. Let/ have the value rj l at e^ the value 
i/ 2 at 2 for the points of @ defined in 2. Then the complex 

0?r ^2 -0 

completely determines /. But this complex determines also a 
point 77 in 9?^ whose coordinates are tj n . We now associate/ with 

77. Hence c 

' 



^ = c. 

On the other hand, f>c, since in $ there is the function 
/(.rj r m ) = $r in 21, where # is any veal number. 

312. Let S3 denote the class of complete or perfect subsets lying in 
the infinite set 21, which latter contains at least one complete set. 



For let tfj, a 2 , == a, all these points lying in 21. Then 



But for t x we may take any number in 3' 1 = (1 9 2, 3, ) ; for 
we may take any number in Q 2 == (t x -f 1, tj + 2, ), etc. 



330 POINT SETS 

Obviously the cardinal number of the class of these sequences 

1) is e c = C. But (a a a a ..-^ 

^a, c* t p ^t,^ t*i 8 ) 

is a complete set in 21. Hence 6>.c. On the other hand, 6<c. 

Forlet A>*>2>- (2 

be a sequence of superimposed cubical division of norms = 0. 
Each D n embraces an enumerable set of cells. Thus the set of 
divisions gives an enumerable set of cells. Each cell shall have 
assigned to it, for a given set in 33, the sign + or according as 
S3 is exterior to this cell or not. This determines a distribution 
of two things over an enumerable set of compartments. 

The cardinal number of the class of these distributions is 2 e = c. 
But each 93 determines a distribution. Hence b< c. 



Transfinite Derivatives 
313. 1. We have seen, I, 266, that 

Thus y M 

Let now 21 be a limited point aggregate of the second species. 
It has then derivatives of every finite order. Therefore by 18, 

D<2T, 21", '", ) (2 

contains at least one point, and in analogy with 1), we call the 
set 2) the derivative of order o> <?/* 21, and denote it by 

2l (w) , 
or more shortly by 

2K 

Now we may reason on 2l w as on any point set. If it is infinite, 
it must have at least one limiting point, and may of course have 
more. In any case its derivative is denoted by 

((0,4.1) or att +i B 

The derivative of 21"" 1 " 1 is denoted by 

51(0,4-2) or ^4-2 ^ etCt 

Making use of co we can now state the theorem : 



TRANSFINITE DERIVATIVES 331 

In order that the point set 21 is of the first species it is necessary 
and sufficient that 21 r " ; = 0. 

2. We have seen in 18 that 21" is complete. The reasoning 1 of 
I, 266 shows that 2l u)+1 , 2l w+2 , , when they exist, are also complete. 
Then 18 shows that, if 2l w+n n = 1, 2, ... exist, 

Dv($? >a w+1 >8l w+2 > ) (3 

exists and is complete. The set 3) is called the derivative of order 

ft> 2 and is denoted by 

2((a>2) or 3 j w 2 > 

Obviously we may continue in this way indefinitely until we 
reach a derivative of order a containing only a finite number of 
points. Then ^ +1 = Q 

That this process of derivation may never stop is illustrated by 
taking for 21 any limited perfect set, for then 



3. We may generalize as follows : Let a denote a limitary ordi- 
nal number. If each 2F > 0, /3 < a, we set 



when it exists. 

4. If 2l a > 0, while 21 +1 = 0, we say 21 is of order a. 

314. 1. Let a be a limiting point of 21. Let 
5 =Card F^a). 

Obviously is monotone decreasing with 8. Suppose that 
there exists an a and a & > 0, such that for all < 8 < & 

= Card V(a). 

We shall say that a is a limiting point of rank a. 
If every 6 > a, we shall say that 

Rank a > . 
If every a$ > , we shall say that 

Rank a > a. 



332 POINT SETS 

2. Let 2( be a limited aggregate of cardinal number a. Then there 
is at least one limiting point of 21, of rank . 

The demonstration is entirely similar to I, 264. Let 8 X > 
S 2 > ... ~ 0. Let us effect a cubical division of 21 of norm 8j. In 
at least one cell lies an aggregate 2lj having the cardinal num- 
ber a. Let us effect a cubical division of 21 a of norm S 2 . In at 
least one cell lies an aggregate 21 2 having the cardinal number , 
etc. These cells converge to a point a, such that 

Card r a (a) = a, 
however small 8 is taken. 

3. 7/Card 21 > e, there exists a limiting point 0/21 of rank > e. 
The demonstration is similar to that of '2. 

4. If there is no limiting point o/2l of rank > e, 21 is enumerable. 
This follows from 3. 

5. Let Card 21 be > e. Let $ denote the limiting points of 21 
whose ranks are > e. Then 33 is perfect. 

For obviously 33 is complete. \Ve need therefore only to show 
that it is dense. To this end let b be a point of 3J. About b h j t 
us describe a sequence of concentric spheres of radii r n = 0. These 
spheres determine a sequence of spherical shells \8 n \, no two of 
which have a point in common. If 2l n denote the points of 21 in 8 n , 
we have y = ^ (ft) = ^ + ^ + ^ + ... 

Thus if eacli 2l m were enumerable, V is enumerable and hence 
Rank b is not > e. Thus there is one set 2l m which is not enu- 
merable, and hence by 3 there exists a point of 33 in 8 m . Hut then 
there are points of 33 in any T ;r r *(ft), and b is not isolated. 

6. A set 21 which contains no dense component ix enumerable. 
For suppose 21 were not enumerable. Let $ denote the proper 

limiting points of 91. Then ^jj contains a point whose rank is > e. 
But the set of these points is dense. This contradicts the hy- 
pothesis of the theorem. 

315. Let a lie in Z n . If 2l a > 0, it is complete. 
For if a is non-limitary, reasoning similar to I, 266 shows that 
2l a is complete. Suppose then that a is limitary, and 2l a is not 



TRANSFINITE DERIVATIVES 333 

complete. The derivatives of 21 of order < a which are not com- 
plete, form a well ordered set and have therefore a first element 
21^, where ft is necessarily a limitary number. Then 

V = Dw(v) , 7 < ft. 

But every point of 21^ lies in each 21?. Hence every limiting 
point of 21^ is a limiting point of each 2l v and hence lies in 21^. 
Hence 2l 3 is complete, which is a contradiction. 

316. Let a be a limitary number in Z n . If 21^ > for each 
yS < , 2l a exists. 

For there exists an K m , m < >t 2, sequence 

7< 8 <e< 7; < ... = a. (1 

Let c be a point of 2l>, d a point of 2l 5 ,-e a point of 21% etc. 

Then the set , -, - 

(6% d, e, f, ) 

has at least one limiting point I of rank S m . Let be any number 
in 1). Then I is a limiting point of rank N m of the set 

o,/, ) 

Thus I is a limiting point of every 21^, /3 < , and hence of 2l a . 

317. Let us show how we may form point sets whose order a 
is any number in Z l or Z 2 . 

We take the unit interval 21 = (0, 1) as the base of our con- 
siderations. 

In 21, take the points 



Obviously / = 1, $; ; = 0. Hence t is of order 1. The set 
SBj divides 21 into a set of intervals 

!,*,, 31 3 - (2 

In 2l t = (0, J) take a set of points similar to 1) which has as 
single limiting point, the point . In 21 2 = Q, |) take a set of 
points similar to 1) which has as single limiting point, the point 
|, etc. Let us cull the resulting set of points 53 2 . 



334 POINT SETS 

Obviously $'._. i 3 i ... -.to 
Hence ^ = %, = l and g,,, = 

A I a 

Thus 93 2 is of order 2. 

In eacli of the intervals 2) we may place a set of points similar 
to 33 2 , such that the right-hand end point of each interval 2l n is a 
limiting point of the set. The resulting set $) 8 is of order 3, etc. 

This shows that we may form sets of every finite order. 

Let us now place a set of order 1 in 2^, a set of order 2 in 21 2 , 
etc. The resulting set 3L is of order co. For 33^ n) has no points 
in 2l p 21 2 2l n -;p while the point 1 lies in every 93<, n) . 

Thus gw w) ___ -j 

Hence oi(a,+i) n 

^u> = U > 

and 33o, is of order co. 

Let us now place in each 2l n a set similar to 93o,, having the 
right-hand end point of 2l n as limiting point. The resulting set 
33(u+i * 8 ^ or( l er w -f 1. In this way we may proceed to form sets 
of order co -{- 2, co -f- 3, just as we did for orders 2, 3, We 
may also form now a set of order o>2, as we before formed a set 
of order co. 

Thus we may form sets of order 

co , co 2 , co 3 , o> 4 

and hence of order o> 2 , etc. 

318. 1. Let 21 be limited or not, and let 2l t (/3) denote the isolated 
points ofW. Then 



a o ^ y8=l, 2, ...<fl. (1 



For r = a; + a" , a" = a/ + a /f/ - 

Thus a/ = ?I , + a// + ... + a(n -D + 5j ( n) . 

that is, 21' is the sum of the points of 2T not in 21", of the points 
of 21" not in 21'", etc. If now there are points common to every 
W we have r . 2a w + a ., , w==1 , 2 , .. 



TRANSFINITE DERIVATIVES 335 

On 8l w we can reason as on 81', and in general for any a < 1 we 
have ^, 

/3<a 

which gives 1). 



2. -#*3l = 0, 21 <md 8' are enumerable. 
For not every 



Hence there is a first a, call it 7, such that 8l v = 0. Then 1) 
reduces to ^ ^ ^ ^ ... <<y> 

ft 

But the summation extends over an enumerable set of terms, 
each of which is enumerable by 289. Hence 81' is enumerable. 
But then 81 is also enumerable by 237, 2. 

3. Conversely, if 81' is enumerable, 81 = 0. 

For if 81 > 0, there is a non-enumerable set of terms in 1), if 
no 3l (/3) is perfect ; and as each term contains at least one point, 
81' is not enumerable. If some 8l (/3) is perfect, 81' contains a per- 
fect partial set and is therefore not enumerable by 245. 

4. From 2, 3, we have : 

For 21' to be enumerable, it is necessary and sufficient that there 
exists a number a in Z 1 or Z% such that 2l a = 0. 

5. If 81 is complete, it is necessary and sufficient in order that 81 
be enumerable, that there exists an a in Z 1 or Z 2 such that 3l a = 0. 

For 8l=3l t + 3l', 

and the first term is enumerable. 

6. If 810 = for some /3 < fl, we say 31 is reducible, otherwise it 
is irreducible. 

319. If 81 > 0, it is perfect. 

By 315 it is complete. We therefore have only to show that 
its isolated points SlJ 1 = 0. Suppose the contrary ; let a be an 
isolated point of 31". 

Let us describe a sphere 8 of radius r about a, containing no 
other point of 31. Let 39 denote the points of 81' in S. Let 

r >r l >r z > = 0. 



POINT SETS 



Let S n denote a sphere about a of radius r n . Let 93 n denote the 
points of 93 lying between S n ^. 1 ^ S n , including those points which 
may lie on S n ^ l . Then 

33= 1 + 93 a +9} 8 + + a. 

Each 93 m is enumerable. For any point of 93" is a point of 
93 n = a. Hence 93" = and 93 m is enumerable by 818, 2. 

Thus 93 is enumerable. This, however, is impossible since 
33" = a, and is thus > 0. 

320. 1. In the relation 

' = 221^ + 2P 0= 1, 2, ... < H, 


rw on the right is enumerable. 

For let US set 



alsolet r^n... =0. 

Let 93 n denote the points of 93 whose distance S from 31 satis- 

fies the relation . ^ ^ 

YU > o > ?*n+i 

Then the distance of any point of 93!, from 21" is > r n+1 . If $ 
includes all points of 93 whose distance from 31 is > r x , we have 

93 = 930 + 93! 4- 93 2 -I- - 

Each 93 n is enumerable. For if not, 93jf > 0. Any point of 
$3% as 6 lies in 21. Hence 

Dist (J, ?I n ) = 0. 

On the other hand, as b lies in 93J,., its distance from 2( is 
> r n+1 , which is a contradiction. 

2. If W is not enumerable, there exists a first number a in Z l or 
Z 2 such that ?T is perfect. 
This is a corollary of 1. 

8. If 21 is complete and not enumerable, there exists a first number 
a in Z^ -f Z% such that 2l a is perfect. 

4. If 21 is complete, Qr ^ , en 

* 21 = Vs -f- -P J 

where @ is enumerable, and ^J is perfect. If 21 i* enumerable, ^ = 0. 



COMPLETE SETS 337 

Complete Sets 

321. Let us study now some of the properties of complete point 
sets. We begin by considering limited perfect rectilinear sets. 
Let 21 be such a set. It has a first point a and a last point b. It 
therefore lies in the interval /=(#, b). If 21 is pantactic in any 
partial interval J~ (a, /3) of 7, 21 embraces all the points of J, 
since 21 is perfect. Let us therefore suppose that 21 is apantactic 
in /. An example of such sets is the Cantor set of I, 272. 

Let D = \ 8 \ be a set of intervals no two of which have a point 
in common. We say D is pantactic in an interval /, when 1 con- 
tains no interval which does not contain some interval 8, or at 
least a part of some 8. 

It is separated when no two of its intervals have a point in 
common. 

322. 1 . Every limited rectilinear apantactic perfect set 21 deter- 
mines an enumerable pantactic set of separated intervals J) = jSj, 
whose end points alone lie in 21. 

For let 21 lie in /=(, y8), where a, /3 are the first and last 
points of 21. Let 33 = / 21. Each point b of 33 falls in some in- 
terval 8 whose end points lie in 21. For otherwise we could 
approach b as near as \ve chose, ranging over a set of points of 21. 
But then b is a point of 21, as this is perfect. Let us therefore 
take these intervals as large as possible and call them 8. 

The intervals 8 are pantactic in /, for otherwise 21 could not be 
apantactic. They are enumerable, for but a finite set can have 
lengths > I/n 4- 1 and < JT/w, n = 1, 2 

It is separated, since 2( contains no isolated points. 

2. The set of intervals .Z) = *Sj just considered are said to be 
adjoint to 21, or determined by 21, or belonging to 21. 

323. Let 21 be an apantactic limited rectilinear perfect point set, to 
which belongs the set of intervals D = |8$. TJien 21 is formed of the 
end points E\t\ of these intervals, and their limiting points JS f . 

For we have just seen that the end points e belong to 21. More- 
over, 21 being perfect, JS f must be a part of 21. 



338 POINT SETS 

21 contains no other points. For let a be a point of 31 not in E, 
E' . Let a be another point of 21. In the interval (a, a) lies an 
end point e of some interval of D. In the interval (a, e) lies an- 
other end point e r In the interval (a, e^) lies another end point 
f? 2 , etc. The set of points 0, e^ #% == a. Hence a lies in E 1 ', 
which is a contradiction. 

324. Conversely, the end points E= \e\ and the limiting points of 
the end points of a pantactic enumerable set of separated intervals 
D == jgj form a perfect apantactic set 21. 

For in the first place, 21 is complete, since 21 = (J?, IS'). 21 can 
contain no isolated points, since the intervals S are separated. 
Hence 21 is perfect. It is apantactic, since otherwise 21 would em- 
brace all the points of some interval, which is impossible, as D is 
pantactic. 

325. Since the adjoint set of intervals D = \B\ is enumerable, it 
can be arranged in a 1, 2, 3, order according to size as follows. 

Let S be the largest interval, or if several are equally large, one 
of them. The interval 8 causes /to fall into two other intervals. 
The interval to the left of 8, call I Q , that to the right of 8, call I r 
The largest interval in J , call S , that in / r call S r In this way 
we may continue without end, getting a sequence of intervals 

8, 8 , 8 X , 00 , S 01 , S 10 , S n --- (1 

and a similar series of intervals 

A AP A' AXP -4n *** 

The lengths of the intervals in 1) form a monotone decreasing 
sequence which == 0. 

If v denote a complex of indices i/f/c 

D=!,,} = {V..J, 
and J, = J^+S F + /, 1 . 

326. 1. The cardinal number of every perfect limited rectilinear 
point set 21 is c. 

For if 21 is not apantactic, it embraces all the points of some in- 
terval, and hence Card 21 = c. Let it be therefore apantactic. 



COMPLETE SETS 339 

Let J9= {} be its adjoint set of intervals, arranged as in 325. 
Let be the Cantor set of I, 272. Let its adjoint set of intervals 
be H= \i] v \, arranged also as in 325. If we set S v ^ ?;, we have 
D^ff. Hence Card 21 = Card 6. 

But Card 6 = c by 244, 4. 

2. The cardinal number of every limited rectilinear complete set 21 
is either e or c. 

For we have seen, 320, 4, that 



where ( is enumerable and $ is perfect, 

If $ = 0, Card 21 = e. 

If ^>0, Card2l = c. 

For Card 21 = Card <g + Card $ = e + c = c. 

327. The cardinal number of every limited complete set 21 in 9t n is 
either e or c. It is c, ^J2l has a perfect component. 

The proof may be made by induction. 

For simplicity take m = 2. By a transformation of space [242], 
we may bring 21 into a unit square S. Let us therefore suppose 
21 were in S originally. Then Card 21 < c by 247, 2. 

Let be the projection of 21 on one of the sides of $, and 53 the 
points of 21 lying on a parallel to the other side passing through a 
point of . If -83 has a perfect component, Card 3$ = c, and hence 
Card 21 = c. If 53 does not have a perfect component, the cardinal 
number of each 53 is e. Now S is complete by I, 717, 4. Hence 
if S contains a perfect component, Card S = c, otherwise Card 
g = e. In the first case Card 21 = c, in the second it is e. 

328. 1. Let 21 be a complete set lying within the cube Q. Let 
J) l > J9 2 > denote a set of superimposed cubical divisions of Q 
of norms = 0. Let d l be the set of those cubes of D l containing 
no point of 21. Let rf 2 be the set of those cubes of D 2 not in d l , 
which contain no point of 21. In this way we may continue. Let 
53 = [d n ] . Then every point of A = Q - 21 lies in 53. For 21 being 



340 POINT SETS 

complete, any point a of A is an inner point of A. Hence /> p (a) 
lies in .A, for some p sufficiently small. Hence a lies in some d m . 
We have thus the result : 

Any limited complete set is uniquely determined by an enumerable 
set of cubes \d n \, each of which is exterior to it. 

We may call S3 = \d n \ the border of 31, and the cells d n , border 
cells. 

2. The totality of all limited perfect or complete sets has the car- 
dinal number c. 

For any limited complete set is completely determined by its 
border \d n \. The totality of such sets has a cardinal number 
< c c = c. Hence Card 5J < c. Since among the sets g is a c-set 
of segments, Card (5 > c. 

329. If 3l t denote the isolated points of 31, and 31 x its proper 
limiting points, we may write 

a = a t + v 

Similarly we have 

HA=3L+A., 

31 A = 3U + 31 AS , etc. 
We thus have 

31 = t + 3I At + 3U + - + 3I A -i t + 31 A . 

At the end of each step, certain points of 31 are sifted out. They 
may be considered as adhering loosely to 31, while the part which 
remains may be regarded as cohering more closely to the set. Thus 
we may call 3I A -i t , the n th adherent, and 3l A n the w th coherent. 

If the n th coherent is 0, 31 is enumerable. 

If the above process does not stop after a finite number of steps, 
let 3L 



If 3l w > 0, we call it the coherent of order CD. 
Then obviously w 



We may now sift 3L as we did 31. 



COMPLETE SETS 841 

If a is a limitary number, defined by 

we set 2l a = Dv\$\ x an ( 

and call it, when it exists, the coherent of order a. Thus we can 

write w vr _i_w 10^/0 /1 

21 =s z 2l A a t -f ?i A /3 a=l, J, <# (1 

a 

where j8 is a number in Z 2 . 



330. 1. TT/^71 21 is enumerable, 

31 = 2 A ., + 31x3 a = 1, 2, .- 

a 

= + 1) ; (1 

where $ is the sum of an enumerable set of isolated sets, and J), when 
it exists, is dense. 

For the adherences of different orders have no point in common 
with those of any other order. They are thus distinct. Thus the 
sum -3 1 can contain but an enumerable set of adherents, for other- 
wise 21 could not be enumerable. Thus there is a first ordinal 
number /3 for which 

2lA = 0. 
As now in general 

21^= ?U 
we have ^ = ^ +1 = = _ 

As SI A 3 thus contains no isolated points, it is dense, when not 0, 
by I, 270. 

2. When 21 is not enumerable, > 0. For if not, 21 = Q, and $ 
is enumerable. 

331. g = I'. (1 

For let J9 be a cubical division of space. As usual let 



denote those cells of D containing a point of 21, 21' respectively. 
The cells of 2l/> not in 2l/> will be adjacent to those of 2l#, and 



342 POINT SETS 

these may be consolidated with the cells of jD, forming a new di- 
vision A of norm 8 which in general will not be cubical. Then 

9? 9?' 4- 9? * 

^*A **A ' ^*A * 

The last term is formed of cells that contain only a finite number 
of points of 21. These cells may be subdivided, forming a new 
division E such that in 

a* = i + a** (2 

the last term is < e/3 Now if 8 is sufficiently small, 



Hence from 2), 3) we have 1). 

332. IfK >0, Card 21 = c. 

For let 33 denote the sifted set of 31 [I, 712], Then $ is per- 
fect. Hence Card 93 = c, hence Card 21 = c. 

333. Let 21 = Jttj, where each a s's metric and not discrete. If no 
two of the cCs have more than their frontiers in common, 21 is an 
enumerable set in the a .s*. 21 may he unlimited. 

Let us first suppose that 21 lies in a cube Q. Let a denote a on 
removing its proper frontier points. Then no two of the a's have 
a point in common. Let 



where the first term q l = Q. There can be but a finite number of 
sets , such that their contents lie between two successive ^'s. 

For if -, 

S . i,->& 
we have - , ~ , . ^ 

4 + a i+ " +t n >wg',. 

But the sum on the left is < Q, for any n. 

/> 

As n may == oo, this makes Q = oo, which is absurd. 

If 21 is not limited, we may effect a cubical division of 3J m . 
This in general will split some of the a's into smaller sets b. In 
each cube of this division there is but an enumerable set of the b's 
by what has just been proved. 



CHAPTER XI 
MEASURE 

Upper Measure 

334. 1. Let 21 be a limited point set. An enumerable set of 
metric sets D= \d L \^ such that each point of 31 lies in some c? t , is 
called an enclosure of 21. If each point of 21 lies within some c? t , D 
is called an outer enclosure. The sets d? t are called cells. To each 
enclosure corresponds the finite or infinite series 



which may or may not converge. In any case the minimum of all 
the numbers 1) is finite and <. 0. For let A be a cubical division 
of space, 21 A is obviously an enclosure and the corresponding sum 
1) is also 2lA> since we have agreed to read this last symbol either 
as a point set or as its content. 

We call M . v . 

Mm 2a t> 

with respect to the class of all possible enclosures D, the upper 
measure of 21, and write 



2. The minimum of the sums 1) is the same when we restrict our- 
selves to the class of all outer enclosures. 

For let J9= \d t \ be any enclosure. For each d L , there exists a 
cubical division of space such that those of its cells, call them d lK , 

containing points of d t have a content differing from d t by < . 

A 1 

Obviously the cells \d iK \ form an outer enclosure of 21, and 



343 



344 MEASURE 

As e is small at pleasure, Min 2c? t over the class of outer en- 
closures = Min 2d t over the class of all enclosures. 

3. Two metric sets whose common points lie on their frontiers 
are called non-overlapping. The enclosure D = 2rf t is called non- 
overlapping, when any two of its cells are non-overlapping. 

Any enclosure D may be replaced by a non-overlapping enclosure. 

For let U(d l , d^) = d l + e< 2 , 



2 d s d) = d 1 + e 2 + e 3 + e 4 , etc. 

Obviously each e n is metric. For uniformity let us set d l = e r 
Then E '= {e n j is a non-overlapping enclosure of 21. As 

2? n <2rf B 

we see that the minimum of the sums 1) is the same, ivhen we restrict 
ourselves to the class of non- overlapping enclosures. 

Obviously we may adjoin to any cell e n , any or all of its 
improper limiting points. 

4. In the enclosure H== \e n \ found in 3, no two of its cells 
have a point in common. Such enclosures may be called distinct. 

335. 1. Let D = jc?J, J?= \e K \ be two non-overlapping enclosures 
of 81. Let 

^ K =T)v(d^e K ). 
Then 

A=?S tM >, ,* = !, 2, . 

is a non- overlapping enclosure of 31. 

For 8 IIC is metric by 22, 2. Two of the S's are obviously non- 
overlapping. Each point of 31 lies in some d t and in some e K , 
hence a lies in S^. 

2. We say A is the divisor of the enclosures D, H. 



336. 

For let J?= [ej be an enclosure of $}. Those of its cells cZ t con- 
taining a point of 31 form an enclosure D= \d t \ of 31. Now the 
class of all enclosures A = ^Sj of 31 contains the class D as a sub- 
class. 



As 

we have 



UPPER MEASURE 345 



Min 2S t < Min 2d t < Min 2? t , 

A D E 



from which 1) follows at once. 
337. 7/31 is metric, 

3i =t. 

For let D be a cubical division of space such that 



(2 

Let us set 33 = 2l/). I>et J?=|6 t j be an outer enclosure of S3. 
Since 83 is complete, there exists a finite set of cells in E which 
contain all the points of 93 by 301. The volume of this set is 

obviously > 33; hence a fortiori 

2>8. 
Hence = ^ 

33>93. 
But = = 

31 > 33, by 336, 

> = /> 

>3l-e, by 2). (3 

On the other hand, __ _ _ 

6, by 2). (4 



From 3), 4) we have 1), since e is arbitrarily small. 

338. If 31 is complete, = _ 
For by definition 

with respect to all outer enclosures D = \d t \. But 21 being com- 
plete, we can replace D by a finite set of cells F= \f,\ lying in D, 
such that F is an enclosure of 31. Finally the enclosure F can be 
replaced by a non-overlapping enclosure Gr = \g,\ by 334, 3. 

Thus 

31 = Min 2# t , 

with respect to the class of enclosures Gr. But this minimum 
value is also 31 by 2, 8. 



346 MEASURE 

339. Let the limited set H = {21J be the union of a finite or infinite 
enumerable set of sets 2l n . Then 



For to each 2f n corresponds an enclosure D n = \d ni ] such that 
Sc?m < 2l n + ~ ^ > 0, arbitrarily small. 

But the cells of all the enclosures Z> n , also form an enclosure. 
Hence 



This gives 1), as e is small at pleasure. 

340. Let 31 lie in the metric set 9ft. Let A = 9K 21, Je 
complementary set. Then 



For from SR = a + 4 

follows -= = = 

2K<a + -A, by 339. 

But = ^ 

SR^aW, by 337. 

341. If 21 = 93 4- S, anc? 33, & are exterior to each other , 

1 = I + f . (1 

For, if any enclosure D = \d L l of 21 embraces a cell containing 
a point of S3 and (, it may be split up into two metric cells rf[, 
rfj', each containing points of S3 only, or of S only. Then 



Thus we may suppose the cells of D embrace only cells 
D 1 = \d(\ containing no point of (, and cells D n = {d'!\ con- 
taining no point of S3. Then 

s5 t * s5; + 2rf['. (2 



UPPER MEASURE 347 

By properly choosing D, we may crowd the sum on the left 
down toward its minimum. Now the class of enclosures D f is 
included in the class of all enclosures of 93, and a similar remark 
holds for D". 

Thus from 2) follows that 



This with 339 gives 1). 

342. If 21 = + 3R, 3R being metric, 



For let D be a cubical division of norm d. Let tt denote points 
of 3ft in the cells containing points of Front 3ft. Let m denote 
the other points of 3ft- Then m and 33 are exterior to each other, 
and by 337 and 341, 



As a = + m + n, 

Meas (33 + m) < I by 336. 
Al8 l<i + m + n by 339. 

Thus 5 + in<|(<5 + ft + fi . (2 

Now if d is sufficiently small, 

3ft-e<m ; n<. 
Thus 2) gives, as m<3ft, 



which gives 1), as e>0 is arbitrarily small. 

343. 1. Let ?l lie in the metric set 93, and also in the metric set 

- Let 5=53-21 , tf=e-si. 

Then 8-5-i-ff. 

For let 

, 6) , iB = S) + 23i , < 



348 MEASURE 

Thus 



8-5*8 + !-(! + 5) = 8--J5 



2. If 2l<33, the complement of 21 with respect to 93 will 
frequently be denoted by the corresponding English letter. Thus 

/I = 6X81), Mod 



Lower Measure 

344. 1. We are now in position to define the notion of lower 
measure. Let 31 lie in a metric set 2)?. The complementary set 
A = 2)? 31 has ;m upper measure A. We say now that 2ft A 
is the lower measure of 31, and write 



By 343 this definition is independent of the set 9M chosen. 

When & M 

a = n 

we say 21 is measurable, and write 

a = Ia. 

A set whose measure is is called a null set. 

2. Let .'= [e^ be an enclosure of A. 

Then H=Max(aH-2g t ). 

wrc'tfA respect to the class of all enclosures E. 

3. If (S = Je t j is an enclosure of 31, the enclosures E and ( may 
obviously, without loss of generality, be restricted to metric cells 
which contain no points not in 2W. If this is the case, and if @, 
JPare each non-overlapping, we shall say they are normal enclosures. 

If (, g are two normal enclosures of a set 21, obviously their 
divisor is also normal. 



LOWER MEASURE 349 

345. 1. 2[>0. 

For let SI lie in the metric set 9K. 

Then 2=-I. 

But by 330, 

llonce 



For let SI lie in the metric set 9)?. 

rheu fi + vi>9tt by 340. 

Hence ?l = SW- 



346. 1. For any limited set 91, 

[ < a < i < S- (i 

For let J9= \d t \ be an enclosure of 31. Then 

S = Min 2<f,, 

i) 

when 2) ranges over the class ^P of all finite* enclosures. On the 
other hand, 



D 

when D ranges over the class E of all enumerable enclosures. 
But the class E includes the class F. Hence S < 21. 

To show that < w (2 

we observe that as just shown 

A>A. 

Hence, _ ^ - 

^ - A < SIR - vt = 91. (3 

Z+ = 1, by 16. 
This with 3) gives 2). 



350 MEASURE 

2. y 21 is metric, it iv measurable, and 

8=5. 
This follows at once from 1). 

347. Let 21 be measurable and lie in the metric set 3D?. Then A 

is measurable, and ** ** ^ 

% + A**m. (1 

For ^ 

A = m-%. (2 

a=<w-3 = 5, 

since 21 is measurable. This last gives 

-! = Z-8. 

This with 2) shows that J. = J. ; hence ^4 is measurable. From 
2) now follows 1). 

348. If 21 < 93, then 8 < 8. (1 

For as usual let A, B be the complements of 21, S3 with respect 
to a metric set 2ft. Since 21 < 53, A > B. 

Hence, by 336, = = 

A. ^_ ./>. 

Thus, ^ = " ^ = 

<m-A<<m-B, 

which gives 1). 

349. For 21 to be measurable, it is necessary and sufficient that 



where 2ft is any metric set > 21, and A = 2W 21. 
It is sufficient, for then 1) shows that 

i=$m-Z 

But the right side is by definition 21 ; hence 21 = 21. 
It is necessary as 347 shows. 

350. Let 21 = \a n ] be the union of an enumerable set of non- 
overlapping metric sets. Then 21 is measurable, and 



LOWER MEASURE 351 

Let S denote the infinite series on the right of 1). As usual 
let S n denote the sum of the first n terms. Let 2l n = (aj, a n ). 
Then 2l n < 21 and by 336, 

in = S n < S , for any n. (2 

Thus S is convergent and 

#<I. (3 

On the other hand, by 339, 

1 < 8. (4 

From 3), 4) follows that 

S = 1 = lira S n = lira . (5 

We show now that 21 is measurable. To this end, let 3ft be a 
metric set > 21, and 21 B + A n = 3ft as usual. 

Then ^ ^ 

2l n + A n = m. (6 

But A < A n , hence A < A n . 

Thus 6) gives = ^ ^^ 

A + 2i n < a, 

for any n. Hence 

I + lim27 n <2Jh 
or using 5), l + f<^. 

Hence by 339, 1 +S ,^. 

Thus by 349, 31 is measurable. 



351. Let 
then 



-f 6 < . 



For let 90? be a metric set > 21. Let A, B, O be the comple- 
ments of 21, S3, S, with reference to 2R. 

Let -tf={ej , F=\f n \ 

be normal enclosures of B, C. Let 

d mn = Dv(* 
and D = {d mn | the divisor of ^?, F. 



352 MEASURE 

As all the points of A are in , and also in (7, they are in both 
E and F, and hence in the cells of D, which thus forms a normal 
enclosure of A. Let 

7m = 0* ml , 'C 2 " ) > Vn = (din, &<& "') 

Let us set . ft / . ? 

^m = 7m + 9m , / = *7n + ^n 

Then by 350, ^ 3 - v^ 

7m = ^mn , ^7n = ^mn - 

By 347, ^ .'. ? 1 T 

e m = 7m + ,^m ^ /n = ^7n + /l n ' 

Hence ^ ^ ^, ^ ^ 



, 

Hence adding, 

(SW-^) 

+ 2 n + Srf mn )] . (2 



Now 9W = f7j.(/ m , A., d mn \ m, n = 1, 2, ... 

Thus by 339, the term in [ ] is < 0. Thus 2) gives 

s /n) < ^- 2 ^n < a. 



But ^ 

S = Max ( 9W - 2^ m ) 

e = Max (W- 2^). 
Thus 3) gives 1) at once. 

Measurable Sets 

352. 1. e 31 = 33 + 6. // , S ar^ measurable, then 21 i* 
measurable, and ^ 

= + 6. (1 

F r + 6<a , by 351 

<S<S + i , by 339. 



LOWER MEASURE 353 

U. Let 81 = SB + S. If 8f, 53 are measurable, so it 6 <m<2 

5-a-. (2 

For let 81 lie in the metric set 3W. Then 

S - 8 = SB - (8 + <.) = (SW - C) - . 
Thus A = <?-8; 

Hence C'= + A 

Thus (7 is measurable by 1. Hence S is measurable by 347, 
and 

a = S + s. 

From this follows 2) at once. 

353. 1. Let 21 = 22l n 60 ^ ,<mm of an enumerable set of measur- 
able sets. Then 21 is measurable and 



If 21 is the sum of a finite number of sets, the theorem is obvi- 
ously true by 352, 1. In case 21 embraces an infinite number of 
sets, the reasoning of 350 may be employed. 

2. Let 31 = \9l n \ be the union of an enumerable set of null sets. 
Then 31 is a null set. 

Follows at once from 1. 

3. Let 21= |2l n J be the union of an enumerable set of measurable 
scfs whose common points two and two, form null sets. Then 21 is 
measurable and 

i = 22l n . 

4. Let @= Je n J be a non-overlapping enclosure 0/21. Then @ is 
measurable, and 

i = s? n . 

5. Let 33 < 21. Those cells of (g containing a point of S3 may 
be denoted by S3(g, and their measure will then be of course 

**' 

If S3 = 21, this will be @. This notation is analogous to that 
used in volume I when treating content. 



8f>4 MEASURE 

6. If g= \\ n \ is another non-overlapping enclosure of some set 
then 

S> 
f* measurable. 

For the cells of 35 are 

&< 
Thus S l<c is metric, and 

S = sS u . 

354. 1. Harnack Sets. Let 21 be an interval of length I. Let 



be a positive term series whose sum X > is <_ I. As in defining 
Cantor's set, I, 272, let us place a black interval of length ^ in the 
middle of 31. In a similar manner let us place in each of the re- 
maining or white intervals, a black interval, whose total lengths 
= Z 2 . Let us continue in this way; we get an enumerable set of 
black intervals 93, and obviously 



If we omit the end points from each of the black intervals we get 
a set S3*, and obviously 



The set = 91 - 93* 

we call a Harnack set. This is complete by 324 ; and by 338, 347, 

= = I - \. 

When X = Z, ^> is discrete, and the set reduces to a set similar 
to Cantor's set. When \ < I, we get an apantactic perfect set 
whose upper content is I \ > 0, and whose lower content is 0. 

2. Within each of the black intervals let us put a set of points 
having the end points for its first derivative. The totality of 
these points form an isolated set Q and Q r == . But by 331, 
$ = $' H now $ is not discrete, $ is not. We have thus the 
theorem : 

There exist isolated point set* which are not discrete. 



LOWER MEASURE ;55f> 

3. It is easy to extend Harnack sets to 5R n . For example, in 9J 2 , 
let S be the unit square. On two of its adjacent sides let us place 
congruent Harnack sets . We now draw lines through the end 
points of the black intervals parallel to the sides. There results 
an enumerable set of black squares @ = \8 n \. The sides of the 
squares @ and their limiting points form obviously an apantactic 
perfect set $. 

Let a\ + 02+ ... = m . 

be a series whose sum < m< 1. 

We can choose $& such that the square corresponding to its larg- 
est black interval has the area a\ ; the four squares corresponding 
to the next two largest black intervals have the total area a$, etc. 

Then 



= 2ai = m. 

Hence i = i-,=5. 

355. 1. If S = \t m \ is an enclosure of 21 such that 



it is called an ^-enclosure. Let A be the complement of 31 with 
respect to the metric set 9JJ. Let E = \e n \ be an e-enclosure of A. 
We call @, E complementary e-enclosures belonging to 21. 

2. If 21 is measurable, then each pair of complementary e/2 
normal enclosures @, E, whose divisor <D = Z)#(@, E), is such that 



35 < e, sma// at pleasure. (1 

For let @, J57 be any pair of complementary e/2 normal enclo- 
sures. Then 



Adding, we get Q 



< + j _ ( + ) < e; 



or 0<i + J-Z<e. (2 

But the points of 3ft fall into one of three classes : 1 the points 
of 3) ; 2 those of @ not in J) ; 3 those of ^ not in 2). Thus 

i + J = m + ix 

This in 2) gives 1). 



356 MEASURE 

356. 1. Up to the present we have used only metric enclosures 
of a set 21. If the cells enclosing 21 are measurable, we call the 
enclosure measurable. 

Let @ = \t n \ be a measurable enclosure. If the points common 
to any two of its cells form a null set, we say (S is non- 
overlapping. The terms distinct, normal, go over without 
change. 

2. We prove now that 



with respect to the class of non-overlapping measurable enclosures. 

For, as in 339, there exists a metric enclosure m n = \d nK \ of 
each e n such that 2d nK differs from e n by < e/2 n . But the set 

K 

jm n ( forms a metric enclosure of 21. Thus 



which establishes 1). 

357. Let (5 be a distinct measurable enclosure of 21. Let f denote 
those cells containing points of the complement A. If for each e > 
there exists an S such that f < e, then 21 is measurable. 

For let @ = e + f. Then e < 21. Hence e < 21 by 348. But 



Hence 
and thus 



358. 1. 2%e divisor 25 0/* too measurable sets 21, 93 i* #Z*0 meas- 
urable. 

For let (, E be a pair of complementary e/4 normal enclosures 
belonging to 21 ; let , F be similar enclosures of S3. Let 



e = Dt>((g, E) , f = 
Then 

e<e/2 , T<*A by 355, 2. 



LOWER MEASURE 357 

Now = Dv((, 3?) is a normal metric enclosure of 35. More- 
over its cells g which contain points of 35 and (7(35) lie among 
the cells of e, f. Hence 



Thus by 357, 35 is measurable. 
2. i0 91, 53 be measurable. 

Let 5D = Dt;(a, 93) , U = (31,93). 



For 

Hence 



359. Let 31 = Z7 { St m | Je tAe union of an enumerable set of 
measurable cells ; moreover let ?l be limited. Then 21 is measurable. 
If we set 



For S) = DyCSlj, S1 2 ) is measurable by 358. 

Lefc x = 3) + ^ , 2l 

Then a v a a are measurable by 352, 2. 

As U = (?I 1 ,2I 2 ) = 

U is measurable. As U and SSj are measurable, so is 2 In a 
similar manner we show that 93 3 , 93 4 are measurable. As 



21 is measurable by 353, 1, and the relation 1) holds by the same 
theorem. 

360. Let SIj < 2I 2 < be a set of measurable aggregates whose 
union 21 is limited. Then 21 is measurable, and 



358 MEASURE 

For let w w g r 

02 = 212-21! > 8 "s ~" 

For uniformity let us set a x = 21. Then 

2r = 2a m . 
As each o n is measurable 



* lira n . 

361. Let 2lj, 21 2 -" 6e measurable and their union 21 limited. If 
3) = Dv j2l n j > 0, i measurable. 

For let 21 lie in the metric set 93? ; 

let s) + D = gw,a n + ^i ll = aK 

as usual. 

Now 3) denoting the points common to all the 2l n , no point of 
D can lie in all of the 2l n , hence it lies in some one or more of the 
A n . Thus D<]A n }. (1 

On the other hand, a point of \A n \ lies in some A m , hence it 
does not lie in 2l m . Hence it does not lie in 33. Thus it lies in 
D. Hence \A n \<D. (2 

From 1), 2) we have fi= $A \ 

As each A n is measurable, so is D. Hence 35 is. 

362. If 21 1 >2I 2 > '* fl^ enumerable set of measurable aggre- 
gates, their divisor 3) is measurable, and 



For as usual let D, A n be the complements of 2), 2l n with respect 
to some metric set 2ft. 

Then 



Hence by 360, 



-lim.. 



LOWER MEASURE 359 

As 5> = 3tt-D, 

we have 



363. 1. The points # = (x l z m ) such that 



form a standard rectangular cell, whose edges have the lengths 
e l = b l a l , , e m =b m a m . 

When e l = e% = = e m , the cell is a standard cube. A normal 
enclosure of the limited set 91, whose cells (S = Je n f are standard 
cells, is called a standard enclosure. 

k l. For each e > 0, there are standard e-enclosures of any limited 
set%. 

For let @ = \t n \ be any ^/-enclosure of SI. Then 

2e n -i<7;. (2 

Each e n being metric, may be enclosed in the cells of a finite 
standard outer enclosure F n , such that 

F n -tt<T,/-2* , n=l, 2,-. 
Then ^ = S ^n5 i an enclosure of SI, and 



<l + 2i;, by 2). 

But the enclosure jP can be replaced by a non-overlapping 
standard enclosure = |fl}, as in 834, 3. But < 
Hence if 2 ?; is taken < e, 


and is an e-enclosure. 

3. Let @ = ie m {, g={W 

be two non-overlapping enclosures of the same or of different 
sets. Let e mn = Z>v(e m , f n ). 



360 MEASURE 

Let e m =(e m<1 , e m , 2 , e,,^. ..)+* m , (3 

then e m is measurable. By this process the metric or measurable 
cell c m falls into an enumerable set of non-overlapping measur- 
able cells, as indicated in 3). If we suppose this decomposition to 
take place for each cell of @, we shall say we have superimposed $ 
on @. 

364. (W. ff. Young.} Let S be any complete set in limited 21. 
Then 

H = Max 6. (1 

For let 31 lie within a cube 2K, and let A = 9W - 21, (7= 9W - 6 
be as usual the complementary sets. 

Let 93 = jbj be a border set of ( [328]. It is also a non- 
overlapping enclosure of 0; we may suppose it is a standard en- 
closure of O. Let E be a standard e-enclosure of A. Let us 
superimpose U on 93, getting a measurable enclosure A of both 
and A. Then 

tf= <7 A >A. 
Hence 

= <m - (7= 9 - (7 A < 9W - ^ A . 
Thus 

6 = g, by 338 

<Meas (2ft- A,) 

! A , by 352, 2 



Hence 

and thus _ - 

MaxS<3l- (2 

On the other hand, it is easy to show that 

MaxS>|. (3 

For let A D be an e-outer enclosure of A, formed of standard 
non-overlapping cells all of which, after having discarded certain 
parts, lie in 2ft. 



LOWER MEASURE 361 

Let $ = 9W-^ Z) + & (4 

where g denotes the frontier points of A D lying in 21. Obviously 
$ is complete. Since each face of D is a null set, g is a null set. 
Thus each set on the right of 4) is measurable, hence 

= m - A D + 

= m-I D 

= aft-I-e' , 0<e'<e 



Thus Max < > ffi ' = I > | - e, 

from which follows 3), since is small at pleasure. 

365. 1. If 21 i$ complete, it is measurable, and 

8 = a. 

For by 364, 

| = 21. 
On the other hand, 

|=2l, by 338. 
2. Let S3 ie any measurable set in the limited set 2(. Then 

= Max . (1 

For g>93 = & 

Hence, >Max. (2 

But the class of measurable components of 21 embraces the 
class of complete components (, since each K is measurable by 1. 

Thus Maxi>Max(f. (3 

From 2), 3) we have 1), on using 364. 

366. Van Vleck Sets. Let 6 denote the unit interval (0, 1), 
whose middle point call M. Let 3 denote the irrational points of 
g. Let the division D n , n = 1, 2, divide @ into equal intervals 
8 n of length l/2 n . 



362 MEASURE 



We throw the points 3 into two classes 21 = j#{, 33 = \b\ having 
the following properties : 

1 To each a corresponds a point b symmetrical with respect 
to M, and conversely. 

2 If a falls in the segment S of J9 n , each of the other seg- 
ments & of D n shall contain a point a' of 21 such that a' is situated 
in S f as a is situated in 8. 

3 Each 8 of D n shall contain a point a f of 21 such that it is 
situated in S, as any given point a of 21 is situated in (. 

4 21 shall contain a point a situated in @ as any given point 
a' of 21 is in any 6 n . 

The 1 condition states that 21 goes over into 33 on rotating ( 
about M. The 2 condition states that 21 falls into n = 1, 2, 2 2 , 
2 3 , congruent subsets. The 3 condition states that the subset 
2l n of 21 in & n goes over into 21 on stretching it in the ratio 2 n : 1. 
The condition 4 states that 21 goes over into 2l n on contracting it 
in the ratio 1 : 2 n . 

We show now that 21, and therefore 33 are not measurable. In 
the first place, we note that _ _ 

2US, 

by 1. As 3 = 21 + 33, if 21 or 33 were measurable, the other would 

be, and 

2I = 5B = i 

Thus if we show 21 or 33 = 1, neither 21 nor 33 is measurable. 
We show this by proving that if 21 = < 1, then $8 is a measurable 

/Sv / 

set, and 33 = 1. But when 33 is measurable, 33 = | as we saw, and 
we are led to a contradiction. 

Let = e l -f e 2 -f be a positive term series whose sum e is 
small at pleasure. Let Sj = \e n \ be a non-overlapping Cj-enclosure 
of 21, lying in @. Then 



Let SBj = 3 - (g x ; then 33 t < 33, and 



LOWER MEASURE 363 

Each interval e n contains one or more intervals ?; nl , 77^, of 

some D,, such that 

27/ wm = e n - <7 n , 0<<r n 

where v 

<T = Z(T n 

may be taken small at pleasure. 

Now each rj nm has a subset 2l nm of $1 entirely similar to 31. 
Hence there exists an enclosure @ n/n of 2I n/;M whose measure nm is 

such that 

a ntl . ?;. 

_n,n _. J^n_ ^ ()r ^ _ ^ ^ 

1 1 

But S 2 = {S n //J is a non-overlapping enclosure of 21, whose 

measure v ~ v/ ~ >. 

" 



if a is taken sufficiently small. 

Let 33 2 denote the irrational points in (S l @ 2 . It is a part of 
33, and 33 2 has no point in common with 33 t . We have 



In this way we may continue. Thus 93 contains the measurable 
component 5^ + 3^+... 

whose measure is 



As e is small at pleasure, SB = 1. 
367. (F. .ff. rbwn^.) Let 



,,,, ,- (1 

6e an infinite enumerable set of point sets whose union 21 is limited. 
Let 2l n > > , w = 1, 2 ?%*w ^rfj ^^s a 8^ of points each 
of which belongs to an infinity of the sets 1) and of lower measure > a. 



864 MEASURE 

For by 365, 2, there exists in the sets 1), measurable sets 

EX , 6, , 6 8 ... (2 

each of whose measures S n > . Let us consider the first n of 

these sets, viz. : 

&J , & 2 n - (o 

The points common to any two of the sets 3) form a measurable 
set J)tK by 858, 1. Hence the union S ln = { J) IK | is measurable, by 
359. The difference of one of the sets 3), as Sj and Dv^ v S ln ), 
is a measurable set c x which contains no point in common with the 
remaining sets of 3). Moreover 



In the same way we may reason with the other sets ( 2 , & 3 
of 3). Thus 31 contains n measurable sets c x , C 2 c n no two of 
which have a common point. 

Hence 

c = Cl + + c n 

is a measurable set and 



The first and last members give 

f ln >-*!. 

n 
Thus however small > may be, there exists a /i such that 

Si,, !-- (4 



Let us now group the sets 2) in sets of p. These sets give rise 
to a sequence of measurable sets 

&I M > ^2fj, , Sa^ (5 

such that the points of each set in 5) belong to at least two of the 
sets J.) and such that the measure of each is > the right side of 4). 
We may now reason on the sets 5) as we did on those in 2). 
We would thus be led to a sequence of measurable sets 

Ci, , 6^ , <* - (6 



ASSOCIATE SETS 365 

such that the points of each set in 6) lie in at least two of the sets 
5), and hence in at least 2 2 of the sets 1), and such that their 
measures are. 



In this way we may continue indefinitely. Let now 93 X be the 
union of all the points of 21, common to at least two of the sets 1). 
Let S3 2 he the union of the points of 31 common to at least 2 2 of 
the sets 1), etc. In this way we get the sequence 

i>a-^ ' 
each of which contains a measurable set whose measure is 



We have now only to apply 25 and 364. 

368. As corollaries of 367 we have: 

1. Let jQj, Q 2 be an infinite enumerable set of non-overlapping 
cubes whose union is limited. Let each Q n > a > 0. Then there 
exists a set of points b whose cardinal number is c, lying in an infin- 
ity of the Q n and such that b > a. 

2. (Arzeld.) Let y 1 ^ y% ==17. On each line y n there exists an 
enumerable set of intervals of length &, r Should the number of inter- 
vals v n on the lines y n be finite, let v n = GO. In any case S n > a > 0, 
w = l, 2, and the projections of these intervals lie in 31 = (a, b). 
Then there exists at least one point x = in 21, such that the ordinate 
through is cut by an infinity of these intervals. 

Associate Sets 

369. 1. Let l > 2 > 3 =0. (1 
Let (g n be a standard e n -enclosure of 2l n . If the cells of g n+1 lie in 
@ n , we write l > 2 >'~ ( 2 
and call 2) a standard sequence of enclosures belonging to 1). 

Obviously such sequences exist. The set 

3l P = -D^S@ n S 
is called an outer associated set of 21. Obviously 



360 MEASURE 

2. Each outer associated set 21, is measurable, and 

a- a. -lime.. a 

fUEOO 

For each ( w is measurable; hence 3l is measurable by 362, and 



= , ase n =0. 

370. 1. Let A be the complement of 21 with respect to some 
cube Q containing 21. Let A f be an outer associated set of A. 

Then a,-c-^ f 

is called an inner associated set of 21. Obviously 



2. The inner associated set 2l t is measurable, and 

21, = 21. 

For A e is measurable by 369, 2. Hence 2l t =Q A, is meas- 
urable. But 

-A.f> s= -A. 
by 369, 2. Hence 



Separated Sets 

371. Let 21, 93 be two limited point sets. If there exist 
measurable enclosures @, % of 21, 93 such that 3)= 7)??((S, 5) ^ s Jl 
null set, we say 21, 93 are separated. 

If we superimpose g on & we R e ^ an enclosure of ( = (21, 93) 
such that those cells containing points of both 21, 93 form a null 
set, since these cells are precisely 35. We shall call such an en- 
closure of ( a null enclosure. 

Let 2l = {2l n } ; we shall call this a separated division of 21 into 
the subsets 2l n , if each pair 2l m , 2l n is separated. We shall also 
say the 2l n are separated. 



SEPARATED SETS M7 

372. For 21, S3 to be separated, it is necessary and sufficient that 

= Dv(%,, S3.) 
t a null set. 

It is sufficient. For let 



Then <S = (a,b,S 

is a measurable enclosure of &, consisting of three measurable 
cells. Of these only 35 contains points of both 21, S3. But by 
hypothesis 35 is a null set. Hence 21, S3 are separated. 

It is necessary. For let 9JZ be a null distinct enclosure of , 
such that those of its cells 9i, containing points of 21, 93 form a 
null set. Let us superimpose 9Dt on the enclosure @ above, get- 
ting an enclosure 5 of 21. 

The cells of arising from a contain no point of S3 ; similarly 
the cells arising from b contain no point of 21. On the other 
hand, the cells arising from 35, split up into three classes 



The first contains no point of S3, the second no point of 21, the 
cells of the last contain both points of 21, S3. As 35 a ,6^ % 

$.. = 0. (1 

On the other hand, 

a. = a + a>>a; 

hence a + 3> a + *>*>. 

Thus n +>>!, (2 

byl). Also i=a + ^=fl by 369, 2. 

This with 2) gives ^ ^ ^ ^ 

a + T) a > / a -hi). 

Hence ^ = ^ (3 

But 3:)_>35a + 3V 

This with 3) gives %) b = 0. 

A 

In a similar manner we find that 35 a = 0. Hence 3) is a null 
set by 3). 



368 MEASURE 

373. 1. 7/21, SJ are separated, then J) = Dv($l, 93) is a null set. 
For SD e = J9i> (21,, 93,) is a null set by 372. But 2) < $),. 

2. Let 21, 93 be the Van Vleck sets in 366. We saw there that 
| = g = 1. Then by 369, 2, 21, = % e = 1. The divisor of 2I, 93 e is 
not a null set. Hence by 372, 21, 53 are not separated. Thus the 
condition that J) be a null set is necessary, but not sufficient. 

374. 1. Let J2l n {, {93J be separated divisions of 21. Let 
S IK = Di>(2l t , 53* ) ^Aera j t *j is a separated division o/2l afo0. 

We have to show there exists a null enclosure of any two of the 
sets @ IK , S mn . Now ( l(C lies in 2l t and 93*; also'6 mn lies in 2l m , 93 n . 
By hypothesis there exists a null enclosure S of 2l t , 2l m ; and a null 
enclosure $ of 93*, 93 n . Then = -Z>v(@, g) is a nu ^ enclosure of 
2l t , 2l m and of 93*, 93 n . Thus those cells of , call them , con- 
taining points of both 2l t , 2l m form a null set; and those of its cells 
6 , containing points of both 93*, 93 n also form a null set. 

Let #= \g\ denote the cells of that contain points of both 
Sue, S mn . Then a cell g contains points of 2l t 2l m 33* 93 n . Thus g 
lies in a or 6 . Thus in either case Q- is a null set. Hence {S t *} 
form a separated division of 21. 

