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INTRODUCTION
INFINITESIMAL ANALYSIS
FUNCTIONS OF ONE REAL VARIABLE
BY
OSWALD VEBLEN
Preceptor in Mathematics, Frznceton University
AND
N. J. LENNES
Instructor in Malhemaiia m the WejidelL PktUipa High Schoolt Chicago
FIRST EDITION
FIRST THOUSAND
NEW YORK
JOHN WILEY & SONS
London: CHAPMAN & HALL, Limited
1907
Copyrigdt, 1907
BI
OSWALD VEBLEN awd N. J. LENNES
ROBERT DRCHHOHD, PRINTER, NEW TORE
PREFACE.
A COURSE dealing with the fundamental theorems of infini-
tesimal calculus in a rigorous manner is now recognized as an
essential part of the training of a mathematician. It appears
in the curriculum of nearly every university, and is taken by
students as "Advanced Calculus " in their last collegiate year,
or as part of "Theory of Functions " in the first year of graduate
work. This little volume is designed as a convenient reference
book for such courses; the examples which may be considered
necessary being supplied from other sources. The book may
also be used as a basis for a rather short theoretical course ou
real functions, such as is now given from time to time in some
of our universities.
The general aim has been to obtain rigor of logic with a
minimum of elaborate machinery. It is hoped that the system-
atic use of the Heine-Borel theorem has helped materially
toward this end, since by means of this theorem it is possible
to avoid almost entirely the sequential division or " pinching "
process so common in discussions of this kind. The definition
of a Umit by means of the notion "value approached" has
simplified the proofs of theorems, such as those giving necessary
and sufficient conditions for the existence of _ limits, and ia
general has largely decreased the number of e's and d's. The
theory of limits is developed for multiple-valued functions,
which gives certain advantages in the treatment of the definite
integral.
In each chapter the more abstract subjects ?ind those which
can be omitted on a first reading are placed in the concluding
IV PREFACE.
sections. The last chapter of the book is more advanced in
character than the other chapters and is intended as an intro-
duction to the study of a special subject. The index at the
end of the book contains references to the pages where technical
terms are first defined.
When this work was undertaken there was no convenient
source in English containing a rigorous and systematic treat-
ment of the body of theorems usually included in even an ele-
mentary course on real functions, and it was necessary to refer
to the French and German treatises. Since then one treatise,
at least, has appeared in English on the Theory of Functions
of Real Variables. Nevertheless it is hoped that the present
volume, on account of its conciseness, will supply a real want.
The authors are much indebted to Professor E. H. Moore
of the University of Chicago for many helpful criticisms and
suggestions; to Mr. E. B. Morrow of Princeton University for
reading the manuscript and helping prepare the cuts; and to
Professor G. A. Bliss of Princeton, who has suggested several
desirable changes while reading the proof-sheets.
CONTENTS.
CHAPTER I.
The System of Real Numbers.
PAoa
{ 1. Rational and Irrational Numbers 1
J 2. Axiom of Continuity 3
§ 3. Addition and Multiplication of Irrationals 7
J 4. General Remarks on the Number System 11
§ 5. Axioms for the Real Number System 13
§ 6. The Number e 15
§ 7. Algebraic and Transcendental Numbers 18
§ 8. The Transcendence of e 19
§ 9. The Transcendence of ;: 25
CHAPTER II.
Sets of Points and of Segments.
§ 1. Correspondence of Numbers and Points 30
§ 2. Segments and Intervals. Theorem of Borel 32
§ 3. Limit Points. Theorem of Weierstrass ; 38
§ 4. A Second Proof of Theorem 15 42
CHAPTER III.
PoNcnoNS IN General. Spbclal Cases of Functions.
§ 1. Definition of a Function 44
5 2. Bounded Functions 47
{ 3. Monotonic Functions. Inverse Functions 49
{ 4. Rational, Exponential, and Logarithmic Functions 63
CHAPTER IV.
Theory of Limits.
