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

PAGES MISSING WITHIN THE BOOK ONLY CO >; 00 162961 ^ 71 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)