2. Let D be a separated division of 21 into the cells d^ d 2 
Let E be another separated division of 21 into the cells e^ e 2 
We have seen that JP = \f tK \ where / t *= Dv(d t , e^) is also a sepa- 
rated division of 21. We shall say that F is obtained by superim- 
posing E on D or D on .#, and write F=D + JE= U+ D. 

3. Let J? be a separated division of the separated component 93 
of 21, while D is a separated division of 21. If d t is a cell of D, e K 
a cell of JE, and c?^ = Dv(d L , e K ), then 

4 = OC <** -)+*.- 

Thus superposing E on D causes each cell d t to fall into sepa- 
rated cells <2 tl , d ta S t . The union of all these cells, arising from 
different d^ gives a separated division of 21 which we also denote 
by D + E. 

375. Let }2l n | be a separated division of 21. Let 93 < 21, and let 
93 n denote the points of 93 in 2l n . Then *93 n | is a separated division 
of 9. 



SEPARATED SETS 369 

For let 2) be a null enclosure of 2l m , 2l n . Let 3X& denote the 
cells of 35 containing points of both 2l m , 2l n . Let S denote the 
cells of ) containing points of 93; let @ 0<6 denote the cells con- 
taining points of both 93 m , 93 n . Then 



As 33^ is a null set, so is (& . 

376. 1. Let 21 = (93, 6) fo a separated division of 21. 

S = i + 1. (1 

For let j > 2 > = 0. There exist e n -measurable enclosures 
of 21, 93, 6 ; call them respectively A n , B n , O n . Then g n = A n + 
J? n H- (7 n is an n -enclosure of 21, 93, simultaneously. 

Since 93, are separated, there exist enclosures jB, (7 of 93, 
such that those cells of D = B -f- containing points of both 93 
and ( form a null set. Let us now superpose D on @ n getting 
an e n -enclosure JE n ~ le ns \ of 21, 93, S simultaneously. Let e bn 
denote the cells of E n containing points of 93 alone ; e cn those 
cells containing only points of ; and e^. those cells containing 
points of both 93, . Then 

2e ns = 2? 6n + 2*T cn + S^c . (2 

5 

As 2# 6c = 0, we see that as n == oo, 

s; n ,=l , s; 6n =i , 2r cn =i. 

Hence passing to the limit n= oo, in 2) we get 1). 

2. .Le 21 = 593 n | be a separated division of limited 21. Then 

l = 2S n . (1 

For in the first place, the series 

.B = 2i n (2 

is convergent. In fact let 2l n == (S^ 93 2 93 n ). 
Then 2l n < 21, and hence I n < I. 



870 MEASl'KK 

On the other hand, by 1 

I n = S 1 + ... + i n = jE? n , 

the sum of the first n terms of the series 2). Thus 

A,<i, 

and hence B is convergent by 80, 4, Thus 

B < n. 

On the other hand, by 339, 

^>I- 
The last two relations give 1). 



CHAPTER XII 
LEBESGUE INTEGRALS 

General Theory 

377. In the foregoing chapters we have developed a theory of 
integration which rests 011 the notion of content. In this chapter 
we propose to develop a theory of integration due to Lebesgue, 
which rests on the notion of measure. The presentation here 
given differs considerably from that of Lebesgue. As the reader 
will see, the theory of Lebesgue integrals as here presented differs 
from that of the theory of ordinary integrals only in employing 
an infinite number of cells instead of a finite number. 

378. In the following we shall suppose the field of integration 
31 to be limited, as also the integrand 31 lies in 9J m and for brevity 
we set f(x) f(x l x m ). Let us effect a separated division of 
31 into cells Sj, S 2 . If each cell S t lies in a cube of side d> we 
shall say D is a separated division of norm d. 

As * Before, let 

, *>, = Osc/= M t m, in S t . 



the summation extending over all the cells of 31, are called the 
upper and lower sums off over 31 with respect to D. 

The sum v 5 

Q, D f = 2fl) t 8 t 

is called the oscillatory sum with respect to D. 
379. If m = Min /, M = Max / in 31, then 



For 

m<m L < 

371 



372 LEBESGUE INTEGRALS 

Hence 2wil t < 2w A < 2 J 

Thus ai28 t <S D <S D < 

But 28 t = I, 

by 376, 2. 

380. 1. Since /is limited in SI, 

Max #/> , Min /S^ 

with respect to the class of all separated divisions D of 31, are 
finite. We call them respectively the lower and upper Lebesgue 
integrals of /over the field 21, and write 



; Cf=UinS D . 
JL<& 

In order to distinguish these new integrals from the old ones, 
we have slightly modified the old symbol I to resemble somewhat 
script L, or I , in honor of the author of these integrals. 



we say /is L-integrable over 21, and denote the common value by 



which we call the L-integral. 

The integrals treated of in Vol. I we will call R-integrals, i.e. 
integrals in the sense of Riemann. 

2. Letf be limited over the null set 21. Then/ is L-integrable in 
21, and 



This is obvious from 379. 

381. Let 21 be metric or complete. Then 



GENERAL THEORY 373 

For let d v c? 2 be an unmixed metric or complete division of 
of norm d. Let each cell d k be split up into the separated cells 
tl , d a - 
Then since c? 4 is complete or metric, 

^ = (^ = 2^. 
Hence using the customary notation, 

^l.< 
Thus summing over , 

m& < 
Summing over i gives 

2mA 
Thus by definition, 



Letting now d = 0, we get 1). 

2. Let 21 be metric or complete. If f is R-integrable in 21, it is 
L-inteqrable and 

(2 



3. In case that 21 is not metric or complete, the relations 1), 2) 
may not hold. 

Example 1. Let 21 denote the rational points in the interval 
(0, 1). 

Let 

/ = 1, for x = , n even 
n 

= 2, when n is odd. 
Then 



while 



since 21 is a null set. Thus 1) does not hold. 



374 LEBESGtJE INTEGRALS 

Example 2. Let/ = 1 at the rational points 2f in (0, 1). Then 

j[/=: , jV-o , *./<./ (=> 

Let#=- 1 in 21. Then 

" 1 ' and 



Thus in 3) the Z-integral is less than the 72-integral, while in 
4) it is greater. 

Example, 3. Let /= 1 at the irrational points 21 in (0, 1). 
Then 



although 21 is neither metric nor complete. 

382. Let Z), A be separated divisions of 91. Let 

# 

Then 



For any cell d^ of D splits up into d^ d lt on superimposing 

A, and = = 

^. = s<.. 

But = ' 

M,A. 
and = 



383. 1. Extremal Sequences. There exists a sequence of sepa- 
rated divisions n n n ^i 

X/ 1 , /> 2 , ^3 '" C A 

each D n +! being obtained from D n by superposition, such that 

^>,>^>- = Cf, (2 

-ta 

^<^<- = C f. (3 

~ 4r 



GENERAL THEORY 375 

For let j > e 2 > ^=0. For each n , there exists a division 
E n such that 



Let E Z + D, = I) Z , ^ 3 + D 2 = 

and for uniformity set 7^ = D r Then by 382, 

$i> n+l <8j> n , #/>,.< tf*,.. 
Hence 



Letting n == oo we get 2). 

Thus there exists a sequence \D' n } of the type 1) for 2), and a 
sequence {!) j of the same type for 3). Let now D n = D' n + D". 
Obviously 2), 3) liold simultaneously for the sequence \D n j. 

2. The sequence 1) is called an extremal sequence. 

3. Let \D n ] be an extremal sequence, and E any separated divi- 
sion of 21. Let E n = D n + E. Then E r E 2 is an extremal 
sequence also. 

384. Let f be L-integrable in 21. Then for any extremal sequence 
J-D.I, 



are the cells of /), awrf 4 aw// 



Hence ^ 

Passing to the limit we get 1). 

385. 1. Let m Min/, df =* Max /in 



This follows at once from 379 and 383, 1. 



376 LEBESGUE INTEGRALS 

2. Let F = Max \f\in 21, then 



This follows from 1. 

386. In order that fbe L-integrdble in 21, it is necessary that, for 
each extremal sequence \ D n \ , 



and it is sufficient if there exists a sequence of superimposed separated 
divisions \E n \, such that 



71=00 

It is necessary. For 

/ = lim S D , / = lim S D . 

eta " Jin 

As /is jk-integrable, 

= / - /= lim (S Dn - ^ n ) = lim Q Dn f. 

o^SC 55^21 

It is sufficient. For _ 



Both \S En \, {Sg n \ are limited monotone sequences. Their 
limits therefore exist. Hence 

= lim fl E = lim S E lim S E . 
Thus 



387. In order that f be L-integrable, it is necessary and sufficient 
that for each e > 0, there exists a separated division D of 21, for 



It is necessary. For by 386, there exists an extremal sequence 
\D n \, such that 

:< lj) n f< e , for any n > some m. 
Thus we may take D m for D. 



GENERAL THEORY 377 

It is sufficient. For let e 1 >e 2 > = 0. Let \D n \ be an 
extremal sequence for which 



Let A! = Dj , A 2 = A x + # 2 , A 3 = A 2 + D z Then { A n | is a 
set of superimposed separated divisions, and obviously 



Hence / is ir-integrable by 386. 

388. In order that f be L-intecjrable, it is necessary and sufficient 
that, for each pair of positive numbers a), cr there exists a separated 
division D of 31, such that if T/ X , ?; 2 , are those cells in which 
Osc/> ft>, then 

2^ t < a: (1 

It is necessary. For by 387 there exists a separated division 
D= \8,\ for which 

Ha/ = 2o> t S t < wo-. (2 

If ^j, ^ 3 denote the cells of D in which Osc/ <. co, 

fl^/ = 2ft) t ^ t -f 2o> J t > o)2^ t . (3 

This in 2) gives 1). 

It is sufficient. For taking e > small at pleasure, let us then 

take 

<r-^ , = -4=, (4 

2fi 231 

where II = Osc /in 31. 

From 1), 3), and 4) we have, since o> t <. ^ 

&0/< 2Hf t -|- 2o) t l t < crfl + 2o)l t < <rfi -f ft>I = . 
We now apply 387. 

389. 1. Iff is L-integrable in 31, it is in 95 < 31. 

For let \D n \ be an extremal sequence of /relative to 31. Then 
by 386, 

0. (t 



878 LEBESGUE INTEGRALS 

Hut the sequence \D n \ defines a sequence of superposed sepa- 
rated divisions of 93, which we denote by \E n \. Obviously 



Hence by 1), 

ft* n /=o, 

and / is .//-integrable in 33 by 386. 

2. Iff is L-integrable in 21, so is \f\. 

The proof is analogous to I, 507, using an extremal sequence 
for /. 

390. 1. Let j2l n j be a separated division of 31 into a finite or in- 
finite number of subsets. Letf be limited in 21. Then 

f /=/'/+ f/+- a 

4^i *LMi <^2i 2 

For let us 1 suppose that the subsets 2l x 2l r are finite in num- 
ber. Let \D n \ bo an extremal sequence of/ relative to 21, and 
}D mn \ an extremal sequence relative to 2l m . Let 



Then \E n \ is an extremal sequence of /relative to 21, and also 
relative to each 2l m . 

Now _ _ 

%*.=!?*!. *.+ +^ r ,*n- 

Letting TI = oo, we get 1), for this case. 
e /io^ r 6e infinite. We have 

I=|ln. (2 

Let S3 = (2l 1 -3I n ) , e n = Sl-S8 B . 

Then S3 B , S B form a se[)aruti'd division of ?t, and 

l=C+i. 

If v is taken large enough, 2) shows that 

in . 



GENERAL THEORY 379 

Thus by case 1, 



fe in <L& n 

= /'+ 4- f+', (3 

<4^i *L%n 



where by 385, 2 

| ' | < M^ n < e , n > v. 



Thus 1) follows from 3) in this case. 

2. Let |2l n J be a separated division of 21. Then 

!''-*' 

cLft cL^ln 

if f is L inte<jrable in 21, or if it i.s in each 2I n , and limited in 21. 

391. 1. Letf = (j in 21 except at the points of a null set 91. 
Then 



(1 

For let =+. Then 

V- (2 

e 

Similarly /"* ___ /' ^o 

But/ = (7 in S3. Thus 2), 3) give 1). 

392. 1. //<r>0; /V=* f/- 
X ^ 



The proof is similar to 3, 8, using extremal sequences. 
2. Iffis L-integrable in 21, ^o i* /*> 



where c is a constant. 



380 LEBESGUE INTEGRALS 



393. 1. Let ^(V^/iO) + +/ n <, each f m being limited 
in%. Then 

n /* / n /* 

z /*< f<* /.- a 

1 sta 0&9 * ^21 

For let f/> n } be an extremal sequence common to -F,/p /, In 
each cell 

^nl ^n 3 ' 

of D n we have 

2 Min/ w < Min F < Max F < 2 Max/ m . 

Multiplying by df na , summing over s and then letting w=oo, 
gives 1). 



2. Iff-^x), /nCz) are ea^A L-integrable in #, so Z 



and 



394. \. /I f* 7+ jn 

I (/+*)</ f+ ff< 
4^21 ^21 ^21 ci2l 

For using the notation of 393, 

Min (/+#) < Min/-f Max^r < Max(/-f ^) 
in each cell c? n , of D n . 

2. J/<7 f* L-integrable in H, 



Reasoning similar to 3, 4, using extremal sequences. 



GENERAL THEORY 381 

For x, / 7* / / 

/ (/-*> < / /+ / (-.</) < / /- J r; 

a&9L _ / jQ| aw91 rrf. '91 _/ .QI 

etc. 

4. Ijf/, ^r are L-integrable in 21, so isf g, and 

9* 



-ff)= ff~ f 

ci;H ciil 



395. J^/, # are L-integrable in 21, so isf-y. 

Also their quotient f/g is L-integrable provided it is limited in SL 

The proof of the first part of the theorem is analogous to I, 
505, using extremal sequences common to both f and g. The 
proof of the second half is obvious and is left to the reader. 

396. 1. Let f, g be limited in 21, andf<^g, except possibly in a 
null set 31. Then -~ 7* 

f< # a 

4/51 ^21 

Let us suppose first that/<. $r everywhere in 21. 
Let \D n \ be an extremal sequence common to both/ and g* 

Then s l)n f<s D ^. 

Letting n = oo , we get 1). 

We consider now the general case. Let 21 = S3 4- 9t. Then 



since 



But in $8,f<ff without exception. We may therefore use the 
result of case 1. 

2. Letf> in 21. Then 



For 



r/= ff , r</= 

4?%. 4^B ^a ' 
Jw Jaw 



382 LEBESGUK INTEGRALS 

397. The relations of 4 also hold for L-integrals^ viz. ; 



I/ 






(2 



-f I/I <//</!/]. (4 

<X2l <^21 d-'H 

The proof is analogous to that employed for the 72-integnils, 
using extremal sequences. 



398. Let 2l = 08to SM) be a separated division for each u == 0. 
M = 0. Then 

Urn f / = f /. 
M> XB W ^a 
For by 390, l, 



I-M- 

4^21 a^SBtt ^S,/ 



But by 385, 2, the last integral == 0, since S M = 0, and since/ is 
limited. 

399. Let f be limited and continuous in 31, except possibly at the 
points of a null set Sft. Then f is L-integrable in 21. 

Let us first take 9i = 0. Then/ is continuous in 31. Let 21 He 
in a standard cube Q. If Osc/ is not < e in 21, let us divide Q 
into 2 n cubes. If in one of these cubes 

Osc/< e, (1 

let us call it a black cube. A cube in which 1) does not hold we 
will call white. Each white cube we now divide in 2 n cubes. 
These we call black or white according as 1) holds for them or 
does not. In this way we continue until we reach a stage where 
all cubes are black, or if not we continue indefinitely. In the 
latter case, we get an infinite enumerable set of cubes 

Hi* Hv <k (2 



GENERAL THEORY 388 

Each point a of 81 lies in at least one cube 2). For since / is 
continuous at x * a, 

l/0*)-/<0|</2 , * in r,(a). 

Thus when the process of division has been carried so far that 
the diagonals of the corresponding cubes are < S, the inequality 
1) holds for a cube containing a. This cube is a black cube. 

Thus, in either case, each point of 21 lies in a black cube. 

Now the cubes 2) effect a separated division D of 31, and in 
each of its cells 1) holds. Hence/ is i-integrable in 21. 

Let us now suppose 31 > 0. We set 

2l = + 9J. 

Then /is i-integrable in & by case 1. It is i-integrable in 31 
by 380, 2. Then it is i-integrable in 21 by 390, 1. 

2. If / is Jv-integrable in 21, we cannot say that the points of 
discontinuity of /form a null set. 

Example. Let/= 1 at the irrational points ^ in 21 = (0, 1) ; 

= at the other points $R, in 21. 
Then each point of 21 is a point of discontinuity. But here 



since 9? is a null set. Thus /is Z-integrable. 



400. If f(x^ # w ) has limited variation in 21, it is L-integrable. 

For let D be a cubical division of space of norm d. Then by I, 
709, there exists a fixed number V, such that 

^o)4 m ~ l < V 

for any D. Let a>, <r be any pair of positive numbers. We take 
d such that 

d<^. a 

Let d( denote those cells in which Osc/>o>, and let the number 
of these cells be i>. Let ?/ t denote the points of 21 in d( . Then 

va>d m - 1 < So)^- 1 < V. 



384 LEBESGUE INTEGRALS 

Hence T/- 

< -f- (2 

tad" 1 " 1 

Thus v = ., ,_^ Ftf m , ON 

^.<w/<-^- , by 2), 

<r , byl). 

ft) 

Hence /is .L-integrable by 388. 



401. Jta <=/, w3l<#; 

= 0, m JL = 53 - 21. 



^ x, 

//=/*, 

<J/Sl at 



a 



if 1, </> is L-integrable in 93 ; or 2, / is L-integrable in 31, flftd 31, ^1 
ar6? separated parts of S3. 

On the 1 hypothesis let !S 8 j be an extremal sequence of <. 
Let the cells of @ a be e x , 2 They effect a separated division 
of 31 into cells d^, d% Let m t , J^ be the extremes off in c? t and 
n t , JV" t the extremes of <f> in e t . Then for those cells containing at 
least a point of 31, 

nfr < m t J t < M& < Nfr (2 

is obviously true when e t = d t . Let d^ < e t . If ra t j<, 0, 

n t F <^ m t c? t , since m t = n t . (3 

If m t > 0, w t = 0, and 3) holds. 

If M L < 0, MI d, < N& , since N L = 0. (4 

If M^ > 0, 4) still holds, since M^N C 

Thus 2) holds in all these cases. Summing 2) gives 



for the division (g,, since in a cell e of @, containing no point of 31, 
= 0. Letting s = <x>, we get 1), since the end members 



INTEGRAND SETS 385 

On the 2 hypothesis, 

r <*>= r </>+ r *= r <M r/> 

XSB .Xa ^ JL% <JL% 
since < being = in A, is Z-integrable, and we can apply 390. 

402. 1. If 



we call /a null function in 21. 

2. If f^ f8 a null function in 21, ^e points ^} where /> 



a 

For let 2T = 3 + $, so that/= in . 
By 401, 



0= //=//. d 



Let l > e 2 > = 0. Let ^3 n denote the points of $ where 
/ > n . Then 



Each ^} n is a null set. For 



f >*Jn=0. 
cL^n 



Hence ^ n = 0. 

Then , ? = }^}= d + a + - 

where ^^ C a =%-*i, ^3=^3-^3- 

As each Q n is a null set, $ is a null set. 

Integrand Sets 

403. Let 21 be a limited point set lying in an w-way space 9J TO . 
Let f(x l x m ) be a limited function defined over 21. Any 
point of 21 may be represented by 



380 LEBKSGUE !KTE(iKALS 

The point x = (aj a^+j) 

lies in an w + 1 way space 3? m + r The set of points \x\ in which 
x m+\ ranges from oo to -t-oo is called an ordinate through a. If 
x m+l is restricted by Q < < ^ 

we shall call the ordinate a positive ordinate of length I ; if it is re- 
stricted by _ i< Xm+1 <Q, 

it is a negative ordinate. The set of ordinates through all the 
points a of 21, each having a length =/(#), and taken positively 
or negatively, as f(a) in ^ 0, form a point set $ in 3f mfl which 
we call an integrand set. The points of <J for which # m+1 has a 
fixed value x m+l = c form a section of 3, and is denoted by 3(0 r 
by a- 

404. Let 21= jaj Je a limited point set in 9? m . Through each 
point a, /e w ^r^<?^ a positive ordinate of constant length /, getting a 
set ), m 3l m+l . Then g = ^ 

For let @j > @ 2 > form a standard sequence of enclosures of 
), such that g = > (2 

Let us project each section of @ n corresponding to a given value 
of x n+l on SR m , and let 2l n be their divisor. Then 2l n > 21. Thus 



Letting n == oo , and using 2), we get 

5 = i *. 

To prove Me res of 1), let be the complement of O with re- 
spect to some standard cube Q in 9J m+1 , of base Q in SR m . 
Then, as just shown, 

5 = IA , where ^1 = Q - 21. 
Hence 



INTEGRAND SETS 387 

405. Letf>0 be L-integrable in %. Then 



where Q is the integrand set corresponding to f. 

For let \8 t \ be a separated division D of 31. On each cell S t 
erect a cylinder & t of height M, = Max /in S t . Then by 404, 



Let = JS t } ; the E t are separated. Hence, e>0 being small 
at pleasure, 



for a properly chosen D. Thus 

3< f / (2 

Xa 
Similarly we find 



From 2), 3) follows 1). 

406. Letf>Q be L-integrable over the measurable field 21. 
e corresponding integrand set 3 /s measurable, and 



3= / (1 

L% 
For by 2) in 405, 

s< f /. 

Xm 

Using the notation of 405, let c n be a cylinder erected on B n of 
height w n = Min / in S n . Let c = \c n \ . Then c < 3, and hence 

c<3- ( 2 

But 31 being measurable, each c n is measurable, by 404, Hence 
c is by 359, Thus 2) gives 

c<3- (3 

Now for a properly chosen D, 






388 LEBESGUE INTEGRALS 

Hence 

as e is arbitrarily small. From 2), 3), 4) 

r = f* 
I f<$<3< I f, 

<Xa "= JL^i 

from which follows 1). 



Measurable Functions 

407. Let /(^ # m ) be limited in the limited measurable set SI. 
Let Six,* denote the points of SI at which 



If each 3l Af * is measurable, we say f is measurable in 31. 

We should bear in mind that when f is measurable in 31, neces- 
sarily 31 itself is measurable, by hypothesis. 

408. 1. Iff is measurable in 31, the points & of 31, at which f = C, 
form a measurable set. 

For let 3l denote the points where 

-<n 
where 1s ...-. 

Then by hypothesis, 3l n is measurable. But & = 
Hence S is measurable by 361. 

2. If f is measurable in 31, the set of points where 



is measurable, and conversely. 
Follows from 1, and 407. 

3. If the points 3{* in 31 where /> A, form a measurable set for 
each X, f is measurable in 31. 

For Slxpi having the same meaning as in 407, 

Six/* = Six 31 M . 
Each set on the right being measurable, so is Slx M 



MEASURABLE FUNCTIONS 389 

409. 1. . r ffis measurable in 81, it is L-integrable. 

For setting m = Min /, M = Max / in 21, let us effect a division 
D of the interval g = (w, J!f ) of norm J, by interpolating a finite 

number of points 
r 



Let us call the resulting segments, as well as their lengths, 

rfj, rf 2 , <2 3 

Let 2l t denote the points of 21 in which 

m i-\<f< m i i * = 1> 2, ; ra = 7w. 
We now form the sums 

?/, 
Obviously 



= , as ci! = 0. (2 

We may now apply 387. 
2. //'/ is measurable in 21 

= lira Sw^a, = lim 2m t S t , (3 



using the notation in 1. 

This follows from 1), 2) in 1. 

3. The relation 3) is taken by Lebesgue as definition of his 
integrals. His theory is restricted to measurable fields and to 
measurable functions. For Lebesgue's own development of his 
theory the reader is referred to his paper, Intfyrale, Longueur, 
Aire, Annali di Mat., Ser. 3, vol. 7 (1902) ; and to his book, 
Lefons sur V Integration. Paris, 1904. He may also consult the 
excellent account of it in ffobson's book, The Theory of Functions 
of a Real Variable. Cambridge, England, 1907. 



390 LEBESUCE INTEGRALS 

Semi-Divisors and Quasi-Divisors 

410. 1. The convergence of infinite series leads to the two 
following classes of point sets. 

Let F= s/.c*! ... *) = i/, + i/ = F n + F n , (i 

1 n+l 

each/ t being defined in 21. 

Let us take > small at pleasure, and then fix it. 
Let us denote by 2l n the points of 21 at which 

n\ y V 

Of course 2l n may not exist. We are thus led in general to the 

se ^ s 21 21 21 (3 

The complementary set A n = 21 2l n will denote the points 

where I ^Y ^ I (k 

I n\^/ | ^ *' v 

If now F is convergent at #, there exists a v such that this point 
lies in 21 21 21 (5 

The totality of the points of convergence forms a set which has 
this property: corresponding to each of its points x, there exists 
a v such that x lies in the set 5). A set having this property is 
called the semi-divixor of the sets 3), and is denoted by 



Suppose now, on the other hand, that 1) does not converge at 
the point x in 21. Then there exists an infinite set of indices 

n i< n z < * == 
such that , ~ 

I^n 8 (*)|>e. 

Thus, the point x lies in an infinity of the sets 

A l , A 2 , A B (6 

The totality of points such that each lies in an infinity of the 
sets 6) is called the quasi-divisor of 6) and is denoted by 

QdvMJ. 
Obviously, 

U + Q<lvMn} = 2l. (7 



SEMI-DIVISORS AND QUASI-DIVISORS 391 

We may generalize these remarks at once. Since F(x) is 
nothing but 



we can apply these notions to the case that the f unctions / t (#j # m ) 
are defined in 21, and that 

lim/ t = <f>. 

2. We may go still farther and proceed in the following abstract 
manner. 

The divisor 35 of the point sets 

*i , v- a 

is the set of points lying in all the sets 1). 

The totality of points each of which lies in an infinity of the sets 
1) is called the quasi- divisor and is denoted by 

QdvfSU. (2 

The totality of points a, to each of which correspond an index m a , 
such that a lies in 

. , L.n,- 

forms a set called the semi-divisor of 1), and is denoted by 

SdvfSU- (3 

If we denote 2), 3) by Q and @ respectively, we have, obviously, 

35 < @ < Q. (4 

3. In the special case that Jlj >21 2 > we have 

Q = @ = 3). (5 

For denoting the complementary sets by the corresponding 
Roman letters, we have 



But Q has precisely the same expression. 
Thus O = 3), and hence by 4), @ = 5). 



392 LEBESGUE INTEGRALS 

4. Let 2l n 4- A n = 93, n = 1, 2, ... Then 



For each point b of S3 lies 

either 1 only in a finite number of 2l n , or in none at all, 
or 2 in an infinite number of 2l n . 

In the 1 case, b does not lie in 21,, 2l, +1 ; hence it lies in 
A t , A 8+l In the 2 case b lies obviously in Qdv J21J. 

5. 7/Slj, 21 2 are measurable, and their union is limited, 



are measurable. 

For let ) n = Di;(2l n , 2l n+1 ) . Then @ = {SD n } . 

But @ is measurable, as each S) w is. Thus Sdv } J. n j is measur- 
able, and hence Q is by 4. 

6. Let O = Qdv 2l n ( , eac*A 3[ n 6em{/ measurable, and their union 
limited. If there are an infinity of the 21 n , sat/ 



whose measure is > a, then ^ 

Q>a. (6 

For let 93 n = (2t ln , 3t ln+1 -.), then S n >. 



(7 
by 362. As Q>93 we have 6) at once, from 7). 

Limit Functions 
411. Let 



a x ranges over 21, r finite or infinite. Let f be measurable in 21 
and numerically <M,for each t near r. Then <f> is measurable in 
21 also. 

To prove this we show that the points 53 of 21 where 



LIMIT FUNCTIONS 393 

form a measurable set for each X, /*. For simplicity let T be finite. 
Let t v 2 -.. =T; also let 1 >e 2 > ==0. Let S n ,, denote the 
points of 21 where 

(2 



Then for each point x of 93, there is an S Q such that 2) holds for 
any if 8 > 8 Q . Let g n = Sdv \ S n ,} . Then 93 < 6 n . But the S n , 

being measurable, & n is by 410, 5. Finally $=> Dv jS n } , and hence 
93 is measurable. 

412. Let 



or x in 81, anrf T finite or infinite. Let t 1 , t" = r. 
, =/(a;, i (a) ) 6^ measurable, and numerically < Jtfl l>ef < =/ a 
, denote the points where 

i&i>. 

TOen foreache>0, U m @. = 0. (1 

*=< 

For by 411, </> is measurable, hence #, is measurable in 81, hence 
, is measurable. 

Suppose now that 1) does not hold. Then 

fim . = I > 0. 

5=00 

Then there are an infinity of the ,, as ^ ,, whose 
measures are >X>0. Then by 410, 6, the measure of 

=Qdv{ f | 
is >\. But this is not so, since/, = <^>, at each point of 81. 

413. 1. Let . 



for x in 81, awrf T finite or infinite. 



Ifeachf 8 =f(x, # a) ) is measurable, and numerically <Min Hfor 
each sequence 1), then 

/./. 

i <f> = lini I / (x, t). (2 

4,21 '= T J/2l' 



394 LEBESGUE INTEGRALS 

For set 



, * , 

<t>=fs +# 

an( l let I // I <^ AT * 1 9 

IffflS^^f > s= i, ^ 

Then as in 412, $ and #, are measurable in SI. Then by 409, 
they are i-integrable, and 

/%= ff.+ /V (3 

Jin <Asi <Xsi 

Let 93, denote the points of SI, at which 



and let 53, -|- B 9 = SI. Then 93 a , ^, are measurable, since g 9 is. 
Thus by 390, r T T 

I ff*= I 9*+ I ff*- 
^21 JL%, J*B. 

Hence 



By 412, , = 0. Thus 

lim 

^" 

Hence passing to the limit in 3), we get 2), for the sequence 
1). Since we can do this for every sequence of points t which 
= T, the relation 2) holds. 

2. 



converge in SI. If each term f t is measurable, and each \ F K \ < M, 
then F is L-integrdble, and 



Iterated Integrals 
414. In Vol. I, 732, seq. we have seen that the relation, 



holds when /is J2-integrable in the metric field St. This result 
was extended to iterable fields in 14 of the present volume. We 



ITERATED INTEGRALS 395 

wish now to generalize still further to the case that / is i-inte- 
grable in the measurable field 21. The method employed is due to 
Dr. W. A. Wilson,* and is essentially simpler than that employed 
by Lebesgue. 

1. Let x = (Zj z,) denote a point in s-way space 3? a , s = m + n. 
If we denote the first m coordinates by x 1 - # w , and the remaining 
coordinates by y^ y n , we have 



The points x== (^ ... ^ ~. 0) 



range over an ra-way space 9? m , when 2; ranges over 8?,. We call 
x the projection of z on SK OT . 

Let z range over a point set 21 lying in 3J a , then x will range 
over a set 53 in 3? w > called tfAe projection of 21 on 9J W . The points 
of 21 whose projection is x is called the section of 21 corresponding 
to x. We may denote it by 

2l(#), or more shortly by S. 
We write = < 

to denote that 21 is conceived of as formed of the sections , cor- 
responding to the different points of its projection 95. 

2. Let O denote a standard cube containing 21, let q denote its 

projection on $R m . Then S<_q. Suppose each section 21 (#) is 

/> 

measurable. It will be convenient to let 2l(#) denote a function 
of x defined over q such that 

2l(V) = Meas 2l(V) = S when # lies in S3, 

= when # lies in q 93. 

This function therefore is equal to the measure of the section of 
21 corresponding to the point x, when such a section exists ; and 
when not, the function = 0. 

When each section 2l(#) is not measurable, we can introduce 
the functions 



* Dr. Wilson's results were obtained in August, 1909, and were presented by me 
in the course of an address which I had the honor to give at the Second Decennial 
Celebration of Clark University, September, 1909. 



896 LEBESGUE INTEGRALS 

Here the first = S when a section exists, otherwise it = 0, in q. 
A similar definition holds for the other function. 

3. Let us note that the sections 



where 2l c , 2l t are the outer and inner associated sets belonging to 21. 
are always measurable. 

For 21, = .Z)tf{( n j, where each (g n is a standard enclosure, each 
of whose cells e nm is rectangular. But the sections e nm (V) are 
also rectangular. Hence 



being the divisor of measurable sets, is measurable. 

415. Let 2l c be an outer associated set of 21, both lying in the stand- 

/i 

ard cube Q. Then 2l c (#) is L-integrable in q, and 

<. a 

For let j( n S be a sequence of standard enclosures of 21, and 
=Je Bm S. Then 

<S* = 2e Bro (2 



Now c nm being a standard cell, e nm (#) has a constant value > 
for all x contained in the projection of c nm on q. It is thus con- 
tinuous in q except for a discrete set. It thus has an JS-integral, 
and 

e n m= j e nm (. 

^ 
This in 2) gives 



f 2e nm <, by 413, 2, 

,(*), (4 



by 3). 



ITERATED INTEGRALS 397 

On the other hand, @(#) is a measurable function by 411. Also 



= Aim l n (V), by 413, 1. (5 

aL/c( 

pJr\Ti^ ^ '*** 

Thus this in 5) gives 1). 

416. Let 21 lie in the standard cube Q. Let 2l t be an inner asso- 
ciated set. Then 2^0*0 L-integrable in q, and 

21= /'Hoe). 

~~ <Xq 

o = a. + A e . 



21: 

For 



Hence S,(a;) is i-integrable in q, and 

Ate*) = /"Sea:) - 

oLq owq < 

= Q-A , by 415, 
= 5 t = by 370, 2. 

417. e measurable ?l Ke tw ^A standard cube O- 



Hence < x( x )< .(*)- ( 2 



using 396, l, and 415, 416. From 2) we conclude 1) at once. 
418. Let 21 = SB S fo measurable. Then ( are L-integrable in 



398 LEBESGUE INTEGRALS 

For by 417, 



by 401. 

419. If 21 = S3 & is measurable, the points of $$ at which & is 
measurable form a null set 31. 

For by 418, 



Hence 



0= f (6-). 

1 oa 

e^fijj 



Thus < = 6 - 6 

is a null function in 93, and by 402, 2, points where <f> > form a 
null set. 



420. i0 21 = 93 & be measurable. Let b denote the points of 
/or which the corresponding sections are measurable. Then 



For by 419, 33=b + 5tt, 

and 5ft is a null set. Hence by 418, 

/= /= / = 
31= /<=/+/ 

^^3 J-b ^ 



421. ie^ /> in SI. ^ ^/^ integrand set Q, corresponding to f 
be measurable, then f is L-integrable in 31, and 

*-/ 

For the points of Q lying in an m -f 1 way space 5R m+1 may be 

denoted by <r (<u...<u ^ 

j x {y l *" y m ,z), 

where y = (yi"-ym) ranges over 5R m , in which 21 lies. Thus 21 
may be regarded as the projection of Q on 5R m . To each point y 



ITERATED INTEGRALS 399 



of 21 corresponds a section 3(y), which for brevity may be denoted 
by ft. Thus we may write 

<*=:. 

As ft is nothing but an ordinate through y of length /(y), we 
have by 419, ^ /^ 

3= = / 



422. .Z/e / Je L-integrable over the measurable field 21 = S3 . 
b denote those points of 93, /or wAi<?A / is L-integrable over the 
corresponding sections . 2%en 

/ / 
/-/ // a 

[ <Xfc*6@ 

Moreover $1 = $8 b is a null set. 

Let us 1 suppose /> 0. Then by 406, 3 is measurable and 

. (2 



Let y8 denote the points of 93 for which 3 (x) is measurable. 
Then by 420, 

3 = f 300- (3 

<Xp 
By 419, the points 

$ = 93 - y8 (4 

form a null set. 

On the other hand, 3K#) is the integrand set of/, for 3l(V) = 6. 
Hence by 421, for any x in /3, 

(^)=r/, (5 

ele 

and ^8 < b. (6 

From 2), 3), 5) we have 

JL% JL? JL< 
From 6) we have 

9ft = 93-b<-/3 = $, 

a null set by 4). Let us set 

b = + it- 



400 LEBESGUE INTEGRALS 

Then n lying in the null set $, is a null set. Hence 

Jut JL& JLn JL& JLb JL& 

This with 7) gives 1). 

Let f be now unrestricted as to sign. We take C > 0, such 
that the auxiliary function 



, n . 

Then /, g are simultaneously L-integrable over any section S. 
Thus by case 1 

f(/+tf)= /' f (/+#) (8 

aLw cLb ~L& 

Now 



, r s* 

(f+0)= f+ 0= I /+(?, (9 

yi JLw i% <% 



= 

By 418, is i-integrable in S3, and hence in b. Thus 
/ / / / /== 

/ /(/+<?)= / / f+c <& 

Xb <?6< <Xb <^(S 04/b 

As b differs from 93 by a null set, 



by 418. From 8), 9), 10), 11), 12) we have 1). 

423. If f is L-integrable over the measurable set 31 = 93 S, then 



For by 422, 

/* /^ 
=/ J . (2 

Jit Jits, 

As SB b = 91 is a null set, 



/=o 

e 



ITERATED INTEGRALS 401 

may be added to the right side of 2) without altering its value. 
Thus 



JL<& JLbJLo, JLwJL& JL<&JL< 



424. 1. (TF. A. Wilson.) Iff(x l -^x m ) is L-inteffrable in 
measurable 21, f is measurable in 21. 

Let us first suppose that/> 0. We begin by showing that the 
set of points 21 A of 21 at which / > X, is measurable. Then by 
408, 3, f is measurable in 21. 

Now/ being i-integrable in 2t, its integrand set 3 is measur- 
able by 406. Let 3> A be the section of 3> corresponding to x m+l =* \. 
Then the projection of 3? A on 9t w is 21 A . Since 3> is measurable, the 
sections $A are measurable, except at most over a null set L of 
values of X, by 419. Thus there exists a sequence 



none of whose terms lies in L. Hence each ^A,, is measurable, and 
hence 21 An is also. 

As 2l An ^j < 2l An , each point of 21 A lies in 



so that ^ < 2, (2 

On the other hand, each point d of $) lies in $l x . For if not, 

/(<*)< X. 

There thus exists an s such that 



< X. < X. (3 

But then d does not lie in 21 A ,, for otherwise f (d) > X., which 
contradicts 3). But not lying in 21 A ., d cannot lie in ), and this 
contradicts our hypothesis. Thus 

)<21 A . (4 

From 2), 4) we have 

5) = Six- 

But then from 1), 21 A is measurable. 
Let the sign off be now unrestricted. 



402 IMPROPER L-INTEGRALS 

Since /is limited, we may choose the constant (7, such that 



Then g is Z-integrable, and hence, by case 1, g is measurable. 
Hence/, differing only by a constant from g, is also measurable. 

2. Let 21 be measurable. Iff is L-integrable in 21, it is measur* 
able in 21, and conversely. 

This follows from 1 and 409, l. 

3. From 2 and 409, 3, we have at once the theorem : 

When the field of integration is measurable, an L-integrable func- 
tion is integrable in Lebesgue's sense, and conversely; moreover, both 
have the same value. 

Remark. In the theory which has been developed in the fore- 
going pages, the reader will note that neither the field of integra- 
tion nor the integrand needs to be measurable. This is not so in 
Lebesgue's theory. In removing this restriction, we have been 
able to develop a theory entirely analogous to Riemaim's theory of 
integration, and to extend this to a theory of upper and lower in- 
tegration. We have thus a perfect counterpart of the theory 
developed in Chapter XIII of vol. I. 

4. Let 21 be metric or complete. If f(x l # m ) is limited and 
R-integrable, it is a measurable function in 21. 

For by 381, 2, it is .L-integrable. Also since 21 is metric or 
complete, 21 is measurable. We now apply 1. 



IMPROPER L-INTEGRALS 

Upper and Lower Integrals 

425. 1. We propose now to consider the case that the integrand 
f(x l x m ~) is not limited in the limited field of integration 21- In 
chapter II we have treated this case for jB-integrals. To extend 
the definitions and theorems there given to .//-integrals, we have 
in general only to replace metric or complete sets by measurable 
sets; discrete sets by null sets; unmixed sets by separated sets ; 



UPPER AND LOWER INTEGRALS 408 

finite divisions by separated divisions; sequences of superposed 
cubical divisions by extremal sequences; etc. 

As in 28 we may define an improper i-integral in any of the 
three ways there given, making such changes as just indicated. 
In the following we shall employ only the 3 Type of definition. 
To be explicit we define as follows : 

Let/(a; 1 a^) be defined for each point of the limited set 31. 
Let 2l a /s denote the points of 21 at which 



The limits ,- /r 

lim / / , lira / / (2 

, 0= eLnap , 0= at 2la|3 

in case they exist, we call the lower and upper (improper) L-in- 
tegrah, and denote them by 



In case the two limits 2) exist and are equal, we denote their 
common value by 

ff 

<% 
and say/ is (improperly) L-integrdble in 21, etc. 

2. In order to use the demonstrations of Chapter II without too 
much trouble, we introduce the term separated function. A func- 
tion f is such a function when the fields 2l a /s defined by 1) are 
separated parts of 21. 

We have defined measurable functions in 407 in the case that 
/ is limited in 21. We may extend it to unlimited functions by 
requiring that the fields 2l a j8 are measurable however large a, $ are 
taken. 

This being so, we see that measurable functions are special cases 
of separated functions. 

In case the field 21 of integration is measurable, 2f a /3 is a meas- 
urable part of 21, if it is a separated part. From this follows the 
important result : 

Iff is a separated function in the measurable field 21, it is It-in- 
tegrate in each 2l a p. 



404 IMPROPER L-INTEGRALS 

From this follows also the theorem : 

Let f be a separated function in the measurable field 21. If either 
the lower or upper integral of f over 21 is convergent^ f is L-integrable 

in 2k and /- /- 

/ /= lira / /. 

Jb^a , fl^aoatjkp 

426. To illustrate how the theorems on improper jR-integrals 
give rise to analogous theorems on improper i-integrals, which 
may be demonstrated along the same lines as used in Chapter II, 
let us consider the analogue of 38, 2, viz. : 

If f is a separated function such that / f converges^ so do I f. 

ia v 

Let { JEfJ'be an extremal sequence common to both 



Let e denote the cells of E n containing a point of ty ft ; e' those 
cells containing a point of typ ; S those cells containing a point of 
2l but none of $: Then 



= lira &MI e 4- 2JK?, - e' -f 



In this manner we may continue using the proof of 38, and so 
establish our theorem. 

427. As another illustration let us prove the theorem analogous 
to 46, viz. : 

Let 2lj, 21 2 , 2l n form a separated division of 21. If f is a 
separated function in 21, then 

T f* r* 

f = I /++ / /, 

__ tt ^2li Ja&n 

provided the integral on the left exists, or all the integrals on the 
right exist. 

For let 21,, denote the points of 2( a/3 in 21,. Then by 390, 1, 



<8l, a/3 

In this way we continue with the reasoning of 46. 



L-INTEGRALS 405 

428. In this way we can proceed with the other theorems ; in 
each case the requisite modification is quite obvious, by a con- 
sideration of the demonstration of the corresponding theorem in 
JS-integrals given in Chapter II. 

This is also true when we come to treat of iterated integrals 
along the lines of 70-78. We have seen, in 425, 2, that if 21 is 
measurable, upper and lower integrals of separated functions do 
not exist as such ; they reduce to i-integrals. We may still 
have a theory analogous to iterated /^-integrals, by extending the 
notion of iterable fields, using the notion of upper measure. To 
this end we define : 

A limited point set at 21 = 33 S i-s submeasurable with respect 
to 33, when 

1= l. 



1= f 
J. 



We do not care to urge this point at present, but prefer to pass 
on at once to the much more interesting case of J>-integrals over 
measurable fields. 

L-Integrals 

429. These we may define for our purpose as follows : 
'Letf(x l x m ) be defined over the limited measurable set 21. 
As usual let 2l a /s denote the points of 21 at which 

-</<& , /3>0. 

Let each 2l a/3 be measurable, and let / have a proper i-integral 
in each 2l a p. Then the improper integral of /over 21 is 

/= lim f/, (1 



when this limit exists. We shall also say that the integral on 
the left of 1) is convergent. 

On this hypothesis, the reader will note at once that the dem- 
onstrations of Chapter II admit ready adaptation ; in fact some 
of the theorems require no demonstration, as they follow easily 
from results already obtained. 



406 IMPROPER L-INTEGRALS 

430. Let us group together for reference the following theo- 
rems, analogous to those on improper J2-integrals. 

1. Iffis (improperly^) L-integrable in 21, it is in any measurable 
part of 21. 

2. Ifg,h denote as usual the non-negative functions associated 
withf, then 

L'-k-fr (1 

3. If I f is convergent, so is I \f\, and conversely. 

* JLw JL%* 

4. When convergent, 

i /* r* 

//<//. (2 

lota L% 

t 
f is convergent, then 
' 

e > 0, a- > 0, 

for any measurable 93 < 21, such that 93 < a-. 

6. Let 2l = (2l x , 21 2 2l n ) be a separated division of 21, each 2l t 
measurable. Then 



provided the integral on the left exists, or all the integrals on the 
right exist. 

7. Let 21 = J2l n ^ be a separated division of 21, into an enumerable 
infinite set of measurable sets 2l n . TJien 



provided the integral on the left exists. 

8. Iff<g in 21, except possibly at a null set, then 

L^L* < 6 

when convergent. 



L-JNTEGRALS 407 

431. 1. To show how simple the proofs run in the present 
case, let us consider, in the first place, the theorem analogous to 
38, 2, viz. : 



If I f converges^ so do if and I f. 
JLw JLy *Lm 



The rather difficult proof of 38, 2 can be replaced by the follow- 
ing simpler one. Since 

a 



is a teparated division of 2T tt|3 , we have 



Hence 



I/ -I -I/-/ 

\eiXafl tJL^aft' \Jdpfi JU$ 



But the left side is < e, for a sufficiently large a, and #, #' > 
some @ Q . This shows that I is convergent. Similarly we show 

the other integral converges. 

2. This form of proof could not be used in 38, 2, since 1) in 
general is not an unmixed division of 2f a/3 . 

3. In a similar manner we may establish the theorem analo- 
gous to 39, viz. : 



If I f and I f converge, so does I /. 
JLy eLw t* 



4. Let us look at the demonstration of the theorem analogous 
to 43, 1, viz. : 



f <7= ff ; f*-- f/, 

% oLy eL% JxW 

provided the integral on either side of these equations converges. 



408 IMPROPER L-INTEGRALS 

Let us prove the first relation. Let 53^ denote the points of 81 
at which /< ft. Then 

, = Stt + % 

is a separated division of 93^, and hence 

f #= / ff+ Cff= I 9= //I etc - 
i0 <X^ JLy ft a6$j9 JU$p 

5. It is now obvious that the analogue of 44, l is the relation 1) 
in 430. 

6. The analogue of 46 is the relation 3) in 430. Its demon- 
stration is precisely similar to that in 46. 

7. We now establish 430, 7. Let 

=(i, V-a-.)- 

Then % = % m + Bm 

is a separated division of 21, and we may take m so large that 
B m < cr, an arbitrarily small positive number. Hence by 430, 6, 
we may take m so large that 



\L 



f 

Km 



r/= r /+ c 

tsi *L% m JL,B 



. 



From this our theorem follows at once. 

Iterated Integrals 

432. 1. Let us see how the reasoning of Chapter II may be 
extended to this case. We will of course suppose that the field 
of integration 21 = 33 is measurable. Then by 419, the points 
of 33 for which the sections are not measurable form a null set. 
Since the integral of any function over a null set is zero, we may 
therefore in our reasoning suppose that every S is measurable. 

Since ?l is measurable, there exists a sequence of complete com- 
ponents A m = B m O m in 21, such that the measure of A = \A m \ is 21. 



ITERATED INTEGRALS 409 

Since A m is complete, its projection B m is complete, by I, 717, 4. 
The points of B m for which the corresponding sections O m are not 
measurable form a null set v m . Hence the union \v m \ is a null 
set. Thus we may suppose, without loss of generality in our 
demonstrations, that 21 is such that every section in each A m is 
measurable. 
Now from 



-I= Ti- f<7= f ((- 

<Z<B rJLB JL 



<* r**. 

we see that those points of 93 where S > O form a null set. We 
may therefore suppose that S = everywhere. Then O is a 
null set at each point ; we may thus adjoin them to C. Thus we 
may suppose that = G at each point of 93, and that 93 = B is the 
union of an enumerable set of complete sets B m * 
As we shall suppose that 



is convergent, let 

!< 2 < = oo, 

!<&<.. =00. 

Let us look at the sets 2l an > 930 n , which we shall denote by 2l n . 
These are measurable by 429. Moreover, the reasoning of 72, 2 
shows that without loss of generality we may suppose that 21 is 
such that 93 n = 93. We may also suppose that each & n is measur- 
able, as above. 

2. Let us finally consider the integrals 

/ (i 



These may not exist at every point of 93, because / does not 
admit a proper or an improper integral at this point. It will 
suffice for our purpose to suppose that 1) does not exist at a null 
set in 93. Then without loss of generality we may suppose in our 
demonstrations that 1) converges at each point of 93. 

On these assumptions let us see how the theorems 73, 74, 75, 
and 76 are to be modified, in order that the proofs there given 
may be adapted to the present case. 



410 IMPROPER L-INTEGRALS 

433. 1 The first of these may be replaced by this : 
Let B^ n denote the points of 33 at which F n > <r. Then 

HmJ a , n = 0. 
For by 419, 



as by hypothesis the sections S are measurable. Moreover, by 
hypothesis 

S=(Sn + C n 

is a separated division of 6, each set on the right being measur- 
able. Thus the proof in 73 applies at once. 

2. The theorem of 74 becomes : 
Let the integrals 



be limited in the complete set 93. Let (5 n denote the points of $8 at 
which 



Then 

lim @ n = 93. 

n=oo 

The proof is analogous to that in 74. Instead of a cubical 
division of the space 9t p , we use a standard enclosure. The sets 
93 n are now measurable, and thus 



is measurable. Thus 6 n = b. The rest of the proof is as in 74. 

3. The theorem of 75 becomes : 
Let the integral 




f 

<X< 



be limited in complete 53. Then 

lim 




ITERATED INTEGRALS 411 

The proof is entirely similar to that in 75, except that we use 
extremal sequences, instead of cubical divisions. 

4. As a corollary of 3 we have 
Let the integral 

/ /2 

be limited and L-integrable in 33. Let JB = {J3 m } the union of an 
enumerable set of complete sets. Then 

lim f f/=0. 

n =*JL%JLc n 

For if S3 m = (^, B^ ... J5 m ), and 33 = 23 m + , we have 



JL JLc n JL^ m *\ JL 2>, n JLt n 
But for m sufficiently large, SD ro is small at pleasure. Hence 



We h^v^ now only to apply 3. 

434. 1. We are now in position to prove the analogue of 
76, viz. : 

Let ?{ = 33 - S be measurable. Let I f be convergent. Let the 

r . 

integrals I f converge in 93, except possibly at a null set. Then 
JLa 

'-{' (1 

L2i JL<% at 
provided the integral on the right is convergent. 

We follow along the line of proof in 76, and begin by taking 
/ > in 21- By 423, we have 



/=lim f Cf. (2 

^^^SB.iQL 



412 IMPROPER L-INTEGRALS 

Now > being small at pleasure, 

-e+ C /'/< I Cf * f r & > some # , 

JL% aid %<,' <X< 



" 



*+ 

J*% << <^93 *tc n 



Since we have seen that we may regard 33 as the union of an 
enumerable set of complete sets, we see that the last term on the 
right = 0, as n = oo, by 433, 4. Thus 

/ / /* /" /* 

/ / < lim / / = / , O 

<A$ *l JL JLa n JL< 

by 2). On the other hand, 



From 3) and 4) we have 1), when/> 0. 
The general case is now obviously true. For 

21 = <$ + Sft, 

where/ > in *ijj, and < in 9t. Here $ and 92 are measurable. 
We have therefore only to use 1) for each of these fields and add 
the results. 

2. The theorem 1 states that if 

i / / 

/,///, 



both converge, they are equal. Hobson* in a remarkable paper on 
Lebesgue Integrals has shown that it is only necessary to assume 
the convergence of the first integral ; the convergence of the second 
follows then as a necessary consequence. 

* Proceedings of the London Mathematical Society, Ser. 2, vol. 8 (1909), 
p. 31. 



ITERATED INTEGRALS 413 

435. We close this chapter by proving a theorem due to 
Lebesgue, which is of fundamental importance in the theory of 
Fourier's Series. 

Letf(x^) be properly or improperly L-integrable in the interval 
2l = a<6). Then 



For in the first place, 

J f< fV(* + *)!<** + \f\dx<2\f\dx. (2 

fi/a 

Next we note that 



Hence 



- 

JL>a 



or J t -J a <J t _^ (3 

From 2), 3) we have 

Jt<J. + z\f-g\dx. (4 

Let now g ^ f fa\f\<a, 

= for |/|>(?. 
Then by 4), J,<J. + * C\f - g \te 

JLa 
<^ + ^ 

where e' is small at pleasure, for Gr sufficiently large. Thus the 
theorem is established, if we prove it for a limited function, 

\ff(*)\<0> 

Let us therefore effect a division of the interval F = ( #, #), 
of norm d, by interpolating the points 

-<?<<?!< f a < <(?, 
causing F to fall into the intervals 

7r 7 2 ' 73 



414 IMPROPER L-INTEGRALS 

Let h m = c m for those values of x for which g(x) falls in the in- 
terval 7 m , and = elsewhere in 51. Then 



e', c f small at pleasure, 
for d sufficiently small. 

Thus we have reduced the demonstration of our theorem to a 
function h(x) which takes on but two values in 21, say and 7. 

Let @ be a <r/4 enclosure of the points where h = 7, while $ may 
denote a finite number of intervals of ( such that g @ < cr/4. 

Let <f> = 7 in (g, and elsewhere = ; let i/r = 7 in 5, and else- 
where = 0. Thus using 4), 



since 7i =<^ in (a, y8), except at points of measure < <r/4. Similarly 



Thus J h <J* + (r r i<Jt + , 

for cr sufficiently small. 

Thus the demonstration is reduced to proving it for a i/r which 
is continuous, except at a finite number of points. But for such a 
function, it is obviously true. 