§ 1. Definitions. Limits of Monotonic Functions 60
§ 2. The Existence of Limits 65
vi CONTENTS.
PAGS
§ 3. Application to Infinite Series 70
§ 4. Infinitesimals. Computation of Limits 74
§ 5. Further Theorems on Limits 81
§ 6. Bounds of Indetermination. Oscillation .,.,,., 83
CHAPTER V.
CoNTiNuoDs Functions.
§ 1. Continuity at a Point 87
§ 2. Continuity of a Function on an Interval 88
J 3. Functions Continuous on an Everywhere Dense Set 94
§ 4. The Exponential Function 97
CHAPTER VI.
Infinitesimals and Infinites.
§ 1. The Order of a Function at a Point 101
§ 2. The Limit of a Quotient 105
§ 3. Indeterminate Forms 108
§ 4. Rank of Infinitesimals and Infinites 114
CHAPTER VII.
Derivatives and Differentials.
{ 1. Definition and Illustration of Derivatives 117
§ 2. Formulas of Differentiation itq
§ 3. Differential Notations 128
§ 4. Mean-value Theorems joq
% 5. Taylor's Series , , .
I 6. Indeterminate Forms , og
J 7. General Theorems on Derivatives 144
CHAPTER VIII.
Definite Integrals.
{ 1. Definition of the Definite Integral 151
§ 2. Integrability of Functions , c^
5 3. Computation of Definite Integrals . cq
§ 4. Elementary Properties of Definite Integrab IfM
{ 5. The Definite Integral as a Function of the Limits of Integration 171
§ 6. Integration by Parts and by Substitution j^ •
§ 7. General Conditions for Integrability ,-_
CONTENTS. vii
CHAPTER IX.
Improper Definite Integrals.
PAGE
5 1. The Improper Definite Integral on a Finite Interval 191
§ 2. The Definite Integral on an Infinite Interval 201
§ 3. Properties of the Simple Improper Definite Integral 205
§ 4. A More General Improper Definite Integral 210
§ 5. Special Theorems on the Criteria of the Existence of the Improper
Definite Integral on a Finite Interval 218
} 6. Special Theorems on the Criteria of the Existence of the Improper
Definite Integral on an Infinite Interval 223
INFINITESIMAL ANALYSIS.
CHAPTER I.
THE SYSTEM OF REAL NUMBERS.
§ 1. Rational and Irrational Numbers.
- The real number system may be classified as follows:
(1) All integral numbers, both positive and negative, in-
cluding zero.
(2) All numbers — , where m and n are integers (n^^O).
(3) Numbers not included in either of the above classes,
such as V2 and w.f
Numbers of classes (1) and (2) are called rational or com-
mensurable numbers, while the numbers of class (3) are called
irrational or incommensurable numbers.
As an illustration of an irrational number consider the
square root of 2. One ordinarily says that \/2 is 1.4 +, or
tit i8 clear that there is no number — such that — 7=2, for if — r = 2.
n n' ' n' '
then m'=2n', where m' and 2n' are integral numbers, and 2n' is the square
of the integral number m. Since in the square of an integral number every
prime factor occurs an even number of times, the factor 2 must occur an
even number of times both in n' and 2n', which is impossible because of
the theorem that an integral number has only one set of prime factors.
2 INFINITESIMAL ANALYSIS.
1.41 + , or 1.414 + , etc. The exact meaning of these statements
is expressed by the following inequalities: t
(1.4)2 <2< (1.5)2,
(1.41)2 <2< (1.42)2,
(1.414)2 <2< (1.415)2,
etc.
Moreover, by the foot-note above no terminating decimal is
equal to the square root of 2. Hence Horner's Method, or
the usual algorithm for extracting the square root, leads to an
infinite sequence of rational numbers which may be denoted
by Oi, a2, as, . . . , a„, . . . (where ai=1.4, 02 = 1.41, etc.), and
which has the property that for every positive integral value
of n
a„<o„+i, a„2<2<
("- + 1^)'
Suppose, now, that there is a least number o greater than
every o„. We easily see that if the ordinary laws of arith-
metic as to equality and inequality and addition, subtraction,
and multiplication hold for a and a^, then a^ is the rational
nimiber 2. For if a2<2, let 2-o2 = e, whence 2=a? + e. II n
were so taken that jjc<-r, we should have from the last in-
2 £ e
<0„2 + 4g + g<02+£.
equality {
so that we should have both 2=a^ + e and 2<a2 + e. On the
t a < 6 signifies that a is less than b. a>b signifies that a is greater than 6.
t This involves the assumption that for eveiy nvimber, e, however small
there is a positive integrer n such that ttt-K t- This is of course obvious
10** o
when e is a rational number. If c is an irrational number, however, the
statement will have a definite meaning only after the irrational number
has been fully defined.
THE SYSTEM OF REAL NUMBERS. 3
other hand, if o^ > 2, let a^ - 2 = e' or 2 + e' =a. Taking n such
that r-r- < — , we should have
lU" 5
(a« + J^)' < (fln^) + e' < 2 + e' < a;
and since Cn + iT^ is greater than a* for all values of k, this
would contradict the hypothesis that a is the least number
greater than every number of the sequence a\, 02, 03, . . . We
also see without difficulty that o is the only nimiber such that
«2 = 2.
§ 2. Axiom of Continuity.
The essential step in passing from ordinary raticwial num-
bers to the number corresponding to the sjonbol \/2 is thus
made to depend upon an assumption of the existence of a
number o bearing the unique relation just described to the
sequence a^, a2, a„, . . . In order to state this hypothesis in
general form we introduce the following definitions:
Definition. — ^The notation [x] denotes a set,'f any element
of which is denoted by x alone, with or without an index or
subscript.
A set of numbers [x] is said to have an upjxr bound,
M, if there exists a number M such that there is no number
of the set greater than M. This may be denoted by M^[x].
A set of numbers [x] is said to have a lower bound, m, if
there exists a number m such that no number of the set is
less than m. This we denote by m^[x}.
Following are examples of sets of numbers:
(1) 1, 2, 3.
(2) 2, 4, 6, . . . , 2i, . . .
(3) 1/2, 1/22, 1/23, . . . , i/2», . . .
(4) All rational numbers less than 1.
(5) All rational numbers whose squares are less than 2.
t Synonyms of set are class, aggregate, collection, assemblage, etc.
4 INFINITESIMAL ANALYSIS.
Of the first set 1, or any smaller number, is a lower bound;,
and 3, or any larger number, is an upper bound. The second
set has no upper bound, but 2, or any smaller nimiber, is a
lower bound. The nimiber 3 is the least upper boimd of the
first set, that is, the smallest niraiber which is an upper bound.
The least upper and the greatest lower bounds of a set of num-
bers [x] are called by some writers the upper and lower limits
respectively. We shall denote them by B[x] and 5[x] respect-
ively. By what precedes, the set (5) would have no least upper
bound xmless V2 were coimted as a number.
We now state our hjrpothesis of continuity in the following
form:
Axiom K. If a set [r] of rational numbers having an upper
bound has no rational least upper bound, then there exists one and
only one number B[r] such that
(a) B[r] > /, where / is any number of [r] or any rational rmmr-
ber less than some number of [r].
(6) B[r]<r", where r" is anyrational upper bound of [rj.f
Definition. — ^The number ^r\ of axiom K is called the
least upper bound of [r], and as it cannot be a rational number
it is called an irrational nimiber. The set of aU rational and
irrational numbers so defined is called the cordinuxms real num-
ber system. It is also called the linear continuum. The set of
all real numbers between any two real numbers is likewise
called a linear continuum.
Theorem i. // two sets of rational numbers [r] and [s], having
upper bounds, are such that no r is greater than every s and no s
greater than every r, then B[r] and B[s] are the same; that is, in
symbols.