CHAPTER XIII 
FOURIER'S SERIES 

Preliminary Remarks 

436. 1. Let us suppose that the limited function f '(V) can be 
developed into a series of the type 

f(x) = a 4- ! cos x 4- # 2 cos 2 a; -{- 3 cos 3 a: 4- 

4- b 1 sin a; 4- 6 2 sin 2 x 4- J 3 sin 3 a? 4- (1 

which is valid in the interval 21 = ( TT, TT). If it is also known 
that this series can be integrated term wise, the coefficients a n , b n 
can be found at once as follows. By hypothesis 

/*rr /ir 

fdx = a G I dx 4- a* / cos a;cfe 4- 
<X-T X-^ 

4- b l I silica; 4- 
As the terms on the right all vanish except the first, we have 

*. (2' 

Let us now multiply 1) by cos nx and integrate. 

f*7T /** 

f\x) cos nxdx = a I cos nxdx 4- a^ I cos a; cos nxctx 4- 
r cX-w <X-^ 

7 T^ 

4- o l I sin a; cos : 
I cos wa; cos nxdx =0 , TTI = /, 

r 

cos 2 wa;c?a; = TT, 



Now 



sin wa; cosna:=0. 

r 

416 



410 FOURIER'S SERIES 

Thus all the terms on the right of the last series vanish except 
the one containing a n . Hence 

a n = - rV(*Ocos nxdx. (2" 



Finally multiplying 1) by sin nx, integrating, and using the 

relations ,-* 

I sin mx sin nxdx = , m^n, 
JL-* 



* 
- 



sin 2 nxclx = TT, 

-n- 

get 



~ 

b n = - / f(x) sin nxdx. (2 

TTeL-V 

Thus under our present hypothesis, 

1 /' 1 /* 7r 

f(x) = - / f(u)du H 2 cos w # I /*( w ) cos 

" 



H 2 s ^ n wa; / /( w ) s ^ n nu du. (3 

7T 1 eX- ff 

The series on the right is known as Fourier s series ; the coeffi- 
cients 2) are called Fourier's coefficients or constants. When the 
relation 3) holds for a set of points 93, we say^(a?) can be de- 
veloped in a Fourier's series in 33, or Fourier's development is valid 
in 93. 

2. Fourier thought that every continuous function in 31 could 
be developed into a trigonometric series of the type 3). The 
demonstration he gave is not rigorous. Later Dirichlet showed 
that such a development is possible, provided the continuous 
function has only a finite number of oscillations in 8. The func- 
tion still regarded as limited may also have a finite number of 
discontinuities of the first kind, i.e. where 

/O + O) , /O-O) (4 

exist, but one at least is ^=/(a). 

At such a point a, Fourier's series converges to 



PRELIMINARY REMARKS 417 

Jordan has extended Dirichlet's results to functions having 
limited variation in 21. Thus Fourier's development is valid in 
certain cases when f has an infinite number of oscillations or 
points of discontinuity. Fourier's development is also valid in 
certain cases when f is not limited in 21, as we shall see in the 
following sections. 

We have supposed that f(x) is given in the interval 
21 = ( TT, TT). This restriction was made only for convenience. 
For if f(x) is given in the interval 3 = (a < 6), we have only to 
change the variable by means of the relation 

u ^7r(2x-a-b) 

b a 

Then when x ranges over $, u will range over 21. 
Suppose /is an even function in 21; its development in Fourier's 
series will contain only cosine terms. For 

00 

/(a?) = 2(a n cos nx -f- b n sin nx), 

o 

CO 

/( - x) = 2(a n cos nx b n sin nx). 

o 

Adding and remembering that f(x) =/( x) in 21, we get 

00 

/OB) = ^a n cos nx, f even. 
o 

Similarly if / is odd, its development in Fourier's series will 
contain only sine terms ; 



f(x) = ^2S n sin nx, f odd. 
i 

Let us note that if f(x) is given only in 93 = (0, TT), and has 
limited variation in 93, we may develop f either as a sine or a 
cosine series in 93. For let 

#O)=/O) , a: in 93 

=/(-#) , zinC-Tr, 0). 

Then g is an even function in 21 and has limited variation. 
Using Jordan's result, we see g can be developed in a cosine 
series valid in 21. Hence / can be developed in a cosine series 
valid in 93. 



418 FOURIER'S SERIES 

In a similar manner, let 



= -/(-*) , -7r<*<0. 

Then h is an odd function in 21, and Fourier's development 
contains only sine terms. 

Unless /(0) = 0, the Fourier series will not converge to /(O) 
but to 0, on account of the discontinuity at x = 0. The same is 
true for X=TT. 

If /can be developed in Fourier's series valid in 2l = ( TT, TT), 
the series 3) will converge for all x, since its terms admit the 
period 2 TT. Thus 3) will represent f(x) in SI, but will not 
represent it unless f also admits the period 2 TT. The series 3) 
defines a periodic function admitting 2 TT as a period. 

EXAMPLES 

437. We give now some examples. They may be verified by 
the reader under the assumption made in 436. Their justifica- 
tion will be given later 



1. /0*0 = x *> f r 

Then 

sin x sin 2x , sin 



If we set x = ^, we get Leibnitz's formula, 

4 = l~3~ f 5~~7* '" 
Example 2. /() = a? > < x< TT 

Then ' ~ ~~ 

f( ^ ^^^I CO8 ^ i_ COS ^ ^ 4_ COS ^ x 4. 1 

If we set x = 0, we get 

"s "r^" 1 "^ 4 - ^ + " - 



PRELIMINARY REMARKS 419 

Example S. /(X) = 1 , < x < TT 

= , X ac 0, 7T 

= 1 , TT < # < 0. 
Then 

*/ ^ 4 f sin # , sin 3 x , sin 5 x . 



Example 4. f(x) = x , 0<z<^ 

7T ^ ^ 

= ^r-^ , 2<*<->r- 

By defining / as an odd function, it can be developed in a sine 
series, valid in (0, TT). We find 

n x sin 3 x , sin 5 x 



Examples. 



By defining / as an even function, we get a development in 
cosines, 

cos x cos 3a; cos 



zW in (0, TT). 
Example 6. f (x) = J(TT x) , < # < TT. 

By defining / as an odd function we get a development in 

sines, 

f(x) = sin x -f- % sin 2# -h ^ sin 3# -f- 



vaZid m ( TT, TT). 

Example 7. Let/(o?) = - , < x < ^ 

3 3 



7T 



27T 



420 FOURIER'S SERIES 

Developing/ as a sine series, we get 

/., N . o , sin 4x , sin 8x , 
/(a;) = sm 2aH -- - + T + 

valid in (0, TT). 

Example 8. f(x)=*e x , in (- TT, TT). 

We find 



j oil it rj* sin. 2 x I - sin o x 

valid for TT < # < TT. 

Example 9. We find 

__ 2 /u. ._ f 1 cos # , cos 2 # cos 3 # 

vafoW/0r 



. 

cos //,# = -- sin - 

TT I 2 



Let us set x = TT, and replace /x by x ; we get 
TT . 1,1,1,1 

-COt 7TX =--_+ 



' O 7-2 ^ ~2 _ 12 ' ~2 _ s>2 ^ ~2 ___ Q2 ^ 

* JU JU J. JU ^^ ~t Jl' ^^ _> 

a decomposition of cot TTX into partial fractions, a result already 
found in 216. 

Example 10. We find 

2 f , 2 cos 2 z 2 cos 4 a; 2 cos 6 x } 

Oj |i /y I I __ _ ______^__ * . * L 

7^1 1.3 3.5 5.7 r 

valid for < # < TT. 

Summation of Fourier's Series 

438. In order to justify the development of/(V) in Fourier's 
series F, we will actually sum the F series and show that it con- 
verges to /(#) in certain cases. To this end let us suppose that 
f(x) is given in the interval 21 = ( TT, TT), arid let us extend /by 
giving it the period 2 TT. Moreover, at the points of discontinuity 
of the first kind, let us suppose 



SUMMATION OF FOURIER'S SERIES 
Then the function 



421 



<K<0 =/O + 2 w) 4/(* - 2 v) - 2/O) 

is continuous at w = 0, and has the value 0, at points of continuity, 
and at points of discontinuity of 1 kind of/. Finally let us sup- 
pose that / is (properly or improperly) Z-integrable in 31 ; this 
last condition being necessary, in order to make the Fourier co- 
efficients a n , b n have a sense. 
Let 



-f <z 2 cos 
4- 6j sin # 4- J 2 s ^ n 2 a; 4- 



= |~ a 4 2(a n cos w# + 6 n sin nx), 
where we will now write 



1 /c 



a n = - 



I /(a?) 

oLc 



(2' 



sn 



Since /(a?) is periodic, the coefficients a n , & n have the same value 
however c is chosen. If we make <?= TT, these integrals reduce 
to those given in 436. 

We may write 



1 /e+2 jr co 

_F= I / ()dt \ I 4 E(cos nx cos n^ -h sin 712; sm 

TTXc 1 

f r+ | cos n(* - x)\f(C)dt. 



Thus 

where 

Provided 

we may write 

1 

p= 



P. -I + 2 cos (-*). 
sin * -Wo, 



(3 

(4 
( 5 



. 
^j sin A"(c 



sn 



- *) +22 sin J( - a;) cos m(t - as) 



422 FOURIER'S SERIES 

P _ sin K2n + !)(*-*) 
n ~ ' ^ 



if 5) holds. Let us see what happens when 5) does not hold. 
In this case %(t x) is a multiple of TT. As both t and x lie in 
(0, c -h 2 TT), this is only possible for three singular values : 

t = x ; = <?, x = c -f 2 TT ; = <? -f- 2 TT, # = <?. 
For these singular values 4) gives 

As P n is a continuous function of , #, the expression on the 
right of 6) must converge to the value 7) as #, t converge to these 
singuLar values. We will therefore assign to the expression on 
the right of 6) the value 7), for the above singular values. Then 

in all cases . x _ 

Fm _ 1 p^8inK2 + l)Q- g ) /(0<ft> 

7T<X C 2 sin | ( 2:) 

Let us set 24-1 = t = 

Then !/*-*)+, sinw 

^ n = I f(x -f 2 w) c?w. 

Kj*\(.c-x) SI 111* 

Let us choose c so that 

c x = TT, 

then m /.o /.I 1 

^n=J = / +/ 

c^_7r X_E eC'O 

2 2 

Replacing w by u in the first integral on the right, it becomes 







~, N sin vu 7 

7(2? 2 u) au. 

smu 



Thus we get 

7T 

"1 /^2 * 

7rJ,x) * sin u 

Let us now introduce the term 2/(V) under the sign of inte- 
gration in order to replace the brace by </>(w). To this end let us 



VALIDITY OF FOURIER'S DEVELOPMENT 425 

where S3' is S3 r or S3 r _i 4- S3 r , depending on the parity of r. Now 

9\* (2 



du 



I xrv r f ( TT\ 1 

l^L^ {K)-^ + -Jj-n 



< 



'2*-l 



V + n) ~ 



*f 

oL/ 7 



du. 



< 



\9\- 



(4 



Thus J n = 0, if the three integrals 2), 8), 4) =^ 0. Moreover, 
if these three integrals are uniformly evanescent with respect to 
some point set & < 53, J n is also uniformly evanescent in &. In 
particular we note the theorem 

J n = 0, if g is L-integrable in 33. 

We are now in a position to draw some important conclusions 
with respect to Fourier's series. 

440. 1. Let f(x) be L-integrable in (c, <?H-27r). Then the 
Fourier constants a n , b n = 0, as n = oo. 



For 



= - / 1+- /(^ 

TTrtLc 



cos 



is a special case of the 7 n integral. Af is i-integrable, we need 
only apply the theorem at the close of the last article. Similar 
reasoning applies to b n . 

2. For a given value of x in 21 = ( TT, TT) let 



sin u 



be L-intecjrable in S3 = f 0, ]. Then Fourier s development is valid 
at the point x. 



426 FOURIER'S SERIES 

For by 438, Fourier's series =/(#) at the point #, if D n (x) 0. 
But D n is a special case of J n for which the g function is in- 
tegrable. We thus need only apply 439. 

3. For a given x in 21 = ( TT, TT), let 

X00 = *> (2 

be L-integrable in S3 = f 0, ^J. Then Fourier's development is valid 
at the point x. 

For let 8 > 0, then 

< 



= , as 8 = , by hypothesis. 
4. .For a ^iv6?i x in 21 = ( TT, TT), Z^^ 

^) (3 



6^ L-integrable in 21. ^Ae/i Fourier's development is valid at the 
point x. 






Thus x is Jv-integrable in f 0, - ), as it is the difference of two 
integrable functions. 

441. (Lebesgue). For a given x in 21 = ( TT, TT) let 
1 limn 

n= 

2 lim / '|^(M -f 8) - ^(w) | du = 

^f- 



VALIDITY OF FOURIER'S DEVELOPMENT 



427 



Then Fourier's development is valid at the point x. 
For as we have seen, 



- sin vu 



sin 



du 



du 



, i *' J \<t> (y) - 

-|_ I T. L sill VU 



( 7- 

Jiftn IS'' 



where /3 n is a certain number which = , as u = oc. 



Hence 



tr^ consider D f . Since ()<w< , we have O<J/M<TT 

~ 



sin ^^ 
sin w 



, 

f- T 



0<<7, T< 



^ v*u?l+ av\i\ H W 
1 M 1 *TT 

6 V 4 J 6 

; ^ j, 



6 



< i/, provided s > ^. 
But this is indeed so. For 



Hence 

Thus 
Fe 



-< VGH ^ -t 7T 

i-- f _i- T - 



> 1 >f , if i/>5. 

D f <v I |0| dw = 0, by hypothesis. 

~i/o 

to D' r . We have 



428 FOURIER'S SERIES 

Now / being L-integrable, 

--K) 



is i-integrable in (17, ~J. Thus 



lim = 0. 

n=*>J^ 

But by condition 2, ,. /^ _ n 



Thus lim IX' = 0. 

5=0 



Finally we consider D f ". But the integrand is an integrable 
function in f /3, ^ J . Thus it = as n = x> . 



442. 1. yAe validity of Fourier's development at the point x de- 
pends only on the nature off in a vicinity of x, of norm 8 as small as 
we please. 

For the conditions of the theorem in 441 depend only on the 
value of /in such a vicinity. 

2. Let us call a point x at which the function 

0( tt ) =/( + 2 u) +f(x - 2 u) - 2 /(a?) 
is continuous at u = 0, and has the value 0, a regular point. 

In 438, we saw that if # is a point of discontinuity of the first 
kind for /(#), then # is a regular point. 

3. Fourier's development is valid at a regular point x, provided 
for some rj 



lim 

3=0 






For at a regular point #, <j>(u) is continuous at w = 0, and = 
for u = 0. Now 

lim J 

A=0 A 



LIMITED VARIATION 429 

tr rr 

Thus 



C n 1 /** 

n I | <(V) | dw = TT I | <f> | 



Hence condition 1 of 441 is satisfied. 

Limited Variation 

443. 1. Before going farther we must introduce a few notions 
relative to the variation of a function f(x) defined over an interval 
21= (a < i). Let us effect a division D of 21 into subintervals, 
by interpolating a finite number of points a 1 < a%< The sum 

F,- 2 |/(a,) -/(+,) | (1 

is called the variation off in 21 for the division D. If 

Max V D (2 

is finite with respect to the class of all finite divisions of 21, we say 
f has finite variation in 21. When 2) is finite, we denote its value by 

Var/, or FJ,, or V 
and call it the variation off in 21. 

We shall show in 5 that finite variation means the same thing 
as limited variation introduced in I, 509. We use the term finite 
variation in sections 1 to 4 only for clearness. 

2. A most important property of functions having finite vari- 
ation is brought out by the following geometric consideration. 

Let us take two monotone increasing curves A, B such that one 
of them crosses the other a finite or infinite number of times. If 
/(#), g(x) are the continuous functions having these curves as 
graphs, it is obvious that 

d(x)=f(x)-g(x) 

is a continuous function which changes its sign, when the curves 
A, B cross each other. Thus we can construct functions in infinite 
variety, which oscillate infinitely often in a given interval, and 
which are the difference of two monotone increasing functions. 



430 FOURIER'S SERIES 

For simplicity we have taken the curves A, B continuous. A 
moment's reflection will show that this is not necessary. 

Since d(x) is the difference of two monotone increasing functions, 
its variation is obviously finite. Jordan has proved the following 
fundamental theorem. 

8. If f(x) has finite variation in the interval 31 = (a < 6), there 
exists an infinity of limited monotone increasing functions </(#), h(x) 
such that f = ff-h. (1 

For let D be a finite division of 21. Let 

P D = sum of terms 5/(0 OT+1 ) /()! which are > 0, 



Then +1 ) - /> \=P D + N D . (2 



Also 

(i) -/() ! + !/(> -/Oh) ! + - 

On the left the sum is telescopic, hence 

W)-M=PD-N D . (a 

From 2), 3) we have 

Fi = 2 P,, +/(;-/()= 2 ^+/(6) -/(a). (4 



Let now Mftx p^ = p ^ Max ^ = N 

with respect to the class of finite divisions D. 

We call them the positive and negative variation of /(#) in 3(. 
Then 4) shows that 



Adding these, we get j^ = p + j^ n; 

From 5) we have 

/(ft) _/() = P-JK (7 

Instead of the interval 21 = (<&), let us take the interval 
(a < re), where x lies in 21. Replacing f> by a; in 7), we have 

=/() + P(x) - N(x). (8 



LIMITED VARIATION 431 

Obviously P(v), N(x) are monotone increasing functions. 
Let v*(x) be a monotone increasing function in 21. If we set 



(9 



we get 1) from 8) at once. 
4. From 8) we have 

1/00 1 < 



5. We can now show that when f(x) has finite variation in the 
interval 21 = (a < 6) i Aas limited variation and conversely. 

For if / has finite variation in 21 we can set 



where </>, ^ are monotone increasing in 21. Then if 21 is divided 
into the intervals Sj, & 2 we have 

Osc/ < Osc <j) -\- Osc -^ , in S t . 

OSC < = A< , OSC i/r = Ai/r , ill S t 



since these functions are monotone. Hence summing over all the 
intervals S t , 



< some ^If, for any division. 
Hence / has limited variation. 
If f has limited variation in 21, 

A/ 1 < Osc/ , in 8 t . 

Hence 2 | A/ 1 < 2 Osc/ < some J!f. 

Hence /has finite variation. 

6. If f(x) has limited variation in the interval 21, its points of 
continuity form a pantactic set in 21. 

This follows from 5, and I, 508. 



432 FOURIER'S SERIES 

7. Let a<b<c; then iff has finite variation in (a, <?), 

V a<b f+ V b<c f= V a , c f, (11 

where V a% b means the variation off in the interval (a, i), etc. 
F r V M f= Max V D f 

with respect to the class of all linite divisions D of (a, <?). The 
divisions D fall into two classes : 

1 those divisions E containing the point 6, 

2 the divisions F which do not. 

Let A be a division obtained by interpolating one or more 
points in the interval. Obviously 

> V D f. 



Let now Gr be obtained from a division F by adding the point 

* Then v a f>v F f. 

Hence Max F A ->Max V F . 



E F 



Hence to find F^ c /, we may consider only the class E. Let 
now E l be a division of (a, J), and E^ a division of (J, c). Then 
j?! -f E^ is a division of class J?. Conversely each division of class 
E gives a division of (a, 6), (J, c?). Now 



From this 11) follows at once. 

444. We establish now a few simple relations concerning the 
variation of two functions in an interval SI = (a < i). 



For 

where for brevity we set f f< \ 

2 ' F(6f)=|t- Vf. (2 

For 



LIMITED VARIATION 



433 



3. Letf, g be monotone increasing functions in 21. Then 

- Vf+ Vg. 



(3 



4. _#V any two functions f, g having limited variation, 

V(f+g)<Vf+Vg. 

5. Letf,^ have limited variation in 21 = (a, 6). 



For by 443, 8) we have 
/=P 

where 
Thus 



- NP 



Hence by 2, 4, 

Vff\< 



- NAi + AP l - 

^ + VPA l +... 
...) , by 3 



But 



^(P + N+ a)(P, + ^ + 
Vf=P + N , hence, etc. 



(4 



(5 



445. Fourier's development is valid at the regular point x, if there 
exists a < f < , such that in (0, f) <Ae variation V(u) o 

e'w any (w, f ) z limited, and such that u V(u) ==0, u = 0. 
By 442, we have only to show that 



is evanescent with 8. 



484 FOUKIKR'S SERIES 



Let us first suppose that i/r(w) is monotone in some (0, f), say 
monotone increasing. Similar reasoning- will apply, if it is mono- 
tone decreasing. Then, taking < ?; -f & < f, 

^ =j\ty(u + )-ty(u)\du= I Vo + S )^- / VOO**- 
In the second integral from the end, set v = w -f- 8. 

/*? /Vf5 

Then I ^r(w -f B)du = I ty(v)dv. 

Hence, ,,,,+a ,,, 

^ = J ty(u)du - / ^(u)du 



< T 25 ^ | du+r**\ + \ du = ^ 4- %. 



Thus 

We will consider the integrals on the right separately. Let 

<H in (S, 28). 



smw 

Now . A 9 A / i 

sin n = u ar'u* , < o-' < ^ . 

Hence, ^ ^ 

= - + (TV , I o- 1 < some Jf. 



sin u u 



= , ;is S = , since <f>(u) = 0, 
as x is a regular point. 

We turn now to ^ 2 . In (/, 77 + 8), 5, 77 sufficiently small, 
sin w> w ^ w 8 > i/(l i; 2 ). 



LIMITED VARIATION 435 

Thus, if fa = Max j <f> \ in (17, rj + S), 



with 8. 

Thus, when i/r is monotone in some (0, f), Fourier's develop- 
ment is valid. But obviously when i/r is monotone, the condition 
that uV(u)^=Q is satisfied. Our theorem is thus established in 
this case. 

Let us now consider the case that the variation V(u) of ^ is 
limited in (u, f). 

From 443, 10), we have 



As before we have 



By hypothesis there exists for each e > 0, a 8 > 0, such that 
uV(u)<e , for any 0< u<S Q . 

Hence, 

V(u)<J- 
u 

1 liUS, 



Let us turn now to W^. Since V(u) is the sum of two limited 
monotone decreasing functions P, ^Vin (w, ^), it is integrable. 
Thus, 

f7?+6 /T)+6 

du + / V(u)du < 8 j I f (?) I + 
aLn 



is evanescent with S. 



436 FOURIER'S SERIES 

446. 1. Fourier s development is valid at the regular point x, if 
<}>(u) has limited variation in some interval (0 < f), 
For let < u < 7 < f, then 



NOW 



sin 

Hence V uy ty <_ J V uy <j> -f- 
But sin u being monotone, 



si i) w sin u sin 7 



Similarly, rr i , 

> yfty^ shl^ry = F 2 " 

Now 

0< u <M , in (0*, f). 
sin w 

The theorem now follows by 445. For we may take 7 so small 
that p 

Thus for any u < 7, 



On the other hand, Wl being sufficiently large, and 7 chosen as 
in 1) and then fixed, 

F 2 <2tt. 
Thus 



for w < some S r . Hence 

w 
for < u < some S. 

2. (Jordan.) Fourier's development is valid at the regular point 
X, iff (x) has limited variation in some domain of x. 



OTHER CRITERIA 437 

For 



+ 2 u)-/(u)j +{/(*- 2 )-/()} 

has limited variation also. 

3. Fourier's development is valid at every point of 31 = (0, 2 TT), 
iff is limited and has only a finite number of oscillations in 21. 



Other Criteria 
447. Let X= 



If X = as & == 0, so does "SP, awe? conversely. 

For x . ^ N xx .^ , 5,, sin(i6 -f S) 

- - 



where _ gm , 



Obviously X and "9 are simultaneously evanescent with 
provided 



Let 

/TV N sin 



u 
Then p = ^(u) \ Z(u + 8) - Z(u) I 

Now 

. , x v cos v sin 



Thus \Z'(v)\<Mv<M. 2. 



438 FOURIER'S SERIES 

Hence * i \ n/r 



" ' smw 

AS | *| <!/( + 2 w) | + |/(s - 2 w) | +2 

J?< 282ft M | =0 , with 8. 

448. (Lipschitz-Dini.^) At the regular point x, Fourier's devel- 
opment is valid, if for each e > 0, there exists a 8 > 0, such that for 
each < 8 < S , 

| <f>(u + 8) - <(w) | < ^_L_, /or any w in (8, 8 ). 

| log 8 | 

For 

(u + 8) - 



Now a: being a regular point, there exists an 17' such that 

| <(w) | < e, for u in any (8, 77'). 

Thus taking ^ , 

* 



- 

I log 8 I 8 rj 

< 2 6, for any 8 < 77. 

Thus v A s A 

X = 0, as o = 0. 



Uniqueness of Fourier's Development 
449. Suppose/ (x) can be developed in Fourier's series 

00 

/(a?) = a + 2(a n cos nz -f- t> n *\n nx), (1 

a n = - I /(^) ^os wa:rfa? , J n = _ / " f(x) sin n^dz, (2 

Tji-Tr 7TX-7T " 



UNIQUENESS OF FOURIER'S DEVELOPMENT 439 

valid in 21 = ( TT, TT). We ask can/(V) be developed in a simi- 
f (x) = I 0Q + 2(a n ' cos nx + b^ sin nz), (3 

also valid in H, where the coefficients are not Fourier's coefficients, 
at least not all of them. 

Suppose this were true. Subtracting 1), 3) we get 

= 2 ( a O a o) + 2 I ( a n #n) COS TiZ + (6 n - ftj) Sill nx \ = 0, 

c' -f- {tf n cos wo; 4- ^ n si* 1 n #J = 0, in 81. (4 

Thus it would be possible for a trigonometric series of the type 
4) to vanish without all the coefficients <? m , d m vanishing. 
For a power series 

Po + Pi* + Pv* + " ( 5 



to vanish in an interval about the origin, however small, we know 
that all the coefficients p m in 5) must = 0. 

We propose to show now that a similar theorem holds for a 
trigonometric series. In fact we shall prove the fundamental 

Theorem 1. Suppose it is known that the series 4) converges to 
for all the points of 3{ = ( TT, TT), except at a reducible set 3?. 
Then the coefficients c m , d m are all 0, and the series 4) = at all the 
points of 21. 

From this we deduce at once as corollaries : 

Theorem 2. Let 3J be a reducible set in 21. Let the series 

4- 2{a n cos nx 4- /3 n sin nx\ (6 

converge in 21, except possibly at the points 3?. Then 6) defines a 
function F(x) in 21 9?. 

If the series / , v^c / , m > 

J KQ + 2, \ cos nx + fin sin nx\ 

converges to F(jx) in 21 9i, its coefficients are respectively equal to 
those in 6). 

Theorem 3. If /(a?) admits a development in Fourier s series for 
the set 21 3t, any other development off(x} of the tt/pe 6), valid in 
9( 3? is necessarily Fourier's series, i.e. the coefficients m , j3 m have 
the values given in 2). 



440 FOURIER'S SERIES 

In order to establish the fundamental theorem, we shall make 
use of some results due to Riemann, Q-. Cantor, Harnack and 
Schwarz as extended by later writers. Before doing this let us 
prove the easy 

Theorem 4* If f(v) admits a development in Fourier's series 
which is uniformly convergent in 21 = ( TT, TT), it admits no other 
development of the type 3), which is also uniformly convergent in 21. 

For then the corresponding series 4) is uniformly convergent 
in 21, and may be integrated termwise. Thus making use of the 
method employed in 436, we see that all the coefficients in 4) 
vanish. 

450. 1. Before attempting to prove the fundamental theorem 
which states that the coefficients n , b n are 0, we will first show 
that the coefficients of any trigonometric series which converges 
in 21, except possibly at a point set of a certain type, must be such 
that they == 0, as n = oc. We have already seen, in 440, 1, that 
this is indeed so in the case of Fourier's series, whether it con- 
verges or not. It is not the case with every trigonometric series 
as the following example shows, viz. : 

2 sin n ! x. (1 

i 

When x = all the terms, beginning with the r I th , vanish, 
r \ 

and hence 1) is convergent at such points. Thus 1) is conver- 
gent at a pantactic set of points. In this series the coefficients a n 
of the cosine terms are all 0, while the coefficients of the sine 
terms b n , are or 1. Thus b n does not = 0, as n = oo. 

2. Before enunciating the theorem on the convergence of the 
coefficients of a trigonometric series to 0, we need the notion of 
divergence of a series due to Harnack. 

Let A = a l a 2 -\ (2 

be a series of real terms. Let# n , & n be the minimum and maxi- 
mum of all the terms 

A A 

^n+1 * -"-n+2 i * 

where as usual A n is the sum of the first n terms of 2). Obviously 



UNIQUENESS OF FOURIER'S DEVELOPMENT 441 

Thus the two sequences f# n }, { Q- n \ are monotone, and if limited, 
their terms converge to fixed values. Let us say 

ff* = ff , #n = #- 
The difference 

\> = &-ff 

is called the divergence of the series 2). 

3. For the series 2) to converge it is necessary and sufficient that 
its divergence b = 0. 

For if A is convergent, 

- + A< A n+p <A + e , ^ = 1,2... 
Thus -e + A<g n <G n <A + . 

Thus the limits Q-, g exist, and 

#-#<2e ; or# = #, 
as > is small at pleasure. 

Suppose now b = 0. Then by hypothesis, #, g exist and are 
equal. There exists, therefore, an n, such that 

#-*<<? n <# n <+, 
or #n~<7n<2e. 

Thus \A n+p -A n \<2e , p=l, 2 - 

and A is convergent. 

451, Let the series 

00 

2 (# n cos w# 4- b n sin w#) 



6e such that for each B > 0, there exists a subinterval of 

l = (-7r, TT) 

a^ eac?A jt>oir^ q/ ^AwA tY divergence b < S. TAett a n , i n = 0, as 

n = QO. 

For, as in 450, there exists for each x an m x , such that 

* 



cos M# + n sn 



m x 



442 FOURIER'S SERIES 

for any point x in some interval 93 of 21. Thus if b is an inner 
point of 23, x = b + /3 will lie in 33, if /3 lies in some interval 
B = (p, q). Now 

a n cos w ( b + /3; -f- b n sin n(b + @) 

= (a n cos rc6 -f & sin nb) cos w/3 (a n sin nb J n cos nb) sin ny8. 

a n cos n(b /3) -f # sin n(b yS) 

= (a n cos w6 -}- 6 n sin nb) cos n/3 4- (#n y i n ^^ ^n cos n &) 8 ^ n w ^- 
Adding and subtracting these equations, and using 1) we have 



| (a n cos nb -f b n sin 116) cos wy8 | < -, 

g 
| (a n sin nb 5 n cos w6) sin n/3 \ < , 

for all n>m x . Let us multiply the first of these inequalities by 
cos nb sin n/3, and the second by sin 716 cos n/3, and add. We get 

\a n sinn/3 l \<S , ^ = 2/3 , n > m x . (2 

Again if we multiply the first inequality by sin nb sin n/3, and 
the second by cos nb cos n/3, and subtract, we get 

| b n sin n/3 l \ < 8 , n>m x . (3 

From 2), 3), we can infer that for any e > 

| a n \ < e , \ b n | < e , n > some m, (4 

or what is the same, that a n , b n = 0. 

For suppose that the first inequality of 4) did not hold. Then 
there exists a sequence 

n i < n i < = (5 

such that on setting 

loJ^+Si, , e-8 = S' 
we will have 

S Wr > '. (6 

If this be so, we can show that there exists a sequence 

v \ < V 2 < " ^ 
in 5), such that for some /3 f in B, 

| a, f sin vfl | > 8, (7 



UNIQUENESS OF FOURIER'S DEVELOPMENT 443 

which contradicts 2). To this end we note that 7 > may be 
chosen so small that for any r and any | 7 | < 7 , 

I , | cos 7 > (S + 8') cos 7 > S. (8 

Let us take the integer i/ a so that 

" + 2 y 



. (9 

q-p 



Then 



Thus at least one odd integer lies in the interval determined by 

the two numbers 

o 9 

. O"i + 7o) -(?"!- 7o>- 

7T 7T 

Let mj be such an integer. Then 

9 2 

~O"i + 7o) < i < -(?"i - 7 ) 

7T 7T 

If we set 

\ 

' ft== v 



... 

( 



we see that the interval ^ 1 = (jt> 1 , q^) lies in 5. Tlie length of 
S l is 2 7 /J> r Then for any ^ in J5 X , 



Thus by 8), 

| a Vi sin ^/3 | = | a Vl | cos 7j > 8. (12 

But we may reason on B l as we have on JB. We determine v^ 
by 9), replacing JP, ^ by p^ q l . We determine the odd integer m 2 
by 10), replacing jp, ^, ^ t by p l ^ q l ^ v^. The relation 11) deter- 
mines the new interval J? 2 = (jt? 2 , <? 2 ), on replacing mj, ^j by w a , ^ 2 * 
The length of J5 2 is 27 /^ 2 , and JS 2 lies in J5 1B For this relation 
of 1/2 and for any y3 in 2 we have, similar to 12), 

a v sin j//9 > 8. 



In this way we may continue indefinitely. The intervals 
B l > J9 2 > = to a point ', and obviously for this /8', the rela- 



444 FOURIER'S SERIES 

tion 7) holds for any x. In a similar manner we see that if b n does 
not = 0, the relation 3) cannot hold. 

452. As corollaries of the last theorem we have : 
1. Let the series 

00 

2(# n cos nx + b n sin nx) (1 

be such that for each S > 0, the points in Sl = ( TT, TT) at which 
the divergence of 1) is >S, form an apantactic set in 21. Then 
a n , # n = 0, as ft = 00. 



2. Jvg A# series 1) converge in 21, except possibly at the points of 
a reducible set 5K. TA0w- a n , 5 n ^=0. 



For $R being reducible [318, 6], there exists in 21 an interval 93 
in which 1) converges at every point. We now apply 451. 

453. Let -r, / -. / . 7 x 

Jf(x) = 2(a n cos nx + 6 n sin w#) 

a^ ^Ae point* of 21 = ( TT, TT), where the series is convergent. At the 
other points of 21, let F(x) have an arbitrarily assigned value, lying 
between the two limits of indetermination g, Gr of the series. If F is 
R-integrable in 21, the coefficients a n , i n = 0. 

For there exists a division of 21, such that the sum of those in- 
tervals in which Osc F > co is < a. There is therefore an interval 
3 in which Osc F < G>. If $ is an inner interval of 3, the di- 
vergence of the above series is < o> at each point of $. We now 
apply 451. 

454. Riemanrfs Theorem. 

00 

Let F(x) = # + 2(a n cos nx -f- b n sin nx) = 2^4 n converge at 
each point of 21 = ( TT, ?r), except possibly at the points of a redu- 
cible set 9Z. The series obtained by integrating this series termwise, 
we denote by 



- (a n cos wa; + 5 B sin r) = 
Gr is continuous in 21. 



UNIQUENESS OF FOURIER'S DEVELOPMENT 445 

Let <E>(V) = G(x + 2 u) + G(x - 2 w) - 2 G(x). (1 

Then at each point of S3 = 31 9?, 

lim ^& = F(x) ; (2 

M=O 4 u* 

and at each point of 31, 

w=0 16 

For, in the first place, since 9? is a reducible set, # n , 5 n = 0. The 
series Gr is therefore uniformly convergent in 31, and is thus a 
continuous function. 

Let us now compute <>. We have 

a n cos n(x -h 2 u) -f # n cos n(# 2 M) 2 a n cos nx 
= 2a n cos 7i# (cos 2nu - 1) 
= 4 a n cos Tta: sin 2 m^. 

Also J n sin 7^(2; -f 2 w) -f ^ n sin n(x - 2 w) - 2 6 ft sin wa; 

97 /- ^-v -| -v 

^ Q sin ny\ cos ij wti I ) 

= 4 b n sin wa; sin 2 nu. 

Thus (^=2^ f^iiL^ 

4 y? o n V nu 

if we agree to give the coefficient of A$ the value 1. Let us 
give x an arbitrary but fixed value in S3. Then for each > 0, 
there exists an m such that 

A Q -f A 1 -f 4- Ai-i = -^0*0 4- e n , | e n | < , n>_m. 

Thus ^4 n = n+1 - n . 

Hence ^ 



, S /- N /sinnw\ 2 

X + 2 (6 n4 .j - n ) 

i \ nu J 

, ^ ffsin (ti l)w"l 2 rsium/l 2 ] .r,, 

-f 2 e n ^ } - (4 

i IL (w-l> J L WM J I 



446 FOURIER'S SERIES 

The index m being determined as above, let us take u such that 

u < , so that m < ; 
m u 

and break S into three parts 



1 m+l K+I 

where K is the greatest integer < TT/U, and then consider each sum 
separately, as u = 0. 

Obviously lim S l = 0. 

u=0 

As to the second sum, the number of its terms increases indefin- 
itely as u = 0. 
For any ?/, 



. 

< 



fTsin wwH 2 ("si 
IL mu J L 

fsin mu~] 2 ^ 

- < , 

L mu J 



since each term in the brace is positive. In fact 

sin v 
v 

is a decreasing function of v as v ranges from to TT, and 
nu<KU<_ f jr , n = m, m 4- 1, tc. 

Finally we consider S%. We may write the general term as 
follows : 



jTsin (ft 1)?/T 2 _ fsin (n l)wT 

iL~c-i)M J ~L "J 

ML "nn "JL"^]}' 



sin 2 (n l)w sin 2 rm __ sin (2 n l)w sin u 

"" 



UNIQUENESS OF FOURIER'S DEVELOPMENT 447 

Thus w f -, 1 x -, 

\s <-T f l l } 4-ivJL 

' 8 ~ 






since 



But # >. 1 , or feu > TT u. 

u 

Thus 

< 6 



I I t 

(TT ?*) 2 TT wj 
Hence S=S, + ^+ S t = 0, as u = 0, 

which proves the limit 2), on using 4). 
To prove the limit 8), we have 

f. \2^ ^ 



nu 



Let us give u a definite value and break T into three suras. 

w 

I 
where m is chosen so that 

| A n < , n > m ; 

A 

flry y 

where X is the greatest integer such that 

Xw <. 1 ; 

and 

A+l 

Obviously for some M, 

T 2 | <. U\ < , 

since 

/sin m,ii. \- 



448 FOURIER'S SERIES 

As to the last sum, 



<eX.l , sincel<X, 



<. 

Thus 



455. Schwarz-Luroth Theorem. 

In 3( =(a < J) fo the continuous function f (x) be such that 

S(x, u) =^ x + M) +/( V M > ~ 2/( -^ 0, as u = 0, (1 

I/' 

except possibly at an enumerable set (S i?i 21- At the points (, let 

uS(x, u) = as u = 0. (2 

2%e/& / i a linear function in 21. 

Let us first suppose with Schwarz that (g = 0. We introduce 
the auxiliary function, 

g(x) = 7/i(V) -\c(x- a)(x - J), 
where 

Z(*) =/(*) -/(a) - |/(6) -/(a) I, 



?/ = 1, and c is an arbitrary constant. 

The function g(x) is continuous in 21, and </() = <7(&) = 0. 

Moreover 



Thus for all < w < some S, 

(3 



From this follows that g(x)<Q in 21. For if #(V)>0, at any 
point in 21, it takes on its maximum value at some point within 21. 

Thus 



for < ^ < 8, & being sufficiently small. Adding these two in- 
equalities gives #<.0, which contradicts 3). Thus#j<0 in 21. 
Let us now suppose L ^= f or some x in 21. We take c so small 

that T r 

= sgn rjL = 77 sgn L. 



UNIQUENESS OF FOURIER'S DEVELOPMENT 449 

But rj is at pleasure 1, hence the supposition that L 3= is 
not admissible. Hence L = in 31, or 

/("O-'/OO-f {/(*)-/() I (4 

is indeed a linear function. 



8 now suppose with Liiroth that (>0. We introduce the 
auxiliary continuous function. 



Thus A(a) = , A(6) = <?( J - a) 2 . 

Suppose at some inner point of 1 

Z,(0. (5 

This leads to a contradiction, as we proceed to show. For then 



provided 

O= 



We shall take c so that this inequality is satisfied, i.e. c lies in 
the interval 6 = (0*, (7*). Thus 



Hence A(V) takes on its maximum value at some inner point e 
of 21. Hence for 8 > sufficiently small, 

<S. (6 



u 2 
Now if ^ is a point of 31 @, 

lira JI(e, u) = 2 c > 0. 

M = 

But this contradicts 7), which requires that 



u=0 



450 FOURIER'S SERIES 

Hence 6 is a point of @. Hence by 2), 



By 6), both terms have the same sign. Hence each term = 0. 
Thus for u > 

= lim A?J^J^ZZL.. .\?2 = lim S^ e u ' 

+ 2c(e a). 
H puff 

/0) = ft _^ + 6'O-a). ( 

Thus to each c in the interval E, corresponds an in @, at which 
point the derivative of f(x) exists and has the value given on the 
right of 8). On the other hand, two different c's, say c and c r , in 
cannot correspond to the same e in @. 

For then 8) shows that 

c \e ""*"" ct j c \e ~" a i^ 
or as ^ , __ 

Thus there is a uniform correspondence between S whose cardi- 
nal number is c, and ( whose cardinal number is e, which is absurd. 
Thus the supposition 5) is impossible. In a similar manner, the 
assumption that L < at some point in 21, leads to a contradiction. 
Hence L = in 21, and 4) again holds, which proves the theorem. 

456. Cantor's Theorem. Let 

00 

^ a -f- 2(a n cos nx + b n sin nx) (1 

i 

converge to in 2l = ( TT, TT), except possibly at a reducible set 9?, 
where nothing is asserted regarding its convergence. Then it con- 
verges to at every point in 21, and all its coefficients 

111 ___ A 

For by 452, 2, a n , b n = 0. Then Riemann's function 

/ O) = 4 V 2 - 2 "^ ( a n COS Wir + J n Sin WSC) 

f n ^ 



UNIQUENESS OF FOURIER'S DEVELOPMENT 451 

satisfies the conditions of the Schwarz-Liiroth theorem, 455, since 9? 
is enumerable. Thus f(x) is a linear function of x in SI, and has 
the form a + /3x. Hence 

1 
a + fix | a z 2 = 2} (a n cos w# + b n sin nz). (2 

The right side admits the period 2 TT, and is therefore periodic. 

Its period o> must be 0. For if <o > 0, the left side has this 
period, which is absurd. Hence <w = 0, and the left side reduces 
to a constant, which gives /3=0, # = 0. But in 21 9i, the right 
side of 1) has the sum 0. Hence a= 0. Thus the right side of 
2) vanishes in 21. As it converges uniformly in 31, we may deter- 
mine its coefficients as in 436. This gives 



CHAPTER XIV 
DISCONTINUOUS FUNCTIONS 

Properties of Continuous Functions 

457. 1. In Chapter VII of Volume I we have discussed some 
of the elementary properties of continuous and discontinuous 
functions. In the present chapter further developments will be 
given, paying particular attention to discontinuous functions. 
Here the results of Baire * are of foremost importance. Le- 
besgue f has shown how some of these may be obtained by sim- 
pler considerations, and we have accordingly adopted them. 

2. Let us begin by observing that the definition of a continu- 
ous function given in I, 339, may be extended to sets having iso- 
lated points, if we use I, 339, 2 as definition. 

Let therefore/^ # w ) be denned over 21, being either limited 
or unlimited. Let a be any point of 21. If for each e > 0, there 
exists a S > 0, such that 

I/O) /O) I < > for any x in F 6 (a), 
we say fis continuous <at a. 

By the definition it follows at once that f is continuous at each . 
isolated point of 21. Moreover, when a is a proper limiting point 
of 21, the definition here given coincides with that given in I, 339. 
If /is continuous at each point of 21, we say it is continuous in 21. 
The definition of discontinuity given in I, 347, shall still hold, 
except that we must regard isolated points as points of con- 
tinuity. 

* " Sur les Functions de Variables reeles" Annali di Mat., Ser. 3, vol. 3 
(1899). 

Also his Lemons sur les Functions Discontinues. Paris, 1906. 
t Bulletin de la Societe Mathematique de France, vol. 32 (1904), p. 229. 

452 



PROPERTIES OF CONTINUOUS FUNCTIONS 453 

3. The reader will observe that the theorems I, 350 to 354 
inclusive, are valid not only for limited perfect domains, but also 
for limited complete sets. 

458. 1. If f(v\ "- %m) * 8 continuous in the limited set 21, and its 
values are known at the points of S3 < 21, then f is known at all 
points of S3' lying in 21. 

For let 6 1 , # 2 , 6 3 be points of S3, whose limiting point b lies 
in 21. Then 



2. If f is known for a dense set S3 in 21, and is continuous in 21, 
f is known throughout 21. 

For 9 ,>^ 

3. If f(&i "- %m) continuous in the complete set 21, the points 
S3 in 21 where f= c, a constant, form a complete set. If 21 is an 
interval, there is a first and a last point of S3. 

For/= c at # = a v 2 which = ; we have therefore 



459. The points of continuity S of ./(^i #) in 21 lie in a 
deleted enclosure . If 21 is complete, $ . 

For let e x > e 2 > == 0. For each e n , and for each point of 
continuity c in 21, there exists a cube O whose center is c, such that 

Osc/< e n , in O. 

Thus the points of continuity of / lie in an enumerable non- 
overlapping set of complete metric cells, in each of which 
Osc/< e n . Let O n be the inner points of this enclosure. Then 
each point of the deleted enclosure 

= Dv\& n \ 

which lies in 21 is a point of continuity of /. For such a point c 
lies within each Q n . 

HenCC Osc/<e, inF.O* 

for S > sufficiently small and n sufficiently great. 



454 DISCONTINUOUS FUNCTIONS 

Oscillation 
46 - Let a> s = Osc/iX ... * m ) in Fa (a). 

This is a monotone decreasing function of 8. Hence if co$ is 

finite, for some S > 0, 

co = lim o) 6 

5=0 

exists. We call co the oscillation off at x = a, and write 

a) = Osc/. 

ar=a 

Should o> 5 = -f- oo, however small 8 > is taken, we say co = -f- oo. 
When co = 0, / is continuous at a: = a, if a is a point in the domain 
of definition of f. When co > 0, f is discontinuous at this point. 
It is a measure of the discontinuity off at x = a ; we write 



461. 1. 



# = a. 

| d - e | < Disc (f ^) < rf 4- e. 

For in F fi (a), 



| Osc/ - Osc // | < Osc (fff)< Ow/ -f Osc #. 

2. Iffis continuous at x = a, while Disc ^7 = d, then 
Disc (/ + g) = d. 

x=a 

For /being continuous at a, Disc/= 0. 
Hence 



< Disc ^ rf- 

3. If c is a constant, 

Disc ((?f ) = | c? | Disc/ , at z = a. 
For Osc ((?/) =1*1 Osc/ , in any F 6 ( 

4. When the limits 



OSCILLATION 455 

exist and at least one of them is different from /(#), the point x 
is a discontinuity of the first kind, as we have already said. 
When at least one of the above limits does not exist, the point x 
is a point of discontinuity of the second kind. 

462. 1. The points of infinite discontinuity 3? of f, defined over 
a limited set 2l,/0rm a complete set. 

For let *], * 2 be points of 3, having k as limiting point. 
Then in any V(k) there are an infinity of the points * n and hence 
in any V(k), OSC/= + OQ. The point k does not of course 
need to lie in 21. 

2. We cannot say, however, that the points of discontinuity of 
a function form a complete set as is shown by the following 

Example. In 21 = (0, 1), let /(V) = x when x is irrational, and 
= when x is rational. Then each point of 21 is a point of dis- 
continuity except the point x = 0. Hence the points of disconti- 
nuity of /do not form a complete set. 

3. Let f be limited or unlimited in the limited complete set 21. 
The points & 0/21 at which Ose/>. k form a complete set. 

For let a v a 2 be points of $ which = a. However small 
S >0 is taken, there are an infinity of the a n lying in V^a). But 
at any one of these points, Osc/_> &. Hence Osc/>.& in Fa (a) 9 
and thus a lies in $. 

4. Letf(x^ - XM) be limited and R-integrable in the limited set 21. 
The points $ at which OSG f >_k form a discrete set. 

For let D be a rectangular division of space. Let us suppose 
St D > some constant c > 0, however D is chosen. In each cell 8 
of D, 

Osc/>*. 

Hence the sum of the cells in which the oscillation is :> k can- 
not be made small at pleasure, since this sum is $ D > But this 
contradicts I, 700, 5. 

5. Let /(#! x m } be limited in the complete set 21. If the points 
$ in 21 at 'which Osc/> k form a discrete set, for each k, then f is 
R-integrable in 21. 



456 DISCONTINUOUS FUNCTIONS 

For about each point of 21 $ as center, we can describe a cube 
& of varying size, such that Osc/< 2 k in (. Let D be a cubical 
division of space of norm d. We may take d so small that 
$t D = 2d t is as small as we please. The points of 21 lie now within 
the cubes S and the set formed of the cubes d,. By Borel's 
theorem there are a finite number of cubes, say 



such that all the points of 21 lie within these T/'S. If we prolong 
the faces of these rfs, we effect a rectangular division such that 
the sum of those cells in which the oscillation is > 2 k is as small 
as we choose, since this sum is obviously < $ D . Hence by I, 700, 
5, f is J2-integrable. 

6. Letf(x l x m ) be limited in 21; let its points of discontinuity 
in 21 be 33. If f is R-integrable, 33 is a null set. If 21 is complete 
and 5) is a null set, f is .R-inte</rable. 

Let / be R-integrable. Then 3) is a null set. For let l > e 2 
> ... = 0. Let J) n denote the points at which Osc/> e n . Then 
33 = {33 W |. But since/ is .R-integrable, each S) n is discrete by 4. 
Hence 33 is a null set. 

Let 21 be complete and 3) a null set. Then each ) n is complete 
by 3. Hence by 365, ) n = SD n . As ) = 0, we see 33 n is discrete. 
Hence by 5, /is jR-integrable. 

If 21 is not complete, / does not need to be 72-integrable when 
3) is a null set. 

Example. Let2l 1 =(-l , n= 1, 2 ... ; m< 2 n . 
n 

' "=1,2...; r<3'. 



Let /(*) = : ' at *=f; 

= 1 in a,. 



Then each point of 21 is a point of discontinuity, and ?l = 3D. 
But 2lj , 212 are null sets, hence 21 is a null set. 



POINTWISE AND TOTAL DISCONTINUITY 457 

On the other hand, 



and / is not ./2-integrable in 21. 



Pointwise and Total Discontinuity 



463. Let/Oj x m ) be defined over 31. If each point of 21 is a 
point of discontinuity, we say /is totally discontinuous in 21. 

We say f is pointwise discontinuous in 21, if f is not continuous 
in 21= \a\, but has in any V(a) a point of continuity. If/ is 
continuous or pointwise discontinuous, we may say it is at most 
pointwise discontinuous. 

Example 1. A function/^ # m ) having only a finite number 
of points of discontinuity in 21 is pointwise discontinuous in 21. 

Example 2. Let 

f(x) = , for irrational x in 21 = (0, 1) 

1 /. m 

= - , lor x = 
n n 

= 1 , for x =0,1. 

Obviously /is continuous at each irrational #, and discontinuous 
at the other points of 21. Hence / is pointwise discontinuous 
in 21. 

Example 3. Let 3) be a Harnack set in the unit interval 
21 = (0, 1). In the associate set of intervals, end points included, 
let/(V)=l. At the other points of 21, let /= 0. As > is 
apantactic in 21, /is pointwise discontinuous. 

Example 4- In Ex. 3, let 35 = (g -f gs where S is the set of end 
points of the associate set of intervals. Let/=l/?i at the end 
points of these intervals belonging to the n th stage. Let/= in 
g. Here / is defined only over 35. The points g are points of 
continuity in 3). Hence/ is pointwise discontinuous in 3). 

Example 5. Let/(#) be Dirichlet's function, i.e. /= 0, for the 
irrational points 3 in 21 = (0, 1), and = 1 for the rational points. 



458 DISCONTINUOUS FUNCTIONS 

As each point of 21 is a point of discontinuity,/ is totally discon- 
tinuous in 21. Let us remove the rational points in 21 ; the deleted 
domain is 3- I n this domain/ is continuous. Thus on removing 
certain points, a discontinuous function becomes a continuous 
function in the remaining point set. 

This is not always the case. For if in Ex. 4 we remove the 
points g-, retaining only the points (, we get a function which is 
totally discontinuous in @, whereas before / was only pointwise 
discontinuous. 

464. 1. Iff(x l XM) is totally discontinuous in the infinite com- 
plete set 21, then the points b w where 

Disc/>o> , o>>0, 
form an infinite set, if a> is taken sufficiently small. 

For suppose b w were finite however small co is taken. Let 
tt) 1 >o) 2 >.-- =0. Let Dj, D 2 , be a sequence of superposed 
cubical divisions of space of norms d n = 0. We shall only con- 
sider cells containing points of 21. Then if d l is taken sufficiently 
small, D l contains a cell Sj, containing an infinite number of 
points of 21, but no point at which Disc/>a> r If d? 2 is taken 
sufficiently small, 7> 2 contains a cell S 2 <Sj, containing no point 
at which Disc/>o> 2 . In this way we get a sequence of cells, 



which == a point p. As 21 is complete, p lies in 21. But / is 
obviously continuous at p. Hence / is not totally discontinuous 
in 21. 

2. If 21 is not complete, b w does not need to be infinite for 
any o> > 0. 

Example. Let 21 = j [ , n = 1, 2, and m odd and <2 n . At 

1 

~, let/= Then each point of 2l is a point of discontinuity. 

*j Zi 

But b w is finite, however small &>>0 is taken. 

3. We cannot say /is not pointwise discontinuous in complete 
2l, when bo, is infinite. 



EXAMPLES OF DISCONTINUOUS FUNCTION'S 459 

Example. At the points | - t = % let / = ; at the other 

\.n J 

points of l = (0, 1), let/=l. 

Obviously / is pointwise discontinuous in 21. But b w is an 
infinite set for co < 1, as in this case it is formed of 5ft, and the 
point 0. 

Examples of Discontinuous Functions 

465. In volume I, 330 seq. and 348 seq.> we have given ex- 
amples of discontinuous functions. We shall now consider a few 
more. 

Example 1. Riemanrfs Function. 

Let (x) be the difference between x and the nearest integer; 
and when x has the form n + |, let (x) = 0. Obviously (x) has 
the period 1. 

It can be represented by Fourier's series thus : 

s *. If sin 2 TTX sin 2 2 TTX , sin 3 2 TTX 



Riemanri 8 function is now 

This series is obviously uniformly convergent in 21 = ( oo, oo). 
Since (#) has the period 1 and is continuous within (--o> |)> 

we see that (nx) has the period -, and is continuous within 

n 

( , ]. The points of discontinuity of (nx) are thus 

\ 2 n 2 nj 

, = 0, 1, 2, ... 



n 

Let (= S@ n J. Then at any x not in (g, each term of 2) is a con- 
tinuous function of x. Hence F(x) is continuous at this point. 

On the other hand, F is discontinuous at any point e of @. For 
F being uniformly convergent, 

m^ (3 

* 



L lira F(x) = 2 lira - (4 

x=e x=e n 



460 DISCONTINUOUS FUNCTIONS 

We show now that 3) has the value 

^(0) _ JlL, for e = ^ l , e irreducible. (5 

16w 2 2 n 

and 4) the value 

^> + i^ (6 

Hence 2 

-- (7 

* 



To this end let us see when two of the numbers 

1 , r A 1 . s 

+ -, and - + - m*n 
2mm 2 n n 

are equal. If equal, we have 

2r-f 1 2s + 1 



m n 

Thus if we take 2 s -f- 1 relatively prime to n, no two of the num- 
bers in @ n are equal. Let us do this for each n. Then no two of 
the numbers in S are equal. 

1 8 

Let now x = e = h - Then (mx) is continuous at this point, 

2 n n 

unless 8) holds; i.e. unless m is a multiple of w, say m= In. Ju 
this case, 8) gives 



Thus I must be odd ; l^ 1, 8, 5 ... In this case (mx) = at 0, 
while jRlim (T?IX)= . When w is not an odd multiple of n, 

jr=e 

obviously 72 lim (w#) = (me). 

xe 

Thus when m = /n, Z odd, 

^ lim (m:r) = 1 -L=:^)-1I 1. 
^=, m 2 2 

When m is not a multiple of n, 



EXAMPLES OF DISCONTINUOUS FUNCTIONS 461 

Hence 

--l k +l + l + ...} 



This establishes 5). Similarly we prove 6). Thus F(x) is 
discontinuous at each point of @. As 



F is limited. As the points (S form an enumerable set, F is 
.R-integrable in any finite interval. 

466. Example 2. Let/(V)=0 at the points of a Cantor set 
(7 = m a^ - ; ra = 0, or a positive or negative integer, and the 
a's = or 2. Let /(#) = 1 elsewhere. Since /(#) admits the 

period l,/(3 nx) admits the period -- Let O l be the points of 

o n 

G which fall in 21 = (0, 1). Let D l be the corresponding set of 
intervals. Let (7 2 = Cj -f Fj, where F x is obtained by putting a 
O l set in each interval of D l . Let Z> 2 be the intervals correspond- 
ing to <7 2 . Let (7 3 = 6 7 2 + F 2 where F 2 is obtained by putting a <7 2 
set in each interval of Z> 2 , etc. 

The zeros of/(3naO are obviously the points of C Y n . Let 



The zeros of F are the points of g = { C n \. Since each C n is a null 
set, & is also a null set. Let A = 21 &. The points -4, S are 
each pantactic in 31. Obviously F converges uniformly in 21, 
since 0</(3 nx) <1. Since / n (V) is continuous at each point a 
of A, F is continuous at a, and 



462 DISCONTINUOUS FUNCTIONS 

We show now that F is discontinuous at each point of . For 
let e m be an end point of one of the intervals of D m+l but not of 
D m . Then 



Hence F(e) = H m = + - + 2 

Jr m* 

As the points A are pantactic in 31, there exists a sequence in 
.4 which = e. For this sequence F = H. Hence 



Similarly, if rj m is not an end point of the intervals D m+v but a 
limiting point of such end points, 



The function F is Jf2-integrable in 21 since its points of discon- 
tinuity form a null set. 

467. Let @ = i0 tl ...,J ^ an enumerable set of points lying in the 
limited or unlimited set 31, which lies in 9t m . For any x in 31 and 
any e^ in (, let x e l lie in 33. Let g(x l # m ) J^ limited in 33 
continuous, except at x = 0, 



= b. 
(7= 2c ... converge absolutely. Then 



is continuous in A = 21 (g, aradf at x= e^ 

Disc.F( = ^- 

For when a; ranges over 21, x e t remains in S3, and g is limited 
in 93. Hence F is uniformly and absolutely convergent in 21. 

Now each g(x e^) is continuous in A ; hence F is continuous 
in A by 147, 2. 



EXAMPLES OF DISCONTINUOUS FUNCTIONS 463 

On the other hand, -Fis discontinuous at #= e K . For 

where If is the series F after removing the term on the right of 
the last equation. But JET, as has just been shown, is continuous 

at x = e K . 

468. Example 1. Let @=j0 n j denote the rational numbers. 

Let <V) = sin- x 

x 

= , x=0. 
Then F(x}=V(x-e^ >1 

^fJL 

is continuous, except at the points S. At x = e n , 

Disc F = ^- 

Example 2. Let @ = J^ n } denote the rational numbers. 
Let xx v nx -i 



- n=00 i + nx 

= , ^ = 0, 
which we considered in I, 331. 

Then -r,/ ^ 



is continuous, except at the rational points, and at x = e m , 

(x)= - 
ml 

469. In the foregoing g(x) is limited. This restriction may be 
removed in many cases, as the reader will see from the following 
theorem, given as an example. 

Let JE = Se tl ... t J be an enumerable apantactic set in 2(. Let (g = 
(j?, _Z7'). For any x in 81, and any e L in J?, let x e L lie within a 
cube S3. Let g(x l x m ) be continuous in 53 except at # = 0, where 
g = -f- QO, as x == 0. Let ^Lc^.., it be a positive term convergent series. 



464 DISCONTINUOUS FUNCTIONS 

Then , ~ . 



is continuous in A = 21 G. <9>? /*e of Aer Aand, <m'7j, jt>0m 0/ S is a 
point of infinite discontinuity. 

For any given point x a of A lies at a distance >0 from (g. 
Thus 



as x ranges over some F^a), and e t over E. 

I fPHPf* i i -mm- 

1 u ct | #O - ^ ) | < some M, 

for 2* in F^a), and t in #. Thus jF 7 is uniformly convergent at 
x = a. Ax each //(a? e k ) is continuous at x = a, J 7 is continuous 
at a. 

l/tf n^xf #= ^. Then there exists a sequence 

x', x" -. = e K (1 

whose points lie in A. Thus the term g(x e^) = -f oo as a; ranges 
over 1). Hence a fortiori ^ = 4-00. Thus each point of J? is a 
point of infinite discontinuity. 

Finally any limit point of E is a point of infinite discontinuity, 
by 462, l. 

470. Example. Let g(x) = , a n = , a>l. 

y> . 

n ~^ y 

Then -n/ N ^ / >, 



is a continuous function, except at the points 

- 1 - 1 - 1 ... 

v, ~, ~, 

a a* a 6 
which are points of infinite discontinuity. 

471. Let us show how to construct functions by limiting 
processes, whose points of discontinuity are any given complete 
limited apantactic set & in an m-way space 3J m . 



EXAMPLES OF DISCONTINUOUS FUNCTIONS 465 

1. Let us for simplicity take m = 2, and call the coordinates of 
a point a?, y. 

Let Q denote the square whose center is the origin, and one of 
whose vertices is the point (1, 0). 

The edge of Q is given by the points x, y satisfying 

\x\+\y\ = l. (1 

Thus 1 f|, on the edge 



i 
, outside 

of the square Q. Hence 

- (I: 



Th " 



2. We next show how to construct a function g which shall = 
on one or more of the edges of Q. Let us call these sides e 1 ^ e%, 3 , 4 , 
as we go around the edge of Q beginning with the first quadrant. 
If 6? = on e t , let us denote it by Gr t ; if Gr = on e t , e K let us 
denote it by Q- il( , etc. We begin by constructing Q- r We observe 

that 

-. ,. n\t\ fl, for^ = 0, 

1 lim - ' = 4 

n=00 1 + n 1 1 \ 10, for t ^ 0. 

Now the equation of a right line I may be given the form 

x cos a -f- y sin = p 
where < a < 2 TJ% p > 0. Hence 

Z( X , v) = 1 - lim n l* CQS( * + ff sin -y | = |1, on Z, 
n= 1 -h ?i | a; cos a + y sin a p \ (0, off L 

If now we make I coincide with e we see that 



E, (x, y) = 2Z(x,y)L (*, y} = 
Hence 



> n . 

1 0, on j and without Q. 



466 DISCONTINUOUS FUNCTIONS 

In the same way, 



#1234 = 9 ~ (#! + ^2 + ^S + ^4>- 

By introducing a constant factor we can replace Q by a square 
Q e whose sides are in the ratio c : 1 to those of Q. 

the ed g e of Qc, 



Q (*, y) = li 




*= 1 + M i + ) 10, outside. 
\ <? c J ' 

We can replace the square Q by a similar square whose center 
is a, b on replacing | a; |, | y | by | 2; a , | y J |. 

We have thus this result : by a limiting process, we can con- 
struct a function g(x, y) having the value 1 inside Q, and on any 
of its edges, and = outside $, ail( i on the remaining edges. 
Q has any point a, b as center, its edges have any length, and its 
sides are tipped at an angle of 45 to the axes. 

We may take them parallel to the axes, if we wish, by replacing 



x 



in our fundamental relation 1) by 



\ x -y\ -> \* + y\* 

Finally let us remark that we may pass to m-way space, by re- 
placing 1) by 

Kl + l*al + - + |^| = 1. 

3. Let now Q = jq n | be a border set [328], of non-overlapping 
squares belonging to the complete apantactic set E, such that 
Q -|- g = 3t the whole plane. We mark these squares in the 
plane and note which sides q n has in common with the preceding 
q's. We take the g n (xy) function so that it is = l in q n , except 
on these sides, and there 0. Then 



G(x, y) = 

has for each point only one term ^ 0, if x, y lies in Q, and no 
term = if it lies in . 



0, for each point of S. 



EXAMPLES OF DISCONTINUOUS FUNCTIONS 467 

Since E is apantactic, each point of & is a point of disconti- 
nuity of the 2 kind ; each point of Q is a point of continuity. 

4. Let /(#/) be a function defined over 21 which contains the 
complete apantactic set S. 

Then 9 - * 



472. 1. Let 21 = (0, 1), S8 n = the points m i n . 

^ n 

Then 93 n , S3,, have no points in common. 
Let/ n (z) = 1 in 93 n , and = in n = 21 - S3 n . 
Let93={93 n J. Then 

*<>-*> -(!:*-- 

The function F is totally discontinuous in 93, oscillating be- 
tween and 1. The series F does not converge uniformly in 
any subinterval of 21. 

2. Keeping the notation in 1, let 



At each point of 93 n , Gr= -, while # = in J5. 

w 



The function 6r is discontinuous at the points of S3, but con- 
tinuous at the points JB. The series 6r converges uniformly in 
21, yet an infinity of terms are discontinuous in any interval in 21. 

473. Let the limited set 21 be the union of an enumerable set 
of complete sets S2l n j. We show how to construct a function/, 
which is discontinuous at the points of 21, but continuous else- 
where in an w-way space. 

Let us suppose first that 21 consists of but one set and is com- 
plete. A point all of whose coordinates are rational, let us call 
rational, the other points of space we will call non-rational. If 21 
has an inner rational point, let /= 1 at this point, on the frontier 
of 21 let /= 1 also ; at all other points of space let /= 0. Then 
each point a of 21 is a point of discontinuity. For if x is a fron- 



468 DISCONTINUOUS FUNCTIONS 

tier or an inner rational point of 2l,/(#) = 1, while in any V(x) 
there are points where /= 0. If x is not in 21, all the points of 
some D(x) are also not in 21. At these points /= 0. Hence /is 
continuous at such points. 

We turn now to the general case. We have 

81 = ^+^2 + ^3 + ... 

where A 1 = $l 1 i A% = points of 21 2 not in 2lj, etc. Let/j = 1 at the 
rational inner points of A, and at the frontier points of 2lj ; at all 
other points let /j = 0. Let / 2 = at the rational inner points of 
-4 2 , and at the frontier points of A^ not in A l ; at all other points 
let/ 2 = 0. At similar points of A Q let/ 3 = |, and elsewhere = 0, 
etc. 

Consider now & *?* , \ 

* = VnOV'-Zm)- 

Let x = a be a point of 21. If it is an inner point of some A t , 
it is obviously a point of discontinuity of F. If not, it is a proper 
frontier point of one of the A* a. Then in any D(a) there are points 
of space not in 21, or there are points of an infinite number of the 
As. In either case a is a point of discontinuity. Similarly we 
see F is continuous at a point not in 21. 

2. We can obviously generalize the preceding problem by sup- 
posing 21 to lie in a complete set S3, such that each frontier point 
of 21 is a limit point of A = S3 21. 

For we have only to replace our m-way space by 83. 

Functions of Class I 

474. 1. Baire has introduced an important classification of 
functions as follows : 

Let /(#!#) be defined over 21; /and 21 limited or unlimited. 
If /is continuous in 21, we say its class is in 21, and write 

Class /=0 , orCl/=0 , Mod 21. 
If 



each/ n being of class in 21, we say its class is 1, if/ does not lie 
in class 0, mod 21. 



FUNCTIONS OF CLASS 1 469 

2. Let the series F(x) = TLf n (x) 

converge in 21, each term/ n being continuous in 21. Since 



we see F is of class 0, or class 1, according as F is continuous, or 
not continuous in 31. A similar remark holds for infinite prod- 

ucts 



3. The derivatives of a function f(x) give rise to functions of 
class or 1. For let f(x) have a unilateral differential coeffi- 
cient g(x) at each point of 21. Both / and 21 may be unlimited. 
To fix the ideas, suppose the right-hand differential coefficient 
exists. Let 7^ > 7^ 2 > = 0. Then 



n 



is a continuous function of x in 21. But 

2O)=lim9 n < 

flnoo 

exists at each x in 21 by hypothesis. 

A similar remark applies to the partial derivatives 

&L, ... J. 

dx 1 ' dx m 
of a function /(a?j a; n ). 

4 - Let 



each/ n being of class 1 in 21. Then we say, Cl/= 2 if /does not 
lie in a lower class. In this way we may continue. It is of 
course necessary to show that such functions actually exist. 

475. Example 1. 

Let f(x \ ^ lim __^_ = I !' for * > ' 

J ^ ' =. 1 + nx I 0, for x = 0. 

This function was considered in I, 331. In any interval 
21 = (0 < b) containing the origin x = 0, Cl/= 1 ; in any inter- 
val (a < i), a > 0, not containing the origin, Cl/= 0. 



470 DISCONTINUOUS FUNCTIONS 

Example 2. 

Let /(*) = lirnM j = 0, in - (-00,00). 

n=oo 

The class of f(x) is in 21. Although each f n is limited in 21, 
the graphs of f n have peaks near x = which == oo, as n = oo. 

Example 3. If we combine the two functions in Ex. 1, 2, we 

get */- \ r f 1 ,11 f 1* forz^O, 

f(x) = hm ^ \nx = 1 ' 

' v y nss30 1 l + nx e nx * I ( 0, for x=0. 

Hence C\f(x) = 1 for any set 33 embracing the origin; =0 
for any other set. 

Example 4> 

Let j?/ \ v *+r^ -or ^A -IN 

/ (a?) = Inn are n , in 21 = (0, 1). 

n=<x> 

Then /(a?) = , for x = 

i 
= a:^ 2 , for x > 0. 

We see thus that / is continuous in (0*, 1), and has a point of 
infinite discontinuity at x = 0. 

Hence Class /(af)= 1, in 21 

= 0, in(OM). 
Example 5. 

Let / O) = Hm -^r in 2T = (0, oo). 

w ~ '>* -L 

n 
Then ..... = 1 fora . >0 



= -h oo , for x = 0. 
Here lim/ n (V) 



does not exist at x = 0. We cannot therefore speak of the class 
of /(#) in 21 since it is not defined at the point x = 0. It is 
defined in 93 = (0*, oo), and its class is obviously 0, mod 33. 



FUNCTIONS OF CLASS 1 



471 



Example 6. 
Let 



f(x) = sin - , for x & 
x 



= a constant c , for x = 0. 
We show that Cl/= 1 in 21 = (- 0, oo). For let 



-f nxj 



\ , nx 
+ ~ - 8in 



o, 



r * 
lim A n 



Now by Ex. 1, 



while 

im n a; = 

0, for o;=0. 
As each f n is continuous in 3, and 

lim / n (*)=/(*) in , 

we see its class is <_ 1. As / is discontinuous at = 0, its class 
is not in 21. 



Example 7. Let , ^ ,. 1 .1 
r f(x) = lim - sin - 

= n x 



Here the functions f n (x) under the limit sign are not defined 
for x = 0. Thus /is not defined at this point. We cannot there- 
fore speak of the class of / with respect to any set embracing the 
point =0. For any set S3 not containing this point, Cl /= 0, 
since /(x) = in SB. 

Let us set 



. . N . 1 . A 

<f>(x) =s sin - , for x * 
x 



= a constant c 



for x 0. 



Let 



g(x) = lim -<t>(x) = lim 



472 DISCONTINUOUS FUNCTIONS 

Here g is a continuous function in 21 = ( oo, oo). Its class is 
thus in 31. On the other hand, the functions < n are each of 
class 1 in 31. 

Example 8. 



is defined at all the points of (00, oo) except 0, 1, 2, 
These latter are points of infinite discontinuity. In its domain 
of definition, F is a continuous function. Hence Cl F(V) = 
with respect to this domain. 

476. 1. If 31, limited or unlimited, is the union of an enumerable 
set of complete sets, we say 31 is hyper complete. 

Example 1. The points S* within a unit sphere S, form a 
hypercomplete set. For let S r have the same center as S, and 
radius r<\. Obviously each 2 r is complete, while J2 r j = /S Y *, r 
ranging over r l < r% < = 1. 

Example 2. An enumerable set of points a l , a 2 form a hyper- 
complete set. For each a n may be regarded as a complete set, 
embracing but a single point. 

2. 7/31J, 31 2 --- are limited hypercomplete sets, so is their union 

;3u = 3i. 

For each 3l m is the union of an enumerable set of complete sets 
Sl w , n . Thus 31 = j3l ;n , n S m, n = 1, 2 .- is hypercomplete. 

Let 31 be complete. If S3 is a complete part of 31, A = 31 93 is 
hypercomplete. 

For let O= Jq n | be a border set of 93, as in 328. The points 
A n of A in each q n are complete, since 31 is complete. Thus 
A=\A n \, and A is hypercomplete. 

Let 21= \tyi n l be hypercomplete, each 3l n being complete. If $8 is 
a complete part of 31, A = 31 93 is hypercomplete. 

For let A n denote the points of 3l n not in 93- Then as above, 
A n is hypercomplete. As A = \A n \, A is also hypercomplete. 



FUNCTIONS OF CLASS 1 473 

477. 1. @ e Sets. If the limited or unlimited set 21 is the union 
of an enumerable set of limited complete sets, in each of which 
Osc/<e, we shall say 21 is an (g e set. If, however small e>0 is 
taken, 21 is an @ set, we shall say 21 is an (g c set, e = 0, which we 
may also express by @^o' 

2. Let /(x!'" XK) be continuous in the limited complete set 21. 
Then 21 is an S set, e == 0. 

For let e > be taken small at pleasure and fixed. By I, 353, 
there exists a cubical division of space D, such that if 2l n denote 
the points of 21 in one of the cells of D, Osc/< e in 2l n . As 2l n is 
complete, since 21 is, 21 is an ( e set. 

3. An enumerable set of points 21 = \ a n \ is an (S^ set. 

For each a n may be regarded as a complete set, embracing but 
a single point. But in a set embracing but one point, Osc/= 0. 

4. The union of an enumerable set of @ e sets 21 = J2l m | is an (g e set. 
For each 2t m is the union of an enumerable set of limited sets 

2l m = J2l m , n {,n=l, 2,... and Osc/< e in each 2U- 

Thus a = j8U} , , rc=l, 2,-.. 

But an enumerable set of enumerable sets is an enumerable set. 
Hence 21 is an S e set. 

5. Letf(x l # w ) be continuous in the complete set 21, except at the 
points 3) = dj, rf 2 d a . Then 21 is an &=M> set. 

For let 6>0 be taken small at pleasure and fixed. About each 
point of 3) we describe a sphere of radius p. Let 2l p denote the 
points of 21 not within one of these spheres. Obviously 2l p is com- 
plete. Let p range over r l > r 2 > - = 0. If we set 21 = A + 3), 
obviously ^ = {2i r J. As/ is continuous in 2l rw , it is an ( set. 
Hence 21, being the union of A and 35, is an @ set. 

478. 1. Let 21 be an (:,, set. The points 3) of 21 common to the 
limited complete set $8 form an @ e set. 

For 21 is the union of the complete sets 2l n , in each of which 
Osc/<. But the points of 2l n in 93 form a complete set -4 n , and 
of course Osc/< e in A n . As 3) = \ A n \, it is an (g e set. 



474 DISCONTINUOUS FUNCTIONS 

2. Let 21 be a limited @ e set. Let 93 be a complete part of 21. 
Then A = 21 - $ is an @ e *tf. 

For 21 is the union of the complete sets 2l n , in each of which 
Osc /<. The points of 2l n not in 33 form a set J. n , such that 
Osc /< in A n also. But A = \A n \, arid each J[ n being hyper- 
complete, is an S e set. 

3. Let/(^ x # m ) be defined over 21, either/ or 21 being limited 
or unlimited. The points of 21 at which 

<*</< (1 

may be denoted by 

(</<) (2 

If in 1) one of the equality signs is missing, it will of course be 
dropped in 2). 



479. 1. Letfi,/^, "-be continuous in the limited complete set H. 
If at each point of 21, Urn f n exists, 21 is an @ e=M) set and so is any 

complete 33 < 21. 

For let lirn f n (x l z m ) =f(x l # m ) in 21. Let us effect a 

n=oo 

division of norm e/2 of the interval ( 00, oo ) by interpolating 
the points m_ 2 , m_ : , w = 0, m l , m 2 
Let 2t t = (m t </< m t+2 ), then 21 = |21J. 

Next let ^ TS f , 1 / ^ ^ 11 

>n, P = ^ w, +-</,< ^ l+2 - - 
>P I n n) 

Then 2l t =535 n , P S , ^^ = 1,2- (1 

For let a be a point of 2l t , and say f(a) = a. Then 

m t < a < w t + 2 . 

But a e</ g (a) <a-f e , j>somejt?, 

and we may take e and n so that 



Hence a is in S) nip . 

Conversely, let a be a point of {$) n , p }. Then a lies in some 
nfp . Hence, 



FUNCTIONS OF CLASS 1 475 

But as/ n (a) ==/(#), we have 



Hence if e is sufficiently small, 



and thus a is in 2l t . 

Thus 1) is established. But ) np is a divisor of complete sets, 
and is therefore complete. Thus 21 is the union of an enumerable 
set of complete sets J93 t j, in each of which Osc/<e, e small at 
pleasure. 

Let now 93 be any complete part of 21. Let a t = Dv J93, 93 t }. 
Then a t is complete, and Osc/<e, in a t - Moreover, 93 = {a t |. 

Hence 93 is an @ =M) set. 

2. // Class /< 1 m limited complete 21, / limited or unlimited, 
21 is ^w (S se. 

This is an obvious result from 1. 

3. Let /(#! # m ) ^ # totally discontinuous function in the non- 
enumerable set 21. Then Class /is ## or 1 iw- 21, i/* b = Disc/a^ 
^(?7i point is < k > 0. 

For in any subset 93 of 21 containing the point #, Osc / > k. 
Hence Osc/is not <e, in any part of SI, if e < &. Thus 21 cannot 
be an ( e set. 

4. J/ Class /(#! a^ m )<. 1 fw the limited complete set 21, the set 
93 = (#</< 5) is a hyper complete set, a, b being arbitrary numbers. 

For we have only to take a = m t , S = m l+2 . Then 93 = 2l t , which, 
as in 1, is hypercomplete. 

480. {Lebesgue.} Let the limited or unlimited function f (x^ # m ) 
be defined over the limited set 21. If 21 may be regarded as an 
(Se^o set with respect to /, the class of f is < 3 . 

For let o) 1 > G> 2 >-.== 0. By hypothesis 21 is the union of a 
sequence of complete sets 

2l u , 2I 12 , 5ffi3*" ($1 

in each of which Osc / <_&> r 21 is also the union of a sequence 
of complete sets 

u - < %- (1 



476 DISCONTINUOUS FUNCTIONS 

in each of which Osc/< <o 2 . If we superpose the division 1) of 
?I on the division S l ^ each 2l tlt will fall into an enumerable set 
of complete sets, and together they will form an enumerable 
sequence 

2l a i * 2122 > IM*" (^2 

in each of which Osc/<LG> 2 . Continuing in this way we see that 
21 is the union of the complete sets 



such that in each set of $ n , ()sc/< o> n , and such that each set lies 
in some set of the preceding sequence S n _ lf 

With each 2l w , , we associate a constant (7 njt , such that 

!/(*)- C^|<*> n , in?l n ,, (2 

and call C nt the corresponding field constant. 

We show now how to define a sequence of continuous functions 
/i'/2 '" which =/. To this end we effect a sequence of super- 
imposed divisions of space Dj, D^ of norms = 0. The vertices 
of the cubes of D n we call the lattice points L n . The cells of D n 
containing a given lattice point I of L n form a cube Q. Let 3l lti 
be the first set of S l containing a point of Q. Let 2l 2 t 2 be the first 
set of # 2 containing a point of Q lying in 3l ltl . Continuing in 
this way we get 

2l ltl >i2l 2l ,>...>2l nln . 

To 2l nln belongs the field constant O nln ; this we associate with 
the lattice point I and call it the corresponding lattice constant. 

Let now S be a cell of D n containing a point of 21. It has 2 n 
vertices or lattice points. Let P 9 denote any product of & differ- 
ent factors a; n , x r ^ x rg . We consider the polynomial 



<f> = AP n + ^BP n ^ + 2 (7P n _ 2 4- - 4- 

the summation in each case extending over all the distinct 
products of that type. The number of terms in </> is, by I, 96, 



FUNCTIONS OF CLASS 1 477 

We can thus determine the 2 n coefficients of <f> so that the values 
of (f> at the lattice points of are the corresponding lattice con- 
stants. Thus <f> is a continuous function in , whose greatest and 
least values are the greatest and least lattice constants belonging 
to . Each cube containing a point of 21 has associated with it 
a <f> function. 

We now define /(#!-#,) ^y stating that its value in any 
cube of 7) n , containing a point of 21, is that of the correspond- 
ing <f> function. Since <f> is linear in each variable, two </>'s belong- 
ing to adjacent cubes have the same values along their common 
points. 

We show now that/ n (>) ==f(x) at any point x of 21, or that 

e >0, v, |/O) -f n (x) | < , n > v. (3 

Let co e < e/8. Let 3I ltl be the first set in /Si containing the point x, 
2l 2l , the first set of S 2 lying in 2l ltl and containing x. Continuing 
we get ^ > ^ > ^ ^ > ^ 

Let ty e be the union of the sets in &\ preceding 2l u ; of the sets in 
$2 preceding 2l 2t and lying in 2l lt , and so on, finally the sets of 
S e preceding 2l et , and lying in 2l e _ liV _ 1 . Their number being 
finite, 8= Dist (2l eta , $*) is obviously > 0. We may therefore 
take v > e so large that cubes of D v about the point x lie wholly 
in !>(, rj < S. 

Consider now/ n (#), n > v, and let us suppose first that x is not 
a lattice point of /> n . Let it lie within the cell of D n . Then 
f n (x) is a mean of the values of 



where Z is any one of the 2 n vertices of , and C njn is the corre- 
sponding lattice constant, which we know is associated with the 

**,*. 

We observe now that each of the 



For each set in S n is a part of some set in any of the preceding 
sequences. Now 2l n?n cannot be a part of 2l 1Jk , k < ij, for none of 



478 DISCONTINUOUS FUNCTIONS 

these points lie in A,(X). Hence 2l n; - n is a part of 2l ltl . For the 
same reason it is a part of 8l 2l2 , etc., which establishes 4). 
Let now x' be a point of 21^ . Then 

I c nja - t\. \<\o nia -/<>') | + !/(*')- o ett i 

<a, B +o, e <l , by 2). (5 

From this follows, since / (a;) is a mean of these C njii , that 

l/n(0-QJ<|- (6 

But now 

\f O) -MX) | < \f O) - C njn | + G nin -/< | . (7 

As x lies in 8k ta , 

I/O) - C njn | < /(*) - O^ I + I (7 ft . - (7 n;n | 

<.+|<|, (8 

by 2), 5). From 6), 8) we have 3) for the present case. 

The case that a; is a lattice point for some division and hence 
for all following, has really been established by the foregoing 
reasoning. 

481. 1. Let fie defined over the limited set 21. If for arbitrary 
a, 5, the sets 93 = (a </< 6) are hyper 'complete , then Class /< 1. 

For let us effect a division of norm e/2 of (00, oo) as in 
479,1. Then 2t=J2lJ, where as before 2l t = (m t </< ra t + 2 ). 
But as Osc/<e in 2l t , and as each 2l t is hypercomplete by 
hypothesis, our theorem is a corollary of 480. 

2. For f(x l # m ) to be of class < 1 in the limited complete set 
21, it is necessary and sufficient that the sets (a <f < 6) are hyper- 
complete, a, b being arbitrary. 

This follows from 1 and 479, 2. 

3. Let limited 21 be the union of an enumerable set of complete sets 
j, such that Cl/< 1 in each 2l n , then Cl/< 1 in 21. 



FUNCTIONS OF CLASS 1 479 

For by 479, 1, ?l n is the union of an enumerable set of complete 
sets in each of which Osc/ < e. Thus SI is also such a set, i.e. an 
iS e set. We now apply 480, 1. 

4. If Class/ < 1 in the limited complete set 31, its class is < 1, 
in any complete part 33 of SI. 

This follows from 479, 1 and 480, 1. 

482. 1. Let f(x l x m ) be defined over the complete set SI, and 
have only an enumerable set S of points of discontinuity in 31. 
Then Class/ = 1 in 31. 

For the points JE of 31 at which Osc/ > e/2 form a complete 
part of 31, by 462, 3. Bat E, being a part of (, is enumerable 
and is hence an @ g set by 477, 3. Let us turn to 33 = 31 E. For 
each of its points b, there exists a 8 > 0, such that Osc/ < in 
the set b of points of 93 lying in -Z> 5 (6). As 31 is complete, so is b. 
As E is complete, there is an enumerable set of these b, call them 
bj, b 2 , such that 33 = \b a \. As 31 = S3 4- E, it is the union of 
an enumerable set of complete sets, in each of which Osc/< e. 
This is true however small e>0 is taken. We apply now 480, 1. 

2. We can now construct functions of class 2. 

Example. Let f n (x l x m )= 1 at the rational points in the 
unit cube }, whose coordinates have denominators < n. Else- 
where let/ n = 0. Since f n has only a finite number of discontinu- 
ities in Q, Cl/ n = 1 in Q. Let now 



At a non-rational point, each f n = 0, .-. /=0. At a rational 
point, / n = l for all w > some s. Hence at such a point /= 1. 
Thus each point of Q is a point of discontinuity and Disc/= 1. 
Hence Cl/ is not 1. As / is the limit of functions of class 1, its 
class is 2. 

483. Let f(x l x m ) be continuous with respect to each # t , at each 
point of a limited set 31, each of whose points is an inner point. 
Then Class /<!. 



480 DISCONTINUOUS FUNCTIONS 

For let 21 lie witliin a cube Q. Then A = Q. 21 is complete. 
We may therefore regard 21 as a border set of A ; that is, a set of 
non-overlapping cubes }q n |. We show now that C1/<1 in any 
one of these cubes as q. To this end we show that the points 93 m 
of q at which 

a+-<f<b- 
m ' m 

form a complete set. For let b l , b% be points of 33 m , which = /8. 
We wish to show that /3 lies in S3 m . Suppose first that i,, b s+1 
have all their coordinates except one, say x, the same as the coordi- 
nates of /3. Since 

* + -</(. + P)<*--, 

m m 

therefore - - 



m p=*> m 



As/ is continuous in a^, and as only the coordinate x l varies in 
< we have 



m m 

Hence lies in 33 m . 

We suppose next that b 8 < b g+l have all their coordinates the 
same as /3 except two, say x l , # 2 . 

We may place each b n at the center of an interval t of length S, 
parallel to the x l axis, such that 

+* -*</<>)<&-- + , 
m m 

since /is uniformly continuous in x^, by I, 352. These intervals 
cut an ordinate in the x, x 2 plane through y8, in a set of points 
c t+p which == ft. Then as before, 



m m 



As is small at pleasure, /3 lies in S3 m . In this way we may 
continue. 

As Cl/< 1 in eacli q n , it is in 21, by 481, 3. 



FUNCTIONS OF CLASS 1 481 

484. (Volterra. ) Let J\,f ti be at most point wist' discontinuous 
in the limited complete set 21. Then there exists a point of ?I at 
which all thef n are continuous. 

For if 21 contains an isolated point, the theorem is obviously 
true, since every function is continuous at an isolated point. Let 
us therefore suppose that 21 is perfect. 

Let e 1 > 2 >---=0. Let a l be a point of continuity of / r 

Then Osc/^6 , insome2I 1 =F 5l (a 1 ). 

In 2^ there is a point b of continuity of f r Hence Osc/j < e 2 
in some F^J), and we may take b so that K,(6)<2l r But in 
Fif(J) there is a point a 2 at which / 2 is continuous. Hence 

Osc/! < 6 2 , O*c/ 2 < e l , in some 21 2 == F,( a a)' 

and we may take <z 2 such that 21 2 < 2lj . Similarly there exists a 
point a 3 in 21 2 , such that 

Osc/t < 3 , Osc/ 2 < e 2 , Osc/ 3 < l , in some 21 3 = ^(^3)^ 

and we may take 8 so that 21 8 < 21 2 . 

In this way we may continue. As the sets 2l n are obviously 
complete, Dv\ty. n \ contains at least one point a of 21. But at this 
point each/ m is continuous. 

485. 1. Let 21 = 33 + & be complete, let 33, & be pantactic with 
reference to 21. Then there exists no pair of functions /, g defined 
over 21, such that if 33 are the points of discontinuity of f in 21, then 
33 shall be the points of continuity of g in 21. 

This is a corollary of Volterra's theorem. For in any Fi(a) of 
a point of 21, there are points of 33 and of @. Hence there are 
points of continuity of /and g. Hence/, g are at most pointwise 
discontinuous in 21. Then by 484, there is a point in 21 where/ 
and g are both continuous, which contradicts the hypothesis. 

2. Let 21= 33 -hS be complete, and let 33, & each be pantactic with 
reference to 21. If 33 is hypercomplete, is not. 

For if 33, ( were the union of an enumerable set of complete 
sets, 473 shows that there exists a function / defined over 21 
which has 33 as its points of discontinuity ; and also a function g 



482 DISCONTINUOUS FUNCTIONS 

which has 6 as its points of discontinuity. But no such pair of 
functions can exist by 1. 

3. The non-rational points $ in any cube Q cannot be hyper- 
complete. 

For the rational points in jQ are hypercomplete. 

4. As an application of 2 we can state : 

The limited function /(^ 1 --.^ OT ) which is < at the irrational 
points of a cube Q, and > at the other points 3 of ' Q, cannot be 
of class or 1 in Q. 

For if Cl/ < 1, the points of O where/ > must form a hyper- 
complete set, by 479, 4. But these are the points 3>. 



486. 1. (Saire.) If the class off^x^-'-x^) is 1 in the com- 
plete set 21, it is at most pointwise discontinuous in any complete 

<a. 

If Cl/= 1 in 31, it is < 1 in any complete 53 < 21 by 481, 4 ; we 
may therefore take 95 = 21. Let a be any point of 21. We shall 
show that in any V= Fi(a) there is a point c of continuity of f. 
Let e l > e 2 > = 0. Using the notation of 479, i, we saw that 
the sets 2l l = (m l </< w t + 2 ) are hypercomplete. By 473, we can 
construct a function ^>,(x l # m ), defined over the w-way space 
9{ w which is discontinuous at the points 2I t , and continuous else- 
where in 9t m . These functions c^, < 2 are not all at most point- 
wise discontinuous in V. For then, by 484, there exists in F a 
point of continuity J, common to all the <'s. This point b must 
lie in some 2l t , whose points are points of discontinuity of < t . 

Let us therefore suppose that fy is not at most pointwise dis- 
continuous in V. Then there exists a point c l in F", and an ^ 
such that V^ = ^(^j) contains no point of continuity of <y. 
Thus Fi<8k. But in 21,- and hence in V^ Osc /<e r The 
same reasoning shows that in V 1 there exists a F^= ^,(^2)' suc ^ 
that Osc/< 2 in F^. As 21 is complete, V l > F 2 > defines a 
point <? in Fat which /is continuous. 

2. If the class off(x l # m ) is 1 in the complete set 21, its points 
of discontinuity Qform <t set of the first category. 



FUNCTIONS OF CLASS 1 483 

For by 462, 3, the points ) n of 3D at which Osc/> - form a 

n 

complete set. Each ) n is apantactic, since / is at most pointwise 
discontinuous, and O n is complete. Hence 35 = \ O ft } is the union 
of an enumerable set of apantactic sets, and is therefore of the 1 
category. 

487. 1. Let f be defined over the limited complete set 21. If 
Class / is not < 1, there exists a perfect set 35 in 21, such that f is 
totally discontinuous in ). 

For if G\f is not <1 there exists, by 480, an e such that for 
this e, 31 is not an S e set. Let now c be a point of 21 such that 
the points a of 21 which lie within some cube q, whose center is tf, 
form an (g e set. Let 93 = JaJ, 6 = \c\. 

Then 93 = S. For obviously (<93, since each c is in some 
a. On the other hand, 93 < S. For any point b of 93 lies within 
some q. Thus b is the center of a cube q' within q. Obviously 
the points of 21 within q' form an < e set. 

By Borel's theorem, each point c lies within an enumerable set 
of cubes {c n |, such that each c lies within some q. Thus the 
points a n of 21 in c n , form an @ e set. As S = }a w j, is an S e set. 

Let 35 = 21 - (. If 35 were 0, 21 = and 21 would be an @ e set 
contrary to hypothesis. Thus 3) > 0. 

3) is complete. For if I were a limiting point of 3) in 6, I must 
lie in some c. But every point of 21 in c is a point of ( as we saw. 
Thus I cannot lie in g. 

We show finally that at any point d of 35, 

Osc/>, with respect to . 

If not, Osc/< with respect to the points b of 35 within 
some cube q whose center is d. Then b is an S e set. Also the 
points c of in q form an S e set. Thus the points b -f e, that is, 
the points of 21 in q form an (< set. Hence d belongs to , and 
not to 3). As Osc/>e at each point of 3D, each point of 35 is a 
point of discontinuity with respect to 3). Thus/ is totally discon- 
tinuous in 35. 

This shows that 35 can contain no isolated points. Hence 35 is 
perfect. 



484 DISCONTINUOUS FUNCTIONS 

2. Let f be defined over the limited complete set 21. If f is at 
most pointwise discontinuous in any perfect 93 < 21, its class is < 1 
w2l. 

This is a corollary of 1. For if Class / were not 0, or 1, there 
exists a perfect set 33 such that /is totally discontinuous in 33. 

488. If the class of f, g < 1 in the limited complete set 21, the class 
of their sum, difference, or product is < 1 . If f > in 21, the class 
of <f> = 



For example, let us consider the product h =fg. If Cl h is not 
< 1, there exists a perfect set 33 in 21, as we saw in 487, 1, such 
that A is totally discontinuous in 33. But/, g being of class <, 1, 
are at most pointwise discontinuous in 33 by 486. Then by 484, 
there exists a point of 33 at which/, g are both continuous. Then 
h is continuous at this point, and is therefore not totally discon- 
tinous in 33. 

Let us consider now the quotient <f>. If Cl (j> is not < 1, <f> is 
totally discontinuous in some perfect set ) in 21. But since /> 
in ), / must also be totally discontinuous in >. This contradicts 
486. 

489. 1. Let F = 2/ ti ...<,(#! # TO ) converge uniformly in the com- 
plete set 21. Let the class of each termf L be <, 1, then Class F < 1 

MI a. 

For setting as usual [117], 



there exists for each e > 0. a fixed rectangular cell R^ such that 
| JF\ | < e, as x ranges over 21. (2 

As the class of each term in F K is < 1, Cl JP A < 1 in 21. Hence 
21 is an @ set with respect to F^ 

From 1), 2) it follows that 21 is an (g e set with respect to F. 

2. Let F = n/^...^^ # m ) converge uniformly in the complete 
set $. If the class of eachf, is < 1, then Cl F < 1 in 21. 



SEMICONTINUOUS FUNCTIONS 485 

Semicontinuous Functions 

490. Let /(#! Xm) be defined over 21. If a is a point of SI, 
Max/ in Fi(a) exists, finite or infinite, and may be regarded as a 
function of S. When finite, it is a monotone decreasing function 
of 8. Thus its limit as S = exists, finite or infinite. We call 
this limit the maximum off at x = a, and we denote it by 

Max/. 

x=a 

Similar remarks apply to the minimum of /in F^(a). Its limit, 
finite or infinite, as 8 == 0, we call the minimum of f at x = a, and 
we denote it by 

Min/ 

x=a 

The maximum and minimum of /in F$(a) may be denoted by 
Max/ , Min/. 

a, a, 6 

Obviously, Max ( _ /} = _ Min/, 

:r=a .r=a 

Min (-/)=- Max/ 

.r=a #=a 

491. Example 1. -, 

/(*0 = iin(-l, 1) , fo 

x 

= , for x =0. 
Then Max/= -f oo , Min/= - oo. 

ar=0 ^=0 

Example 2. - 

/(^) = sin - in (1,1) , f 
x 

= , for x = 0. 
Then Max/=l , Min/=-l. 

X=0 X=Q 

Examples. /(a; ) = 1 in (- 1, 1) , for**0 

= 2 , for a; = 0. 

Then Max/=2 , Min/=l. 



486 DISCONTINUOUS FUNCTIONS 

We observe that in Exs. 1 and 2, 

Hm/==Max/ , lim/==Min/; 

r=0 x=0 #=0 x-Q 

while in Ex. 3, _ _ 

lim/= 1 , arid hence Max/> lira/. 



Also lim/= Min/. 



#=0 



Example 4* i 

/(&) = (^ + 1) sin- in (-1,1) , for 



= - 2 , f or rr = 0. 
Here =l , Min/ =-2, 



x=Q -c=0 

Examples. Lei f(x ) = x , for rational * in (0, 1) 

= 1 , for irrational x. 
Here Max/=l , Min/=0, 

lhn/=l. 

492. 1. For M to be the maximum of f at x = a< it is necessary 
and sufficient that 

1 e > 0, S > 0, /< < M+ e, for any x in F(a) ; 
2 there exists for each e > 0, and in any V&(a), a point a such 

M- </(). 

These conditions are necessary. For M is the limit of Max/ 
in Fi(a), as S =^= 0. Hence 

> 0, o > 0, Max/ < 

a, 

But for any x in Fi(a), 

/O) < Max/. 

a, 



SEMICONTINUOUS FUNCTIONS 487 

Hence f(x)<M+e , x in F 5 (a), 

which is condition 1. 

As to 2, we remark that for each e > 0, and in any F$(a), 
there is a point a, such that 

- +Max/ </(). 

a, 6 

But M< Max/. 

a, 8 

Hence 



which is 2. 

These conditions are sufficient. For from 1 we have 

Max/ < 7tf + e, 

a, 6 

and hence letting 8 = 0, 



since e > is small at pleasure. 
From 2 we have 

Max/> M-e, 

a, 6 

and hence letting 8=0, 

Max/ > M. (2 

From 1), 2) we have M = Max/. 



2. J^or m to be the minimum of f at x = a, i is necessary and 
sufficient that 

1 e > 0, 8 > 0, m e </(.r), /or any x in T 5 (a) ; 

2 ^a there exists for each e > 0, rtm# in any Fi(a), a point a 
such that 

/(a) < m -f . 

493. When Max/ = /(a), we say / is supracontinuous at x = a. 

ar=a 

When Min/ = /(a), we say / is mfracontinuous at a. When/ is 

ar=a 

supra (infra) continuous at each point of SI, we say / is supra 
(infra) continuous in 21. When /is either supra or infracontinu- 
ous at a and we do not care to specify which, we say it is semi- 
continuous at a. 



488 DISCONTINUOUS FUNCTIONS 

The function which is equal to Max /at each point x of 21 we 
call the maximal function of/ and denote it by a dash above, viz. 
/(. Similarly the minimal function /(V) is defined as the value 
of Min /at each point of 21. ~~ 

Obviously Ogc/ = Max/ _ Min/ = Digc/ 

xa xa xa x=a 

We call 7 

(*)=/()-/() 

the oscillatory function. 

We have at once the theorem : 

For f to be continuous at x = a, it is necessary and sufficient that 

/(a) =/()= /(a). 
F r Min / < /() < Max /. 

a, 6 a, 6 

Passing to the limit x = a, we have 

Min /</(<*)< Max/, 

x^a x=a 

or 



/oo < 

But for / to be continuous at x = a, it is necessary and suffi- 
cient that 

(V) = Osc/= 0. 

x=a 

494. 1. For f to be supracontinuous at x = a, it is necessary and 
sufficient that for each e > 0, there exists a S > 0, such that 

/(V) < /(a) -f e , for any x in F 6 (a). (1 

Similarly the condition for infracontinuity is 

/(a) e < f(x) , for any x in F fi (a). ' (2 

Let us prove 1). It is necessary. For when /is supracontinu- 
ous at a, 



Then by 492, 1, 

>0 , S>0 , f(x) < f(a) + e , for any x in F fi (a), 
which is 1). 



SEMICONTINUOUS FUNCTIONS 489 

It is sufficient. For 1) is condition 1 of 492, i. The condition 
2 is satisfied, since for a we may take the point a. 

2. The maximal function f(x) is supracontinuous ; the minimal 
function f(x) is infracontinuous, in 21. 

To prove that /is supracontinuous we use 1, showing that 

f(x) < /(#) + e -> for any x in some Fa 00- 
Now by 492, 1, 
e' > 0, 8 > , f(x) < 700 4- e' , for any x in Fi(a). 

Thus if e' < e 

8 

/(V) < /(a) + , f or any x in V^ (a) , 17 = - . 

3. The sum of two supra (infra) continuous functions in 21 is a 
supra (infra) continuous function in 31. 

For let/, g be supracontinuous in 21 ; let/+ y = h. Then by 1, 



for any a; in some Fa (a) ; hence 



This, by 1, shows that h is supracontinuous at #= a. 

4. If f(x) is supra (infra) continuous at x = a, g(x)~--f(x) 
is infra (supra) continuous. 

Let us suppose that /is supracontinuous. Then by 1, 

/(#)</()+ e , for any x in some V*(a). 
Hence 



or g(a)e<g(x) , for any a: in 

Thus bv 1, Q is infracontinuous at a. 



490 DISCONTINUOUS FUNCTIONS 

495. Iff\x l "-x m ) is supracontinuous in the limited complete 
set 31, the points 93 of 31 at which /> c an arbitrary constant form a 
complete set. 

For let /> c at b 1 , h 2 which = b ; we wish to show that b lies 
in S3. 

Since/ is supracontinuous, by 494, 1, 

/(V)</(6) + e , for any x in some F 5 (5)= V. 

But <?</(6 n ), by hypothesis ; and b n lies in FJ for n> some w 
Hence 



*-</(). 

As e > is small at pleasure, 
and 5 lies in 33. 



496. 1. TA0 oscillatory function a>(x) is supracontinuous. 

For by 493, ,, \ TVT ^ \f * 

J ' a)(a;)= Max/ Mm/ 

= Max/+ Max (-/). 

But these two maximal functions are supracontinuous by 494, 2. 
Hence by 494, 3, their sum o> is supracontinuous. 

2. The oscillatory function o> is not necessarily infracon- 
tinuous, as is shown by the following 

Example. /= 1 in (1, 1), except for x = 0, where /= 2. 
Then &(x) = 0, except at x = 0, where CD = 1. Thus 

Min o>(z) = , while o>(0) = 1. 

x=Q 

Hence w(x) is not infracontinuous at x = 0. 

3. Let fc>(#) J# ^A^ oscillatory function of f(x l rr m ) in 31. -#W 
/ <o be at most pointwise discontinuous in 31, z{ zs necessary that 
Min o> = at each point of 31. /f 31 z complete, this condition is 
sufficient. 



SEMICONTINUOUS FUNCTIONS 491 

It is necessary. For let a be a point of 21. As f is at most 
pointwise discontinuous, there exists a point of continuity in any 
Hence Min <*>(x) = 0, in Fi(a). Hence Min o>(z) = 0. 



It is sufficient. For let 1 >e 2 > =0. Since Min (.*:) = 0, 

.r= 

there exists in any Fi(a) a point j such that o>( 1 )<| 1 . 
Hence o>(#) < j in some Fi,( a i) < ^V ' n ^61 ^ nere exists a point 
2 such that ct>(#)<e 2 in some F 6 /a) < F^, etc. Since 21 is com- 
plete and since we may let 8 n = 0, 

Pi, > ^ a >== a point of 21, 

at which f is obviously continuous. Thus in each ^(a) is a point 
of continuity of/. Hence /is at most pointwise discontinuous. 

497. 1 . At each point x of 21, 

<f> = Min \f(x) -f(x-)\,and^= Min \f(x) -f(x)\ 
are both == 0. 

Let us show that <f> = at an arbitrary point a of 21. By 494, 
2, f(x) is supracontinuous ; hence by 494, l, 

f(x) </(a) -he , for any x in some Fa(a) = F. (1 

Also there exists a point a in Fsuch that 

. (2 



Also by definition 

/() 

If in 1) we replace ^ by a \\e 
/()< 
From 2), 3), 4) we have 



or 



As >0 is small at pleasure, this gives 

<K<0 = o. 



492 DISCONTINUOUS FUNCTIONS 

2. If/is semicontinuous in the complete set ?l, it is at most point- 
wise discontinuous in 21. 

For K*) =/(*)-/(*) 

-/(*)] + [/GO -f(*y\ (1 



To fix the ideas let / be supracontinuous. Then <f> = in 51. 
Hence 1) gives 

Min to(x) = Min ty(x) = 0, by 1. 

Thus by 496, 3, / is at most pointwise discontinuous in 21. 



CHAPTER XV 
DERIVATES, EXTREMES, VARIATION 

Derivates 

498. Suppose we have given a one-valued continuous function 
f(x) spread over an interval 21= (a<6). We can state various 
properties which it enjoys. For example, it is limited, it takes 
on its extreme values, it is integrable. On the other hand, we 
do not know 1 how it oscillates in 21, or 2 if it has a differ- 
ential coefficient at each point of 21. In this chapter we wish to 
study the behavior of continuous functions with reference to these 
last two properties. In Chapters VIII and XI of volume I this 
subject was touched upon ; we wish here to develop it farther. 

499. In I, 363, 364, we have defined the terms difference quo- 
tient, differential coefficient, derivative, right- and left-hand dif- 
ferential coefficients and derivatives, unilateral differential coeffi- 
cients and derivatives. The corresponding symbols are 



Lf(a) , Rf'(x) , Lf'(x). 

The unilateral differential coefficient and derivative may be de- 
noted by 

Z7/'() , Uf'(x). (1 

When A , 

lim= 

A=0 A# 

does not exist, finite or infinite, we may introduce its upper and 
lower limits. Thus 

(2 



always exist, finite or infinite. We call them the upper and lower 
differential coefficients at the point x = a. The aggregate of values 

493 



494 DERIVATES, EXTREMES, VARIATION 

that 2) take on define the upper and lower derivatives of ./(#), as 
in I, 363. 

In a similar manner we introduce the upper and lower right- 
and left-hand differential coefficients and derivatives, 

Rf , Rf , Lf , Lf. (8 

Thus, for example, 



finite or infinite. Of. I, 336 seq. 



is defined only in 21 = ( < /3), the points a, a -f- h must 
lie in 31. Thus there is no upper or lower right-hand differential 
coefficient at x /3 ; also no upper or lower left-hand differential 
coefficient at x = a. This fact must be borne in mind. We call 
the functions 3) derivates to distinguish them from the deriva- 
tives Rf, Lf. When Rf ( )= Rf\a ), finite or infinite, 
Rf (a) exists also finite or infinite, and has the same value. A 
similar remark applies to the left-hand differential coefficient. 

To avoid such repetition as just made, it is convenient to in- 
troduce the terms upper and lower unilateral differential coeffi- 
cients and derivatives, which may be denoted by 

Vf' , V?'. (4 

The symbol U should of course refer to the same side, if it is 
used more than once in an investigation. 

When no ambiguity can arise, we may abbreviate the symbols 
3), 4) thus: 

R , R , L , L , U , U. 

The value of one of these derivates as R at a point x = a may 
similarly be denoted by 

5(). 

The difference quotient 

/(>=/(*) 

a-b 
may be denoted by 

A(a, A). 



DERIVATES 495 

Example 1. f(x) = xs'm - , z=0 in ( 1, 1) 

x 

= , 2=0. 

j 

A sin - 

Here for x = 0, -~^- = = = sin T . 
Ax li h 

Hence *fo- + i . 

Z/'(0)= + i , 



757/yvyyMflo7*> ^? ~F ( r r^l T' yi TI ^ -/- in ( 1 1 *\ 

JliJUViiiVUijG & J \^Ji> ) Ji> nlll ^ JL' =f- \f 111 ^ A} J. J 

X 

= , z = 0. 

A * Sln I 

A/ a 

Here for a: = , -~- = = 

A^ A f 

Hence .#/'(0)= 4- oo , J2f(0)=-oo, 



: + 00 , /'(0)=-00. 

Example 3. f(x) = 2: sin - , for < x < 1 

x 

= x^ sin - , for 1 < x < 
x 

= , f or x = 0. 
Here 



500. 1. Before taking up the general theory it will be well 
for the reader to have a few examples in mind to show him how 
complicated matters may get. In I, 367 seq., we have exhibited 
functions which oscillate infinitely often about the points of a set 



496 DERIVATES, EXTREMES, VARIATION 

of the 1 species, and which may or may not have differential co- 
efficients at these points. 

The following theorem enables us to construct functions which 
do not possess a differential coefficient at the points of an enumer- 
able set. 

2. Let S = \e n \ be an enumerable set lying in the interval 21. For 
each x in 21, and e n in S, let x e n lie in an interval S3 containing 
the origin. Let g(x) be continuous in S3. Let g 1 (x} exist and be 
numerically <. M in S3, except at x = 0, where the difference quotients 
are numerically < M. Let A = 2a n converge absolutely. Then 



is a continuous function in 21, having a derivative in ( = 21 . 
At the points of @, the difference quotient of F behaves essentially as 
that of g at the origin. 

For g(x) being continuous in S3, it is numerically < some con- 
stant in 21. Thus F converges uniformly in 21. As each term 
g(x # n ) is continuous in 21, F is continuous in 21. 