Proof. — ^If B[r] is rational, it is evident, and if ^r] is irra-
tional, it is a consequence of Axiom K that
B[r]>s',
t This axiom implies that the new (irrational) numbers have relations
of order with all the rational numbers, but does not explicitly state rela-
tions of order among the irrational numbers themselves. Cf. Theorem 2.
THE SYSTEM OF REAL NUMBERS. 5
where s' is any rational number not an upper bound of [s].
Moreover, if s" is rational and greater than every s, it is greater
than every r. Hence
where s" is any rational upper bound of [s]. Then, by the
definition of B[s],
B[r]=m
Definition. — If a number x (in particular an irrational
■number) is the least upper bound of a set of rational numbers
[r], then the set [r] is said to determine the number x.
Corollary 1. The irrational numbers i and i' determined by
the two sets [r] and [/] are equal if and only if there is no num-
ber in either set greater than every number in the other set.
Corollary 2. Every irrational number is determined by some
set of rational numbers.
Definition. — If i and i' are two irrational numbers deter-
mined respectively by sets of rational numbers [r] and [/]
and if some number of [r] is greater than every niunber of [/],
then
i>i' and i'<i.
From these definitions and the order relations among the
rational numbers we prove the following theorem:
Theorem 2. If a and b are any two distinct real numbers, then
a<b orb<a; if a<b, then not b<a; if a<b and b<c,then a<c.
Proof.— Let o, 6, c all be irrational and let [x], [y], [2] be sets
of rational numbers determining a, b, c. In the two sets [x]
and [y] there is either a number in one set greater than every
number of the other or there is not. If there is no number in
either set greater than every number in the other, then, by
Theorem 1, a=b. If there is a number in [x] greater than
every number in [y], then no number in [y] is greater than
•every number in [x]. Hence the first part of the theorem is
6 INFINITESIMAL ANALYSIS.
proved, that is, either a=b or a<b orb <a, and if one of these^.
then neither of the other two. If a number yi of [y] is greater
than every number of [x], and a number zi of [z] is greater
than every number of [y], then zi is greater than every num-
ber of [x]. Therefore if o<6 and b<c, then a<c.
We leave to the reader the proof in case one or two of the
numbers a, b, and c are rational.
Lemma. — If [r] is a set of rational numbers determining an
irrational number, then there is no number ri of the set [r]
which is greater than every other number of the set.
This is an immediate consequence of axiom K.
Theorem 3. If a and b are any two distinct numbers, then
there exists a rational number c such that a<c and c<b, or b<e
and c<a.
Proof. — Suppose a<b. When a and b are both rational
—^ is a nimiber of the required type. If a is rational and b
irrational, then the theorem follows from the lemma and Corol-
lary 2, page 5. If a and b are both irrational, it follows from
Corollary 1, page 5. 11 p, is irrational and b rational, then
there are rational numbers less than b and greater than every
number of the set [x] which determines a, since otherwise b
would be the smallest rational nimiber which is an upper bound
of [x], whereas by definition there is no least upper bound
of [x] in the set of rational numbers.
Corollary. A rational number r is the least upper bound of
the set of all numbers which are less than r, as well as of the
set of all rational numbers less than r.
Theorem 4. Every set of numbers [x] which has an upper
bound, has a least upper bound.
Proof. — Let [r] be the set of all rational mmabers such that
no number of the set [r] is greater than every number of the
set [x]. Then B[r] is an upper boimd of [x], since if there were
a number Xi of [x] greater than B[r], then, by Theorem 3, there
would be a rational nmnber less than Xi and greater than
B[r], which would be contrary to the definition of [r] and B[r].
THE SYSTEM OF REAL NUMBERS. 7
Further, B[r]js the least upper bound of [x], smce if a number
N less than B[r] were an upper bound of [x], then by Theorem 3
there would be rational numbers greater than N and less than
B[r], which again is contrary to the definition of [r].
Theorem 5. Every set [x] of numbers which has a lower
bound has a greatest lower bound.
Proof. — ^The proof may be made by considering the least
upper bound of the set [y] of all numbers, such that every num-
ber of [y] is less than every niunber of [x]. The details are
left to the reader.