Let us consider its differential coefficient at a point x of S. 
Since g'(x e n ) exists and is numerically < M, 

^'(z)==2a n y(*--* n ) , by 156, 2. 
Let now x = e m , a point of (, 

F(x) = a m g(x - e m ) 4- 



The summation in 2* extends over all nj=m. Hence by what 
has just been shown, Gr has a differential coefficient at x = e m . 

Thus - behaves at x e m , essentially as -^ at x = 0. Hence 

J 



501. Example 1. Let 

g(x) = ax , x > 



b < < a. 

x < 0, 



DERIVATES 497 

Then , 



is continuous in any interval 31, and has a derivative 

n*)=S-Wo*-o 

it/ 
at the points of 21 not in (5. At the point e m , 



Let @ denote the rational points in 21. The graph of F(x) is a 
continuous curve having tangents at a pantactic set of points ; 
and at another pantactic set, viz. the set @, angular points (I, 366). 

A simple example of a g function is 



Example 2. Let g(x) = x* sin , x = 

x 



This function has a derivative 

g f (x) = 2x sin TT cos , 
x x 

= , x = Q. 

Thus if 2e? n is an absolutely convergent series, and (g = \e n \ an 
enumerable set in the interval 21 = (0, 1), 

F(x) =2^0- O 
is a continuous function whose derivative in 21 is 



Thus F has a derivative which is continuous in 21 (, and at 
the point # = e m 

Disc F r = 2 c m 7r, 

since 



498 DKRIVATKS, KXTRKMKS, VARIATION 

If S is the set of rational points in 21, the graph of F(x) is a 
continuous curve having at each point of 21 a tangent which does 
not turn continuously as the point of contact ranges over the 
curve; indeed the points of abrupt change in the direction of the 
tangent are pantactic in 21. 

Example 8. Let g(x) = x sin log x 2 , x ^ 

Then #0*0 = s in 1M' - r2 4- ^ <' ( >s log # 2 , x^O. 

At x = 0, = sin log A 2 

which oscillates infinitely often between 1, as h = Ax == 0. Let 
@ = j^ n j denote the rational points in an interval 21. The series 



satisfies the condition of our theorem. Hence F(oi) is a continu- 
ous function in 21 which has a derivative in 21 @. At #= e m , 



Thus the graph of F is a continuous curve which has tangents at 
a pantactic set of points in 21, and at another pantactic set it has 
neither right- nor left-hand tangents. 

502. Weierstrass" Function. For a long time mathematicians 
thought that a continuous function of x must have a derivative, at 
least after removing certain points. The examples just given 
show that these exceptional points may he pantactic. Weierstrass 
called attention to a continuous function which has at no point a 
differential coefficient. This celebrated function is defined by the 
series 

F (x) = 2 a n cos b n irx = cos irx + a cos birx + a 2 cos WTTX -j- (1 
where < a < 1 ; J is an odd integer so chosen that 

a6>l + *7r. (2 



DERIVATKS 499 

The series F converges absolutely and uniformly in any interval 

21, since . _ ln , ^ _ 

' |0 n cos b n 7Tx I < a n . 

Hence F is a continuous function in 21. Let us now consider 
the series obtained by differentiating 1) term wise, 



If ab < 1, this series also converges absolutely and uniformly, 
and 



by 155, 1. In this case the function has a finite derivative in 21. 
Let us suppose, however, that the condition 2) holds. We have 



^?= Q = V {cos ft"7r(2; + A) - cos ft-TT^S = Q m + ~Q m . (8 
A# Y ^ 

Now w _ n 



~r J cos n 7r(x + i) cos 
A 



sin b n 7rudu. 
Since 



I /*^-h^ > I /^+/t 

I sin b n 7rudu, < I 



- 

Consider now _ < n 

Q m = 2 - f Jcos b n Tr(x 4- A) cos 

m * 

Up to the present we have taken h arbitrary. Let us now 
take it as follows ; the reason for this choice will be evident in a 
moment. 

Let 



where i m is the nearest integer to b m x. Thus 

-*<&.<*. 

Then " > 



500 DERIVATES, EXTREMES, VARIATION 

We choose h so that 

^m = m 4- hb m is 1, at pleasure. 

Then _ *. 

h = -^~r~^ === 0, as ra == oo ; 

moreover sgn A - sgn r, m , and |, ra - f. | < |. 

This established, we note that 

cos b n jr(x -f- A) = cos b n ~ m 7r - b m (x -f h) = cos # n ~ w (* m -h ?/ m )7r 
= cos (t w 4-7/ m )7r , since b is odd 
= ( l) l w+i , since r; m is odd. 

S COS 6 W 7T^ = COS 6 n ~ m (t m -h f m)7T 

Thus _ oo an 

m 'I 
W ' 16re ( \\ m +\ 

Now each } } > and in particular the first is > 0. Thus 
sgn Q m = sgn ^ = sgn e m ij m , 



Thus if 2) holds, | Q m \ > \ Q m |. Hence from 3), 

sgn Q = sgn Q m = sgn e m r) m , 
and 



Let now m = QO . Since i/ m = 1 at pleasure, we can make 
Q = -f QO, or to QO , or oscillate between GO, without becoming 
definitely infinite. Thus F (x) has at no point a finite or infinite 
differential coefficient. This does not say that the graph of F does 
not have tangents; but when they exist, they must be cuspidal tangents. 



DERIVATES 501 

503. 1. VolterrcCs Function. 

In the interval 21 = (0, 1), let <Q = \<r)\ be a Harnack set of 
measure 0<A<1. Let A = {S n j be the associate set of black 
intervals. In each of the intervals S n = (a < y8), we define an 
auxiliary function f n as follows : 

/nO) = O- ) 2 sin , in (a*, 7), (1 

where 7 is the largest value of x corresponding to a maximum of 
the function on the right of 1), such that 7 lies to the left of the 
middle point /i of S n . If the value of f n (x) at 7 is #, we now 



* /" x 

= # > HI (7, /i). 

Filially f n (a)= 0. This defines / (#) for one half of the inter- 
val 8 n . We define f n (x) for the other half of S n by saying that if 
x<x r are two points of S n at equal distances from the middle 
point* then /(*)=/(*') 

With Volterra we now define a f unction f(x) in 31 as follows: 
f(x) = f n (x) , inS n , n = l, 2, .- 

= , in . 

Obviously /(#) is continuous in 21. 
At a point x of 1 not in &,f(jz) behaves as 

2 a; sin -- cos-, 
x x 

as is seen from 1). Thus as x converges in 8 n toward one of its 
end points a, /?, we see that f f (x) oscillates infinitely often be- 
tween limits which = 1. Thus 



R lim/'<= + 1,7? lira/ (a?) = - 1 ; 

*= a 1^. 

similar limits exist for the points 0. 

Let us now consider the differential coefficient at a point rj of 
. We have 



502 DERIVATES, EXTREMES, VARIATION 



If r) -f k is a point of ,/0? -4- &) = 0. If not, 77 -f- A lies in some 
interval 8 m . Let x = e be the end point of 8 m nearest 1; + A. 
Then 



Thus/' (97)= 0. Hence Volterra's function /(a?) has a differen- 
tial coefficient at each point of 21 ; moreover f (x) is limited in 21. 
Each point rj of is a point of discontinuity of /'(#), and 

Disc/' (a?) > 2. 



Hence /'(#) is not /2-integrable, as 

We have seen, in I, 549, that not every limited .R-integrable 
function has a primitive. Volterra's function illustrates con- 
versely the remarkable fact that Not every limited derivative is 
R-integrable. 

2. It is easy to show, however, that The derivative of Volterra's 
function is L-integrable. 

For let 21 A denote the points of 21 at which /'(a?) >X. Then 
when X>l/w, w=l, 2, 21 A consists of an enumerable set of 
intervals. Hence in this case 21 A is measurable. Hence 21 A , X>0, 
is measurable. Now 21 , X>0, differs from the foregoing by add- 
ing the points n in each S n at which/' {x) = 0, and the points ^>. 
But each $ n is enumerable, and hence a null set, and |j is measur- 
able, as it is perfect. Thus 21 A , X>0, is measurable. In the 
same way we see 21 A is measurable when X is negative. Thus 21 A 
is measurable for any X, and hence i-integrable. 

504. 1. We turn now to general considerations and begin by 
considering the upper and lower limits of the sum, difference, prod- 
uct, and quotient of two functions at a point x = a. 

Let us note first the following theorem : 

Letf(x^ Xjn) be limited or not in 21 which has x = a as a limiting 
point. Let&= Max /, < 6 = Min / in V** (a) . Then 

lim/=lim<^5 , lim / = lim 4>$ . 

=5- 5=0 #=a fi=o 

This follows at once from I, 338. 



DERI VAXES 508 

2. Letf(x l > x m ), g(^x l x m ) be limited or not in 21 which has 
x = a as limiting point. 

Let lim/= , lim cj = ft 

lim/ = A , lim g = B 
as x^= a. Then, these limits being finite, 

a + < Hm (/ + //) < A + A (1 

a-B< lim (/ - </) < ^ - ft. (2 

For in any FS*(), 
Min / -f Min g < Min (/ 4- //) < Max (/ + </) < Max/ + Max #. 

Lulling 8 = 0, we gel 1 ). 
Also in Fi*(), 

Min/ Max // < Min (./' #) < Max(/ ^) < Max / Min^. 

Let ling 8 = 0, we gel 2). 



Imi ./// < ^1 J5. (3 



(4 



/ -,- - - f ^ A ,r 

<hm f :-< (5 

- - 



a<0<A , ,^(^)> 

^- l"^ f ^ A 



st* 

(6 



The relations 3), 4), 5), 6) may be proved as in 2. For exam- 
ple, lo prove 5), we observe Ibal in 



A,. /^AT f ^ Max/ 
Mm -^ < Max tL. < _ _y_ . 



-- . _^ _ 

Max// /y ^ Min 



504 DERIVATES, EXTREMES, VARIATION 

5. + /3< lim (/ + #)< + . (7 

. (8 
(9 

. (10 

If /(*)><> , </(*)> o, 

a/3 < limfff < , (11 

A^<\^nfg<AB. (12 
V g(x) > k > 0, 

f<l^. (13 

|<E/<|. (14 

6. If lim/ exists, 

lim (/ -h ^) = lim / -f lim g> (15 

(16 



If \\rng exists, 

lim(/-50 = lmi/-limflr, 

(18 



f(x) > 0, ^r(rr) > 0. Let lim # ri^. 

lim fg = lim / lim g, (19 

lim fg = lim / lim g. (20 

g(x) > k > 0, 

(21 



iim //gr = lim //lim #. (22 

505. The preceding results can be used to obtain relations be- 
tween the derivates of the sum, difference, product, and quotient 
of two functions as in I, 373 seq. 



DERIVATES 505 

1. Let w (V) = u (x) -f v (V) . Then 

A^y__Aw Av ,^ 

Ax A# Az 

Thus from 504, 1), we get the theorem : 

Uu f + v'U<Uw'< Uu f + Uv'. (2 

// u has a unilateral derivative Uu', 

Uw f = Uu r + Uv', (3 

Uw' = ZTw' + Uv'. (4 

We get 3), 4) from 1), using 504, 15), 16). 

2. In the interval 31, M, v are continuous, u is monotone increasing, 



v is > 0, awe? v' exists. Then, if w = MV, 

Uiv' = uv' + vUu', (1 

Uw r =uv r +vUu r . (2 

For from A Ai , A 

- = (WH- A?/,) -f w , 

" 



we have ' = ^ + 






which gives 1). Similarly we establish 2). 

506. 1. We show now how we may generalize the Law of the 
Mean, I, 393. 

Let f(x) oe continuous in 21 =(#<&). Let m, M be the mini- 
mum and maximum of one of the four derivates off in SI. Then for 



. 

/3 a 

To fix the ideas let us take Rf'(x) as our derivate. Suppose 
now there exists a pair of points < /3 in 31, such that 



506 DERIVATKS, EXTREMES, VARIATION 



We introduce the auxiliary function 

$(aO=/(aO-Caf+e)*, (2 

where 0<c<e = c+8. 

Then Q8 )- QO = /()-/() (Jf|g)= s. 
Henue 



Consider now the equation 



It is satisfied for # = a. If it is satisfied for any other x in the 
interval (a/3), there is a last point, say x = 7, where it is satisfied, 
by 458, 3. 

Thus for x > 7, <O)i 8 >4>(). 

Hence Jty'(7)>0. (3 

Now from 2) we have 



Hence M is not the maximum of Rf'(x) in 21. Similarly the 
other half of 1) is established. The case that m or M is infinite 
is obviously true. 



2. Let f(x) be defined over 91 = (a < 5). JW rtj < # 2 < < a n fe 
m 31. Let m and M denote the minimum and maximum of the dif- 
ference quotients 

AO&^ag) , A(a 2 , a 8 ) , ... A(a n _ r a n ). 
Then 



For let us first take three points < /3 < 7 in 21. We have iden- 
tically Q Q 

7). 



Now the coefficients of A on the right lie between and 1. 
Hence 1) is true in this case. The general case is now obvious. 



DERIVATES 507 

507. 1. Let f(x) be continuous in 21 = (a < 6). The four deri- 
vates off have the same extremes in 21. 

To fix the ideas let 

Min L =5 m , Min R = /K, in 21. 

We wish to show that m = /*. To this end we 



For there exists an a in 21, such that 

L^oC) < ra -|- e. 
There exists therefore a /3 < a in 21, such that 



, 

a p 
Now by 506, 1, 

fjb = Min R<q. 



Hence 
as >0 is small at pleasure. 

JF0 sAow wo?u ^Aa^ ^ /0 

7M < /I. (2 

For there exists an a in 21, such that 

jR() < fJL + . 

There exists therefore a /9 > a in 21, such that 



C6-/3 

Thus by 506, 1, 

w = Miu L<q. 

Hence as before ra</*. From 1), 2) we have m = /it. 

2. In 499, we emphasized the fact that the left-hand derivates 
are not defined at the left-hand end point of an interval, and the 
right-hand derivates at the right-hand end point of an interval 
for which we are considering the values of a function. The fol- 
lowing example shows that our theorems may be at fault if this 
fact is overlooked. 



508 DERIVATES, EXTREMES, VARIATION 

Example. Let/ (x) = j x \. 

If we restrict x to lie in 21 = (0, 1), the four derivates = 1 when 
they are defined. Thus the theorem 1 holds in this case. If, 
however, we regarded the left-hand derivates as defined at x = 0, 
and to have the value 

= - 1, 



as they would have if we considered values of / to the left of 31, 
the theorem 1 would no longer be true, 

For then Min = - 1 , Min ]B = + 1, 

and the four derivates do not have the same minimum in 21. 

3. Let f '(#) be continuous about the point x= c. If one of its 
four derivates is continuous at x = c, all the derivates defined at this 
point are continuous, and all are equal. 

For their extremes in any Fi(<?) are the same. If now R is 
continuous at x = <?, 

R(c) - e < R(x) < JRO) + e, 
for any x in some V^(ci). 

4. Let f (x) be continuous about the point x c. If one of its 
four derivates is continuous at x = c, the derivative exists at this 
point. 

This follows at once from 3. 

Remark. We must guard against supposing that the derivative 
is continuous at x = <?, or even exists in the vicinity of this point. 

Example. Let F(x) be as in 501, Ex. 1. Let 



21= (0,1) and g= (-). 
I n) 



Let 
Then 



tx) = 2 xF(x) -f 
LH'(x) = 2 xF(x) + 



Obviously both Rff' and Lff f are continuous at x = and 
J5F(0) = 0. But H 1 does not exist at the points of (, and hence 



DERIVATES 509 

does not exist in any vicinity (0, 8) of the origin, however small 
S > is taken. 

9 

5. If one of the derivates of the continuous function f(x) is 
continuous in an interval 21, the derivative /'(#) exists, and is con- 
tinuous in 21. 

This follows from 3. 

6. If one of the four derivates of the continuous function f (x) is 
= in an interval 21, f(x) = const in 21. 

This follows from 3. 

508. 1. If one of the derivates of the continuous function f(x) is 
> in 21 = (a < J), f(x) is monotone increasing in 21. 

For then m = Min Ef f > 0, in (a < x). Thus by 506, i, 



2. If one of the derivates of the continuous function f(x) is _>_ 
in 21, f(x) is monotone decreasing. 

3. If one of the derivates of the continuous function f(x) is > 
in 21, without being constantly in any little interval of 21, /(a?) is 
an increasing function in 21. Similarly f is a decreasing function 
in 21, if one of the derivates is <C 0, without being constantly in any 
little interval of 21. 

The proof is analogous to I, 403. 

509. 1. Letf(x) be continuous in the interval 21, and have a deriv- 
ative, finite or infinite, within 21. Then the points where the deriva- 
tive is finite form a pantactic set in 21. 

For let a < ft be two points of 21. Then by the Law of the 
Mean, 



As the right side has a definite value, the left side must have. 
Thus in any interval (a, ft) in 21, there is a point 7 where the 
differential coefficient is finite. 



510 DERI VAT ES, EXTREMES, VARIATION 

2. Let f(x) be continuous in the interval 2l = (#<6). Then 
Uf* (#) cannot be constantly + GO, or constantly oo in 21. 
For consider 



a 



which is continuous, and vanishes for x = a, x = b. We observe 
that $(V) differs from /(V) only by a linear function. If now 
t/f'(V)= + ac constantly, obviously 7<'(V)= -h oo also. Thus </> 
is a uni variant function in 21. This is not possible, since </> has 
the same value at a and b. 



8. Let f(x) be continuous in 2J[ = (a< b), <mc? Aowtf a derivative, 
finite or infinite, in 2(=(a*, J). TVierc 

Min/ (a) < Rf(a)< Max/ (a;) , m 91. 
For the Law of the Mean holds, hence 



Letting now A = 0, we get the theorem. 

Remark. This theorem answers the question : Can a continu- 
ous curve have a vertical tangent at a point x a, if the deriva- 
tives remain < M in V*(a) ? The answer is, No. 

4. Let f(x) be continuous in 21 = (a < J), and have a derivative, 
finite or infinite, in 21* = (a*, b). Iff'(a) exists, finite or infinite, 
there exists a sequence j > 2 > == a in 21, such that 



- , , . (2 

/I 

Let now A range over Aj > A 2 > = 0. If we set w n = ,, , the 
relation 1) follows at once from 2), since 1 f f (a) exists by 
hypothesis. 

510. 1. A right-hand derivate of a continuous function f(x) 
cannot have a discontinuity of the 1 kind on the riyht. A similar 
statement holds for the other derivates. 



DEB1VATES /ill 

For let R(x) be one of the right-hand derivates. It it has a 
discontinuity of the 1 kind on the right at or = a, there exists a 
number I such that 

I e <_ R(x) <_ I -f c , in some ( " < # -t- 8). 
Then by 500, i, 



Hence R(a)= I, 

and R(V) is continuous on the right at # = a, which is contrary 
to hypothesis. 

2. It can, however, have a discontinuity of the 1 kind on the 
left, as is shown by the following 

Example. Let/(r)== | r |= + Va? , in 31 = (- 1, 1). 
Here R(x)=+\ , for ?.> in 31 

= I , for x < 0. 

Thus at x = 0, R is continuous on the right, but has a discon- 
tinuity of the 1 kind on the left. 

8. Let f(x) be continuous in ?I = (a, ?>), tt/tc? /m^' a derivative, 
finite or infinite, in 31* =(<*, />*). Then tlie discontinuities off'(x) 
in 31, if any exist, must be of the second kind. 

This follows from 1. 

^O in 31 = (0, 1) 



Example. 


/O) = 


a? 2 sin - , f( 






x 




= 


, for x = 


Then 




. 1 



0*0 =2 2; sin --cos , 
a: # 

= , x = 0. 
The discontinuity of ./ v (.r) at a; = 0, is in fact of the 2 kind. 



4. Let f(x) be continuous in 3t = (a<6), except at x a, which 
is a point of discontinuity of the 2 kind. Let f 1 (x) exist ^ finite or 
infinite, in (a*, 6). Then x = a is a point of infinite discontinuity 

<>//'(*) 



512 DERIVATES, EXTREMES, VARIATION 

For if 



there exists a sequence of points 1 > 2 >... =a, such that 
/(a n )=_=jt?; and another sequence ft l >/3 2 > =, such that 
== ? We may suppose 

>&. , ora n </3 n , rc=l, 2, - 
Then the Law of the Mean gives 



where 7 n lies between n , /3 n . Now the numerator = JP ^, while 
the denominator = 0. Hence Q n = -f- oo , or oo , as we choose. 

5. Let f (x) have a finite unilateral differential coefficient U at 
each point of the interval ?l. Then U is at most pointwise discon- 
tinuous in 21. 

For by 474, 3, 7is a function of class 1. Hence, by 486, l, it is 
at most pointwise discontinuous in 21. 

511. Let f (x) be continuous in the interval (a < J). Let R(x) 
denote one of the right-hand derivates of f(x). If R is not con- 
tinuous on the right at a, then 



ujhere _____ 

I = R lim R(x) , m = R lim R(x) , x = a. 

To fix the ideas let R be the upper right-hand derivate. Let us 
suppose that a = Rf f (a) were >ra. Let us choose 77, and c such 
that . 

wi -f V < c < (2 

We introduce the auxiliary function 



Now if > is sufficiently small, 

72/'O)< + >? , for any x in 21* = (a*, a + S). 



DERI VAXES 513 

Thus 2), 3), show that 

R$(X)><T , <r>0. 
Hence </>(#) is an increasing function in 31*. But, on the other 



since a > m. Hence 

Mfi (a) = c - fif' (a) = c - a < 0. 

Hence < is a decreasing function at x = a. This is impossible 
since <f> is continuous at a. Thus <.m. 
Similarly we may show that <.. 

512, 1. Let f(x) be continuous in ?I = (a < 5), <mc? have a 
derivative, finite or infinite. Ifa=f'(a), /9 =/'(&), then f (x) 
takes on all values between a, /3, as x ranges over 81. 

For let a < 7 < /3, and let 

) = /<JLJI/M , A>0 . 



We can take h so small that 

Q(a, A)<7 , and 
N W <?(i, -*) 

Hence (-*,*)> 7- 

If now we fix A, ^ (#, A) is a continuous function of #. As 
is < 7, for x =s a, and > 7, for # = 6 A, it takes on the value 7 
for some x, say for x = , between a, 6 A. Thus 

<?(?, A) = 7- 
But by the Law of the Mean, 

(*)=/' Oi), 



Thus/' (x) = 7, at x == 77 in 31. 

2. ie /(#) 5e continuous in the interval 31, and admit a deriva- 
tive, finite or infinite. If f'(x) = in 31, except possibly at an 
enumerable set @, then f = aZs0 in ( 



514 DER1VATES, EXTREMES, VARIATION 

For if /'() = 0, and /'() = b & 0, then f'(x) ranges over all 
values in (0, b), as x passes from a to ft. But this set of values 
has the cardinal number c. Hence there is a set of values in 
(, /3) whose cardinal number is c, where /'(#) =jt 0. This is 
contrary to the hypothesis. 



8. Let /(#), #(#) ^ continuous and have derivatives, finite or 
infinite, in the interval 81. // in ?{ //ere i an /or which 



/3 /"or 

/ a 7 /0r which 



/' (7) =/(?), 

*(*)=/<*>-*(*> 

Aas a derivative, finite or infinite. 

For by hypothesis 

'()> , S'(/3)<0. 

Hence by 1 there is a point where 8' = 0. 

513. 1. If one of the four derivates of the continuous function 
f(x) is limited in the interval 81, all four are, and they have the 
same upper and lower R-integrals. 

The first part of the theorem is obvious from 507, 1. Let us 
effect a division of 81 of norm d. Then 



R = li 

d=Q 



lim ^Mdi , Mi = Max R, in d,. 



Hut the maximum of the three other derivates in d L is also M^ by 
f)07, l. Hence the last part of the theorem. 

2. Let f(x) be continuous and have a limited unilateral derivate 
as R in 81 = (a < 6). Then 



For let a < aj < a 2 < ... < ft determine a division of 31, of norm d. 



DERIVATES 515 

Then by 506, 1, 

Min R < ^lAz/^n). < Max R, 

tfm+l - m 

in the interval (a m , a mJrl ) = d m . 
Hence 

2d m Min R </6) -/a < 2<k Max 5. 



Letting d == 0, we get 1). 

3. If f(x) is continuous, and Uf is limited and R-integrable in 
, then 



514. 1. Letf(x) be limited in 21 = (a < 5), and 
F(x)= Cfdx , a<a;<6. 

\/a 
Whew _ _ 

- ^ 71 U\\mf<UF(u)<U\imf, (1 

a^=u *=" 

or any u within 31. 
To fix the ideas let us take a right-hand derivate &tz = u. Then 

A Min/ < f/ds < h Max/ , in (w*, u + A), A > 0. 
_ M 

Thus 

Letting A = 0, we get 



R lim / < RF' (u) < R lim /, 

*= ^ M 

which is 1) for this case. 

2. i# /(^) ^^ limited in the interval 21 = (a < 6). If f(x + 0) 



JS derivative I fdx=f(x -f 0) ; 

^a 

and iff(x 0) exists, a<x<b 

L derivative ( fdx f(x 0). 

Ja 



5ir> DKRIVATES, KXTRKMKS, VARIATION 

3. Let f(x) be limited and R-integrable in 2l=(a<S). The 
points where 

F<= Cfdx , a<x<b 

v/O 



does not have a differential coefficient in 31 1 form 

F r J*(:r)=/Cr) by 1,537, 1, 

when / is continuous at #. But by 462, 6, the points where / is 
not continuous form a null set. 

515. In I, 400, we proved the theorem : 

Let /(#) be continuous in 21 = (a < ft), and let its derivative 
= within 21. Then /is a constant in 31. This theorem we have 
extended in 507, 6, to a derivate of f(x). It can be extended still 
farther as follows : 



1. (L. Scheefer). ///(#) is continuous in 31 = (#</>), and if 
one of its derivates = in 21 except possibly at thr points of an 
enumerable set (, then f = constant in 21. 

If /is a constant, the theorem is of course true. We show that 
the contrary case leads to an absurdity,- by showing that Card (5 
would = c, the cardinal number of an interval. 

For if / is not a constant, there is a point c in 21 where 
jp=/(6') /(a) is =^0. To fix the ideas let jt?>0; also let us 
suppose the given derivate is R = Rf'(x). 

Let g(z,f)=f(x)-f(a)-t(x-a) , * > 0. 

Obviously | g \ is the distance/ is above or below the secant line, 



Thus in particular for any , 

g(a, 0=0 , 

Let q > be an arbitrary but fixed number < p. Then 
#0, t) - q = p - q - t(v - a ) 



\it<T, where 

r. 

a 



DER1VATES 517 

Hence a* > , 

for any t in the interval Z = (T, 7), < T < J 7 . We note that 

Card X = c. 

Since for any in I, #(a, = 0, and #O, > 9, let x = e ( be 
the maximum of the points < c where g(x, ) = ? Then e < c, 
and for any h such that e -f 7t lies in (e, 6'), 




Thus for any t in I, 0< lies in @. As ranges over , let ^ 
range over @ x < S. To each point e of (S^ corresponds but one 
point of Z. For 

O-^Cu, 0-^,0 = (*-O(-). 
Hence * = f , as >a. 

Thus Card I = Card (gj < Card @, 

which is absurd. 

2. Let f (x) be continuous in < $t = (a<b). Let S denote the 
points of 31 wAere o?i^ of the derivates has one sign. If S exists, 

Card (5 = c, A# cardinal number of the continuum. 

The proof is entirely similar to that in 1. For let c be a point 
of 6. Then there exists a d > c such that 



We now introduce the function 

g(z, Q =/GO-/(<0- '(*-*) * *>0, 

and reason on this as we did on the corresponding g in 1, using 
here the interval (c, d) instead of (a, ). We get 

Card g 1= =Card = c. 

3. Letf(x), g(x) be continuous in the interval 31. Let a pair of 
corresponding derivates as Rf, Eg 1 be finite and equal, except pos- 
sibly at an enumerable set g. Then f=g + C, in 31, where C is a 
constant. 



518 DERI VAXES, EXTREMES, VALUATION 



Then in 



But if .72<// < at one point in 31, it is < at a set of points 95 
whose cardinal number is c. But 93 lies in @. Hence R<f> is 
never < 0, in 21. The same holds for i/r. Hence, by 508, <f> and 
-*fr are both monotone increasing. This is impossible unless 
<f> = a constant. 

516. The preceding theorem states that the continuous function 
/(#) in the interval 21 is known in 21, aside from a constant, when 
f (x) is finite and known in 21, aside from an enumerable set. 

Thus f(x) is known in 21 when f 1 is finite and known at each 
irrational point of 21. 

This is not the case when/' is finite and known at each rational 
point only in SI. 

For the rational points in 21 being enumerable, let them be 

r n r v r 3- 0- 

Let J=* 1 + J 2 +J 8 +- 

be a positive term series whose sum I is < 21. Let us place r^ 
within an interval S l of length < ? x . Let r t be the first number 
in 1) not in S r Let us place it within- a non-overlapping interval 
S 2 of length < ? 2 , etc. 

We now define a function /(a?) in 21 such that the value of /at 
any x is the length of all the intervals and part of an interval 
lying to the left of x. Obviously f(x) is a continuous function of 
x in 21. At each rational point /' (x) = 1. But f(x) is not de- 
termined aside from a constant. For 28 n < I. Therefore when 
I is small enough we may vary the position and lengths of the 
S-intervals, so that the resulting /'s do not differ from each other 
only by a constant, 

517. 1. Let f(x) be continuous in 21 = (a < J) and have a finite 
derivate, say Ef\ at each point of 21. Let S denote the points of 21 
where R has one sign, say > 0. If S exists, it cannot be a null set. 



DERI VAXES 519 

For let c be a point of g, then there exists a point d > c such 
that 



Let S n denote the points of where 

n-l<Rf'<n. (2 

Then g = 6 X 4- S 2 4- Let < q < p. We take the positive 
constants ft, q% such that 



If now & is a null set, each @ m is also. Hence the points of 6 m 
can be inclosed within a set of intervals S mn such that 2S mn < q m . 

n 

Let now q m (#) be the sum of the intervals and parts of intervals 
^m, m n == 1> ^ which lie in the interval (a < #). Let 



Obviously #(^) is a monotone increasing function, and 

0<<?(aO<9- ( 3 

Consider now 

P(^)=/(^ 

We have at a point of 21 S, 



Hence at such a point 

EP f < Rf < 0. 



But at a point x of (, /ZP' < also. For x must lie in some 
S m , and hence within some S mn . Thus q m (x*) increases by at least 
A# when x is increased to x 4- Aa?. Hence mq m (x), and thus 
#(a; is increased at least wA#. Thus 



Aa; 
ThUS 



' < Rf> - m < 0, by 2), 



520 DERI VAXES, EXTREMES, VARIATION 

since x lies in m . Thus RP' < at any point of 31. Thus P is 
a monotone decreasing function in 2, by 508, 2. Hence 



Hence > 0, 



or using 1), 3) 

jt>-?<0, 
which is not so, as p is > q. 

2. (Lebesgue*) Let f(x)i g (&) be continuous in the interval 81, 
#W6? have a pair of corresponding derivates as Hf 1 * Rg 1 which are 
finite at each point of 21, and also equal, the equality holding except 
possibly at a null set. Thenf(x) g(x) constant in 31. 

The proof is entirely similar to that of 515, 3, the enumerable 
set ( being here replaced by a null set. We then make use of 1. 

518. Letf'(x) be continuous in some interval A = (u 8, u 4- 8). 
Letf"(x) exist, finite or infinite, in A, but he finite at the point x=u. 



A=0 

where 



Let us first suppose that/"(M) = 0. We have for < h < rj < 



_ 
"Al " A -X 

= ^ {/'(a/) -/'(" )j , <*'< + /* , u-h<x"<u 
= |[(a,'-) {/() + e'| - ^"-^{/"do -I e"{J, 

/fc 

where |e'|, | e /; | are < e/2 for T; sufficiently small. 
Now x' u^* \x n ^l <^i 

~Y~~" ~T~-^ 

while /"(ti) = , by hypothesis. 

Hence | <?/!<* , for 

and 1 ) holds in this case. 



MAXIMA AND MINIMA 521 

Suppose now that f"(u) = a = 0. Let 

y(. x ) =/O) - ?O) where f/(.r) = | ax 2 -f- fo + c. 
Since ^"(w) = a , </"( = 0. 

Thus we are in the preceding case, and lim Qg = 0. 
But Q9=Qf-Qq. 

Hence lim Qf= a. 

Maxima and Minima 

519. 1. In I, 466 and 476, we have defined the terms /(V) as 
a maximum or a minimum at a point. Let us extend these terms 
as follows. Let/(^ .r m ) be defined over 91, and let x= a be an 
inner point of 21. 

We say f has a maximum at x = a if 1, / (a) / (#) > 0, for any 
x in some V(a), and 2,/(#) J\x) >0 for some x in any F(a). 
If the sign ^ can be replaced by > in 1, we will say f has a 
proper maximum at a, when we wish to emphasize this fact; and 
when > cannot be replaced by >, we will say / has an improper 
maximum. A similar extension of the old definition holds for 
the minimum. A common term for maximum and minimum is 
extreme. 

2. If f(x) is a constant in some segment 33, lying in the inter- 
val 21, 33 is called a segment of invariability, or a constant segment 
of /in 21. 

Example. Let/(#) be continuous in 21 = (0, 1*). 

Let /1 

x = a^a^a^ (I 



be the expression of a point of 2l in the normal form in the dyadic 

system. Let fc 

J = - 



be expressed in the triadic system, where n = a n , when a n = 0, 
and =2 when a n = l. The points = jj form a Cantor set, 
I, 272. Let j3 n j be the adjoint set of intervals. We associate 



522 DERIVATES, EXTREMES, VARIATION 

now the point 1) with the point 2), which we indicate as usual by 
x~ f . We define now a function g(x) as follows : 

#() =/O) > when x ~ . 

This defines ^ for all the points of . In the interval 3n let 9 
have a constant value. Obviously g is continuous, and has a 
pantactic set of intervals in each of which g is constant. 

3. We have given criteria for maxima and minima in I, 468 
seq., to which we may add the following : 

Let f(x) be continuous in (a , a + ). If Rf 1 '(#) > and 
0, finite or infinite, f (x^) has a minimum at x = a. 

)< and Lf'(a) > 0, finite or infinite, f(x) has a maxi- 
mum at x = a. 

For on the 1 hypothesis, let us take a such that J2--a 
Then there exists a 8' >0 such that 



h 
Hence /(a + *)>/() , a + A in (a, a + 8')- 

Similarly if /S is chosen so that L 4- /? < 0, there exists a S" > 0, 
such that /.. 7x /rx >. 



h 
Hence /(a-A)>/(a) , a + A hi (a - 8", a*). 

520. Example 1. Let/(a;) oscillate between the #-axis and the 
two lines y = x and y = x, similar to 



In any interval about the origin, y oscillates infinitely often, hav- 
ing an infinite number of proper maxima and minima. At the 
point # = 0,/has an improper minimum. 

Example 2. Let us take two parabolas P l , P 2 defined by y = # 2 , 
y = 2 # 2 . Through the points x= |, ^ let us erect ordi- 
nates, and join the points of intersection with P x , P 2 , alternately 
by straight lines, getting a broken line oscillating between the 



MAXIMA AND MINIMA 523 

parabolas P l , P 2 . The resulting graph defines a continuous func- 
tion f(x) which has proper extremes at the points @ = j - 1 

i n ) 

However, unlike Ex. 1, the limit point x of these extremes is 
also a point at which f(x) has a proper extreme. 

Example 3. Let jSj be a set of intervals which determine a 
Harnack set lying in 21 = (0, 1). Over each interval B = (a, /3) 
belonging to the n ih stage, let us erect a curve, like a segment of 
a sine curve, of height h n = 0, as n == oo, and having horizontal 
tangents at a, /3, and at 7, the middle point of the interval 8. At 
the points \%\ of 21 not in any interval S, let/" (x) = 0. The func- 
tion/ is now defined in 21 and is obviously continuous. At the 
points \y\if has a proper maximum ; at points of the type a, /3, 
f,/has an improper minimum. These latter points form the set 
whose cardinal number is c. The function is increasing in each 
interval (a, 7), and decreasing in each (7, /3). It oscillates in- 
finitely often in the vicinity of any point of . 

We note that while the points where / has a proper extreme 
form an enumerable set, the points of improper extreme may form 
a set whose cardinal number is c. 

Example 4. We use the same set of intervals jSj but change 
the curve over S, so that it has a constant segment 77 = (X, /A) in its 
middle portion. As before /=0, at the points not in the 
intervals 8. 

The f unction /(V) has now no proper extremes. At the points 
of ^p, / has an improper minimum ; at the points of the type X, ^, it 
has an improper maximum. 

Example 5. Weierstrass" Function. Let S denote the points in 
an interval 21 of the type 

x = ~ , r, s, positive integers. 

For such an x we have, using the notation of 502, 

b m x = t m 4- m = b m ~'r. 

Hence | m = , for m>_8. 

Thus e m = (- 1> +1 = (- l) r+1 . 



524 DER1VATES, EXTREMES, VARIATION 

Hence sgn = sgn Q = sgu e m y m = sgn ( - 1 yh 



if r is even, and reversed if r is odd. Thus at the points @, the 
curve has a vertical cusp. By 519, 3, F has a maximum at the 
points S, when r is odd, and a minimum when r is even. The 
points ( are pan tactic in 31. 

Weierstrass' function has no eonstant segment fi, for then 
f'(x) = in S. Hut F 1 does not exist at any point. 

521. 1. Let f (JL\ jc m ) be continuous in the limited <>r unlimited 
set 21. Let ($ denote the points of ?{ where f has a proper extreme. 
Then ( is enumerable. 

Let us first suppose that 21 is limited. Let S > be a fixed 
positive number. There can be but a finite number of points in 
31 such that 



For if there were an infinity of such points, let ft be a limiting 
point and 77 < | 8. Then in V^(ff) there exist points ', " such 
that Fi(cc'), V s (a rf ) overlap. Thus in one case 

/(')>/(">, 

and in the other 

/(')</(">, 

which contradicts the first. 

Let now Sj > 8 3 > =0. There are but a finite number of 
points a for which 1) holds for 8 = Sj, only a finite number for 
S = S 2 , etc. Hence ( is enumerable. The case that 21 is unlim- 
ited follows now easily. 

2. We have seen that Weierstrass' function lias a pantactic set 
of proper extremes. However, according to 1, they must be 
enumerable. In Ex. 3, the function has a minimum at each point 
of the non-enumerable set ; but these minima are improper. On 
the other hand, the function has a proper maximum at the points 
}7J, but these form an enumerable set. 



MAXIMA AND MINIMA 525 

522. 1. Let f(x) be continuous in the interval 21. Let f have a 
proper maximum at x = a, and .*; = /3 in 21. Then there is a point 7 
between a, ft where f has a minimum, which need not however be a 
proper minimum. 

For say a < 13. In the vicinity of a, f(x) is </() ; also in 
the vicinity of /3, /(#) is </(/?). Thus there are points S3 in 
(a, /3) where /is < either /() or/(/8). Let /A be the minimum 
of the values of /(#), as # ranges over S3. There is a least value 
of x in (, /3) for which /(V) = /x. We may take this as the 
point in question. Obviously 7 is neither nor /3. 

2. That at the point 7, / docs not need to have a proper mini- 
mum is illustrated by Exs. 1, or 3. 

3. In 21 = (#, 6) /<' /'(.r) esist, finite or infinite. The points 
within 2( fl which f 1m* an extreme proper or improper, lie among 
the zeros off'(x). 

This follows from the proof used in I, 408, 2, if we replace there 
< 0, by <: 0, and > 0, by > 0. 

4. Let /'OO be continuous in the interval 21, and let f(x) have 
no constant segments in 21. The points (5 of 21 where f has an ex- 
treme, form an apantactic set in 21. Let denote the zeros of f (x) 
in ty. If 33 = Jb n ( is the border set of intervals lying in 21 corre- 
sponding to S,f(%) is univariant in each b n . 

For by 3, the points (5 lie in $ As f\x) is continuous, is 
complete and determines the border set 33. Within each b n , 
/'(#) lias one sign. Hence /(a;) is univariant in b n . 

5. Letf(x) be a continuous function having no constant segment 
in the interval 21. If the points ( where f has an extreme form a 
pantarfic set in 21, then the points 53 where /'(#) does not exist or is 
discontinuous, form also a pantactic set in 21. 

For if 93 is not pantactic in 21, there is an interval & in 21 
containing no point of 33. Thus /'(#) is continuous in @. But 
the points of (S in S form an apantactic set in S by 4. This, 
however, contradicts our hypothesis. 

Example. Weierstrass' function satisfies the condition of the 
theorem 5. Hence the points where F f (x) does not exist or is 



526 DERIVATES, EXTREMES, VARIATION 

discontinuous form a pantactic set. This is indeed true, since 
F' exists at no point. 

6. Let f(x) be continuous and have no constant segment in the 
interval 21. Let f'(x) exist, finite or infinite. The points where 
/'(X) i^ finite and is = form a pantactic set in 21. 

For let a < & be any two points in 21. If /() =/(y8), there is 
a point a < 7 < /3 such that /() ^/(Y), since / has no constant 
segment in 21. Then the Law of the Mean gives 



a-y 

Thus in the arbitrary interval (a, /3) there is a point f, where 
f 1 (x) exists and is = 0. 

7. Let f \x) be continuous in the interval 21. Then any interval 
S3 in 21 which is not a constant segment contains a segment @ in which 
f is univariant. 

For since f is not constant in S3, there are two points a, b in S3 
at which f has different values. Then by the Law of the Mean 

/()-/(*)=( -*)AO . eill - 

Hence f f (c) = 0. As f r (x) is continuous, it keeps its sign in 
some interval (c 8, c + ), and/ is therefore univariant. 

523. Letf(x) be continuous in the interval 21, and have in any in- 
terval in 21 a constant segment or a point at which f has an extreme. 
If f(x) exists, finite or infinite, it is discontinuous infinitely often in 
any interval in 21, not a constant segment. At a point of continuity 
of the derivative, /' (#) = 0. 

For if f(x) were continuous in an interval S3, not a constant 
segment, / would be univariant in some interval S:<S3, by 522, 7. 
But this contradicts the hypothesis, which requires that any inter- 
val as has a constant segment. Hence /'(#) is discontinuous 
in any interval, however small. 

Let now x = c be a point of continuity. Then if c lies in a con- 
stant segment, /'(V) = obviously. If not, there is a sequence of 
points e 19 e% = e such that /(#) ^ as & n extreme at e n . But then 
/'(e n )=0, by 522,3. As f(x) is continuous at x = c, /'(c?) = 
also. 



MAXIMA AND MINIMA 527 

524. (Kttnig.) Letf(x) be continuous in 31 and have a pantactic 
set of cuspidal points . Then for any interval 93 of 21, there exists 
a /3 such that f(x) = /3 at an infinite set of points in 33. Moreover, 
there is a pantactic set of points \%\ in 33, such that k being taken at 



For among the points & there is an infinite pantactic set c of 
proper maxima, or of proper minima. To fix the ideas, suppose 
the former. Let x = c be one of these points within 53. Then 
there exists an interval 6:<33, containing (?, such that 



Let p. = Min/(V), in b- 

Then there is a point x where / takes on this minimum value. 
The point c divides the interval b into two intervals. Let I be 
that one of these intervals which contains #, the other interval we 
denote by ttl. Within tit let us take a point c l of c. Then in I 
there is a point c[ such that 



The point c 1 determines an interval b x , just as c determined b. 
Obviously bjfCtn, and bj falls into two segments t x , ttt 1 as before 
b did. Within m t we take a point of c. Then in I there is a 
point c^, and in Ij a point c%, such that 



In this way we may continue indefinitely. Let 






be the points obtained in this way which fall in (. Let c' be a 
limit point of this set. Let 

/" /*" /" ... 

c l 5 C 2 C 3 

be the points obtained above which fall in lj, and let c 11 be a limit 
point of this set. Continuing in this way we get a sequence of 
limiting points c , ^ c n ^ c m ... ^2 

lying respectively in I, I L , I a 



f>28 DEKIVATES, EXTREMES, VARIATION 

Since f is continuous, 

/(O=/('")=/<y")=- (3 

Thus if we set /(V)= /9 we see that f(x) takes on the value y8 at 
the infinite set of points 2), which lie in 33. 

Let 7j, 7 2 ... be a set of points in 2) which = 7. 

Then /Cl) -/(_7l) == /(7)"/(7t) = . . . = o. ( 4 

7 ~ 7i 7 ~ 7 2 

Thus if f'(x) exists at # = 7, the equations 3) show that ^(7) 
= 0. If/' does not exist at 7, they show that 

/' < < /' , at x = 7- 
Let now k be taken at pleasure. Then 

g(x)=f(x)-kx 

is constituted as/, and 

/<>)=/<>)-*. 

This gives 1). 

525. 1. Lineo-Owillatinf/ Functions. The oscillations of a con- 
tinuous function fall into two widely different classes, accord- 
ing as f(x) becomes monotone on adding ii linear function 
l(x)=*ax + b, or does not. 

The former are called lineo-oscillating functions. A continu- 
ous function which does not oscillate in 21, or if it does is lineo- 
oscillating, we say is at most a lineo-oscillating function. 

Example 1. Let */ \ js \ 

r J {x) = sin x , l(x) = x. 

If we set jf f ^ , 7 . . 

y =/()+() 

and plot the graph, we see at once that y is an increasing function. 
At the point # = 77-, the slope of the tangent to /(#)= sin# is 
greatest negatively, i.e. sin # is decreasing here fastest. But the 
angle that the tangent to sin x makes at this point is 45, while 
the slope of the line l(x) is constantly 45. Thus at x = TT, y has 
a point of inflection with horizontal tangent. 

If we take l(x) = ax, a > 1, y is an increasing function, increas- 
ing still faster than before. 



MAXIMA AND MINIMA f>29 

All this can be verified by analysis. For setting 
y = sin x -f ax i # > 1, 



and 



Thus y is a lineo-oscillating function in any interval. 

Example 2. /(#) = :r 2 sin - , xJ=Q 

x 

= , r=0. 

J(#) = ax + 5 , y =/O) + ZO). 
Then 



u' = 2 a; sin cos--f-a , 

XX 

a , x= 0. 

Hence, if a > 1 -h 2 TT, y is an increasing function in 21 = ( TT, TT). 
The function /" oscillates infinitely often in 21, but is a lineo-oscil- 
lating function. 

Example 8. f(x) 

= 



Here 11 1 

^'=sin^ --- cos - 4- a 
a; ^ ^ 

For x=Q, y f does not exist, finitely or infinitely. 

Obviously, however great a is taken, y has an infinity of oscilla- 
tions in any interval about x= 0. Hence/' is not a lineo-oscillat- 
ing function in such an interval. 

2. If one of the four derivates of the continuous function f(x) is 
limited in, the interval 21, /(>) is at most lineo-oscillating in 21. 

For say Rf > - a in ?(. L t < < /3, 
and 



530 DERIVATES, EXTREMES, VARIATION 

Then 



/<=/3 +/'(*)> 0. 
Hence g is monotone increasing by 508, 1. 

3. Letf^x) be at most lineo-oscillating in the interval 21. If Uf f 
does not exist finitely at a point x in 21, it is definitely infinite at the 
point. Moreover, the sign of the GO is the same throughout 21. 

For if / is monotone in 21, the theorem is obviously true. If 

not ' let <,(*>=/(*)+* 

be monotone. Then 

Uf'=Ug'-a, 

and this case is reduced to the preceding. 

Remark. This shows that no continuous function whose graph 
has a vertical cusp can be lineo-oscillating. All its vertical tan- 
gents correspond to points of inflection, as in 



Variation 

526. 1. Letf(x) be continuous in the interval 21, and have limited 
variation. Let D be a division of 21 of norm d. Then usin</ the no- 
tation 0/443, 

\imVjJ=Vf , limP D f=Pf , UmN D f=Nf. (1 

For there exists a division A such that 



where for brevity we have dropped / after the symbol V. Let 
now A divide 21 into v segments whose minimum length call X. 
Let D be a division of 21 of norm d<d <\. Then not more 
than one point of A, say a x , can lie in any interval as (a t , a t+1 ) of 
D. Let E= D + A, the division obtained by superposing A on D. 
Then fi denoting some integer < *>, 



2j 

K 1 



VARIATION 531 

If now c? is taken sufficiently small, Osc/ in any interval of D 
is as small as we choose, say < . Then 



But since E is got by superposing A on D, 

Hence for any D of norm < rf , 

IF Fl < 

which proves the first relation in 1. The other two follow at 
once now from 443. 

527. If f(x) is continuous and has limited variation in the in- 
terval 21 = O<6), then 

POO , N(x) , FOO 
are a?s0 continuous functions of x in 21. 

Let us show that V(x) is continuous ; the rest of the theorem 
follows at once by 443. 

By 526, there exists a d , such that for any division D of norm 
d<d Q , F(6) = F^(J) + e' , 0<e'<e/3. 

Then a fortiori, for any z< b in 21, 

In the division J), we may take # as one of the end points of an 
interval, and x + h as the other end point. Then 

F(a? + A) ^F^C*) + |/(a; + A) -/()| + ^ , 0< 2 </3. (2 
On the other hand, if d Q is taken sufficiently small, 

|/0*+A)-/00|<! , forO<A<S. (3 

d 

From 1), 2), 3) we have 

V(x) < e , for any < A < 8. (4 



532 DERIVATES, EXTREMES, VARIATION 

But in the division J), x is the right-hand end point of some in- 
terval as (x k,x). The same reasoning shows that 

\V(x-K)-V(x)\<e , foranyO<<S. (5 

From 4), 5) we see V(x) is continuous. 

528. 1. If one of the derivates of the continuous function f (a?) is 
numerically < Ttfm Ae interval ?l, i/if? variation Voff is < M^{. 

For by definition 



with respect to all divisions D= \d,\ of 81. Here 

Now by 506, 1, 

* 



Hence 

2. Letf(x) be limited and R-integrahle in 31 = (a< 6). 
F(x)= Cfdx , a< ^< 6 

*^a 

As limited variation in ?I. 

For let D be a division of 31 into the intervals d t = (a t , l+1 ). 

Then 



< 2 l/ j da: < M 2 

*/a t 

Thus Max F,, - F < 

and J 7 has limited variation. 

529. 1. If f (x) has limited variation in the interval 21, the 
points $ where Osc / > A, are finite in number. 

For suppose they were not. Then however large Gr is taken, 
we may take n so large that nk > Gr. There exists a division D 



VARIATION /W3 

of 31, such that there are at least n intervals, each containing a 
point of & within it. Thus for the division D, 



Thus the variation of f is large at pleasure, and therefore is not 
limited. 

2. If f has limited variation in the interval 21, its points of dis- 
continuity form an enumerable set. 

This follows at once from 1. 

530. 1. Let D 19 J9 2 be a sequence of superposed divisions, of 
norms d n = 0, of the interval 21. Let L Dn be the sum of the oscilla- 
tions of f in the intervals of D n . If Max fl Dn is finite, f(x) has 
limited variation in 21. 

For suppose f does not have limited variation in 21. Then 
there exists a sequence of divisions jE^, J? 2 such that if H^ n is 
the sum of the oscillations of /in the intervals of IH n , then 

n^< fi*,< = +QO. (i 

Let us take v so large that no interval of D v contains more than 
one interval of E n or at most parts of two E n intervals. Let 
.F n = E n + D v . Then an interval 8 of D v is split up into at most 
two intervals ', B n in F n . Let a>, a/, (*> n denote the oscillation of 
f in 8, S', S" . Then the term co in D v goes over into 

a> r +a>"<2co 
in n^. Hence if Max fl = M, 



which contradicts 1). 

2. Let V Dn 2 |/( t ) /(X+i) I ^ ie summation extended 
over the intervals (a t , a l+1 ) of the division D n . If Max V Dn is 

n 

tinite with respect to a sequence of superposed divisions \D n \-> we 
cannot say that /has limited variation. 

Example. For let/(V) = 0, at the rational points in the inter- 
val 21 = (0, 1), and = 1, at the irrational points. Let D n be 



534 DERIVATES, EXTREMES, VARIATION 

obtained by interpolating the points m f 7 in & Then /= 

at the end points a t , a t+1 of the intervals of D n . Hence V Dn = 0. 
On the other hand, f(x) has not limited variation in 21 as is 
obvious. 

531. Let F (x) = lim/(#, ), r finite or infinite, for x in the 

t = T 

interval 21. Let Var/(o:, t) <_M for each t near r. 
Then F(x) has limited variation in 21. 
To fix the ideas let r be finite. Let 



Then for a division D of 21, 

V D F<V D f+ V D g. 
But 

V D g = 2 \g(a m ) - ff(a m+1 ) | f 

where (a m , a w+1 ) are the intervals of D. 
But for some t = t' near r, each 



where 8 is the number of intervals in the division D. 
Thus 

Hence 

and J 7 has limited variation. 

532. Let /(#), g(%) have limited variation in the interval 21, then 
their sum, difference, and product have limited variation. 

If also l#l>7>0 , in 21 

thenf/g has limited variation. 

Let us show, for example, that h=fg has limited variation. 

Forlet Min/=m , Min</ = 7* 

in the interval rf t . 
Osc/=a> , Osc g = T 



VARIATION 535 

en / = m + o> , g = n + fir , in d t , 

< a < 1 , < < 1. 
us 



fg = mn -h 7W/3r 4- r&aa> -f a/3a>T. 
Now 



77M W T U ft) WT<g < 77171 + W T + U CO + ft)T. 

Hence ^ = OscA<^2Jr|m| + a> | /i | +O>T{. 

Rnf" 

I m 1 1 I w I T <. some -ff. 
Thus ^ h < 4 JT2a> + 2 JT2r, 

< some 6?, 
and h has limited variation. 

533. 1. Let us see what change will be introduced if we 
replace the finite divisions D employed up to the present by 
divisions JE, which divide the interval 21 = {a < 6) into an infinite 
enumerable set of intervals (a t , # l+1 ). 



and !F 

for the class of finite or infinite enumerable divisions 

Obviously TT> F; 

hence if TFis finite, so is V. 

We show thab if V is finite, so is W. For suppose W were 
infinite. Then for any Q- > 0, there exists a division E, and an 
w, such that the sum of the first n terms in 1) is > (?, or 

Ws, n >&. (2 

Let now D be the finite division determined by the points a^ , 
#2 " a n+i which figure in 2). 

Then v ^^ 

heuee ^=00, which is contrary to our hypothesis. 



53(3 



DERI VAXES, EXTREMES, VARIATION 



We show now that V and W are equal, when finite. For let 
E be so chosen that 



W- -<W E <W. 



Now 



W E = W Et n + e' , | e f < /2 
if w is sufficiently large. 

Let D correspond to the points a l a% in TF^ n . Then 

and hence V + ' > W -4- ' W 

Hence TT-^<6. 

We may therefore state the theorem : 

2. Iff has limited variation in the interval 31 with respect to th<> 
class of finite divisions D, it has with respect to the class of enumer- 
able divisions E, and conversely. Moreover 

Max V D = Max V K . 

534. Let us show that Weierstrass' function F, considered in 
502, does not have limited variation in any interval 21 = (a < /3) 
when ab > 1. Since F is periodic, we may suppose > 0. Let 



Jm Jm frm 

be the fractions of denominator b m which lie in 21. 
These points effect a division D m of 31, and 






> ) 
If I is the minimum of the terms F 3 - under the 2 sign, 

Now ft 1 

Hence - yu -f- 2 

^ ^ fz, ' /^ 



(1 



(2 



NOX-LNTUITIONAL CURVKS 537 

On the other hand, using the notation and results of 502, 



and also 



F(x+h)- F(x) 



> amftm /2 _ TT \ 
" \3 a6-l/ 



Let us now take 



Then _&+/ L 

*--T ' * 

Hence from 3), p > /2 TT 
^~ a U~^= 

ThuB ^>-^(|--^)(^- 2 ) ' byl),2). 

As a < 1, and a6 > 1, we see that 

V Dm = -f QO, as m = QO . 

Non-intuitional Curves 

535. 1. Let /(#) be continuous in the interval 31. The graph 
of/ is a continuous curve C. If / has only a finite number of os- 
cillations in 21, and has a tangent at each point, we would call an 
ordinary or intuitional curve, it might even have a finite num- 
ber of angle points, i.e. points where the right-hand tangent is 
different from the left-hand one [cf. I, 366]. But if there were 
an infinity of such points, or an infinity of points in the vicinity 
of each of which / oscillates infinitely often, the curve grows less 
and less clear to the intuition as these singularities increase in 
number and complexity. Just where the dividing point lies be- 
tween curves whose peculiarities can be clearly seen by the intui- 
tion, and those which cannot, is hard to say. Probably different 
persons would set this point at different places. 

For example, one might ask : Is it possible for a continuous 
curve to have tangents at a pantactic set of points, and no tangent 
at another pantactic set? If one were asked to picture such a 
curve to the imagination, it would probably prove an impossibility. 



538 DERIVATES, EXTREMES, VARIATION 

Yet such curves exist, as Ex. 3 in 501 shows. Such curves might 
properly be called non-intuitional. 

Again we might ask of our intuition : Is it possible for a con- 
tinuous curve to have a tangent at every point of an interval 21, 
which moreover turns abruptly at a pantactic set of points ? Again 
the answer would not be forthcoming. Such curves exist, how- 
ever, as was shown in Ex. 2 in 501. 

We wish now to give other examples of non-intuitional curves. 
Since their singularity depends on their derivatives or the nature 
of their oscillations, they may be considered in this chapter. 

Let us first show how to define curves, which, like Weierstrass' 
curve, have a pantactic set of cusps. To effect this we will extend 
the theorem of 500, 2, so as to allow g(x) to have a cusp at x = 0. 

536. Let (S = \e n \ denote the rational points in the interval 
21 = ( #, a). Let g(x) be continuous in 33 = ( 2 a, 2 #), and 
= 0, at x = 0. Let 93* denote the interval $ after removing the 
point x = 0. Let g have a derivative in 93*, such that 



Then 



A IP 

is a continuous function in 21, and behaves at x = e m essentially 

Ao; 

as does at the origin.* 

LJkX 

To simplify matters, let us suppose that S does not contain the 
origin. Having established this case, it is easy to dispose of the 
general case. We begin by ordering the e n as in 233. Then 
obviously if 

e n = *- , q > , p positive or negative, 

we have ^ 

n > q. 



Let 



s 



ife m =r, 

s 

>!>_!. (2 

qs mn 



* Cf. Dini, Theorie der Functioned etc., p. 192 sea. Leiozic. 1892. 



NON-INTUITIONAL CURVES 539 

Let E(x) be the F series after deleting the m th term. Then 
F (x) = a^(x -~ em ) + JE (x). 

We show that E has a differential coefficient at x = e m , obtained 
by differentiating E termwise. To this end we show that as h = 0, 

(3 



converges to ^ = 2^'CO , m*. (4 

That is, we show 

e>0 , rj > , | D(A) -(?|<e , 0<|A <rj. (5 



Let us break up the sums 3), 4) which figure in 5), into three 

parts r .5 oo 

2 = 2 + 2 + 2. (6 

1 1 r+l *+l 

THUS |J)-fl t |<|7) r -G t ,| + |-Z>,..-fl [ r..| + |A-&.| (7 
< A + B + (7. 

Since g'(e mn ) exists, the first term may be made as small as we 
choose for an arbitrary but fixed r ; thus 

A <\- 

Let us now turn to B. We have 

s<\D n \+ 



provided g' (x) exists in the interval (e mn , e mn + A). 
But by 2), 



if 1 

9<-i-. (8 

2 W8 

Thus by 1), 

I #'Omn+ A') | ^ 2 a Mm a n a < JS/^r , jlfj a constant. 



540 DEK1VATKS, EXTREMES, VARIATION 

Hence a fortiori, 



, ffl ^ , < ^ (9 

Now the sum , 



converges if p > 0. Hence J5T P> 4 and 5,, may be made as small as 
we choose, by taking p sufficiently large. Let us note that by 91, 

ff P <-~. (10 

up* 

Thus if p = Min (, /3), 



for a sufficiently large r. 

We consider finally O. We have 

< | D. | 



< Oi+ (7 2 4- <7 8 . 

From 9) we see that 

<7 3 <^5.<l, 

for * sufficiently large. Since g(x) is continuous in 55, 

\g(.x)\<N. 

HeDC6 /* M ^ < 1 ^ ^ 1 

a - 



if ^.:-J-TI on using 10). 
1*1 

Taking * still larger if necessary, we can make 



Thus 



G v 0, < J. 



NON-INTUITIONAL CURVES 541 

The reader now sees why we broke the sum 6) into three parts. 
As h == 0, the middle term contains an increasing number of terms. 
But whatever given value h has, 8 has a finite value. 

Thus as A, B, O are each < e/3, the relation 5) is established. 

Hence E has a differential coefficient at x = e m , and as 

AJF __ A(0) 

T" ~ am ~T~ 

our theorem is established. 

537. Example 1. Let x 

<7O) 

Then for a; =*= 0, g' (x) = | L Here 

3 Va; 

Forar-O, %/ (a;) = + ^ ? ^ (a; 
Thus , - ------ - 



is a continuous function, and at the rational points e m in the in- 
terval 51, 

RF (x) = 4- oo , J^F (a;) = - oo. 



Hence the graph of F has a pantactic set of cuspidal tangents 
in 21. The curve is not monotone in any interval of 9, however 
small. 

Example 2. Let ^ 

$r (x) = a; sin - , x ^= 
a: 

= , a: = 0. 

Then 111 

g f (x) as sin --- cos - , a: s 0. 

T iC ^ 

Here = 1. For x = 0, 

+! , '*- -1. 



542 DERIVATES, EXTREMES, VARIATION 

Then 



is a continuous function in 31, and at the rational point e m , 



where E is the series obtained from F by deleting the m th term. 

538. Pompeiu Curves.* Let us now show the existence of 
curves which have a tangent at each point, and a paiitactic set of 
vertical inflectional tangents. 

We first prove the theorem (Borel): 

Let B(x) = V ^ = V~ , a n > 0, 

V ? n ^n 

where (5 = \e n \ is an enumerable set in the interval SI, and 

A = 2Va n - 

in convergent. Then B converges absolutely and uniformly in a set 
83 < 2(, and 3} is as near 21 as we choose. 

The points 2) where adjoint B is divergent form a null set. 

For let us enclose each point e n in an interval 8 n of length a " 



k 
with e n as center. 

The sum of these intervals is 



^ e, 

"" ' 1C K 

for k > sufficiently large. Let now k be fixed. A point x of 21 
will not lie in any S n if 

r n = | x - g n | > -~n. 

Then at such a point, 

k 

Adjoint B < ^a n -~i=- = k!,Vc^ = kA. 
Va n 

. Annalen, v. 68 (1907), p. 326. 



NON-INTUITIONAL CURVES 543 

As & > 21 e, the points 3) where B does not converge ab- 
solutely form a null set. 

539. 1. We now consider the function 

f(x) = I a n (x - O* = 2/nO) (1 

where @ = J^ n J is an enumerable pantactic set in an interval 21, and 

A-Sa. (2 

is a convergent positive term series. 

Then F is a continuous function of x in ?(. For | x e n \ 3 is < 
some M in ?{. 

Let us note that each f n (x) is an increasing function and the 
curve corresponding to it has a vertical inflectional tangent at the 
point x = e n . 

We next show that F (x) is an increasing function in 51. For let 
x' < x 1 '. Then 

/.(<> </(*") 



JW*') < 

Thus Ji(^) < 

Hence ^(a/) 

2. Let us now consider the convergence of 



obtained by differentiating F term wise at the points of H (. 
Let 3) denote the points in 31 where 



diverges* We have seen 3) is a null set if 

(5 



544 DERIVATE8, EXTREMES, VARIATION 

is convergent. Lei 21 = 3) -f . Let x be a point of , i.e. a 
point where 4) is convergent. We break 3) into two parts 



such that in JDj, each n < 1. Then J> 2 is obviously convergent, 
since each of its terms 

a n ^ , c. 

, :S'f n , where f n = 

and the series 2) is convergent. 

The series D l is also convergent. For as f n < 1, the term 

t^t 

and the series 4) converges by hypothesis, at a point x in &. 
Hence 7>(.r) /s convergent at any point in (, r///ff G = 2( ?/'/^^ 5) is 

<?0/MW#0?l. 

3. Let C 1 ' denote the points in 21 where 3) converges. Let 
2f= C+ A. 

We next show that F\x) = D(^), for x in C. For taking x at 
pleasure in C but fixed, 



We now apply 156, 2, showing that Q is uniformly convergent 
in (0*, 77). By direct multiplication we find that 



Thus 6 ) gives 
Q( h ) = 



(x + h- c n ) -h Or -h h - OO ~ O + (x - 
Let us set 



Then 



NON-INTUITIONAL (CURVES 545 

for < | h <. ?;, 7; sufficiently small. As the series on the right is 
independent <>f h, Q converges uniformly in (0*, rj). Thus 
by 156, 2 

F r = D , for any x in O. 

4. Let now x be a point of A, not in S. At such a point we show 

that 

/"(*)= + 00, (8 

and thus the curve F has a vertical inflectional tangent. For as 
D is divergent at #, there exists for each AT an m, such that 



But the middle term in 7) shows that for \h\< some ij f each 
term in Q m is > * the corresponding term in D m . Thus 



Since each term of ^ is > 0, as 7) shows, 

Q(h) > M. 
Hence 8) is established. 

5. Let us finally consider the points x = e m . If 4> denotes the 
series obtained from F by deleting the m th term, we have 



, m . 

Ax h i &x 

As jPis increasing, the last term is >.0. 
Hence !"()= + , in @. 

vl a result we see the curve F ha* at each point a tangent. At an 
enumerable pan tactic set F", it has points of inflection with vertical 
tangents. 

7. Let us now consider the inverse of the function F, which we 
denote by 

x=G(t^. (9 

As x in 1) ranges over the interval 21, t =F(x) will range over 
an interval S3, and by I, 381, the inverse function 9) is a one- 
valued continuous function of t in 83 which has a tangent at each 



546 DERIVATES, EXTREMES, VARIATION 

point of 33. If TFare the points in 33 which correspond to the 
points V in 21, then the tangent is parallel to the -axis at the 
points W, or (?'() = 0, at these points. The points TFare pan- 
taetic in 33- 

Let Z denote the points of 33 at which GP '() =0. We show 
that Z is of the 2 category, and therefore 

CardZ=c. 

For Cr ( () being of class <_! in 33> its points of discontinuity 8 
form a set of the 1 category, by 486, 2. On the other hand, the 
points of continuity of (?' form precisely the set Z, since the 
points W are pantactic in 93 arid G- 1 = in W. In passing let us 
note that the points Z in 33 correspond 1-1 to a set of points $ at 
which the series 3) diverges. For at these points the tangent to 
F is vertical. But at any point of convergence of 3), we saw in 
2 that the tangent is not vertical. 

Finally we observe that 3) shows that 



2 , n 

3 <T p 

Hence 



ori 



Summing up, we have this result : 

8. Let the positive term series 2Va n converge. Let (, = \e n \ be 
an enumerable pantactic set in the interval 21. The Pompeiu curves 
defined by 

F(x)=-S.aJx- e rf 
have a tangent at each point in 31, whose, slope, is given by 



when this series is convergent, i.e. for all x in 21 except a null set. 
At a point set of the 2 category which embraces @, the tangents 
are vertical. The ordinates of the curve F increase with x. 

540. 1. Faber Curves.* Let F(x) be continuous in the interval 
21 = (0, 1). Its graph we denote by F. For simplicity let 

* Math. Annalen, v. 66 (1908), p. 81. 



NON-INTUITIONAL CURVES 547 

_F(0) = 0, F(l) = 1 Q . We proceed to construct a sequence of 
broken lines or polygons, 



which converge to the curve F as follows : 

As first line L Q we take the segment joining the end points of 

F. Let us now divide 21 into n^ equal intervals 

* 

8 11' 8 12'" S l,n t (2 

of length ., 1 

o t = , 

1 i 
and having 

n, 12>*18 (3 

as end points. As second line L v we take the broken line or 
polygon joining the points on .F whose abscissae are the points 3). 
We now divide each of the intervals 2) into w 2 equal intervals, 
getting the n^ intervals 

S 21 , S 22 , 8 23 ... (4 

of length ., 1 

On - - 1 

and having 



as end points. In this way we proceed on indefinitely. Let us 
call the points 

-4=Kni 

terminal points. The number of intervals in the r th division is 

v r = n l - Wjj w r . 

If L m (jx) denote the one-valued continuous function in SI whose 
value is the ordinate of a point on L m , we have 



, (6 

since the vertices of L m lie on the curve F. 
2. For each x in 21, 



m (x) = F(x). (7 

m= 

For if # is a terminal point, 7) is true by 6). 



548 DER1VATES, EXTREMES, VARIATION 

If x is not a terminal point, it lies in a sequence of intervals 

S 1 >^>- 

belonging to the 1, 2 division of 21. 
Let r, __ , x 

m C^m, ni a m, n+l) 

Since -F(aO is continuous, there' exists an s, such that 

|^(^)-^(a m , n )|<|, m>* (8 

for any x in S m . As L m (x) is monotone in S m , 

| -Z/ m (z) - L m (a mn ) \ < \ L m (a mn ) - L m (a m<n+l *) \ 



Thus I^C^-^KJI^I. (9 

Hence from 8), 9), 

which is 7). 

8. We can write 7) as a telescopic series. For 

1= :i -f (A-A)) 

L^L^ ( 2 - ij) = L, + (L, - X ) + (i 2 - A) 
etc. Hence 

^(a) = lim i n (2-) = i (^) + f ji^rc) - ^^(^l. 
If we set 



we have jP(ar) - t/ n (a?) , (11 

o ' 

n 11 rl w 

a ^(^) = 2/.(^) = i n ( : r). (12 



The function / n (o;), as 10) shows, is the difference between the 
ordinates of two successive polygons L n _ l , L n at the point x. It 
may be positive or negative. In any case its graph is a polygon 



NON-INTUITIONAL CURVES 549 

f n which has a vertex on the a?-axis at the end point of each 
interval S n ^. Let I n8 be the value of f n (x) at the point x = a M , 
that is, at a point corresponding to one of the vertices of f n . We 
call l na the vertex differences of the polygon L n . 

? n5 , n = Max ? . 



s 



Then l/nC*Ol<9n , in 21. (13 

In the foregoing we have supposed F(x) given. Obviously if 
the vertex differences were given, the polygons 1) could be con- 
structed successively. 

We now show : 

If 2 9n (14 

is converge^ 



is uniformly convergent in 21, and is a continuous function in 21. 

For by 13), 14), F converges uniformly in 21. As each f n (x) 
is continuous, F is continuous in 21. 

The functions so defined may be called Faber functions. 

541. 1. We now investigate the derivatives of Faber $ functions, 
and begin by proving the theorem : 

If 2w r ..^ 8 =S^, (1 

s 

converge, the unilateral derivatives of F(x) exist in 21 = (0, 1 ) . More- 
over they are equal, except possibly at the terminal points A = \a mn \. 

For let x be a point not in A. Let x r , x fr lie in V ' V*(x) ; 
letx'-x=h',x"-x=h". 

Let 0== F(x^-F(xy 

V h' h" 

Then F r (x) exists at x, if 

e>0 , 7;>0 , \Q\< , for any x r , x" in V. (2 



550 


DER1VATES, KXTKKMKS, VARIATION 


Now 

Q\< 


F m (.X>~) - F m (x~) 


F n (x") - F m (x) 


+ 


W (v 1 ^ 

^m\ X j 


> - ^mO) 


h' 


h" 


h' 


h" 



But 



Hence 



Similarly 



/.O') - 



sufficiently large. 



<?.< 



Finally, if 77 is taken sufficiently small, x, x 1 ', r" will correspond 
to the side of the polygon l/ m . Hence using 540, 12), we see 
that Q l = 0. Thus 2) holds, and F'(x) exists at x. 

If # is a terminal point a mn , and the two points x 1 ', rr" are taken 
on the same side of a mn , the same reasoning shows that the uni- 
lateral derivatives exist at a mn . They may, however, be different. 

2. Let Wj = w 2 = =2. For the differential coefficient F f (x) to 
exist at the terminal point x, it is necessary that 



Km 2 n p n = oo, 



(3 
(4 



the points where the differential coefficient does not exist form a 
pantactic set in 31. 

Let us first prove 3). Let b < a< c be terminal points. Then 
they belong to every division after a certain stage. We will 
therefore suppose that 6, c are consecutive points in the n {h 
division, and a is a point of the n 4- 1 st division falling in the 
interval 8 n = (6, c). If a differential coefficient is to exist at a, 



a~c 



must be numerically less than some M, as n = oo, and hence their 
sum Q remains numerically < 2 M. 



NON-INTUITIONAL CURVES 551 

Now 



\a-b\ = \a-c\ = S n = - n+l . 
Thus Q = 2+i j2 L n+l (a) - [i n (6) + L n < 



or | @ | = 4 2 n / n , ^ supposing a = a nt . 

Hence 2 n y n < M, 

which establishes 3). 

Let us now consider 4). By hypothesis there exists a sequence 
n l <n 2 < - = oc, such that 

2 nm pn m > G- , m = 1, 2 -., 

(3 1 being large at pleasure. Hence at least one of the difference 
quotients 5) belonging to this sequence of divisions is numerically 
large at pleasure. 

3< If X = 2C (1 

i absolutely convergent, the functions F(x) have limited variation in 
31. 

Forf m (x) is monotone in each interval .. Hence in S^, 
Var/ m = \l m> - L,. + i I < | l m ,\ + | L,. + i |- 

Hence in 21, Var/.,( a! )<22J B .. 

Hence 

VarJ 7 B (a;)<222Z m . = 2X , in . 

m=l s 

We apply now 531. 



552 



DERIVATES, EXTREMES, VARIATION 



542. Faber Functions without Finite or Infinite Derivatives. 

To simplify matters let us consider the following example. 
The method employed admits easy generalization 
and gives a class of functions of this type. We 
use the notation of the preceding sections. 

Let / { have as graph Fig. 1. We next 
divide 21 = (0, 1) into 2 1! equal parts 8 n , S 12 and 
take /!< as in Fig. 2. We now divide 31 into 
2 2! equal parts S 21 , S 22 , S 23 , S 24 and take / 2 (#) as 

in Fig. 3. The height of the peaks is Z 2 = 








In the m th division SI falls into 2 m! equal parts 



FIG. 1 



FIG. 2 



one of which may be denoted by 



Its length may be denoted by the same letter, 
thus -i 



In Fig. 4, S 
division. 



is an interval of the m 1 s 






FIG. 8 



AAA/ 



Fro. 4 
1 



The maximum ordinate of / m (V) is C=- = - . ^- The 

10 m z 10 W 

part of the curve whose points have an ordinate < 2 l m have been 
marked more heavily. The x of such points, form class 1. The 
other x$> make up class 2. With each x in class 1, we associate 
the points a m < /3 m corresponding to the peaks of f m adjacent to x. 
Thus a m <x<fi m . If x is in class 2, the points m , y8 m are the 
adjacent valley points, where f m = Q. 

Let now # be a point of class 1. The numerators in 



(1 



have like signs, while their denominators are of opposite sign. 
Thus the signs of the quotients 1) are different. Similarly if x 
belongs to class 2, the signs of 1) are opposite. Hence for any a?, 



NON-INTUITIONAL CURVES 



553 



the signs of 1) are opposite. It will be convenient to let e m denote 
either a^ or /3 m . We have 



Hence 



f m (x)f n (e n ) 



m! 



410" 



On the other hand, for any x^:r r in S m , 



2l m 
x'-x ~X" 

Hence setting x 1 = e n , and letting n > w, 

!/(.)-/(*) I < Zm -I . - * ! < ' ^ 

o o 

wt w 

1 2 ! 1 9n-l! 

10"^ ' 2^ 10 TO * 2 nl 

J 1_ 

< 10 n " I0 m ' 
For if Iog 2 a l)e the logarithm of a with the base 2, 



71-1 



Iog 2 10 , for n sufficiently large. 



Hence 
Thus 



n! 



or 



2 n ~ 1! 



(8 



2 n! 10 n ' 
and this establishes 4). 

Let us now extend the definition of the f unctions f n (x) by giv- 
ing them the period 1. The corresponding Faber function F(x) 
defined by 540, 12) will admit 1 as period. We have now 



= 2 4- 



From 2) we have 



77 > i / 
^ i ^ 2 6 n 



554 DERIVATES, EXTREMES, VARIATION 

As to 2g, we have, using 4) and taking n sufficiently large, 

.1 J_ 

' 9 ' 10" 



m-1 



Similarly 



2,|< 



-/.(*) I < 2 



Thus finally 



As 



8gn 
S 



Thus 



e.x 



18*. 



3610" "' 



As e n may be at pleasure B or /9 n , and as the signs of 1) are 
opposite, we see that 



awe? F(x) has neither a finite nor an infinite differential coefficient 
at any point. 



CHAPTER XVI 
SUB- AND INFRA-UNIFORM CONVERGENCE 

Continuity 

543. In many places in the preceding pages we have seen how 
important the notion of uniform convergence is when dealing 
with iterated limits. We wish in this chapter to treat a kind of 
uniform convergence first introduced by Arzeld, and which we 
will call subuniform. By its aid we shall be able to give condi- 
tions for integrating and differentiating series termwise much 
more general than those in Chapter V. 

We refer the reader to Arzela's two papers, u Sulle Serie di 
Funzioni," R. Accad. di Bologna, ser. V, vol. 8 (1899). Also 
to a fundamental paper by Osgood, Am. Journ. of Math., vol. 19 
(1897), and to another by ffobson, Proc. Lond. Math. Sac., ser. 2, 
vol. 1 (1904). 



544. 1. Let/^j ... x m , ^ n )=/(#, t) be a function of two 
sets of variables. Let # = (2^ x m ) range over I in an ra-way 
space, and ^ = (^... n ) range over X in an n-way space. As a; 
ranges over 3E and over !, the point (^ ... ^ )== (a;, ) will 
range over a set 31 lying in a space 3J P , p = m + n. 

Let T, finite or infinite, be a limiting point of X. 

Let lim/(3? * t t ) = AO ... a? ) in I 

*=T m 

Let the point x range over 93<3E, while t remains fixed, then 
the point (#, ) will range over a layer of ordinate t, which we 
will denote by ? e . We say x belongs to or is associated with this 
layer. 

We say now that/= <, subuniformly in X when for each >0, 
t/>0: 

566 



556 SUB- AND INFRA-UNIFORM CONVERGENCE 

1 There exists a finite number of layers % t whose ordinates t 
lie in Vf(r). 

2 Each point # of is associated with one or more of these 
layers. Moreover if x = a belongs to the layer 8 t , all the points 
x in some V^(a) also belong to r 



while (#, ) ranges over any one of the layers 8^. When w= 1, 
that is when there is but a single variable x which ranges over an 
interval, the layers reduce to segments. For this reason Arzela 
calls the convergence uniform in segments. 

2. In case that subuniform convergence is applied to the series 



convergent in 21, we may state the definition as follows : 

F converges subunif ormly in 21 when 

1 For each e > 0, and for each v there exists a finite set of 
layers of ordinates > v, call them 

81, V" (2 

such that each point x of 21 belongs to one or more of them, and if 
x = a belongs to m , then all the points of 21 near a also belong 
tog m . 

2 ^.*-" 



as the point (#, ri) ranges over any one of the layers 2). 
545. ^Example. Let 



Here 



The series converges uniformly in 21, except at x = 0. The 
convergence is therefore not uniform in 21; it is, however, sub- 
uniform. For 

n\x\ 



CONTINUITY 557 

Hence taking m at pleasure and fixed, 

\P m \ <e , s in 8 1 =(~S, 8), 
sufficiently small. On the other hand, 



Thus for w sufficiently large, 



Hence we need only three segments 8 V $ 2 , $ 3 to get subuniform 
convergence. 

546. 1. Let /(a?!---^, ^ n ) = <^(a? 1 # m ) in 3E, as = r, 
finite or infinite. Let f(z* f) be continuous in H for each t near r. 
For (f) to be continuous at the point x = a in X, it is necessary that 
for each e > 0, there exists an tj> 0, and a d t for each t in F T) *(r) 
such that 



for each t in F^ and for any x in V dt (cC). 

It is sufficient if there exists a single t=/3 in F^*(r) for which 
the inequality 1) holds for any x in some F$(a). 

It is necessary. For since < is continuous at x = a, 

| <(#) </>(a) | < | , for any x in some F$(a). 

o 

Also since /= </>, 

|/(a, ~ ^( a ) I < | > for an J t in some ^i*( T )- 

o 

Finally, since /is continuous in x for any near T, 

-/(a, 01 < ^ for an 7 ^ in some V^(a). 



Adding these three inequalities we get 1), on taking 

d t < 8, S| . 



558 SUB- AND INFRA-UNIFORM CONVERGENCE 

It u sufficient. For by hypothesis 

\f(x, /3) - <O) | > I , for any x in some F fi ,(a); 
and hence in particular. 



Also since /(#, /3) is continuous in #, 

a?, /8-, # < > for an * in some 



Thus if S < ', S", these unequalities hold simultaneously. Add- 
ing them we get 

<f>(x) <K)| < i for any X in Ps(a), 
and thus < is continuous at = a. 
2. As a corollary we get : 
Let V( x) = 2/ t ... ,(! - z m ) 



converge in 21, 0t'A /^r/>i 6mt^ continuous in 31. 

tinuous at the point x a in 31, it is necessary that for each e > 0, 

and for any cell 11^ > some /2 A i there exists a S M such that 

\F^(x)\<e , /or awy x in ^(a). 
It is sufficient if there exists an R K and a S > 

a? in F s 



547. 1. />^^ Urn /(^j ^ m , ^ t n ) = ^(^j x m } in J- T finite 

r=x 

or infinite. Letf(.r, t) be continuous in Hi for each t near r. 
1 //"/== <f> sub nn if or ml y in 3E, <j> is continuous in . 
"2 If 3 is complete, and <f> is continuous in , / = </> subuniformly 



n 



7b prove 1. Let ^ = # be a point of 3. Let e > be taken at 
pleasure and fixed. Then there is a layer 8^ to which the point 
a belongs and such that 



CONTINUITY 559 

when (#, ) ranges over the points of S^. But then 1) holds for 
t = ft and x in some V B (a). Thus the condition of 546, 1 is satis- 
fied. 

To prove 2. Since <f> is continuous at x = a, the relation 1 ) 
holds by 546, 1, for each t in F^*(T) and for any x in V dt (a}. 
With the point a let us associate a cube O a ^ lying in D dt (cC) and 
having a as center. Then each point of 36 lies within a cube. 
Hence by Borel's theorem there exists a finite number of these 
cubes (7, such that each point of lies within one of them, say 

0. A . ^,- (2 

But the cubes 2) determine a set of layers 

8,, , V" ( 

such that 1) holds as (x, t) ranges over the points of 21 in each 
layer of 8). Thus the convergence of /to < is sub uniform in J. 

2. As a corollary we have the theorem : 

Let F^x l .-x m ) = ^,.. ln (x 1 ...x m ) 

converge in 26, each f, being continuous in 3. // JP converges sub- 
uniformly in , F is continuous in . // X z's complete and F is 
continuous in 36, J5 7 converges sukuniformly in 3E. 

548. 1. Let ^) = 2f.,., n (*,**.) 

converge in 31. 

j[/? fA^ convergence be uniform in 21 except possibly for the points 
of a complete discrete set 93 = \ b \ . For each 6, let there exist a \ 
such that for any \ > \ , 

lim JF A <) = 0. 



^ converges subuniformly in 21. 

For let D be a cubical division of norm d of the space 9I TO in 
which 21 lies. We may take d so small that $8 D is small at 
pleasure. Let B D denote the cells of D containing points of 21 
but none of S3. Then by hypothesis ^converges uniformly in JB D . 
Thus there exists a /A O such that for any ft > /* , 

I ^0*0 1 < e ' * Qr an y x ^ ^ ^ n BD- 



560 SUB- AND INFRA-UNIFORM CONVERGENCE 

At a point b of S3, there exists by hypothesis a Fs(5) and a X 
such that for each X > X 

| J\< j < , for any x in F 5 (6). 

Let (7 6iA be a cube lying in J9 5 (5), having b as center. Since S3 
is complete there exists a finite number of these cubes 

C^ i C& jA8 (1 

such that each point of S3 lies within one of them. 
Moreover 



for any x of 21 lying in the tc ih cube of 1). 

As B D embraces but a finite number of cubes, and as the same 
is true of 1), there is a finite set of layers such that 



I < in each 2- 
The convergence is thus subunif orm, as X, /* are arbitrarily large. 

2. The reasoning of the preceding section gives us also the 
theorem : 



in 36, r finite or infinite. Let the convergence be uniform in J except 
possibly for the points of a complete discrete set (S \e\. For each 
point e, let there exist an rj such that setting e(#, t) =/(., t) <(#), 

lim e(#, f) = , for any t in F^*(r). 

jce 

Thenf= <f> subuniformly in 36. 

3. As a special case of 1 we have the theorem : 

Let *(*)=/,(*)+/,(*)+- 

converge in 21, and converge uniformly in 21, except at x = j, a:= <t . 
.4^ re = t Ze^ there exist a i> t wcA that 



, n t > v, , t = 1, 2 ... 5. 

^=a t 

F converges subuniformly in 21. 



CONTINUITY 561 

4. When 



, N * x \ 
t) = $(x) 

t=T 

we will often set 

/(*,*) = *(*) + o*o. 

and call e the residual function. 

549. Example 1. 
f(x, n) = --*" = <(X> = , for n ~ oo in 21 = (0 < a), 

M.rP 

a, , \ > , /* > 0. 

The convergence is suburiiform in 31. For x = is the only 
possible point of non-uniform convergence, and for any m, 

I e(>, m) 1=^^=0 , asa:=0. 
1 y e m^ 

/yi A/yd 

.Example 2. /(#, n) = - - - = </> (a;) = , as n = <x>, 



^ in 21 == (0 < a) , a, /3, X, p > , /t > X , <? > 0. 
The convergence is uniform in 93 = (0 < a), where e > 0. For 

| (*,n)|<-? A -^-- , in 33 
1 '""(? + w^e^ 

a a n A 

^ n'* 

< e , for n > some m. 

Thus the convergence is uniform in 21, except possibly at x = 0. 
The convergence is subuniform in 21. For obviously for a given n 

lim/(#, n) = 0. 

x=Q 

550. 1. Let limf(x l x m t l ^ n ) = ^(^ a: m ) iw X, r finite 

<=T 

or infinite. 

Let the convergence be uniform in H except at the points 



562 SUB- AND INFRA-UNIFORM CONVERGENCE 

For the convergence to be sub-uniform in , it is necessary that for 
each b in SB, and for each > 0, there exists a t = near r, such that 

}im\e(x, *)\>c. (1 

x=b 

For if the convergence is subuniform, there exists for each 
and rj > a finite set of layers ?,, t in F' 7? *(r) such that 

| e(#, ) | < e , x in 8,. 

Now the point # = b lies in one of these layers, say in 80 . 
Then 

| e(a?, /8) | < , for all # in some V*(ti). 

But then 1) holds. 

2. Example. Let ^ , ~ /-i x 

r ^v*0 = 2# n (l #). 

o 

This is the series considered in 140, Ex. 2. 

F converges uniformly in 21 = ( 1, 1), except at x = 1. 



we see that ,. TJ s \ -i 

hm F m (x) = - 1. 

Hence F is not subuniformly convergent in 31. 



Integrdbility 

551. 1. Infra-uniform Convergence. It often happens that 
f(x l " x m t^ - n ) = (f>(x l -- x m ) 

subuniformly in J except possibly at certain points (= \e\ form- 
ing a discrete set. To be more specific, let A be a cubical divi- 
sion of $ TO in which J lies, of norm 8. Let X denote those cells 
containing points of J, but none of @. Since (5 is discrete, 
Jf A = J. Suppose now/=< subuniformly in any JST A ; we shall 
say the convergence is infra-uniform in X. When there are no 
exceptional points, infra-uniform convergence goes over into sub- 
uniform convergence. 



INTEGRABTLITY 563 

This kind of convergence Arzela calls uniform convergence by 
segments, in general. 

2. We can make the above definition independent of the set @, 
and this is desirable at times. 

Let H = ( X, j) be an unmixed division of such that may be 
taken small at pleasure. If f=<f> subuniformly in each X, we 
say the convergence is infra-uniform in 3E. 

3. Then to each e, 77 >(), and a given Jf, there exists a set of 
layers I x , t a , t \\\ F^*(T), such that the residual function e(o;, t) 
is numerically < e for each of these layers. As the projections of 
these layers { do not in general embrace all the points of J, we 
call them deleted layers. 

4. The points we shall call the residual points. 

x 2 

5. Example 1. __ V T~T3~* 

This series was studied in 150. We saw that it converges uni- 
formly in Sl = (0, 1), except at #= 0. 

As -, 



and as this == 1 as x = for an arbitrary but fixed n, F does not 
converge subuniformly in 21, by 550. The series converges infra- 
uniformly in 21, obviously. 

6. Example 2. ^^ % x n (l - x} 

o 

This series was considered in 550, 2. Although it does not 
converge subuniformly in an interval containing the point x = 1, 
the convergence is obviously infra-uniform. 

552. 1. Let lim f (x l x m t n ) = </)(^ # TO ) be limited in , 

r finite or infinite. For each t near r, letf be limited and R-inteyrable 
in H. For <f> to be R-integrable in , it is sufficient thatf == <f> infra- 
uniformly in X. If Hi is complete, this condition is necessary. 



564 SUB- AND INFRA-UNIFORM CONVERGENCE 



It is sufficient. We show that for each e, o>> there exists a 
division D of 9} m such that the cells in which 

OSC <f> > Q) (1 

have a volume < <r. For setting as usual 

/=<+, 

we have in any point set, 

Osc<<0sc/+ Osce. 
Using the notation of 551, 



in the finite set of deleted layers I p ( 2 corresponding to 
=}, 2 For each of these ordinates t L ,f(x, t ) is integrable 
in 3. There exists, therefore, a rectangular division D of 9? OT , 
such that those cells in which 



have a content < ^, whichever ordinate t t is used. Let E be a 

A 

division of 9J m such that the cells containing points of the residual 
set y have a content < cr/2. Let F = D + E. Then those cells 
of .F in which 

O>^ or Osc ' e<>, *.) | >| 

J ^J 

t = l, 2 have a content < o-. Hence those cells in which 1) 
holds have a content < a. 

It is necessary, ifH is complete. For let 

*i> 2 == T - 

Since <fr and/(#, ^ n ) are integrable, the points of discontinuity of 
<(V) and of f (x, t n ) are null sets by 462, 6. Hence if S, S< denote 
the points of continuity of <(#) and /(a;, in X, 



since 3 is measurable, as it is complete. 



INTEGRABILITY 565 

Let = Qdv{6/|, 

then = 

by 410, 6. 

Let SD = Dv((E, ), 

then = , (1 

as we proceed to show. For if 6r = I , 



But 6Ms a null set. Hence Meas Dv(, 6r) = 0, and thus 
g = i = ), which is 1). 

Let now be a point of 5), let it lie in S^, Sj, where ^, f 2 
form a monotone sequence = r. Then since 



there is an m such that 

I (& O I < I fo r any n >m. (2 

But ^ lying in ), it lies in S and S <n . 
Thus 



for any ^ in Fad). Hence 

| <>, f n ) - (f , O | < ^ , x in Fi(f ). (3 

Now 

e(x, O = e(a:, n ) - e(f, U + c(f, * n ). 

Hence from 2), 3), 

| e(>, t n )\<e , for any x in F" 6 (|:). 

Thus associated with the point , there is a cube T lying in -Dfi(f), 
having | as center. As D = X 35 is a null set, each of its points 
can be enclosed within cubes (7, such that the resulting enclosure 



566 SUB- AND INFRA- UNIFORM CONVERGENCE 

@ has a measure < <r, small at pleasure. Thus each point of J lies 
within a cube. By Borel's theorem there exists a finite set of 
these cubes 

*1> ^2 '*' r ' 1' 2 '" ^' 

such that each point of 3E lies within one of them. But corre- 
sponding to the F's, are layers 

81, 8,, - 8, 

such that in each of them 



Thus/ = </> subuniformly in X = (T l , T 2 ... T r ). Let y be the 
residual set. Obviously $ < &. Thus the convergence is infra- 
uniform . 

2. As a corollary we have : 

Let F(x) = ^... in (x^^x m } 

converge in 21. Let F be limited, and each f, be limited and R~in- 
tegrable in 21. For F to be R-integrable in ?J, it in sufficient that F 
converges infra-uniformly in 21. 

If 21 is complete* this condition is necessary. 

553. Infinite Peaks. 1. Let lim/(a? t x m t l t n ) = <(V) in X, 

t=T 

T finite or infinite. Although f(x, t) is limited in I for each t 
near r, and although <f>(x) is also limited in I, we cannot say that 

|/('#, | < some JMT (1 

for any x in X and any near r, as is shown by the following 

to 
Example. Let /(a;, O = == 0(a?) = 0, as == oo for 2: in 



= (-- 00, GO). 

It is easy to see that the peak of / becomes infinitely high {is 

n = oo. 

In fact, for x = , / = ^. Thus the peak is at least as high 
V t e 

as , which == oo . 
e 



INTEGRAB1LITY 567 

The origin is thus a point in whose vicinity the peaks of the 
family of curves f(x, t) are infinitely high. In general, if the 
peaks of /v-Wi-Q 

in the vicinity F 6 of x become infinitely high as t = T, however 
small 8 is taken, we say is a point with infinite peaks. 

On the other hand, if the relation 1) holds for all x and t in- 
volved, we shall say /(#, ) is uniformly limited. 

2. If lini /Ov-.a^ ^ 1 ..^ n ) = </)(^ 1 ...^ m ), <wdf i/ /(, w 

<=T 

uniformly limited in 36, Am < / limited in H. 

For a; being taken at pleasure in H and fixed, $(V) is a limit 
point of the points /(#, t) us t = T. But all these points lie in 
some interval ( 6r, 6r) independent of x. Hence <f> lies in this 
interval. 

3. If H is complete, the points $ in 3E with infinite peaks also form 
a complete set. If these points $ are enumerable, they are discrete. 

That $ is complete is obvious. But then $ = ^ = 0, as $ is 
enumerable. 

554. 1 . Let lim / ( x l x m t 1 n ) = ^(^ o; m ) m J, metric or 

tT 

complete. Let f (x, t) be uniformly limited in 3E, and R-integrable 
for each t near r. For the relation 

lim (/(a:, 0= f<K*) 

<=T /X /; 

^o A0W, i^ Z8 sufficient that f '^ (j> infra- uniformly in 3E. If Hi is 
for each t complete, this condition is necessary. 

For by 552, </> is J?-integrable if /= </> infra-uniformly, and when 
X is complete, this condition is necessary. By 424, 4, each/ (x, ) 
is measurable. Thus we may apply 381, 2 and 413, 2. 

2. As a corollary we have the theorem : 

Let FC*)-^ <>!*.) 

converge in the complete or metric field 21- Let the partial sums F^ be, 
uniformly limited in 21- Let each term/, be limited and R-integrable 
in 21. Then for the relation 

f F=2 ft 

J% *V l 



568 SUB- AND INFRA-UNIFORM CONVERGENCE 

to hold it is sufficient that F is infra -uniformly convergent in 21. 
21 is complete, this condition is necessary. 

555. Example 1. Let us reconsider the example of 150, 



We saw that we may integrate termwise in 21 = (0, 1), al- 
thongh jPdoes not converge uniformly in 21. The only point of 
non-iunl'orm convergence is x = 0. In 551, 5, we saw that it con- 
verges, however, infra-uniformly in 21. As 

I ^n(X) I < 1 * f r an y x in 21, and for every n, 

all the conditions of 554 are satisfied and we can integrate the 
series termwise, in accordance with the result already obtained 
in 150. 

Example. Let F(x) = V I ^ - ^ ~ ^ x U 0. 
F fM e nx * e (n ~ w 

Then 



We considered this series in 152, i. We saw there that this 
series cannot be integrated termwise in 21 = (0 < a). It is, how- 
ever, subuniformly convergent in 21 as we saw in 549, Ex. 1. We 
cannot apply 554, however, as F n is not uniformly limited. In 
fact we saw in 152, l, that x = is a point with an infinite peak. 

Example 3. F(x) = i# n (l - x). 

o 

We saw in 551, 6, that F converges infra-uniformly in 21 = (0, 1). 
Here F n (x)\ = \l ~ x\ < some M, 

for any x in 21 = (0 < u), u <_ 1, and any n. Thus the F n are 
uniformly limited in 21. 

We may therefore integrate termwise by 554, 2. We may 
verify this at once. For 



x = 0. 



INTEGRABILITY 569 

Hence C u F(x)dx = u. (1 

/o 
On the other hand, 

X u ?/ n+ l 

F n dx = u -- - = u , as n = oo. (2 

n -f 1 



n -f 1 
From 1), 2) we have 



^n+1 U *M 



- + 1 w 4- 2 J ' 



556. 1. //' l /'^! " x m ^i O == $(#i x m) infra-uniformly 
in the metric or complete field , as t == T, T ^iii^ or infinite ; 

2 /(a?, is uniformly limited in 36 flncif R-integrable for each t 
near r; 

Then ^ 

uniformly with respect to the set of measurable fields ?l in I- 

If H is complete, condition 1 may be replaced by 3 (/>(#) is 
R-integrdble in %. 

For by 552, 1, when 3 holds, 1 holds ; and when 1 holds, </> 
is 7^-integrable in X. 

Now the points & t where 

are such that ^ 

lim <g, = , by 412. 

Let = (, + ,. Then 



But 

which establishes the theorem. 
2. As a corollary we have : 

If 1 .F(V)= 2/ tl ... ln (2i a; m ) converc/es wfra--uniformlt/< and 
each of its terms f^ is R-inteyrable in the metric or complete field 9f ; 



570 SUB- AND INFRA-UNIFORM CONVERGENCE 

2 F\(jx) is uniformly limited in SI/ 

Then 



and the series on the right converges uniformly with respect to all 
measurable 93 _<_ 21. 

3. If 1 lim/(:r, ^ t n )=*<j>(x) is R-integrable in the interval 

t=T ' 

?l = (a < />), T finite or infinite ; 

2 f(x y ix uniformly limited, and R-integrable for each t near r; 

Then lim f J f(x, t^dx = f*4>(x)dx = <S>(x), 

f=T *-<! *^ 

uniformly in ^ and <\>(x) is continuous in 21. 



and also each termf t are R-integrable in the interval 2l = (a< 6); 
2 F^x) is uniformly limited in 21; 

^^ a(rr)= 2 f'f^dx , m 21 

*^a 
i continuous. 

For (? is a uniformly convergent series in 21, each of whose terms 



is a continuous function of x. 

Differentiability 
557. 1 . If 1 lim /Or, ^ n ) == </>(^) m 21 = ( a < 6), r j^m'te or 

tT 

infinite ; 

2fj(x, fy is R-integrable for each t near T, and uniformly limited 
m2l; 

3 fj.(x, ^)= ^(^) infra -uniformly in ?(, as t = r ; 

Then at a point :r of continuity of ty in 21 

4>'(X>=tO> (1 

is the same 

Or, ) = lini./OM)- ( 2 



DIFFERENTIABILITY 571 

For by 554, 

lira f /;(*, f)dx = PV(*)fe (8 

tT ^d *J a 

= liui [/(a:, ) - /(, )] , by I, 538 

t=* 

= <K*)-<Ka) , by 1. 
Now by I, 537, at a point of continuity of ->Jr, 



From 3), 4), we have 1 ), or what is the same 2). 

2. In the interval 31, if 

1 F(x)= /' ti ... lri (#) converges ; (1 

2 ~Eachfl(x) is limited and R-integrable ; 

3 .F( (#) is uniformly limited ; 

4 (?(#)= 2// is infra-uniformly convergent ; 

TA^w dtf a point of continuity of Gr(j') in 31, wv ^//y differ mtiati* 
the series 1) termwise, or F r (x) G(x). 

3. /TI ^e interval 31, if 

1 /(^, ^ O = <O) < === T, r finite or infinite : 

2 f (x, f) is uniformly limited, and a continuous function of x ; 

3 ty(x) = lim/^(a;, t) is continuous ; 

tT 

Then 



or t^Aa^ f the same 

-^lim/O, 0=lim-?-/(rc, Q. (2 

a^ /=T <=T ao; 

For by 547, 1, condition 3 requires that /' = ^ subuniforml y 
in SI. But then the conditions of 1 are satisfied and 1) and 2) 
hold. 

4. In the interval 31 let us suppose that 

1 JF (/) = 2/4 . c n (*0 Converges ; (1 



572 SUB- AND INFRA-UNIFORM CONVERGENCE 

2 Each termf, is continuous; 

3 F((x) is uniformly limited ; 

4 6r(V)= 2/'(#) is continuous ; 

Then we may differentiate 1) termwise, or F'(x) = Q-(x). 

558. Example 1. We saw in 555, Ex. 3 that 

n+l 



The series got by differentiating termwise is 

#(aO=2z n (l--20==l 0<a;<l 

o C2 

= , z = 0. v 

Thus by 557, 4, 



The relation 3) does not hold for x = 0. 

Example 2. 



- ^ gv ^ - arctg ^ 



Vw- -f 1 



Here JP(a;)=arctga;, for any x. (1 

^ )=S ' ** (2 



Hence Cr(x) is continuous in any interval 21, not containing 
x = 0. Thus we should have by 557, 4, 

JF(aO0(a!> * in 21. (3 

This relation is verified by 1), 2). The relation 3) does not 
hold for x = 0, since 

J(0)=l, , 0(0) -0. 



DIFFERENTIABILITY 573 

Example 8. 



= log (1 + a 2 ) , for any x. 



In any interval 31, all the conditions of 557, 4, hold. 

Hence F'(x)=G-(x} , for any x in 21. (3 

In case we did not know the value of the sums 1), 2) we could 

still assert that 3) holds. For by 545, Gr is subuniformly con- 

vergent in 21, and hence is continuous. 



Example 4* 

l+nx 



- J+O + i)* 1 = 1 

(n + iy +* J e 



ne n 
Here 

The series obtained by differentiating F termwise is 
and hence 

e x e nx 



The peaks of the residual function 



are of height = l/e. The convergence of Q- is not uniform at 
x = 0. The conditions of 557, 4, are satisfied and we can differ- 
entiate 1) termwise. This is verified by 2), 3). 



574 SUB- AND INFRA-UNIFORM CONVERGENCE 

559. 1. Ifl lim/(#, t 1 t n ) = <(V) is limited and R-inteyrable 
t-t 

in the, interval 21 = ( a < b) ; 

2 f(x, t) Is limited, and H-integrable in $1, for each t near r; 

8 T/r(a;)= lim (* f(x< f) = lim#O, t) 

t=T J<1 * t=T 

is a continuous function in ?I; 

4 The point* & in 31 in whose vicinity the peaks of f(x, t) a* 
t= r are infinitely high form an enumerable set ; 

Then * x /v 

0(x)= I 0<X) = liiii I /(^ t)dx = ^(x), (1 
^/^ /- T At 

^ /v /** 

lim I /(a;, t)ds = I lim/(^, O^, 

/=T ^^ ' /tt f=T ' 

awrf ^/i6 set @ i complete and discrete. 

For @ is discrete by 553, 3. 

Let a be a point of A 31 (. Then in an interval a about , 

|/(j?, t) | < some Tlf , x in a, any ^ near r. (^ 

Now by 556, 3, taking e>0 small at pleasure, there exists an 
77 > such that 



<e 

for a^y x in a, and in F^ *(T). If we set x = a + h, we have 

^ = o ? o i i^ = ir > ^ ^ + e' 

Aa* h hJ h 

Also by 556, 3, we have 



for any x in a, and t in F^*(T). Thus 

1 i* f < , 7 B(x) v( ) e" A$ 6^ 

I / ( X* 1 )uX - - - - - -j~ -. . ^r . -|- _-_ 

//V" ' h h A.r A 

From M), 4 ) we have 4 

e' A^ . e ;/ 



DIFFERENTIABILITY 575 

Now e may he made small at pleasure, and that independent of 
A. Thus the last relation mves 



Ailr Atf P < 

- , ror ;r in A. 

A.r A# 

As this holds however small h = A# is taken, we have 

f/'V/r d0 f 



Hence by 515, 3, 

(x)=6(x)+ const , in 21. 



For a? = a, ^(a ) 0(a) = ; 

and thus 



in . 



2. As a corollary we have : 

If] F(x) 2/' t1 ... t/t (#) is limited a)id R inteijrable in the inter- 
val 21 = ( a < J> ) ; 

2 JF\(x) in limited and each term f\ i$ R- httet/rable ; 
3 G-(x)= ^ I / t is continuous; 



4 The points 6 m ?l i>^ wlioxe vicinity the peaks of F x (x) are in- 
finitely JiiifJt form an enumerable xet; 






rt t y integrate the F series termwise. 

560. 1 . IfT lim/( ar, t l - ^ n ) = </>(>) * ^ = ( a < fy r finite or 
infinite ; 

2 f' r (x^ f) is limited and R-inteyr able for each t near T; 

3 The points (g o^ 1 ?l i ^Ao^ vicinity f^x, t) has infinite peaks 
as t = r form an enumerable set: 

4 <(#) *'* continuous at the points @; 

5 ty(x) = limj^(r, O / limited and R-integrable in ?l; 



576 SUB- AND INFRA-UNIFORM CONVERGENCE 

Then at a point of continuity of ^(#) in 31 

or what is the same 

- 

ax t= r 

For let 8 = (a < /3) be an interval in 31 containing no point of 
(. Then for any x in 



) , by 2. 

^ 

Hence ~ x 

lira I /;(*, f)dx = \im\f(x, f) -/(, )} 

/=T 'a /=T 

== </><V) - <K) , byl. (2 

By 556, 3, </>(^) is continuous in S. Thus <(#) i y continuous 
at any point not in (g. Hence by 4 it is continuous in 31. 

We may thus apply 559, l, replacing therein f(x, t) by/j(#, t). 
We get 

C x C x C x 



t=T 



Since 2) obviously holds when we replace a by a, this relation 
with 3) gives 



At a point of continuity, this gives 1) on differentiating. 

2. I/* 1 jF(z) = 2/ ti ... tw (^) converges in the interval 21; 

2 (?(a;) = /[(#) anc? eac?A q/ its terms are limited and R- 
integrable in 31; 

8 The points of 31 m whose vicinity (? A (^) Aa infinite peaks as 
\ == (X),form an enumerable set at which F(x) is continuous; 

Then at a point of continuity of Gr(x) we have 



or what is the same 



DIFFERENTIABILITY 577 

561. Example. 

F 

Hence 



<*> , , , i\ ^ a 

(x ^ = y (5^_ (*L+LX l _ 

w T 1 * nx ' * M ** J ~ **' 



The series obtained by differentiating F termwise is 
&(x )== \? [2nx _2n*a* __ 2(n 

nx * nx2 ^ n 



^(71+1)^ J 

Here a- r ^ - - - I ?^- - - 

Hence 9 

is a continuous function of x. 

The convergence of the Q- series is not uniform at x = 0. For 
set a n == I/ft,. Then 

2 

9 

^, 



To get the peaks of the residual function we consider the 
points of extreme of 



We find .-, r o , o 

' ^n 5 war + 2 

"** 



Thus y 1 = when 

2 nV - 5 wa ? + 1 = 0, 



or when x = ~ or - , a, a constants. 

Vw- Vn 

Putting these values in 3), we find that y has the form 

y = c V/i. 

Hence # = is the only point where the residual function has 
an infinite peak. Thus the conditions of 560, 2, are satisfied, and 
we should have F'(x) = &(x) for any x. This is indeed so, as 1 ), 
2) show. 



CHAPTER XVII 
GEOMETRIC NOTIONS 

Plane Curves 

562. In this chapter we propose to examine the notions of 
curve and surface together with other allied geometric concepts. 
Like most of our notions, we shall see that they are vague and 
uncertain as soon as we pass the confines of our daily experience. 
In studying some of their complexities and even paradoxical 
properties, the reader will see how impossible it is to rely on his 
unschooled intuition. He will also learn that the demonstration 
of a theorem in analysis which rests on the evidence of our 
geometric intuition cannot be regarded as binding until the 
geometric notions employed have been clarified and placed on a 
sound basis. 

Let us begin by investigating our ideas of a plane curve. 

563. Without attempting to define a curve we would say on 
looking over those curves most familiar to us that a plane curve 
has the following properties : 

1 It can be generated by the motion of a point. 

2 It is formed by the intersection of two surfaces. 

8 It is continuous. 

4 It has a tangent at each point. 

5 The arc between any two of its points has a length. 

6 A curve is not superficial. 

7 Its equations can be written in any one of the forms 

y =/(*), a 

(2 

(3 

and conversely such equations define curves. 

578 



PLANE CURVES 579 

8 When closed it forms the complete boundary of a region. 

9 This region has an area. 

Of all these properties the first is the most conspicuous and 
characteristic to the naive intuition. Indeed many employ this 
as the definition of a curve. Let us therefore look at our ideas 
of motion. 