Theorem 6. If all numbers are divided into two sets [x] and
[y] such that x<y for every x and y of [x] and [y], then there is
a greatest x or a least y, but not both.
Proof. — The proof is left to the reader.
The proofs of the above theorems are very simple, but ex-
perience has shown that not only the beginner in this kind of
reasoning but even the expert mathematician is likely to make
mistakes. The beginner is advised to write out for himself
every detail which is omitted from the text.
Theorem 4 is a form of the continuity axiom due to Weier-
strass, and 6 is the so-called Dedekind Cut Axiom. Each of
Theorems 4, 5, and 6 expresses the continuity of the real num-
ber system.
§ 3. Addition and Multiplication of Irrationals.
It now remains to show how to perform the operations of
addition, subtraction, multipUcation, and division on these
numbers. A definition of addition of irrational numbers is sug-
gested by the following theorem: "If a and b are rational num-
bers and [x] is the set of all rational niunbers less than a, and
[y] the set of all rational numbers less than b, then [x+y] is
the set of all rational nvmibers less than a+b." The proof of
this theorem is left to the reader.
Definition. — ^If a and b are not both rational and [x] is the
set of aU rationals less than a and [y] the set of all rationals less
8 INFINITESIMAL ANALYSIS.
than b, then a+b is the least upper bound of [x + y], and is
called the sum of o and b.
It is clear that if b is rational, [x+b] is the same set as [x+y];
for a given x+b is equal to x' + (6-(x'-x))=a;'+2/', where x'
is any rational number such that x<x'<a; and conversely,
any x+y is equal to (i - 6 + j/) + 6 = i" + 6. It is also clear that
a+b=b+a, since [x+y] is the same set as [y+x]. Likewise
{a+b) +c=a + {b+c), since [{x+y) +z] is the same as [x + {y+z)].
Furthermore, in case b<a, c=B[x'—y'], where a<x' <b and
a<y'<b, is such that b + c=a, and in case b<a, c=^x' —y'] is
such that b + c==a; c is denoted by o— & and called the differ-
ence between a and b. The negative of a, or — o, is simply 0— a.
We leave the reader to verify that if a>0, then a+b>b, and
that if o<0, then a+b<b for irrational numbers as well as
for rational*'.
The theorems just proved justify the usual method of add-
ing infinite decimals. For example : ;: is the least upper bound
of decimals hke 3.1415, 3.14159, etc. Therefore n:4-2 is the
least upper bound of such nimibers as 5.1415, 5.14159, etc.
Also e is the least upper bound of 2.7182818, etc. Therefore
r+e is the least upper bound of 5, 5.8, 5.85, 5.859, etc.
The definition of multiphcation is suggested by the follow-
ing theorem, the proof of which is also left to the reader.
Let a and b be rational numbers not zero and let [x] be the
set of all rational numbers between and a, and [y] be the set
of aU rationals between and b. Then if
a > 0, 6 > 0, it follows that ab = B[xy] ;
a<0, b<0, " " " ab = B[xy];
a<0, 5>0, " " " a6 = 5[a;j/];
a>0, 6<0, " " " ab = B[.xyl
Definition. — If a and 6 are not both rational and [a;] is the
set of all rational numbers between and o, and [y] the set of
all rationals between and 6,_then if a>0, b>0, ab means
B[xy]; if a<0, 6<0, ab means ^xy]■, if o<0, 6>0, ab means
B[xy]; if a>0, 6<0, ab means B^xj/]. If a or 6 is zero, then
ab=0.
THE SYSTEM OF REAL NUMBERS. 9
It is proved, just as in the case of addition, that ab=ha,
that a{bc) = idb)c, that if a is rational [ay] is the same set as
[xy], that if a>0, b>0, ab>0. Likewise the quotient -r- is
defined as a number c such that ac=b, and it is proved that
in case a>0, 6>0, then c=b\ —/\, where [y'] is the set of all
rationals greater than b. Similarly for the other cases. More-
over, the same sort of reasoning as before justifies the usual
method of multiplying non-terminated decimals.