564. Motion. In this notion, two properties seem to be essen- 
tial. 1 motion is continuous, 2 it takes place at each instant in 
a definite direction and with a definite speed. The direction of 
motion, we agree, shall be given by dy/dx, its speed by ds/dt. 
We see that the notion of motion involves properties 4, 5, and 7. 
Waiving this point, let us notice a few peculiarities which may 
arise. 

Suppose the curve along which the motion takes place has an 
angle point or a cusp as in I, 366. What is the direction of 
motion at such a point? Evidently we must say that motion is 
impossible along such a curve, or admit that the ordinary idea of 
motion is imperfect and must be extended in accordance with the 
notion of right-hand and left-hand derivatives. 

Similarly ds/dt may also give two speeds, a posterior and an 
anterior speed, at a point where the two derivatives of s = </>() 
are different. 

Again we will admit that at any point of the path of motion, 
motion may begin and take place in either direction. Consider 
what happens for a path defined by the continuous function in 
I, 367. This curve has no tangent at the origin. We ask how 
does the point move as it passes this point, or to make the ques- 
tion still more embarassing, suppose the point at the origin. In 
what direction does it start to move? We will admit that no 
such motion is possible, or at least it is not the motion given us 
by our intuition. Still more complicated paths of this nature are 
given in I, 369, 371, and in Chapter XV of the present volume. 

It thus appears that to define a curve as the path of a moving 
point, is to define an unknown term by another unknown term, 
equally if not more obscure. 

565. 2 Property. Intersection of Two Surfaces. This property 
has also been used as the definition of a curve. As the notion 



580 GEOMETRIC NOTIONS 

of a surface is vastly more complicated than that of a curve, it 
hardly seems advisable to define a complicated notion by one still 
more complicated and vague. 

566. 3 Property, Continuity. Over this knotty concept philos- 
ophers have quarreled since the days of Democritus and Aristotle. 
As far as our senses go, we say a magnitude is continuous when 
it can pass from one state to another by imperceptible gradations. 
The minute hand of a clock appears to move continuously, although 
in reality it moves by little jerks corresponding to the beats of the 
pendulum. Its velocity to our senses appears to be continuous. 

We not only say that the magnitude shall pass from one state 
to another by gradations imperceptible to our senses, but we also 
demand that between any two states another state exists and so 
without end. Is such a magnitude continuous ? No less a mathe- 
matician than Bolzano admitted this in his philosophical tract 
Paradoxien des Unendlichen. No one admits it, however, to-day. 
The different states of such a magnitude are pantactic, but their 
ensemble is not a continuum. 

But we are not so much interested in what constitutes a con- 
tinuum in the abstract, as in what constitutes a continuous curve 
or even a continuous straight line or segment. The answer we 
have adopted to these questions is given in the theory of irra- 
tional numbers created by Cantor and Dedekind [see Vol. I, 
Chap. II], and in the notion of a continuous function due to 
Cauchy and Weierstrass [see Vol. I, Chap. VII]. 

These definitions of continuity are analytical. With them we 
can reason with the utmost precision and rigor. The consequences 
we deduce from them are sufficiently in accord with our intuition 
to justify their employment. We can show by purely analytic 
methods that a continuous f unction /(#) does attain its extreme 
values [I, 354], that if such a function takes on the value a at the 
point P, and the value b at the point Q, then it takes on all inter- 
mediary values between a, J, as x ranges from P to Q [I, 357]. 
We can also show that a closed curve without double point does 
form the boundary of a complete region [cf. 576 seq.]. 

567. 4 Property. Tangents. To begin with, what is a tangent ? 
Euclid defines a tangent to a circle as a straight line which meets 



PLANE CURVES 581 

the circle and being produced does not cut it again. In com- 
menting on this definition Casey says, " In modern geometry a 
curve is made up of an infinite number of points which are 
placed in order along the curve, and then the secant through two 
consecutive points is a tangent." If the points on a curve were 
like beads on a string, we might speak of consecutive points. As, 
however, there are always an infinite number of points between any 
two points on a continuous curve, this definition is quite illusory. 

The definition we have chosen is given in I, 365. That property 
3 does not hold at each point of a continuous curve was brought 
out in the discussion of property 1. Not only is it not necessary 
that a curve has a tangent at each of its points, but a curve does 
not need to have a tangent at a pantactic set of points, as we saw 
in Chapter XV. 

For a long time it was supposed that every curve has a tangent 
at each point, or if not at each point, at least in general. Analytic- 
ally, this property would go over into the following : every con- 
tinuous function has a derivative. A celebrated attempt to prove 
this was made by Ampere. 

Mathematicians were greatly surprised when Weierstrass ex- 
hibited the function we have studied in 502 and which has no 
derivative. 

Weierstrass* himself remarks: "Bis auf die neueste Zeit hat 
man allgemein angenommen, dass eine eindeutige und continuir- 
liche Function einer reellen Verlinderlichen auch stets eine erste 
Ableitung habe, deren Werth nur an einzelnen Stellen unbestimmt 
oder unendlich gross werden konne. Selbst in den Schriften von 
Gauss, Cauchy, Dirichlet findet sich meines Wissens keine 
Ausserung, aus der unzweifelhaft hervorginge, dass diese Mathe- 
matiker, welche in ihrer Wissenschaft die strengste Kritik iiberall 
zu iiben gewohnt waren, anderer Ansicht gewesen seien." 

568. Property 5. Length. We think of a curve as having 
length. Indeed we read as the definition of a curve in Euclid's 
Elements : a line is length without breadth. When we see two 
simple curves we can often compare one with the other in regard 
to length without consciously having established a way to measure 

* Werke, vol. 2, p. 71. 



582 GEOMETRIC NOTIONS 

them. Perhaps we unconsciously suppose them described at a 
uniform rate and estimate the time it takes. It may be that we 
regard them as inextensible strings whose length is got by 
straightening them out. A less obvious way to measure their 
lengths would be to roll a straightedge over them and measure 
the distance on the edge between the initial and linal points of 
contact. 

We ask how shall we formulate arithmetically our intuitional 
ideas regarding the length of a curve ? The intuitionist says, a 
curve or the arc of a curve has length. This length is expressed 
by a number L which is obtained by taking a number of points 
Pj, P 2 , P 3 "- on the curve between the end points P, P', and 
forming the sum 



The limit of this sum as the points became pantactic is the 
length L of the arc PP'. 

Our point of view is different. We would say : Whatever 
arithmetic formulation we choose we have no a priori assurance 
that it adequately represents our intuitional ideas of length. 
With the intuitionist we will, however, form the sum 1) and see if 
it has a limit, however the points P t are chosen. If it has, we will 
investigate this number used as a definition of length and see if it 
leads to consequences which are in harmony with our intuition. 

This we now proceed to do. 

569. 1. Let = y = 



be one-valued continuous functions of t in the interval 21 = 
As t ranges over 21 the point a?, y will describe a curve or an arc 
of a curve O. We might agree to call such curves analytic, in 
distinction to those given by our intuition. The interval 21 is 
the interval corresponding to C. 

Let D be a finite division of 21 of norm c?, defined by 



To these values of t will correspond points 

P,P,,P,. (2 



PLANE CURVES 583 

on (7, which may be used to define a polygon P D whose vertices 
are 2). 

Let (m, m -f 1) denote the side P m P m -i-ii as well as its length. 
If we denote the length of P D by the same letter, we have 

P D = 2(m, m 4- 1) = 2 VA^+~A^. 
If lim P D (3 

exists, it is called the length of the arc C, and is rectifiable. 

2. (Jordan. ) For the arc PQ to be rectifiable^ it is necessary and 
sufficient that the functions (/>, i/r in 1) have limited variation in 21. 
j\ n . 

Hence p -> y I A I 

But the sum on the right is the variation of < for the division 7). 
If now <f> does not have limited variation in 21, the limit 3) does 
not exist. The same holds for *\Jr. Hence limited variation is a 
necessary condition. 

The condition is sufficient. For 

P D < 2 I A# 1+21 Ay I = Var <j> -}- Var \fr . 

D D 

As <, ^ have limited variation, this shows that 

P = Max P D 

D 

is finite. We show now that 

For there exists a division A such that 

p _ 1 < p^ < p (5 

Let A cause 21 to fall into v intervals, the smallest of which has 
the length X. Let D be a division of 21 of norm d<d Q <\. . 
Then no interval of D contains more than one point of A. 
Let ED+ A. 

~ ~ or P A . 



Obviously P E >P 



584 GEOMETRIC NOTIONS 

Suppose that the point t K of A falls in the interval ( t , ^) of 
D. Then the chord (/, i -f- 1) in P D is replaced by the two chords 
(t, /c), (/e, 6 4- 1) in P^. Hence 

P E ^P D ^^O. K , * = 1,2 /*<* 

where ^ = ^ ^ + ( ^, + X) _ ^ i + 1 ^ 

Obviously as <, ^ are continuous we may take d Q so small that 
each 



Gf^K<~- * for any d < d . 

Hence p p e fi 

J ^ /'/,<- (^ 

From 5), 6) we have 

P - Pj)< , for any d < rf , 
which gives 4). 

3. If the arc PQ is rectifiable, any arc contained in PQ is also 
rectifiable. 

For 0, ^ having limited variation in interval 21, have a fortiori 
limited variation in any segment of 31. 

4. Let the rectifiable arc G fall into two arcs (7j, (7 2 . If s, s x , $ 2 
are ^Ae lengths of C, <7j, (7 2 , ^/t^Ti 

8=8^ S 3 . (7 

For we saw that Cj, (7 2 are rectifiable since O is. Let Slj, 21 2 
be the intervals in 21 corresponding to 6\, (7 2 . Let D v J9 2 be 
divisions of 2lj , 2I 2 of norm d. Then 

Sj = lim P A , * 2 = lim P D ^ 

But Dj, D 2 effect a division of 21, and since 

s = lim P K (8 

<r=0 

with respect to the class of all divisions of 21, the limit 8) is the 
same when E is restricted to range over divisions of the type of D. 
Now 

PD = PDI 4- Pj) t 

Passing to the limit, we get 7). 



PLANE CURVES 585 

The preceding reasoning also shows that if C l , (7 2 are rectifiable 
curves^ then is, and 7) holds again. 

5. If 1) define a rectifiable curve, its length 8 is a continuous func- 
tion s() of t. 

For </>, -v/r having limited variation, 



where the functions on the right are continuous monotone increas- 
ing functions of t in the interval 91 = (# < i). 

For a division D of norm d of the interval A3! = (t, t + h) we 
have 



+ 2 | Ay | 
4- 



where S^ = (^(^-h A) <(), and similarly for the other func- 
tions. As $j is continuous, 8^ == 0, etc., as A=0. We may 
therefore take rj > so small that S^ , S(f> 2 , SI/TJ , S^|r 2 < e/4, if A < rj. 

Hence As = (^4- A) (0 < Max P^< e , if < h > y. 

Thus s is continuous. 



6. 2%e length s of the rectifiable arc C corresponding to the inter- 
val (a < t) is a monotone increasing function oft. 

This follows from 4. 

7. If x, y do not have simultaneous intervals of invariability, s(f) 
is an increasing function of t. The inverse function is one-valued 
and increasing and the coordinates x, y are one- valued functions of s. 

That the inverse function t (s) is one-valued follows from I, 214. 
We can thus express t in terms of s, and so eliminate t in 1). 

570. 1. If (f> f , yfr r are continuous in the interval 31, 



For 



lim 2VA02 + A-^2. (2 



586 



GEOMETRIC NOTIONS 



Now A0 K = <'(X)Mc A ^* = ^'(OMc (3 

where t' K , tf" lie in the interval At 

As <', -v|r' are continuous they are uniformly continuous. Hence 
for any division D of norm < some d , 



where \ a K \ , | & | < some ?;, small at pleasure, for any K. Thus 



and we may take 

ThuS 
Hence 



= lira 

rf=0 



(.)* 



lira 



8 



< , 



which establishes 1). 

For simplicity we have assumed $ ; , ^ to be continuous in 31. 
This is not necessary, as the following shows. 

2. Let aj, a n , Jj, 4 n >0 but not all = 0. 

rrii 

611 | Vo?+- +o2- V4fT^ + 1 | < S I a m - b m | , 



For 



m = 1, 2 - ra. (4 



)(Vaf+ 



Hence 
VofT^ 
But 



f + - + VAJ + 



This in 5) gives 4). 



PLANE CURVES 587 

Let us apply 4) to prove the following theorem, more general 
than 1. 

3. (2?atV0.) If $ , ^ are limited and R-integrable, then 

8 = 

For by 4), 



or 4> K - V K = 77.' Osc </>'(0 + i/l Osc i/r'<T) , in S K = A, 
where 77*', 77^' are numerically <1. Thus 

| 28,*, - 2S^ | = 28,1k' Osc $' + 28^ Osc <p. (6 

As (//, -^ f are integrable, the right side = 0, as d === 0. Now 

lira 28^ 

rf-o 

Thus passing to the limit in 6), we have 

lim 2A* K V^C^TT^C^) 5 " = f- 

^a 

This with 2), 3) gives 1) at once. 

571. Volterra's Curve. It is interesting to note that there are 
rectifiable curves for which <'(0> ^'(0 are no ^ both M-integr able. 
Such a curve is Volterra's curve, discussed in 503. Let its equa- 
tion be y =/(#). Then f\x) behaves as 

.1 1 

z x sin -- cos - 

^ X 

in the vicinity of a non null set in 21 = (0, 1). Hence f r (x) is 
not jR-integrable in 21. But then it is easy to show that 



does not exist. For suppose that 



588 GEOMETRIC NOTIONS 

were 72-integrable. Then # 2 = 1 -t-/'(X) 2 ^ 8 jR-integrable, and 
hence /'(#) 2 also. But the points of discontinuity of /' 2 in 21 do 
not form a null set. Hence/' 2 is not U-integrable. 

On the other hand, Volterra's curve is rectifiable by 569, 2, and 
528, 1. 

572. Taking the definition of length given in 569, 1, we saw 
that the coordinates 



must have limited variation for the curve to be rectifiable. But we 
have had many examples of functions not having limited variation 
in an interval 21. Thus the curve defined by 

y = x sin - , x = 

x (4 

does not have a length in 21 = ( 1, 1) ; while 

1 

y = x* sin - , x = 

^ * ' (5 

=0 , z=0 
does. 

It certainly astonishes the naive intuition to learn that the 
curve 4) has no length in any interval B about the origin how- 
ever small, or if we like, that this length is infinite, however small 
S is taken. For the same reason we see that 

No arc of Weierstrass' curve has a length (or its length is infinite) 
however near the end points are taken to each other, when ab>\. 

573. 1. 6 Property. Space-filling Curves. We wish now to 
exhibit a curve which passes through every point of a square, i.e. 
which completely fills a square. Having seen how to define one 
such curve, it is easy to construct such curves in great variety, not 
only for the plane but for space. The first to show how this may 
be done was Peano in 1890. The curve we wish now to define is 
due to Hilbert. 

We start with a unit interval 21 = (0, 1) over which t ranges, 
and a unit square 93 over which the point x, y ranges. We define 



PLANE CURVES 



589 



as one-valued continuous functions of t in 21 so that xy ranges over 
93 as t ranges over 21. The analytic curve C defined by 1) thus 
completely fills the square $8. 

We do this as follows. We effect a division of H into four 
equal segments 8J, B'%, S' z , 84, and of 35 into equal squares rj{, rf^ 
y' B , ifv as in Fig. 1. 

We call this the first division or D r The corre- 
spondence between 21 and 93 is given in first 
approximation by saying that to each point P in 
S[ shall correspond some point Q in rj[ . 

We now effect a second division D 2 by dividing 
each interval and square of D^ into four equal 
parts. 

We number them as in Fig. 2, 




FIG. I. 



Si' 



Sn 
16 




Fio. 2. 



As to the numbering of the rfs we observe the 
following two principles : 1 we may pass over the 
squares 1 to 16 continuously without passing the 
same square twice, and 2 in doing this we pass 
over the squares of Dj in the same order as in 
Fig. 1. The correspondence between 21 and 93 is 
given in second approximation by saying that to each point P in 
8[' shall correspond some point Q in ij( f . In this way we continue 
indefinitely. 

To find the point Q in 93 corresponding to P in 21 we observe 
that P lies in a sequence of intervals 

8' >S" >"' >... =0, (2 

to which correspond uniquely a sequence of squares 

if >V /'"> =0. (3 

The sequence 3) determines uniquely a point whose coordinates 
are one-valued functions of t, viz. the functions given in 1). 

The functions 1) are continuous in 21. 

For let t' be a point near t ; it either lies in the same interval as 
t in D n or in the adjacent interval. Thus the point Q r corre- 



590 GEOMETRIC NOTIONS 

spending to t r either lies in the same square of D n as the point Q 
corresponding to , or in an adjacent square. But the diagonal 
of the squares = 0, as n = oo. Thus 



Thus 
both = 0, as t = t. 

As t ranges over 21, the point x, y ranges over every point in the 
square 33. 

For let Q be a given point of 93. It lies in a sequence of 
squares as 3). If Q lies on a side or at a vertex of one of the 77 
squares, there is more than one such sequence. But having taken 
such a sequence, the corresponding sequence 2) is uniquely de- 
termined. Thus to each Q corresponds at least one P. A more 
careful analysis shows that to a given Q never more than four 
points P can correspond. 

2. The method we have used here may obviously be extended 
to space. By passing median planes through a unit cube we 
divide it into 2 3 equal cubes. Thus to get our correspondence 
each division D n should divide each interval and cube of the pre- 
ceding division D n _ l into 2 s equal parts. The cubes of each divi- 
sion should be numbered according to the 1 and 2 principles of 
enumeration mentioned in 1. 

By this process we define 



as one-valued continuous functions of t such that as t ranges over 
the unit interval (0, 1), the point a?, y, z ranges over the unit 
cube. 

574. 1. Hitterf s Curve. We wish now to study in detail the 
correspondence between the unit interval 21 and the unit square 
93 afforded by Hilbert's curve defined in 573. A number of inter- 
esting facts will reward our labor. We begin by seeking the 
points P in 21 which correspond to a given Q in 93- 

To this end let us note how P enters and leaves an rj square. 
Let B be a square of D n . In the next division B falls into four 



PLANE CURVES 591 

squares B l J5 4 and in the w-f 2 d division in 16 squares B^ 3 . 
Of these last, four lie at the vertices of B ; we call them vertex 
squares. The other 12 are median squares. A simple considera- 
tion shows that the rj squares of JD n +2 are so numbered that we 
always enter a square B belonging to 2> n , and also leave it by a 
vertex square. 

Since this is true of every division, we see on passing to the 
limit that the point Q enters and leaves any rj square at the ver- 
tices of 77. We call this the vertex law. 

Let us now classify the points P, Q. 

If P is an end point of some division D n > we call it a terminal 
point, otherwise an inner point, because it lies within a sequence 
of 8 intervals 8' > 8" > = 0. 

The points Q we divide into four classes : 

1 vertex points, when Q is a vertex of some division. 

2 inner points, when Q lies within a sequence of squares 

V>T?"> - =0. 

3 lateral points, when Q lies on a side of some rj square but 
never at a vertex. 

4 points lying on the edge of the original square 93. Points 
of this class also lie in 1, 3. 

We now seek the points P corresponding to a Q lying in one of 
these four classes. 

Class 1. Q a Vertex Point. Let D n be the first division such 
that Q is at a vertex. Then Q lies in four squares rj L , 77,-, rj K , ij t of 

D n . 

There are 5 cases : 

) ij k I are consecutive. 

/3) ij k are consecutive, but not I. 

7) LJ are consecutive, but not k I. 

8) ij, also k /, are consecutive, 
e) no two are consecutive. 

A simple analysis shows that a), #) are not permanent in the 
following divisions ; 7), 8) may or may not be permanent ; e) is 
permanent. 



592 GEOMETRIC NOTIONS 

Now, whenever a case is permanent, we can enclose? Q in a se- 
quence of j] squares whose sides = 0. To this sequence corre- 
sponds uniquely a sequence of 8 intervals of lengths = 0. Thus 
to two consecutive squares will correspond two consecutive inter- 
vals which converge to a single point P in 21. If the squares are 
not consecutive, the corresponding intervals converge to two dis- 
tinct points in 21. Thus we see that when 7) is permanent, to Q 
correspond three points P. When S) is permanent, to Q corre- 
spond two points P. While when Q belongs to e), four points P 
correspond to it. 

Class "2. Q an Tuner Point. Obviously to each Q corresponds 
one point P and only one. 

Class 3. Q a Lateral Point. To fix the ideas let Q lie on a ver- 
tical side of one of the ?/\s. Let it lie between rj^ ijj of D n . There 
are two cases : 

) j = i + 1. 



We see easily that a) is not permanent, while of course /S) is. 
Thus to each Q in class 3, there correspond two points P. 

Class 4. Q lies on the edge of 8. If Q is a vertex point, to it 
may correspond one or two points P. If Q is not a vertex point, 
only one point P corresponds to it. 

To sum up we may say : 

To each inner point Q corresponds one inner point P. 

To each lateral point Q correspond two points P. 

To each edge point Q correspond one or two points P. 

To each vertex point Q, correspond two, three, or four points P. 

2. As a result of the preceding investigation we show easily 
that : 

To the points on a line parallel to one of the sides of S3 correspond 
in 21 an apantactic perfect set. 

3. Let us now consider the tangents to Hilbert's curve which 
we denote by H. 



PLANE CURVES 



593 




Let Q be a vertex point. We saw there were three permanent 
cases 7), 8), e). 

In cases 7), 8) we saw that to two consecutive 8 intervals cor- 
respond permanently two contiguous ver- 
tical or horizontal squares. 

Thus as t ranges over L - j ' ^ ' Q 

8 t i 8 t+1 , the point #, y ranges 

over these squares, and the secant line 

joining Q and this variable point cr, y oscillates through 180. 

There is thus no tangent at Q. In case e) we see similarly that 

the secant line ranges through 90. Again there is no tangent 

at Q. 

In the same way we may treat the three other classes. We find 
that the secant line never converges to a fixed position, and may 
oscillate through 360, viz. when Q is an inner point. As a result 
we see that Hubert's curve has at no point a tangent, nor even a 
unilateral tangent. 

4. Associated with Hilbert's curve IT are two other curves, 



The functions <, i/r being one-valued and continuous in 31, these 
curves are continuous and they do not have a multiple point. A 
very simple consideration shows that they do not have even a 
unilateral tangent at a pantactic set of points in 21. 

575. Property 7. Equations of a Curve. As already remarked, 
it is commonly thought that the equation of a curve may be 
written in any one of the three forms 

y =/(*), (i 

4>(x,y)=0, (2 



and if these functions are continuous, these equations define con- 
tinuous curves. 

Let us look at the Hilbert curve H. We saw its equation 
could be expressed in the form 3). JSTcuts an ordinate at every 
point of it for which < y < 1. Thus if we tried to define H by 



594 GEOMETRIC NOTIONS 

an equation of the type 1), /(V) would have to take on every 
value between and 1 for each value of x in 21 = (0, 1). No such 
functions are considered in analysis. 

Again, we saw that to any value x = a in 21 corresponds a perfect 
apan tactic set of values \t a \ having the cardinal number c. Thus 
the inverse function of x = </>() is a many-valued function of x 
whose different values form a set whose cardinal number is c. 
Such functions have not yet been studied in analysis. 

How is it possible in the light of such facts to say that we may 
pass from 8) to 1) or 2) by eliminating t from 3). And if we 
cannot, how can we say a curve can be represented equally well 
by any of the above three equations, or if the curve is given by 
one of these three equations, we may suppose it replaced by one 
of the other two whenever convenient. Yet this is often done. 

In this connection we may call attention to the loose way 
elimination is treated. Suppose we have a set of equations 



We often see it stated that one can eliminate ^ t n and obtain 
a relation involving the #'s alone. Any reasoning based on such 
a procedure must be regarded as highly unsatisfactory, in view of 
what we have just seen, until this elimination process has been 
established. 

576. Property 8. Closed Curves. A circle, a rectangle, an 
ellipse are examples of closed curves. Our intuition tells us that 
it is impossible to pass from the inside to the outside without 
crossing the curve itself. If we adopt the definition of a closed 
curve without multiple point given in I, 362, we find it no easy 
matter to establish this property which is so obvious for the simple 
closed curves of our daily experience. The first to effect the 
demonstration was Jordan in 1892. We give here * a proof due 
to de la Valle~ Poussin.-f 

Let us call for brevity a continuous curve without double point 

. * The reader is referred to a second proof due to Brouwer and given in 698 seq. 
t Cours # Analyse, Paris, 1903, Vol. 1, p. 307. 



PLANE CURVES 595 

a Jordan curve. A continuous closed curve without double point 
will then be a closed Jordan curve. Cf . I, 362. 

577. Lei C be a closed Jordan curve. However small or> is 
taken, there exists a polygonal ring R containing C and such that 

1 Each point of R is at a distance < cr from C. 

2 Each point of C is at a distance < a from the edges of R. 

For let x = <<T) , y = i/r(0 C 1 

be continuous one-valued functions of in T = (a < 6) defining C. 
Let D = (a, a v a 2 6) be a division of T of norm d. Let 
a, j, 2 be points of corresponding to a, a l If d! is suffi- 
ciently small, the distance between two points on the arc 
(7 t = ( t _i, t ) is <e', small at pleasure. Let A be a quadrate 
division of the a?, y plane of norm S. Let us shade all cells con- 
taining a point of (7 t . These form a connected domain since O t is 
continuous. We can thus go around its outer edge without a 
break.* If this shaded domain contains unshaded cells, let us 
shade these too. We call the result a link A,. It has only one 
edge E t , and the distance between any two points of E, is ob- 
viously < f + 2 V2 8. We can choose d, S so small that 

c' + 2V2~8 < (7, arbitrarily small. (1 

Then the distance between any two points of A. is < <r. Let e" 
be the least distance between non-consecutive arcs (7 4 , We take 
B so small that we also have 

' (2 



Then two non-consecutive links A^ Aj have no point in common. 
For then their edges would have a common point P. As P lies 
on _Z7 t its distance from O, is < V2 S. Its distance from G j is also 
< V2 B. Thus there is a point P t on C t , and a point Pj on 0$ such 
that 



* Here and in the following, intuitional properties of polygons are assumed as 
known. 



59G GEOMETRIC NOTIONS 

But by hypothesis e" < 77. Hence 

e"<2V2S, 
which contradicts 2). 

Thus the union of these links form a ring R whose edges are 
polygons without double point. One of the edges, say (? t , lies 
within the other, which we call Gr e . The curve lies within R. 
The inner polygon (? 4 must exist, since non-consecutive links have 
no point in common. 

578. 1. Interior and Exterior Points. Let tr l > cr 2 > =0. 
Let JSj, /2 2 be the corresponding rings, and let 



be their inner and outer edges. A point P of the plane not on 
which lies inside some 6r t we call an interior or inner point of O. 
If P lies outside some 6r c , we call it an exterior or outer point of C. 

Each point P not on O must belong to one of these two classes. 
For let p = Dist (J, (7); then p is > some er n . It therefore lies 
within 6j^ n) or without (r^ w) , and is thus an inner or an outer point. 
Obviously this definition is independent of the sequence of rings 
\JR n \ employed. The points of the curve (7 are interior to each 
Gr^ and exterior to each Gf-[ n) . 

Inner points must exist, since the inner polygons exist as al- 
ready observed. Let us denote the inner points by 3 and the 
outer points by O. Then the frontiers of $ and Q are the curve C. 

2. We show now that 

1 Two inner points can be joined by a broken line L, lying in 3. 

2 Two outer points can be joined by a broken line L e lying in O. 

3 Any continuous curve $ joining an inner point i and an outer 
point e has a point in common with O. 

To prove 3, let 



be the equations of $, the variable t ranging over an interval 
T=(a</9), t=a corresponding to i and t=zj3 to e. Let t r be 



PLANE CURVES 697 

such that a<t< t f gives inner points, while t = t f does not give an 
inner point. Thus the point corresponding to t = t r is a frontier 
point of 3 and hence a point of O. 

To prove 1. If A, B are inner points, they lie within some Gr, . 
We may join J., JS, 6? t by broken lines L a , L^ meeting (3\ at the 
points .A', J3', say. Let Gr^ be the part of (? t lying between A \ 
B 1 . Then 

La + G-ab + Lb 

is a broken line joining A to jB. 
The proof of 2 is similar. 

579. 1. Let P f , P" correspond to t = t f , t = t n , on the curve 
defined by 577, 1). If '<", we say P f precedes P" and write 



Any set of points on C corresponding to an increasing set of 
values of t is called an increasing set. 

As t ranges from a to 6, the point P ranges over C in a direct 
sense. 

We may thus consider a Jordan curve as an ordered set, in the 
sense of 265. 

2. (I)e la Valise- Poussin.) On each arc (7 t of the curve <7, there 
exists at least one point P t - such that 



may be regarded as the vertices of a closed polygon without double 
point and whose sides are all < e. 

For in the first place we may take 8 > so small that no square 
of A contains a point lying on non-consecutive arcs O t of C. Let 
us also take A so that the point a corresponding to t = a lies 
within a square, call it A^, of A. As t increases from t = a, there 
is a last point P l on where the curve leaves S r The point P l 
lies in another square of A, call it $ 2 , containing other points of 
C. Let -P 2 be the last point of O in S%. In this way we may 
continue, getting a sequence 1). 

There exists at least one point of 1) on each arc 0, . For other- 
wise a square of A would contain points lying on non-consecutive 
arcs O K . The polygon determined by 1) cannot have a double 



598 GEOMETRIC NOTIONS 

point, since each side of it lies in one square. The sides are < e, 
provided we take SV2 < e, since the diagonal is the longest line 
we can draw in a square of side S. 

580. Existence of Inner Points. To show that the links form a 
ring with inner points, Schonfliess* has given a proof which may 
be rendered as follows : 

Let us take the number of links to be even, and call them L^ 
Z 2 , - 2n . T nen A> ^3' -^6'" l* e en ti re ly outside each other. 
Since L^ L% overlap, let P be an inner common point. Simi- 
larly let Q be an inner common point of _L 2 , L 3 . Then P, Q 
lying within J& 2 may be joined by a finite broken line b lying 
within L 2 . Let 5 2 be that part of it lying between the last point 
of leaving L and the following point of meeting Z 8 . In this 
way the pairs of links 

L^L Z ; L 3 L 5 ; 

define finite broken lines 



No two of these can have a common point, since they lie in 
non-consecutive links. The union of the points in the sets 

L l > *2 L S ' *4 " A-l ' b 2n 

we call a ring, and denote it by 9t. The points of the plane not 
in $R fall into two parts, separated by 9t. Let Z denote the part 
which is limited, together with its frontier. We call Z the inte- 
rior of 3t. That Z has inner points is regarded as obvious since 
it is defined by the links 



which pairwise have no point in common, and by the broken lines 

^2 ' J 4 ' *6 " 

each of which latter lies entirely within a link. 
Let S 



* Die Entwickelung der Lehre von den Punktmannigfaltigkeiten. Leipzig, 1908, 
Part 2, p. 170. 



PLANE CURVES 599 

Then these 8 have pairwise no point in common since the J a 
have not. 

Let Z = 8 2 4- 8 4 + + ? 2n + 

Then $ > 0. For let us adjoin L% to 3t, getting a ring 9? 2 whose 
interior call J 2 . That ! 2 has inner points follows from the fact 
that it contains 4 , 8 6 Let us continue adjoining the links 
L^ L Q Finally we reach .L 2n , to which corresponds the 
ring 9? 2n , whose interior, if it exists, is Z 2n - If 2 ^ oes no ^ ex *st, 
5E 2w _ 2 contains only 2n . This is not so, for on the edge of L^ 
bounding , is a point P, such that some -O p (P) contains points 
of no L except L r In fact there is a point P on the edge of L^ 
not in either L 2 or l/ 2n , as otherwise these would have a point in 
common. Now, if however small p > is taken, j?) p (P) contains 
points of some L other than L^ , the point P must lie in L K which 
is absurd, since L^ has only points in common with .Z/ 2 , L% n , and 
P is not in either of these. Thus the adjunction of L 2 , L, 
L 2n produces a ring 9? 2n whose interior 2n does not reduce to ; 
it has inner points. 

581. Property 9. Area. That a figure defined by a closed 
curve without double point, i.e. the interior of a Jordan curve, 
has an area, has long been an accepted fact in intuitional geometry. 
Thus Lindemann, Vorlesun<jen ilber Greometrie, vol. 2, p. 557, says 
" einer allseitig umgrenzten Figur kommt ein bestimmter Flachen- 
inhalt zu." The truth of such a statement rests of course on 
the definition of the term area. In I, 487, 702 we have given a 
definition of area for any limited plane point set 21 which reduces 
to the ordinary definition when 21 becomes an ordinary plane figure. 
In our language 21 has an area when its frontier points form a 
discrete set. Let 



define a Jordan curve 6, as t ranges over 2 T =(a<6). The 

figure 21 defined by this curve has the curve as frontier. In I, 

708, 710, we gave various cases in which ( is discrete. The 
reasoning of I, 710, gives us also this important case : 

If one of the continuous functions <, ^ defining the Jordan curve 
S, has limited variation in T, then is discrete. 



600 



GEOMETRIC NOTIONS 



l(i 17 



It was not known whether would remain discrete if the con- 
dition of limited variation was removed from both coordinates, 
until Osgood * exhibited a Jordan curve which is not discrete. 
This we will now discuss. 

582. 1. Osgood' 8 Curve. We start with a unit segment 
T = (0, 1) on the t axis, and a unit square /S in the xy plane. 

We divide Tin to 17 equal parts 

/Tf /jj rn / -\ 

"M'-*2'"*17' V 

and the square S into 9 equal 
squares 

/Sj, $g, 5 ' A3j 7 , (2 



by drawing 4 bands B l which 
are shaded in the figure. On 
these bands we take 8 segments, 



marked heavy in the figure. 

Then as t is ranging from left 
to right over the even or black 

intervals T^ T^ - T 16 marked heavy in the figure, the point a?, y 
on Osgood's curve, call it ), shall range univariantly over the 
segments 3). 

While t is ranging over the odd or white intervals 7p T 3 T 17 
the point xy on 5 shall range over the squares 2) as determined 
below. 

Eacli of the odd intervals 1) we will now divide into 17 equal 
intervals T tj and in each of the squares 2) we will construct 
horizontal and vertical bands J? 2 as we did in the original square 
S. Thus each square 2) gives rise to 8 new segments on ) 
corresponding to the new black intervals in 7, and 9 new squares 
& L J corresponding to the white intervals. In this way we may 
continue indefinitely. 

The points which finally get in a black interval call & the 
others are limit points of the /3's and we call them X. The point 




* Trans. Am. Math. Soc., vol. 4 (1903), p. 107. 



PLANE CURVES 601 

on O corresponding to a point has been defined. The point of 
O corresponding to a point X is defined to be the point lying in 
the sequence of squares, one inside the other, corresponding to the 
sequence of white intervals, one inside the other, in which X falls, 
in the successive divisions of T. 

Thus to each t in T corresponds a single point #, y in S. The 
aggregate of these points constitutes Osgood's curve. Obviously 
the #, y of one of its points are one-valued functions of t in J 7 , say 

* = <KO , y = t(0- ( 4 

The curve ) has no double point. This is obvious for points of 
O lying in black segments. Any other point falls in a sequence 
of squares 

Si>St>s VK ... 

to which correspond intervals 



in which the corresponding t's lie. But only one point t is thus 
determined. 

The functions 4) are continuous. This is obvious for points ft 
lying within the black intervals of T. It is true for the points X. 
For X lies within a sequence of white intervals, and while t ranges 
over one of these, the point on ) ranges in a square. But these 
squares shut down to a point as the intervals do. Thus </>, ty are 
continuous at t = X. In a similar manner we show they are con- 
tinuous at the end points of the black intervals. 

We note that to t = corresponds the upper left-hand corner 
of /S^ and to t = 1, the diagonally opposite point. 

2. Up to the present we have said nothing as to the width of 
the shaded bands ^ 2> 

&\ ^V" 

introduced in the successive steps. Let 



be a convergent positive term series whose sum A < 1. We 
choose S l so that its area is a r J? 2 so t ' ia ^ ^ ts area ^ s a 2-> e ^ c - 
Then = , = 1-A (5 



602 GEOMETRIC NOTIONS 

as we now show. For ) lias obviously only frontier points ; 
hence O = 0. Since O is complete, it is measurable and 



Let = S- O, and B={B n \. Then (9 < B. For any point 
which does not lie in some B n lies in a sequence of convergent 
squares A^ > S^ > which converge to a point of ). Now 



On the other hand, B contains a null set of points of ), viz. the 
black segments. Thus 

= S = A , and hence 6 = 1 - A 
and 5) is established. 

Thus Osgood's curve is continuous, has no double point, and its 
upper content is 1 A. 

3. To get a continuous closed curve (7 without double point 
we have merely to join the two end points a, /3 of Osgood's curve 
by a broken line which docs not cut itself or have a point in com- 
mon with the square S except of course the end points a, /3. 
Then (7 bounds a figure g whose frontier is not discrete, and $ 
does not have an area. Let us call such curves closed Osgood 
curves. 

Thus we see that there exist regions bounded by Jordan curves 
which do not have area in the sense current since the Greek 
geometers down to the present day. 

Suppose, however, we discard this traditional definition, and 
employ as definition of area its measure. Then wo can say : 

A figure g formed of a closed Jordan curve J and its interior 3 ; 
has an area, viz. Meas g. 

For Front $ = J. Hence g is complete, and is therefore meas- 
ureable. 

We note that a _ f a 

We have seen there are Jordan curves such that 

J>0. 



DETACHED AND CONNECTED SETS 608 

We now have a definition of area which is in accordance with the 
promptings of our geometric intuition. It must be remembered, 
however, that this definition has been only recently discovered, 
and that the definition which for centuries has been accepted leads 
to results which flatly contradict our intuition, which leads us to 
say that a figure bounded by a continuous closed curve has an 
area. 

583. At this point we will break off our discussion of the 
relation between our intuitional notion of a curve, and the con- 
figuration determined by the equations 



where <, ^ are one-valued continuous functions of t in an interval 
T. Let us look back at the list of properties of an intuitional 
curve drawn up in 563. We have seen that the analytic curve 
1) does not need to have tangents at a pantactic set of points on 
it ; no arc on it needs have a finite length ; it may completely fill 
the interior of a square ; its equations cannot always be brought 
in the forms y =/(#) or J 7 (^)=0, if we restrict ourselves to 
functions /or F employed in analysis up to the present; it does 
not need to have an area as that term is ordinarily understood. 

On the other hand, it is continuous, and when closed and with- 
out double point it forms the complete boundary of a region. 

Enough in any case has been said to justify the thesis that 
geometric reasoning in analysis must be used with the greatest 
circumspection. 

Detached and Connected Sets 

584. In the foregoing sections we have studied in detail some 
of the properties of curves defined by the equations 



Now the notion of a curve, like many other geometric notions, is 
independent of an analytic representation. We wish in the fol- 
lowing sections to consider some of these notions from this point 
of view. 



(504 GEOMETRIC NOTIONS 

585. 1. Let 21, 33 be point sets in w-way space 9t m . If 
Dist(2l, 33)>0, 

we say 21, 33 are detached. If 21 cannot be split up into two parts 
S3, such that they are detached, we say 21 has no detached parts. 
If 21 = 33 -}- and Dist (S3, )>0, we say S3, 6 are detached parts 
of 21. 
.Let the set of points, finite or infinite, 

a, a v 2 , b (1 

be such that the distance between two successive ones is < e. We 
call 1) an e-sequence between a, 6 ; or a sequence with segments 
(#i ^ a i+i) f length < . We suppose the segments ordered so 
that we can pass continuously from a to b over the segments without 
retracing. If 1) is a finite set, the sequence is finite, otherwise 
infinite. 

2. Let 21 have no detached parts. Let a, b be two of its points. 
For each e > 0, there exists a finite ^-sequence between a, b, and lying 
in 21. 

For about a describe a sphere of radius e. About each point of 
21 in this sphere describe a sphere of radius e. About each point 
of 21 in each of these spheres describe a sphere of radius e. Let 
this process be repeated indefinitely. Let S3 denote the points of 
21 made use of in this procedure. If S3 < 21, let = 21 - 33. Then 
Dist (S3, )>e, and 21 has detached parts, which is contrary to 
hypothesis. Thus there are sets of e-spheres in 21 joining a and b. 

Among these sets there are finite ones. For let $ denote the 
set of points in 21 which may be joined to a by finite sequences ; let 
= 21 - g. Then Dist (g, )>e. For if <e, there is a point/ 
in $, and a point g in @ whose distance is < e. Then a and g can 
be joined by a finite e-sequence, which is contrary to hypothesis. 

3. If 21 has no detached parts, it is dense. 

For if not dense, it must have at least one isolated point a. 
But then a, and 21 a are detached parts of 21, which contradicts 

the hypothesis. 

4. Let 21, S3, be complete and 21 = ($, ). If 21 has no de- 
tached parts, S3, have at least one common point. 



IMAGES 605 

For if 53, S have no common point, S = Dist (53, S) is > 0. 
But S cannot > 0, since 53, S would then be detached parts of 21. 
Since S = and since S3, are complete, they have a point in 
common. 

5. If 21 is such that any two of its points may be joined by an 
e-sequence lying in 21, where e > is small at pleasure, 21 has n<r 
detached parts. 

For if 21 had 53, ( as detached parts, let Dist (33, <) = S. Then 
8 > 0. Hence there is no sequence joining a point of 53 with a 
point of S with segments < 8. 

6. If 21 is complete and has no detached parts, it is said to be 
connected. We also call 21 a connex. 

As a special case, a point may be regarded as a connex. 

1. If 21 is connected, it is perfect. 

For by 3 it is dense, and by definition it is complete. 

8. If 21 is a rectilinear connex, it has a first point a and a last 
point yS, and contains every point in the interval (a, /:?). 

For being limited and complete its minimum and maximum 
lie in 21 and these are respectively a and /3. Let now 

*i>* 2 > = - 

There exists an e r sequence C between a, /3. Each segment has 
an e 2 -sequence (7 2 . Each segment of <7 2 has an e 3 -sequence (7 3 , 
etc. Let be the union of all these sequences. It is pantactic 
in (a, /3). As 21 is complete, 

21 = (, 0). 

Images 

586. Let a^AOi-'-O *n=/n(*i'"O C 1 

be one-valued functions of t in the point set . As t ranges over 
Z, the point x = (x l # n ) will range over a set 21 in an w-way 
space 3t n . We have called 21 the image of . Cf. I, 238, 3. 
If the functions / are not one-valued, to a point t may correspond 
several images x', x f! finite or infinite in number. Conversely 



606 GEOMETRIC NOTIONS 

to the point x may correspond several values of t. If to each 
point t correspond in general r values of #, and to each x in 
general 8 values of , we say the correspondence between J, 21 is 
r to s. If r = s = 1 the correspondence is 1 to 1 or unifold ; if 
r > 1, it is manifold. If r = 1, 21 is a simple image of J, other- 
wise it is a multiple image. If the functions 1) are one-valued 
and continuous in J, we say 21 is a continuous image of J. 

587. Transformations of the Plane. Example 1. Let 

u = x sin y , v = x cos y. (1 

We have in the first place 

U 2 -f V* = iE 2 . 

This shows that the image of a line x = a, a=(), parallel to the 
*/-axis is a circle whose center is the origin in the u, v plane, and 
whose radius is a. To the y-axis in the x, y plane corresponds 
the origin in the u, v plane. 

From 1) we have, secondly, 

u 

- = tan y. 

v 

This shows that the image of a line y = 6, is a line through the 
origin in the u, v plane. 

From 1) we have finally that u, v are periodic in y, having the 
period 2 TT. Thus as #, y ranges in the band J5, formed by the 
two parallels y TT, or TT < y < TT, the point u, v ranges over 
the entire u, v plane once and once only. 

The correspondence between B and the w, v plane is unifold, 
except, as is obvious, to the origin in the w, v plane corresponds 
the points on the t y-axis. 

Let us apply the theorem of I, 441, on implicit functions. The 
determinant A is here 



= x. 



sin y, cos y 
x cos y, x sin y 

As this is = when x, y is not on the y-axis, we see that the 
correspondence between the domain of any such point and its 
image is 1 to 1. This accords with what we have found above. 



IMAGES 



607 



It is, however, a much more restricted result than we have found ; 
for we have seen that the correspondence between any limited 
point set 21 in B which does not contain a point of the y-axis and 
its image is unifold. 

588. Example 2. Let 



the radical having tlie })ositive sign. 
first quadrant Q in the x, y plane. 
From 1 ) we have at once 

<L u> < 1 , 

Hence the image of Q is a band 
From 1) we get secondly 



2 , (1 

Let us find the image of the 

v > 0. 

parallel to the v-axis. 



Hence 



y = uv 



, # = y Vl 

,2 2 
-f ^ = fl 



(2 



Thus the image of a circle in $ whose center is the origin and 
whose radius is a is a segment of a right line v == a. 

When x = y = 0, the equations 1) do not deline the correspond- 
ing point in the t&, v plane. If we use 2) to define the corre- 
spondence, we may say that to the line v = in B corresponds the 
origin in the #, y plane. With this exception the correspondence 
between Q and B is uniform, as 1), 2) show. 

The determinant A of 1) is, setting 



r = V# 2 -f- # 2 , 

x y * 



d(u, v} _ 



7* 
x 
r 



r 3 



Z + y* 



for any point #, y different from the origin. 

589. Example 8. Reciprocal Radii. Let be the origin in the 
#, y plane and fl the origin in the u, v plane. To any point 
P = (#, ?/) in the #, y plane different from the origin shall cor- 
respond a point Q = (u, v) in the u, v plane such that flQ has 



608 GEOMETRIC NOTIONS 

the same direction as OP and such that OP flQ = 1. Analyti- 
cally we have 

x = \y , u = \v , X > 0, 
and 



From these equations we get 

u 



and also 



x 2 2 2 

ir + v 2 ir -f tr 



The correspondence between the two planes is obviously unifold 
except that no point in either plane corresponds to the origin in 
the other plane. We find for any point a?, y different from the 
origin that . , , ^ 

A = ( Ml v ) = __ 1 
~~ ' 



Obviously from the definition, to a line through the origin in 
the x, y plane corresponds a similar line in the w, v plane. As xy 
moves toward the origin, u, v moves toward infinity. 

Let x, y move on the line x = a = 0. Then 1) shows that u, v 
moves along the circle 

a(w a + v 2 )- u = 

which passes through the origin. A similar remark holds when 
x, y moves along the line y = b = 0. 

590. Such relations between two point sets 81, 93 as defined in 
586 may be formulated independently of the functions f. In fact 
with each point a of 81 we may associate one or more points 6 X , J 2 
of 93 according to some law. Then 93 may be regarded as the 
image of 81. We may now define the terms simple, manifold, etc., 
as in 586. When b corresponds to a we may write b ~ a. 

We shall call 93 a continuous image of 31 when the following con- 
ditions are satisfied. 1 To each a in 81 shall correspond but one 
b in 93, that is, 93 is a simple image of 81. 2 Let b ~ a, let a t , a 2 
be any sequence of points in 81 which = a. Let b n ~ a n . Then 
b n must =s b however the sequence \a n \ is chosen. 



IMAGES 609 

When S3 is a simple image of 21, the law which determines 
which b of 93 is associated with a point a of 21 determines obviously 
n one-valued functions as in 586, 1), where ^ t m are the m co- 
ordinates of a, and x l x n are the n coordinates of 6. We call these 
functions 1) the associated functions. Obviously when S3 is a 
continuous image, the associated functions are continuous in 21. 

591. 1. Let 93 be a simple continuous image of the limited complete 
set 21. Then 1 93 is limited and complete. If 2 21 is perfect and 
only a finite number of points of 21 correspond to any point of 93, then 
93 is perfect. If 3 21 is a connex, so is 93. 

To prove 1. The case that 93 is finite requires no proof. Let 
b 1 , b 2 ... be points of 93 which = /3. We wish to show that ft lies 
in 93. To each b n will correspond one or more points in 21; call 
the union of all these points a. Since 93 is a simple image, a is an 
infinite set. Let a r # 2 -.. be a set of points in a which = a, a 
limiting point of 21. As 21 is complete, lies in 21. Let b ~ a. 
Let b. n ~ a n . As a n =^= a, b, n = /3. But 93 being continuous, b, 
must = b. Thus ft lies in 93. That 93 is limited follows from the 
fact that the associated functions are continuous in the limited 
complete set 21. To prove 2. Suppose that 3} had an isolated 
point b. Let b ~ a. Since 21 is perfect, let a l , 2 = a. Let 
b n ~ a n . Then as $8 is continuous, b n = 6, and b is not an isolated 
point. To prove 3. We have only to show that there exists 
an e-sequence between any two points a, ft of 93, small at pleasure. 
Let a ~ a, ft ~ b. Since 21 is connected there exists an ^-sequence 
between a, b. Also the associated functions are uniformly con- 
tinuous in 21, and hence y may be taken so small that each segment 
of the corresponding sequence in 93 is > e. 

2. Let /(j ... t m } be one-valued and continuous in the connex 21, 
then the image of 21 is an interval including its end points. 

This follows from the above and from 585, 8. 

3. Let the correspondence between 21, 93 be unifold. If 93 is a 
continuous image of 21, then 21 is a continuous image of 93. 

For let \b n \ be a set of points in 93 which = b. Let a n ^ 6 n , 
a ~ b. We have only to show that a n = a. For suppose that it 
does not, suppose in fact that there is a sequence # tl , # l$ which 



610 GEOMETRIC NOTIONS 

= a = a. Let /3 ~ . Then i tl , J tl - = /3. But any partial se- 
quence of \b n \ must = b. Thus b = /3, hence a = a, hence a n = a. 

4. A Jordan curve J is a unifold continuous image of an interval 
T. Conversely if J is a unifold continuous image of an interval T, 
there exist two one-valued continuous functions 



, y = 

such that as t ranges over T, the point x, y ranges over J. In case 
J is closed it may be regarded as the image of a circle F. 

All but the last part of the theorem has been already established. 
To prove the last sentence we have only to remark that if we set 

x r cos t , y = r sin t 

we have a unifold continuous correspondence between the points 
of the interval (0, 2 ?r*) and the points of a circle. 

5. The first part of 4 may be regarded as a geometrical definition 
of a Jordan curve. The image of a segment of the interval T or 
of the circle F, will be called an arc of J. 

592. Side Lights on Jordan Curves. These curves have been 
defined by means of the equations 

y = *() (1 



As t ranges over the interval T = (a < 5), the point P = (#, y) 
ranges over the curve J. This curve is a certain point set in the 
x, y plane. We may now propose this problem : We have given 
a point set & in the #, y plane ; may it be regarded as a Jordan 
curve ? That is, do there exist two continuous one-valued func- 
tions 1) such that as t ranges over some interval T 7 , the point P 
ranges over the given set S without returning on itself, except 
possibly for t = a, t = J, when the curve would be closed? 

Let us look at a number of point sets from this point of view. 

593. Example 1. 

y = gin 1 ^ x in the i nterva i g _ (_ i ? i) 9 but * 

x 

= , for x = 0. 



IMAGES 611 

Is this point set S a Jordan curve ? The answer is, No. For a 
Jordan curve is a continuous image of an interval 21. By 591, 1, 
it is complete. But S is not complete, as all the points on the 
y axis, 1 < y < 1 are limiting points of 6, and only one of them 
belongs to , viz. the origin. 

2. Let us modify S by adjoining to it all these missing limiting 
points, and call the resulting point set C. Is G a Jordan curve ? 
The answer is again, No. For if it were, we can divide the inter- 
val T into intervals 8 so small that the oscillation of <, -fy in any 
one of them is < o>. To the intervals 8, will correspond arcs O t on 
the curve, and two non-consecutive arcs C t are distant from each 
other by an amount > some e, small at pleasure. This shows that 
one of these arcs, say C K , must contain the segment on the ^/-axis 
1 < y < ! Hut then Osc ^ = 2 as t ranges over the correspond- 
ing S K interval. Thus the oscillation of ^r cannot be made < e, 
however small S K is taken. 

3. Let us return to the set & defined in 1. Let A, E be the 
two end points corresponding to x = 1, x = 1. Let us join them 
by an ordinary curve, a polygon if we please, which does not cut 
itself or &. The resulting point set $ divides all the other points 
of the plane into two parts which cannot be joined by a contin- 
uous curve without crossing $. For this point of view $ must be 
regarded as a closed configuration. Yet $ is obviously not complete. 

On the other hand, let us look at the curve formed by removing 
the points on a circle between two given points a, b on it. The 
remaining arc 8 including the end points a, b is a complete set, but 
as it does not divide the other points of the plane into two sepa- 
rated parts, we cannot say 8 is a closed configuration. 

We mention this circumstance because many English writers 
use the term closed set where we have used the term complete. 
Cantor, who first introduced this notion, called such sets abge- 
schlossen, which is quite different from geschlossen = closed. 

_i 
594. Example 2. Let p = e ', for in the interval 21 = (0, 1) 

except 9 = 0, where p = 0. These polar coordinates may easily be 
replaced by Cartesian coordinates 

-i * -1 

*cos0 = e*sin0 > in a > 



612 GEOMETRIC NOTIONS 

except 6 = 0, when x, y both = 0. The curve thus defined is a 
Jordan curve. 

Let us take a second Jordan curve 

J), 



with p = for 6 = 0, If we join the two end points on these 
curves corresponding to 6 = 1 by a straight line, we get a closed 
Jordan curve <7, which has an interior $, and an exterior ). 

The peculiarity of this curve J is the fact that one point of it, 
viz. the origin x y = 0, cannot be joined to an arbitrary point 
f 3 hy a finite broken line lying entirely in Q ; nor can it be 
joined to an arbitrary point in O by such a line lying in ) 

595. 1. It will be convenient to introduce the following terms. 
Let 21 be a limited or unlimited point set in the plane. A set 
of distinct points in 51 

a l , a 2 , a 3 --. (1 

determine a broken line. In case 1) is an infinite sequence, let a n 
converge to a fixed point. If this line has no double point, we call 
it a chain, and the segments of the line links. In case not only the 
points 1) but also the links lie in 31, we call the chain a path. If 
the chain or path has but a finite number of links, it is called 
finite. 

Let us call a precinct a region, i.e. a set all of whose points are 
inner points, limited or unlimited, such than any two of its points 
may be joined by a finite path. 

2. Using the results of 578, we may say that, 

A closed Jordan curve J divides the other points of the plane into 
two precincts, an inner Q and an outer ) Moreover, they have a 
common frontier which is /. 

3. The closed Jordan curve considered in 594 shows that not 
every point of such a closed Jordan curve can always be joined to 
an arbitrary point of 3 r O by a finite path. 

Obviously it can ly an infinite path. For about this point, call 
it.P, we can describe a sequence of circles of radii r = 0. Between 
any two of these circles there lie points of $ and of ), if r is suf- 



IMAGES 613 

ficiently small. In this way we may get a sequence of points in 3, 
viz. /j, I 2 == P. Any two of these I m , I m+l may be joined by a 
path which does not cut the path joining /j to I m . For if a loop 
were formed, it could be omitted. 

4. Any arc $ of a closed Jordan curve J can be joined by a path 
to an arbitrary point of the interior or exterior, which call 21. 

For let J= $ + . Let k be a point of $ not an end point. 
Let S = Dist(&, ), let a be a point of 21 such that Dist(#, &) 

< A S. Then T , . OA 1 ,, 

2 77= Dist(>, 8) > 8. 

Hence the link = (a, A) has no point in common with 8. Let 
b be the first point of I in common with $. Then the link 
m = (a, 6) lies in 21. If now a is any point of 21, it may be joined 
to a by a path p. Then p + m is a path in 21 joining the arbi- 
trary point a to a point b on the arc $. 

596. Example 3. For in 21 = (0*, 1) let 

p = a(l + O, 

and ,, , -( + ik 

p = a(l + e v e/ )- 

These equations in polar coordinates define two non-intersecting 
spirals S^, S 2 which coil about p = a as an asymptotic circle F. 
Let us join the end points of the spirals corresponding to 6 = 1 
by a straight line L. Let & denote the figure formed by the 
spirals S^ /S^, the segment L and the asymptotic circle F. Is 
a closed Jordan curve ? The answer is, No. This may be seen 
in many ways. For example, 6 does not divide the other points 
into two precincts, but into three, one of which is formed of points 
within F. 

Another way is to employ the reasoning of 593, 2. Here the 
circle F takes the place of the segment on the ^-axis which figures 
there. 

Still another way is to observe that no point on F can be joined 
to a point within S by a path. 

597. Example 4. Let S be formed of the edge @ of a unit 
square, together with the ordinates o erected at the points 



GEOMETRIC NOTIONS 

x ~, of length , r&= 1, 2 Although 6 divides the other 

points of the plane into two precincts $ and ), we can say that 
S is not a closed Jordan curve. 

For if it were, 3 and O would have to have S as a common 
frontier. But the frontier of ) is (, while that of 3 is S. 

That & is not a Jordan curve is seen in other ways. For 
example, let 7 be an inner segment of one of the ordinates o. 
Obviously it cannot be reached by a path in D. 



Brouwer's Proof of Jordan's TJieorem 

598. We have already given one proof of this theorem in 577 
seq., based on the fact that the coordinates of the closed curve are 
expressed as one-valued continuous functions 



Brouwer's proof * is entirely geometrical in nature and rests 
on the definition of a closed Jordan curve as the unifold continu- 
ous image of a circle, cf. 591, 5. 

If 21, 33, are point sets in the plane, it will be convenient to 
denote their frontiers by g^, g^ - so that 

5^= Front 21 , etc. 

We admit that any closed polygon $ having a finite number of 
sides, without double point, divides the other points of the plane 
into an inner and an outer precinct ty t , $ e respectively. In the 
following sections we shall call such a polygon simple, and usu- 
ally denote it by ty. 

We shall denote the whole plane by (. 



Let 21 be complete. The complementary set A is formed, as 
we saw in 328, of an enumerable set of precincts, say A = \A n \. 

* Math. Annalen, vol. 69 (1910), p. 169. 



BROUWER'S PROOF OF JORDAN'S THEOREM 615 

599. 1. If a precinct 31 and its complement* A each contain a 
point of the connex S, then $% contains a point of S. 

For in the contrary case c = JDv(8l, S) is complete. In fact 
33 = ?l -f- gf^ is complete. As & is complete, Dv(58, S) is com- 
plete. But if ggi does not contain a point of , c = Dv($8, S). 
Thus on this hypothesis, c is complete. Now c = Dv(A, (5) is 
complete in any case. Thus S = c + c, which contradicts 585, 4. 

2. If ^J t , *Pe, the interior and exterior of a simple polygon $ each 
contain a point of a connex @, then ^ contains a point of S. 

3. Let $ be complete and not connected. There exists a simple 
polygon $ such that no point of $ lies on ^3, while a part of $ lies in 
$,, and another part in ty e . 

For let itj, $ 2 be two non-connected parts of & whose distance 
from each other is p > 0. Let A be a quadrate division of the 
plane of norm S, so small that no cell contains a point of ^ 1 and 
$ 2 . Let A! denote the cells of A containing points of $ r It is 
complete, and the complementary set A 2 = @ AJ is formed of one 
or more precincts. No point of St 1 lies in A 2 or on its frontier. 

Let Pj, P 2 be points in $j, $ 2 respectively. Let D be that 
precinct containing P 2 . Then $ D embraces a simple polygon $ 
which separates P l and P 2 . 

4. Let $j, $ 2 be two detached connexes. There exists a sim.ple 
polygon $ which separates them.. One of them is in *i)3 the other in 
^ and no point of either connex lies on $. 

For the previous theorem shows that there is a simple polygon 
$ which separates a point P l in ^ l from a point P a in $ 2 and no 
point of ff x or $ 2 lies on ty. Call this fact F. 

Let now P l lie in *ip t . Then every point of $j lies in $ t . For 
otherwise ^ and ^ each contain a point of the connex $j . Then 
2 shows that a point of l lies on $, which contradicts J 7 . 

5. Let $8 be a precinct determined by the connex (. Then 
6 = Front 53 is a connex. 

* Since the initial sets are all limited, their complements may be taken with ref- 
erence to a sufficiently large square ; and when dealing with frontier points, points 
on the edge of jQ may be neglected. 



616 GEOMETRIC NOTIONS 

For suppose b is not a connex. Then by 3, there exists a simple 
polygon $ which contains a part of b in ^ and another in ty e , 
while no point of b lies on $. Hence a point /3' of b lies in $ t , 
and another point ft" in $ e . As 35 is a precinct, let us join /3', 
ft" by a path t; in 33. Thus $ contains at least one point of v, 
that is, a point of 33 lies on *ij3. As b and ^ have no point in 
common, and as one point of ty lies in 33i all the points of ty lie 
in S3. Hence Dv($, 6) = 0. (1 

As b is a part of ( and hence some of the points of 6 are in ^ e 
and some in ty L , it follows from 2 that a part of ^ lies in g. This 
contradicts 1). 

6. Let $j, $ 2 be two connexes without double point. By 8 
there exists a simple polygon ty which separates them and has 
one connex inside, the other outside $. 

Now $ = $} 4- $2 i s complete and defines one or more precincts. 
One of these precincts contains ty. 

For say ty lay in two of these precincts as 21 and 33- Then the 
precinct 21 and its complement (in which 33 lies) each contain a 
point of the connex $. Thus $% contains a point of $. But $$ 
is a part of , and no point of $ lies on ty. 

That precinct in Comp $ which contains ty we call the inter- 
mediate precinct determined by $^ U 2 , or more shortly the pre- 
cinct between $ x , $ 2 and denote it by Inter ($ r $ 2 ). 

7. Let $j, $ 2 t> e t wo detached connexes, and let I = Inter ($ r $ 2 ). 
Then $ 19 $ 2 can be joined by a path lying in f, except its end points 
which lie on the frontiers of ^ $ 2 respectively. 

For by hypothesis p = Dist(^ ) 1 , $ 2 )>0. Let P l be a point of 
5^ such that some domain b of P contains only points of $j and 
of f. Let Q l be a point of f in b. Join P^ Q 1 by a right line, let 
it cut 5% fi 1 ' 8 ^ a t the point P 1 . In a similar way we may reason 
on $ 2 , obtaining the points P", Q 2 . Then P' Q^^P" is the path 
in question. If we denote it by v> we may let v* denote this 
path after removing its two end points. 

8. Let $j, $ 2 be two detached connexes. A path v joining $j, 
$ 2 and lying in f = Inter ($j, $ 2 )> en d points excepted, determines 
one and only one precinct in I . 



BKOUWEirS PROOF OF JORDAN'S THEOREM 617 

For from an arbitrary point P in f, let us draw all possible 
paths to v. Those paths ending on the same side (left or right) 
of v certainly lie in one and the same precinct f r or fy in f. Then 
since one end point of v is inside, the other end point outside $, 
there must be a part of $ which is not met by v and which joins 
the right and left sides of v. We take this as an evident property 
of finite broken lines and polygons without double points. 

Thus ti and ! r are not detached ; they are parts of one precinct. 

9. Two paths jjj, # 2 without common point, lying in f and joining 
$j, $' 2 , split t into two precincts. 

Let i = f v l ; this we have just seen is a precinct. From any 
point of it let us draw paths to # 2 . Those paths ending on the 
same side of v% determine precincts t$, i r which may be identical. 
Suppose they are. Then the two sides of v 2 can be joined by a 
path tying in f, which does not touch v 2 (end points excepted), 
has no point in common with Vj, and together with a segment of 
v 2 forms a simple polygon ^ which has one end point of v^ in $ t , 
the other end point in ty e . Thus by 2, ty contains a point of the 
connex v l . This is contrary to hypothesis. 

Similar reasoning shows that 

10. The n paths v l v n pairwise without common point, lying in 
f, and joining the connexes $j, $ 2 split I into n precincts. 

Let us finally note that the reasoning of 595, 4, being independ- 
ent of an analytic representation of a Jordan curve, enables us to 
use the geometric definition of 591, 6, and we have therefore the 
theorem 

11. Let 21 he a precinct whose frontier is a Jordan curve. Then 
there exists a path in 21 joining an arbitrary point of 21 with any arc 



Having established these preliminary theorems, we may now 
take up the body of the proof. 

600. 1. Let 21 be a precinct determined by a closed Jordan curve 
J. Then g = Front 21 is identical with J. 

If J determines but one precinct 21 which is pantactic in (, we 
have obviously g = J. 



618 GEOMETRIC NOTIONS 

Suppose then that SI is a precinct, not pantactic in (. Let S3 
be a precinct =31 determined by ft. Let 6 = Front S3. Then 
b <_ <L </. Suppose now b < J. As Jis a connex by 591, l, g is a 
connex by 599, 6. Similarly since 8? is a connex, b is a connex. 
Since b < <7i let 5 ~ b on the circle F whose image is J. We 
divide b into three arcs J x , 6 2 , ^3 to which ~ b x , b 2 , b 8 in b. 

Let /9 = Inter (b x , b 3 ). 

Then by 599, 11, we can join b p b 3 by a path v 1 in SI, and by a 
path v 2 in S3. By 599, 9, these paths split /3 into two precincts 
$i> /3 2 . We can join v^ v% by a path u^ lying in ySj, and by a 
path u 2 lying in /3 2 . 

Now the precinct S3 and its complement each contain a point of 
the connex u^. Hence by 599, 1, b contains a point of u v Simi- 
larly b contains a point of u%. Thus u^ u 2 cut b, and as they 
do not cut bj, b 3 by hypothesis, they cut b 2 . Thus at least one 
point of fii and one point of /3 2 He in b 2 . 

Let p be a point of /3 l lying in b 2 , let p ~p on the circle. Let 
b 1 be an arc of 6 2 containing p. Let b' ~b f . As the connex b ; 
has no point in common with Front /3 X , b' must lie entirely in /3 X 
by 599, 1. This is independent of the choice of b', hence the 
connex b 2 , except its end points, lies in /3 V Thus /3 2 can contain 
no point of b 2 , which contradicts the result in italics above. 

Thus the supposition that b < J is impossible. Hence b = J, 
and therefore g = *? 

As a corollary we have : 

2. A Jordan curve is apantactic in (. 

3. A closed Jordan curve J cannot determine more than two 
precincts. 

For suppose there were more than two precincts 

!, ^, a, - (i 

Let us divide the circle F into four arcs whose images call <7 X , J%, 

J v J t- 

Then by 1, the frontier of each of the precincts 1) is J. Thus 
by 599, 9, there is a path in each of the precincts 2^, Slg join- 
ing J l and 7 8 . These paths split 



DIMENSIONAL INVARIANCE 619 

I = Inter (J^, Jg) 

into precincts fj, f a 

Now as in 1, we show on the one hand that each f t must contain 
a point of J" 2 or J" 4 , and on the other hand neither 7 2 nor J 4 can 
lie in more than one f t . 

4. A closed Jordan curve J must determine at least two precincts. 

Suppose that J determines but a single precinct 21. From a 
point a of 51 we may draw two non-intersecting paths u^ u% to 
points ij, J 2 of J. 

Since the point a may be regarded as a connex, a and e/are two 
detached connexes. Hence by 599, o, the paths w x , u z split ?l into 
two precincts Sip 21 2 . Let / = (^, M 2 , <7). The points 6 X , 6 2 
divide <7into two arcs Jj, J2, and 



are closed Jordan curves. Regarding a and Jj as two detached 
connexes, we see/! determines two precincts, a^ 0%. By 599, 1, a 
path which joins a point a l of ^ with a point a 2 of 2 must cut j\ 
and hence y. It cannot thus lie altogether in Slj or in 21 2 Thus 
both j, a 2 do not lie in 2lj, nor both in 21 2 . Let us therefore 
say for example that 2^ lies in c^, and SL^ in 2 . Hence by 2, 
2lj is pantactic in c^, and 2T 2 in 2 . By 1, each point of j\ is com- 
mon to the frontiers of ^ and of 03, and hence of ?Ij and of 21 2 , 
as these are pantactic. 

Let P be a point of J^ . It lies either in a t or 0%. Suppose it 
lies in a l . Then it lies neither in 2 nor on Front 2 , and hence 
neither in S1 2 nor on Front 21 2 . But every point of / 2 and also 
every point of j\ lies on Front 1 2 . We are thus brought to a 
contradiction. Hence the supposition that J determines but a 
single precinct is untenable. 

Dimensional Invariance 

601. 1. In 247 we have seen that the points of a unit interval 
/, and of a unit square S may be put in one to one correspondence. 
This fact, due to Cantor, caused great astonishment in the mathe- 
matical world, as it seemed to contradict our intuitional views 



620 GEOMETRIC NOTIONS 

regarding the number of dimensions necessary to define a figure. 
Thus it was thought that a curve required one variable to define 
it, a surface two, and a solid three. 

The correspondence set up by Cantor is not continuous. On 
the other hand the curves invented by Peano, Hilbert, and others 
(cf. 573) establish a continuous correspondence between /and S, 
but this correspondence is not one to one. Various mathemati- 
cians have attempted to prove that a continuous one to one corre- 
spondence between spaces of m and n dimensions cannot exist. 
We give a very simple proof due to Lebesc/ue.* 

It rests on the following theorem : 

2. Let 21 be a point set in 9? m . Let Q < 31 be a standard cube 
0<# t # t <2cr , i=l, 2" m. 

Let Sj, E 2 "* be a finite number of complete sets so small that each 
lies in a standard cube of edge or. If each point of 21 lies in one of 
the S's, there is a point of 21 which lies in at least m -f- 1 of them. 

Suppose first that each 6 t is the union of a finite number of 
standard cubes. Let (Sj denote those GTs containing a point of 
the face f x of Q lying in the plane x 1 = a r The frontier JJi of @i 
is formed of a part of the faces of the CTs. Let F l denote that 
part of ^ which is parallel to fj. Let O 1 = ^(Q, FI). Any 
point of it lies in at least two ('s. 

Let @ 2 denote those of the S's not lying altogether in @j and 
containing a point of the face f 2 of Q determined by x 2 = a 2 . Let 
.F 2 denote that part of Front ( 2 which is parallel to f 2 . Let 
Q 2 = jDt^Qj, .F 2 ). Any point of it lies in at least three of the &'s. 

In this way we may continue, arriving finally at Q m , any point 
of which lies in at least m -f 1 of the S's. 

Let us consider now the general case. We effect a cubical 
division of space of norm d<&. Let 0, denote those cells of D 
which contain a point of & t . Then by the foregoing, there is a 
point of 21 which lies in at least m + 1 of the (7's. As this is true, 
however small d is taken, and as the (Ts are complete, there is at 
least one point of 21 which lies in m 4- 1 of the S's. 

* Math Annalen, vol. 70 (1911), p. 166. 




DIMENSIONAL INVAKIANCE (521 

3. We now note that the space 9? m may be divided into congruent 
cells so that no point is In more than m 4- 1 cells. 

For m = 1 it is obvious. For m = 2 we may 
use a hexagonal pattern. We may also use 
a quadrate division of norm 8 of the plane. 
These squares may be grouped in horizontal 
bands. Let every other band be slid a distance 
^ 8 to the right. Then no point lies in more 
than 3 squares. For m = 3 we may use a 
cubical division of space, etc. 

In each case no point of space is in more than m -f- 1 cells. 

Let us call such a division a reticulation of 9f m . 

4. Let 21 be a point set in 3J m having an inner point a. There is 
no continuous unifold image 33 of 31 in 9i n , w=w, such that l)~a is 
an inner point of $8. 

For let n > m. Let us effect a reticulation H of 9t m of norm p. 
If S > is taken sufficiently small A = J9 2 (a) lies in 21. Let 
E '= -Da(fl) ; if p is taken sufficiently small, the cells 

0^0,- 0. (1 

of R which contain points of E, lie in A. Let the image of E be 
@, and that of the cells 1) be 

Si, 6a <.. (2 

These are complete. Each point of ( lies in one of the sets 2). 
Hence by 2, they contain a point /? which lies in n + 1 of them. 
Then a~/3 lies in n -f 1 of the cells 1). But these, being part of 
the reticulation R, arc such that no point lies in more than m + 1 
of them. Hence the contradiction. 

602. 1. Sehonflies* Theorem. Let 

u =/(, #) , v = g(x, #) (1 

be one- valued and continuous in a unit square A whose center is 
the origin. These equations define a transformation T. If T is 
regular, we have seen in I, 742, that the domain ^0 P (P) of a point 
P = (#, y) within A goes over into a set E such that if Q~P 
then D a (Q) lies in E, if cr >0 is sufficiently small. 



622 GEOMETRIC NOTIONS 

These conditions on /, g which make T regular are sufficient, 
but they are much more than necessary as the following theorem 
due to Schonfliess * shows. 

2. Let A B+cbea unit square in the x, y plane, whose center 
is the origin and whose frontier is c. 

u =/(#, y} , v = g(x, y) 

be one-valued continuous functions in A. As (x, y) ranges over A, 
let (u, v) range over 21 = 93 -f c where c ~ c. Let the correspondence 
between A and 21 be uniform. Then c is a closed Jordan curve and 
the interior c t of c is identical with 93. 

That C is a closed Jordan curve follows from 576 seq., or 598 
seq. Obviously if one point of 93 lies in C all do. For if /3 t , /8 e 
are points of 93, one within c and the other without, let J t ~/3 t , 
J e ~&. Then J t , b e lying in B can be joined by a path in B 
which has no point in common with c. The image of this path is 
a continuous curve which has no point in common with c, which 
contradicts 578, 2. 

Let 



be the equation of c in polar coordinates. 
If < /*, < 1, the equation 

P = rt(P) 

defines a square, call it <?^, concentric with c and whose sides are 
in the ratio fi : 1 with those of c. The equations of C M ~ <v are 

W =/ 50 cos , 



These C M curves have now the following property : 

If a point (p, q) is exterior {interior) to c Mo , it is exterior (in- 
terior) to c^ifor all /z such that 

I A 6 "~ MO I ^ some e > 0. 

For let PH be the distance of (/?, q) from a point (w, v) on c^. 
Then , _ - _ 



*Goettingen Nachrichten, 1899. The demonstration here given is due to Osgood, 
Goett. Nachr., 1900. 



AREA OF CURVED SURFACES 623 

is a continuous function of 0, //. which does not vanish for /x = /A O , 
when < 6 < 2 TT. But being continuous, it is uniformly con- 
tinuous. It therefore does not vanish in the rectangle 

' < < 2 7T. 



We can now show that if 33<C it is identical with c t . To this 
end we need only to show that any point /3 of c t lies on some c^. 
In fact, as /x = 0, c^ contracts to a point. Thus ft is an outer point 
of some c^, and an inner point of others. Let /LC O be the maximum 
of the values of /JL such that /3 is exterior to all c u , if /A</LC O . 
Then /3 lies on c Mo . For if not, is exterior to (V + e , by what we 
have just shown, and /^ is not the maximum of p. 

Let us suppose that 33 lay without c. We show this leads to a 
contradiction. For let us invert with respect to a circle f, lying 
in C . Then c goes over into a curve f, and 31 goes over into 
3) = S + f . Then @ lies inside f . Let , ?? be coordinates of a 
point of 3). Obviously they are continuous functions of z, y in 

^' and A -3) , c~f, uniformly. 

By what we have just proved, ( must fill all the interior of f. 
This is impossible unless 21 is unlimited. 

3. We may obviously extend the theorem 2 to the case 



. u m = 



and A is a cube in ra-way space 5K m , provided we assume that c, the 
image of the boundary of A, divides space into two precincts 
whose frontier is c. 

Area of Curved Surfaces 

603. 1. The Inner Definition. It is natural to define the area of a 
curved surface in a manner analogous to that employed to define 
the length of a plane curve, viz. by inscribing and circumscrib- 
ing the surface with a system of polyhedra, the area of whose 
faces converges to 0. It is natural to expect that the limits of 
the area of these two systems will be identical, and this common 
limit would then forthwith serve as the definition of the area of 
the surface. The consideration of the inner and the outer sys- 



624 



GEOMETRIC NOTIONS 



terns of polyhedra afford thus two types of definitions, which 
may be styled the inner and the outer definitions. Let us look 
first at the inner definition. 

Let the equations of the surface 8 under consideration be 



x = 



y = 



z = 



v), 



(1 



the parameters ranging over a complete metric set 21, and a?, ?/, z 
being one- valued and continuous in 21. 

Let us effect a rectangular division D of norm d of the u, v 
plane. The rectangles fall into triangles t K on drawing the 
diagonals. Such a division of the plane we call quasi rectangular. 

J*Q=(u Q iV Q ) , P 1= O + 8, v) , jP 2 = (V v o + 7 ?) 

be the vertices of ^. To these points in the u, v plane corre- 
spond three points ^ t = (# t , y t , 2 t ), 4=1, 2, 3, of $ which form the 
vertices of one of the triangular faces r K of the inscribed polyhe- 
dron n^ corresponding to the division D. Here, as in the follow- 
ing sections, we consider only triangles lying in 21. We may do 
this since 21 is metric. 

Let X K , Y K , Z K be. the projections of T K on the coordinate planes. 
Then, as is shown in analytic geometry, 



where 

2 X 

*J -*X<C 



y* 



y\ ~ 



i'y , A' 
i"y , A"z 



and similar expressions for Y K , Z K . 
Thus the area of 11^ is 



the summation extending over all the triangles t K lying in the 
set SI. 

Let x, y, z have continuous first derivatives in 31. Then 



OV 



AREA OF CURVED SURFACES 



625 



with similar expressions for the other increments. Let 



dy dz 

du du 

dy dz 

dv dv 



D == 



dx dz 




dx dy 


du du 
dx dz 

dv dv 


, c= 


du du 
dx dy 
dv dt; 


+~~" O \ + *7 f /~f i . 
JK.)V K , J K == ( \S K "f- f 



(2 



Then 

-*,= ( 

where a K j3 K <y K are uniformly evanescent with d in 21- Thus if 
A, B, do not simultaneously vanish at any point of 2k we liave 
as area of the surface 8 

lim '/)= I V^-h^H- O^dudv. (3 

*=o J* 

2. An objection which at once arises to this definition lies in 
the fact that we have taken the faces of our inscribed polyhedra 
in a Very restricted manner. We cannot help asking, Would we 
get the same area for $ if we had chosen a different system of 
polyhedra ? 

To lessen the force of this objection we observe that by replac- 
ing the parameters u, v by two new parameters it', v' we may 
replace the above quasi rectangular divisions which correspond to 
the family of right lines u = constant, v = constant by the infinitely 
richer system of divisions corresponding to the family of curves 
u f = constant, v r = constant. In fact, by subjecting u f , v r to cer- 
tain very general conditions, we may transform the integral 3) 
to the new variables u f , v r without altering its value. 

But even this does not exhaust all possible ways of dividing 21 
into a system of triangles with evanescent sides. Let us there- 
fore take at pleasure a system of points in the u, v plane having 
no limiting points, and join them in such a way as to cover the 
plane without overlapping with a set of triangles t K . If each 
triangle lies in a square of side c?, we may call this a triangular 
division of norm d. We may now inquire if /$/> still converges 
to the limit 3), as d = 0, for this more general system of divisions. 
It was generally believed that such was the case, and standard 
treatises even contained demonstrations to this effect. These 
leinonst rations are wrong ; for Schwarz * has shown that by 

* Werke, vol. 2, p. 309. 



626 



GEOMETRIC NOTIONS 



properly choosing the triangular divisions D, it is possible to 
make S D converge to a value large at pleasure, for an extensive 
class of simple surfaces. 

604. 1. Schwarzs Example. Let C be a right circular cylin- 
der of radius 1 and height 1. A set of planes parallel to the base 

at a distance - apart cuts out a system of circles F x , F 2 Let 

71 

us divide each of these circles into m equal 
arcs, in such a way that the end points of 
the arcs on Fj, F 3 , F 5 lie on the same 
vertical generators, while the end points of 
F 2 , F 4 , F 6 - lie on generators halfway 
between those of the first set. We now 
inscribe a polyhedron so that the base of 
one of the triangular facets lies on one 

circle while the vertex lies on the next circle above or below, as 

in the figure. 

The area t of one of these facets is 




m 



1 

1 COS 



m/ 



Thus 



m * ri* 2m 

There are 2 m such triangles in each layer, and there are n 
layers. Hence the area of the polyhedron corresponding to this 
triangular division D is 



Sj> = 2k = 2 mn sin --v + 4 sin * ~ 
m * n* 2m 

Since the integers 7, n are independent of each other, let us 
consider various relations which may be placed on them. 
Case 1. Let n = \m. Then 



m 



- 

2 m 



= 2 



. 7T / 


7T 


4 


sin / 


sm 




7T ml. i 7T* 


2m 




m TT I X 2 /^ 2 2 4 ?ft 4 

1 


7T 




m V 


2m 




as m = oo. 



AREA OP CURVED SURFACES 



627 



Case 2. Let n = \m*. Then 



m 



Bin 51 



m 



1 * 

4-4 7r -- 


m n 




2 m 






X m 2m 


7T 






2m 





= 27T\[ 



l + X 2 , as m = oo. 
4 



Case 3. Let n = Xm 3 . Then 



sin 



7T 

m 




sin ~~- 



w 2 X 2 



= +oo 



as m == QO. 



2. Thus only in the first case does S D converge to 2 TT, which 
is the area of the cylinder C as universally understood. In the 
2 and 3 cases the ratio h/b = 0. As equations of C we may 

take 

x = cos u , y = sin u , 2 = v. 

Then to a triangular facet of the inscribed polyhedron will cor- 
respond a triangle in the u, v plane. In cases 2 and 3 this tri- 
angle has an angle which converges to TT as m = GO. This is not 
so in case 1. Triangular divisions of this latter type are of great 
importance. Let us call then a triangular division of the u, v 
plane such that no angle of any of its triangles is greater than 
TT e, where e > is small at pleasure but fixed, positive triangu- 
lar divisions. We employ this term since the sine of one of the 
angles is > some fixed positive number. 

605. The Outer Definition. Having seen one of the serious diffi- 
culties which arise from the inner definition, let us consider briefly 
the outer definition. We begin with the simplest case in which 
the equation of the surface S is 

z =/(#, y), (1 

/ being one-valued and having continuous first derivatives. Let 
us effect a metric division A of the x, y plane of norm S, and on 



628 GEOMETRIC NOTIONS 

each cell d K as base, we erect a right cylinder (7, which cuts out an 
element of surface 8*. from S. Let ^ be an arbitrary point of 8 K 
and Z K the tangent plane at this point. The cylinder O cuts out 
of Z K an clement A/S^ . Let V K be the angle that the normal to Z K 
makes with the 2-axis. Then 

1 

COS V K = 



^v 

\dy)* 

and A S r - ^" 

cos V K 

The area of S is now defined to be 

Urn 2A& (2 

when this limit exists. Tlie derivatives being continuous, we have 
at once that this limit is 



) 

Air 



which agrees with the result obtained by the inner definition in 
IJ03, 3). 

The advantages of this form of definition are obvious. In the 
first place, the nature of the divisions A is quite arbitrary ; however 
they are chosen, one and the same limit exists. Secondly, the most 
general type of division is as easy to treat as the most narrow, viz. 
when the cells d K are squares. 

Let us look at its disadvantages. In the first place, the elements 
AiS K do not form a circumscribing polyhedron of S. On the con- 
trary, they are little patches attached to S at the points ty K , and 
having in general no contact with one another. Secondly, let us 
suppose that S has tangent planes parallel to the 2-axis. The de- 
rivatives which enter the integral 603, 3) are no longer continuous, 
and the reasoning employed to establish the existence of the limit 
2) breaks down. Thirdly, we have the case that z is not one- 
valued, or that the tangent planes to S do not turn continuously, 
or do not even exist at certain points. 



AREA OF CURVED SURFACES 629 

To get rid of these disadvantages various other forms of outer 
definitions have been proposed. One of these is given by O-oursat 
in his Cours d* Analyse. Instead of projecting an arbitrary 
element of surface on a fixed plane, the xy plane, it is projected on 
one of the tangent planes belonging to that element. Hereby the 
more general type of surfaces defined by 603, 1) instead of those 
defined by 1) above is considered. The restriction is, however, 
made that the normals to the tangent planes cut the elements of 
surface but once, also the first derivatives of the coordinates are 
assumed to be continuous in 21. Under these conditions we get 
the same value for the area as that given in 603, 3). 

When the first derivatives of a?, /, z are not continuous or do 
not exist, this definition breaks down. To obviate this difficulty 
de la Vallee-Pomsin has proposed a third form of definition in his 
Cours d" Analyse, vol. 2, p. 30 seq. Instead of projecting the 
element of surface on a tangent plane, let us project it on a plane 
for which the projection is a maximum. In case that S has a con- 
tinuously turning tangent plane nowhere parallel to the z-axis, de 
la Vallee-Poussin shows that this definition leads to the same 
value of the area of S as before. He does not consider other cases 
in detail. 

Before leaving this section let us note that Jordan in his Cours 
employs the form of outer definition first noted, using the paramet- 
ric form of the equations of S. In the preface to this treatise the 
author avows that the notion of area is still somewhat obscure, and 
that he has not been able u a definir d'une mani^re satisfaisante 
1'aire d'une surface gauche que dans le cas ou la surface a un plan 
tangent variant suivant une loi continue." 

606. 1. Regular Surfaces. Let us return to the inner definition 
considered in 603. We have seen in 604 that not every system of 
triangular divisions can be employed. Let us see, however, if w r e 
cannot employ divisions much more general than the quasi rec- 
tangular. We suppose the given surface is defined by 



the functions $, i/r, ^ being one-valued, totally differentiable func- 
tions of the parameters u, v which latter range over the complete 



680 



GEOMETRIC NOTIONS 



metric set 31. Surfaces characterized by these conditions we 
shall call regular. Let 



be the vertices of one of the triangles t K , of a triangular division 
D of norm d of 21. As before let $ , ^ $ 2 be the corresponding 
points on the surface S. Then 



r 



and similar expressions hold for the other increments. Also 



dy 



du 



4- 2 Jf ' 
I -' -*** 



where JT denotes the sum of several determinants, involving the 
infinitesimals 

a' a" 8' 8" 

^y t tt y i Pz > Pz * 

Similar expressions hold for Y K , Z*. We get thus 



where A^ B, are the determinants 2) in 603. Then the area of 
the inscribed polyhedron corresponding to this division D is 



Let us suppose that 



as it, v ranges over 1. Also let us assume that 

Vt Vf *7f 

-A* -* K 6* 



(2 



(3 



AREA OF CURVED SURFACES 631 

remain numerically < e for any division D of norm d< d , small 
at pleasure, except in the vicinity of a discrete set of points, that 
is, let 3) be in general uniformly evanescent in 8, as d = 0. Then 



where in general 



n .or 

1 ' Cont 31 
If now J., J5, (7 are limited and .B-integrable in 81, we have at 



once 



lim S D =fdudv^A* + B* + <7 2 
as in 603. 

2. We ask now under what conditions are the expressions 3) 
in general uniformly evanescent in 21 ? The answer is pretty evi- 
dent from the example given by Schwarz. In fact the equation 
of the tangent plane Z at ^Q is 



On the other hand the equation of the plane T= 
is 



x y 



o, 



r 

or finally 



Thus for 3) to converge in general uniformly to zero, it is nec- 
essary and sufficient that the secant planes T converge in general 
uniformly to tangent planes. Let us call divisions such that the 
faces of the corresponding inscribed polyhedra converge in general 
uniformly to tangent planes uniform triangular divisions. For 
such divisions the expressions 3) are in general uniformly evanes- 
cent, as d s= 0. We have therefore the following theorem : 

3. Let W be a limited complete metric set. Let the coordinates 
X) y, z be one-valued totally differentiate functions of the parame- 



632 GEOMETRIC NOTIONS 

ters u, v in 21, such that A 2 + B* + C 2 is greater than some positive 
constant^ and is limited and R-integrable in ?l. Then 



# = lim S D = 
rf-O 

D denoting the class of uniform triangular divisions of norms d. 

This limit we shall call the area of S. From this definition we 
have at once a number of its properties. We mention only the 
following : 

4. Let ?(j, 2l m be unmixed metric sets whose union is 21. Let 
$1, S m be the pieces of S corresponding to them. Then each S K 
has an area and their sum is S. 

5. Let 21 A be a metric part of 21, depending on a parameter \ = 0, 
such that 21 A = 21. Then 

limS^S. 

A = 

(). The area of S remains unaltered when S is subjected to a dis- 
placement or a transformation of the parameters as in I, 744 seq. 

607. 1. Irregular Surfaces. We consider now surfaces which 
do not have tangent planes at every point, that is, surfaces for 
which one or more of the first derivatives of the coordinates rr, #, z 
do not exist, and which may be styled irregular surfaces. We 
prove now the theorem : 

Let the coordinates x, y, z be one-valued functions of u, v having 
limited total difference quotients in the metric set 21. Let D be a 
positive triangular division of norm d<d Q . Then 

Max S D 
is finite and evanescent with 21. 

For let the difference quotients remain < JJL. We have 



But 



cosec 



= 



AREA OF CURVED SURFACES 633 



where K is the angle made by the sides P^P^ P^P^ As D is a 
positive division, one of the angles of t K is such that cosec K is 
numerically less than some positive number M. Thus 



where /A, M are independent of K and d. Similar relations hold 

for \Y K \, \Z K \. Hence 

Sj> < 2 6 fji *M . t K = 6 



where 77 > is small at pleasure, for d Q sufficiently small. 

2. Let 21 awe? a;, y, z be as in 606, 3, except at certain points form- 
ing a discrete set a, the first partial derivatives do not exist. Let 
their total difference quotients be limited in 21. Then 



lim^= f 

rf=0 J 



where D denotes a positive trianyular division of norm d. 

Let us first show that the limit on the left exists. We may 
choose a metric part S3 of 21 such that S = 21 S3 is complete and 
exterior to 21 and such that 93 is as small as we please. Let S 
denote the area of the surface corresponding to S. The triangles 
t K fall into two groups : Q-^ containing points of 93 ; 6? 2 containing 
only points of S. Then 



S D = 2 V-XJ + 17 + 2 = 2 + 2. 

6', G, 

But 93 may be chosen so small that the first sum is < e/4 for 
any d < d Q . Moreover by taking d still smaller if necessary, we 
have 

|2 

03 

Hence 



Similarly for any other division D f of norm 

|<S/>.-fl<s|<e/2 , d f <d Q 
decreasing d Q still farther if necessary. Thus 



03i GEOMETRIC NOTIONS 

Hence lira S D exists, call it S. Since S exists we may take d Q 
so small that 

\S-S0\<e/-2 , d<d . 

This with 1) gives 

|tf 
that is, 

k = lim f 

./g 



by I, 724. 

608. 1. The preceding theorem takes care of a large class of 
irregular surfaces whose total difference quotients are limited. 
In case they are not limited we may treat certain cases as follows: 

Let us effect a quadrate division of the u, v plane of norm d, 
and take the triangles t K so that for any triangular division D 
associated with d, no square contains more than n triangles, and 
no triangle lies in more than v squares; w, v being arbitrarily 
large constants independent of d. Such a division we call a 
quasi quadrate division of norm d. If we replace the quadrate by 
a rectangular division, we get a quasi rectangular division. 

We shall also need to introduce a new classification of functions 
according to their variation in 31, or along lines parallel to the 
u, v axes. Let D be a quadrate division of the w, v plane of norm 
d<d . Let 

W K = Osc/(w, v) , in the cell d*. 

Then Max Sc^c? 

is the variation of / in 81. If this is not only finite, but evanes- 
cent with 81, we say/ has limited fluctuation in SI. Obviously this 
may be extended to any limited point set in w-way space. 

Let us now restrict ourselves to the plane. Let a denote the 
points of 21 on a line parallel to the w-axis. Let us effect a divi- 
sion D f of norm d 1 . Let a>' K = Osc/(w, v) in one of the intervals 
of D'. Then 

rj a = Max 2o> 
is the variation of /in a. 



AREA OF CURVED SURFACES 635 

Let us now consider all the sets a lying on lines parallel to the 
is, and let 



If now there exists a constant Or independent of a such that 



that is, if ?; a is uniformly evanescent with tr, we say that/(w, v) 
has limited fluctuation in 31 with respect to u. 

With the aid of these notions we may state the theorems : 

2. Let the coordinates x, y, z be one-valued limited functions in 
the limited complete set 31. Let x, y have limited total difference 
quotients, while z has limited variation in 31. Let D denote a quasi 
quadratic division of norm d<d Q . Then 

Max S D 

. t D 

is finite. 

For, as before, 

2|jr ic |<|At|.|Ai'| + |Aj,'|.|A;|. 
But p denoting a sufficiently large constant, 
|Ai!, |A^| are </**. 

Let o) t = Osc z in the square s t . If the triangle t K lies in the 
squares * 4 , s v 



Thus, n denoting a sufficiently large constant, 

2 JST. 



the summation extending over those squares containing a triangle 
of D. But z having limited variation, 

2ft> t d < some M. 
Hence 2| ^| ^ 2 ( ^ ( ^ 

Finally, as in 607, 

2 \Z \ <some M r . 

The theorem is thus established. 



636 GEOMETRIC NOTIONS 

3. The coordinates #, y, 2, being as in 2, except that z has limited 
fluctuation in 21, and D denoting a quasi quadrate division of 
norm d < c? , 

Max S D 

_ D 

is finite and evanescent with 21. 

The reasoning is the same as in 2 except that now M, M f are 
evanescent with 21. 

4. Let the coordinates x, y, z have limited total difference quo- 
tients in 21, while the variation of z along any line parallel to the u 
or v axis is < M. Let 21 lie in a square of side s = 0. Then 

Max S D <sGr, 

D 

where G is some constant independent of s, and D is a quasi rectan- 
gular division of norm d < d Q , 

For here 

22 | X K | < 2 | A'y 1 | A"* | + 2 | A"y | | A'* | 



where M 1 denotes a sufficiently large constant ; d u , d v denote the 
length of the sides of one of the triangles t K parallel respectively 
to the u, v axes, and <W M , co v the oscillation of z along these sides. 
Since the variation is < M in both directions, 



Ms. 

V 

Similarly 

2a) v d u < M,. 

The rest of the proof follows as before. 

5. The symbols having the same meaning as before, except that z 
has limited fluctuation with respect to u, v, 



The demonstration is similar to the foregoing. Following the 
line of proof used in establishing 607, 2 and employing the 
theorems just given, we readily prove the following theorems : 



AREA OF CURVED SURFACES 637 

6. Let 21 be a metric set containing the discrete set a. Let b be 
a metric part of 21, containing a such that 33 = 21 b is exterior to a, 
and b == 0. Let the coordinates #, y, z be one-valued totally differ- 
entiable functions in 33, such that A 2 4- B* -h O 2 never sinks below a 
positive constant in any 33, is properly R-integrable in any 93, and 
improperly integrable in 21. Let x, y have limited total difference 
quotients, and z limited fluctuation in b. Then 



lim So = f V2 2 
d=o 'a 



lim S = V2 2 "+ & + C*dudv 



where A, B, are the determinants in 603, 2), and D is any quasi 
quadrate division of norm d. 

7. Let the symbols have the same meaning as in 6, except that 
1 a reduces to a finite set. 

2 z has limited variation along any line parallel to the u, v axes. 
3 D denotes a uniform quasi rectangular division. Then 



= fVZ 2 + & + CPdudv. 

^2l 



8. :7% symbols having the same meaning as in 6, except that 

1 z has limited fluctuation with respect to u, v in b. 

2 D denotes a uniform quasi rectangular division. Then 



lim S D = I V^l 2 + * + C^ttdto. 

d=0 



0. If we call the limits in theorems 6, 7, 8, area, the theorems 
606, 3, 4, 5 still hold. 



INDEX 



(Numbers refer to pages) 



Abel's identity, 87 

series, 87 
Absolutely convergent integrals, 31 

series, 79 

products, 247 
Addition of cardinals, 292 

ordinals, 312 

series, 128 
Adherence, 340 
Adjoint product, 247 

series, 77, 139 

set of intervals, 337 
Aggregates, cardinal number, 278 

definition, 276 

distribution, 295 

enumerable, 280 

equivalence, 276 

eutactic, 304 

exponents, 294 

ordered, 302 

power or potency, 278 

sections, 307 

similar, 303 

transfinite, 278 

uniform or 1-1 correspondence, 276 
Alternate series, 83 
Analytical curve, 582 
Apantactic, 325 
Area of curve, 599, 602 

surface, 623 
Arzela, 365, 555 
Associated simple series, 144 

products, 247 

multiple series, 145 

normal series, 245 

logarithmic series, 243 

inner sets, 365 



Associated, outer sets, 365 
non-negative functions, 41 

Baire, 326, 452, 482, 587 
Bernouillian numbers, 265 
Bertram's test, 104 
Bessel functions, 238 
Beta functions, 267 
Binomial series, 110 
Bocher, 165 
Bonnet's test, 121 
Borel, 324, 542 
Brouwer, 614 

Cahen's test, 340 

Cantor's 1 and 2 principle, 316 

theorem, 450 
Category of a set, 326 
Cauchy's function, 214 

integral test, 99 

radical test, 98 

theorem, 90 
Cell of convergence, 144 

standard rectangular, 359 
Chain, 612 

Class of a function, 468, 469 
Conjugate functions, 238 

series, 147 

products, 249 
Connex, 605 
Connected sets, 605 
Contiguous functions, 231 
Continuity, 452 

infra, 487 

semi, 487 

supra, 487 
Continuous image, 608 



689 



640 



INDEX 



Contraction, 287 

Convergence, infra-uniform, 562 
monotone, 176 
uniform, 156 
at a point, 157 
in segments, 556 
sub-uniform, 555 
Co-product, 212 
Curves, analytical, 582 
area, 599, 602 
Faber, 546 
Jordan, 595, 610 
Hilbert, 590 
length, 579 
non-intuitional, 537 
Osgood, 600 
Pompeiu, 542 
rectifiable, 583 
space-filling, 588 

D'Alembert, 96 
Deleted series, 139 
Derivates, 494 
Derivative of a set, 330 

order of, 331 
Detached sets, 604 
Dilation, 287 
Dini, 176, 185, 438, 538 

series, 86 
Discontinuity, 452 

at a point, 454 

of 1 kind, 416 

of 2 kind, 455 

pointwise, 457 

total, 457 
Displacement, 286 
Distribution, 295 
Divergence of a series, 440 
Division, complete, 30 

separated, 366, 371 

unmited, 2 

of series, 196 

of products, 253 
Divisor of a set, 23 

quasi, 390 



Divisor, semi, 390 
Du Bois Reymond, 103 

< c , (S^osets, 473 

Elimination, 594 

Enclosures, complementary c-, 355 

deleted, 452 

distinct, 344 

divisor of, 344 

e-, 355 

measurable, 356 

non-overlapping, 344 

null, 366 

outer, 343 

standard, 359 
Enumerable, 280 
Equivalent, 276 
Essentially positive series, 78 

negative series, 78 
Euler's constant, 260 
Eutactic, 301 
Exponent*, 29 1 
Exponential series, 96 
Extremal sequence, 374 

Faber curves, 516 
Fluctuation, 63 1, 635 
Fourier's coefficient, 416 
constants, 416 
series, 416 

Function, associated non-negative func- 
tions, 41 
Basel's, 238 
Beta, 267 
Cauchy's, 214 
class of, 468, 469 
conjugate, 233 
contiguous, 231 
continuous, 452 
infra, 487 
semi, 487 
supra, 487 
discontinuous, 452 
of 1 kind, 410 
of 2 kind, 455 



INDEX 



641 



Function, Gamma, 267 
Gauss' II(a;), 238 
hyperbolic, 228 
hypergeometric, 228 
lineo-oscillating, 528 
maximal, 488 
measurable, 338 
minimal, 488 
monotone, 137 
null, 385 
oscillatory, 488 
pointwise discontinuous, 457 
residual, 561 
Riemann's, 459 
totally discontinuous, 457 
truncated, 27 

uniformly limited, 160, 567 
Volterra's, 501, 583 
Weierstrass', 498, 523, 581, 588 

Gamma function, 267 
Gauss' function U(x), 238 

test, 109 
Geometric series, 81, 139 

Harnack, divergence of series, 440 

sets, 354 
Hermite, 300 
Hubert's curves, 590 
Hobson, 389, 412, 555 
Hyperbolic functions, 228 
Hypercotnplete sets, 472 
Hypergeometric functions, 229 

series, 112 

Images , simple, multiple, 606 
unifold, manifold, 606 
continuous, 606, 608 
Integrals, absolutely convergent, 31 
L- or Lebesgue, proper, 372 

improper, 403, 405 
improper, author's, 32 
classical, 20 

de la Vallee-Poussin, 27 
inner, 20 



Integrals, R- or Riemannian, 372 

Integrand set, 385 
Intervals, of convergence, 90 

adjoint set of, 337 

set of, belonging to, 337 
Inversion, geometric, 287 

of a series, 204 
Iterable sets, 14 
Iterated products, 251 

series, 149 

Jordan curves, 595, 610 
variation, 430 
theorem, 436 

Kdnig, 527 

Rummer's test, 106, 124 

Lattice points, 137 

system, 137 

Law of Mean, generalized, 505 
Layers, 555 

deleted, 563 
Lebesque or L- integrals, 372 

theorems, 413, 424, 426, 452, 475 S 

520, 619 

Leibnitz's formula, 226 
Length of curve, 579 
Lindermann, 300, 599 
Lineo-oscillating functions, 528 
Link, 612 

Liouuille numbers, 301 
Lipschitz, 438 
Logarithmic series, 97 
Luroth, 448 

Maclaurin's series, 206 
Maximal, minimal functions, 488 
Maximum, minimum, 521 

at a point, 485 
Measure, 348 

lower, 348 

upper, 343 
Mertens, 130 
Metric sets, 1 



642 



INDEX 



Monotone convergence, 176 

functions, 137 
Moore-Osgood theorem, 170 
Motion, 579 
Multiplication of series, 129 

cardinals, 293 

ordinals, 314 

infinite products, 253 

Normal' form of infinite product, 245 
Null functions, 385 

sets, 348 
Numbers, Bernouillian, 265 

cardinal, 278 

class of ordinal numbers, 318 

limitary, 314. 

Liouville, 301 

ordinal, 310 

rank of limitary numbers, 331 

Ordered sets, 302 

Order of derivative of a set, 331 

Oscillation at a point, 464 

Oscillatory function, 488 

Osfjfood curves, 600 
-Moore theorem, 170 
theorems, etc., 178, 555, 622 

Pantactic, 325 
Path, 612 
Peaks, 179 

infinite, 566 
Poly ant, 153 
Point sets, adherence, 340 

adjoint set of intervals, 337 

apantactic, 325 

associated inner set, 365 
outer set, 365 

Baire sets, 326 

category 1 and 2, 326 

coherence, 340 

conjugate, 51 

connected, 605 

convex, 605 

detached, 604 



Point sets, divisor, 23 

<B., <E = s e ^, 473 

Harnack sets, 354 

hypercomplete, 472 

images, 605, 606 

integrand sets, 385 

iterable, 14 

measurable, 343, 348 

metric, 1 

negative component, 37 

null, 348 

pautactic, 325 

positive component, 37 

potency or power. 278 

projection, 10 

quasi divisor, 390 

reducible, 336 

reticulation, 621 

semidivisor, 390 

separated intervals, 337 

sum, 22 

transfinite derivatives, 330 

union, 27 

well-ordered, 304 
Pointwise discontinuity, 457 
Pompeiu, curves, 542 
Potency or power of a set, 278 
Power series, 89, 144, 187, 191 
Precinct, 612 
Pringsheim, theory of convergence, 113 

theorems, etc., 141, 215, 216, 217, 

220, 273 
Projection, 10 
Products, absolute convergence, 247 

adjoint, 247 

associate simple, 247 

conjugate, 249 

co-product, 242 

iterated, 251 

normal form, 245 

Quasidivisor, 390 

Raabe's test, 107 

Rank of limitary numbers, 331 



INDEX 



643 



Rate of convergence or divergence, 102 
Ratio test, 96 
Reducible sets, 335 
Remainder series, 77 

of Taylor's series, 209, 210 
Rectifiable curves, 583 
Regular points, 428 
Residual function, 561 
Reticulation, 621 
Richardson, 32 
Riemann's function, 459 

theorem, 444 

R- or Riemann integrals, 372 
Rotation, 286 

Scheefer, theorem, 516 
Schoitfliess, theorems, 598, 621 
Schtrarz, theorem, etc., 448, 626 
Section of an aggregate, 307 
Segment, constant, or of invariability, 

521 

Semidivisor, 390 
Separated divisions, 366, 371 

functions, 403 

sets, 366 

of intervals, 337 
Sequence, extremal, 374 

w -tuple, 137 
Series, Abel's, 87 

absolute convergent, 79 

adjoint, 77, 139 

alternate, 83 

associate logarithmic, 243 
normal, 245 
simple, 144 
multiple, 144 

Bessels, 238 

binomial, 110 

cell of convergence, 144 

conjugate, 147 

deleted, 139 

Dini's, 86 

divergence of, 440 

essentially positive or negative, 78 

exponential, 96 



Series, Fourier's, 416 

geometric, 81, 139 

harmonic, 82 
general of exponent p,, 82 

hypergeometric, 112 

interval of convergence, 90 

inverse, 204 

iterated, 149 

logarithmic, 97 

Maclaurin's, 206 

power, 89, 144, 187 

rate of convergence or divergence, 
102 

remainder, 77 

simple convergence, 80 

Taylor's, 206 

tests of convergence, see Tests 

telescopic, 85 

trigonometric, 88 

two-way, 133 
Similar sets, 303 
Similitude, 287 

Simple convergence of series, 80 
Singular points, 26 
Space-filling curves, 588 
Steady convergence, 176 
Submeasurable, 405 
Sum of sets, 22 
Surface, area, 623 

irregular, 632 

regular, 629 

Taylor's series, 206 
Telescopic series, 85 
Tests of convergence, Bertram, 104 

Bonnet, 121 

Cahen, 108 

Cauchy, 98, 99 

d'Alembert, 96 

Gauss, 109 

Rummer, 106, 124 

Pringsheim, 123 

Raabe, 107 

radical, 98 

ratio, 96 



644 



INDEX 



Tests of convergence, tests of 1 and 2 Uniformly limited function, 160, 567 



kind, 120 

Weierstrass, 120 
Theta functions, 135, 184, 256 
Total discontinuity. 457 
Transfinite cardinals, 278 

derivatives, 330 
Translation, 286 
Trigonometric series, 88 
Truncated function, 27 
Two-way series, 133 

Undetermined coefficients, 197 
Unifold image, 606 
Uniform convergence, 156 

at a point, 157 
correspondence, 276 



Union of sets, 22 

Vallee-Poussin (dela), 27, 594 

Van Vleck sets, 361 

Variation, limited or finite, 429, 530 

positive and negative, 430 
Volterra curves, 501, 587 

Wall is formula, 260 

Weierstrass' function, 498, 523, 588 

test, 120 

Well-ordered sets, 304 
Wilson, W. A., vii, 395, 401 

Young, W. H., theorems, 360, 363 
Zeros of power series, 191 



SYMBOLS EMPLOYED IN VOLUME II 



(Numbers refer to pages) 



Frontal. F^, 614 
f*,20 

J* 
ft, i 

U,{ },22 
Dv, 22 

Adj J, 31 

/A,M,31 

SU.0,32. S*/, a , 0,34 

fy, *_., 34 

A H ,A H ,AdjA,77. A n , p , 78 

^ = ^...^,138; 1, = ^...^ 139 

R v - ,..., 139 

31 - 23, 276 ; % & 8, 303 

Card a, 278 

e = ,280; c, 287 

91,, 290 

So, 307 

Ord 31, 311 

w ,311; O, 318 



KpKj-, 318,323 
Z r Z 2 .-.,318 

a<> = 21", 330; a<> = 5l a , 331 

8 = Mea^ , 343 ; g = Meas , 348 

8 = Meas $, 348 

T, /, /', 372, 403, 405 

Sdv, Qdv, 300 
V* 429; Var/= V,, 429 
Osc / = oscillation in a given set. 
Osc/, 454 

Disc/ 454 

xa=a 

e, e.io, 473 

/>),/(*), 488 

/'(*),/'(*), 493 

/', .ff/', /', X/', 7/', Uf, 5(o), 

.R(a), 494 
A(a, )8), 494 



INDEX 



645 



The following symbols are defined in Volume I and are repeated here for 
the convenience of the reader. 



l)ist(a, x) is the distance between 
a and x 

D 6 (), called the domain of the point 
a of norm 8 is the set of points or, 
such that Dist (a, x) < 8 

F (5 (</), called the vicinity of the point 
a of norm 8, refers to some set $, 
and is the set of points in />$() 
which lie in $1 

7)5* (a) 5 TV (a) are the same as the 
above sets, omitting . They are 
called deleted domains, deleted vi- 
cinities 

a n == means a n converges to a 



f(x) = , means /(x) converges to a 

A line of symbols as : 

< 0, w, I - M I < c, n > m 
is of constant occurrence, and is to 
be read : for each c > 0, there exists 
an index /, such that | a a n | < e, 
for every n > m 

Similarly a line of symbols as : 

>0, 8>0, |/(*) -!<,* iii IV() 
is to be read : for each c > 0, there 
exists a 8 > 0, such that 

!/(*) -!<, 

for every x in Fg* (a)