To complete the rules of operation we have to prove what
is known as the distributive law, namely, that
o(6+c)=a6+ac.
To prove this we consider several cases according as a, b, and
c are positive or negative. We shall give in detail only the
case where all the numbers are positive, leaving the other cases
to be proved by the reader. In the first place we easily see
that for positive numbers e and /, if [t] is the set of all the
rationals between and e, and [T] the set of all rationals less
than e, while [u] and [U] are the corresponding sets for /, then
e+f=B[T+U]=B[t+u].
Hence if [x] is the set of all rationals between and a, [y] be-
tween and b, [z] between and c,
b + c=B[y+z] and hence a{b+c) = B[x{y+z)].
On the other hand ab=B[xy], (ic=^^xz], and therefore db+ac=
^{xy+xz}]. But since the distributive law is true for rationals,
x(y+z)=xy+xz. Hence E[x{y+z)]=B[ixy+xz)] and hence
aib + c)'=ab+ac.
We have now proved that the system of rational, and irra-
tional nimabers is not only continuous, but also is such that we
may perform with these nimibers all the operations of arith-
metic. We have indicated the method, and the reader may
10 INFINITESIMAL ANALYSIS.
prove in detail that every rational number may be represented
by a terminated decimal,
aaO*+Oifc_ilO*-i + . . .+ao+~-+ •■■ +^
= afcfflt_i . . . aoffi_ia_2 . . . ffl_n,
or by a circulating decimal,
O'kP'k-l • • • ffloffl-lffl-2 ■ . ■ d-i . . . d- id-i ■ . ■ a_j . . . ,
where i and j are any positive integers such that i<]'; whereas
every irrational nimiber may be represented by a non-repeating
infinite decimal,
aicO-ic-i ■ ■ ■ aoa_ia_2 . . . «_„ . . .
The operations of raising to a power or extracting a root on
irrational numbers wiU be considered in a later chapter (see
page 53). An example of elementary reasoning with the sym-
bol jB[x] is to be found on pages 17 and 18. For the present
we need only that x", where n is an integer, means the number
obtained by multipl3dng x by itself n times.
It should be observed that the essential parts of the defini-
tions and arguments of this section are based on the assumption
of continuity which was made at the outset. A clear under-
standing of the irrational number and its relations to the
rational number was first reached during the latter half of the
last century, and then only after protracted study and much
discussion. We have sketched only in brief outline the usual
treatment, since it is beUeved that the importance and diffi-
culty of a full discussion of such subjects wUl appear more
clearly after reading the following chapters.
Among the good discussions of the irrational number in the
English language are : H. P. Manning, Irrational Numbers and
their Representation hy Sequences and Series, Wiley & Sons, New
York; H. B. Fine, College Algebra, Part I, Ginn & Co., Boston-
THE SYSTEM OF REAL NUMBERS. 11
Dedekind, Essays on the Theory of Number (translated from
the German), Open Court Pub. Co., Chicago; J. Piehpont,
Theory of Functions of Real Variables, Chapters I and II, Ginn
& Co., Boston.
§ 4. General Remarks on the Number System.
Various modes of treatment of the problem of the number
system as a whole are possible. Perhaps the most elegant is
the following : Assume the existence and defining properties of
the positive integers by means of a set of postulates or axioms.
From these postulates it is not possible to argue that if p and
V
q are prime there exists a number a such that ap=q or a = -,
i.e., in the field of positive integers the operation of division is
not always possible. The set of all pairs of integers jm, n),
if \mk, nk\ {k being an integer) is regarded as the same as
{Tn,n\, form an example of a set of objects which can be added,
subtracted, and multiplied according to the laws holding for
positive integers, provided addition, subtraction, and multipli-
cation are defined by the equations,t
{m,n\®{p,q\ = {mp,Tiq}
{m, n!01p, 5! = {mq+np, nq].
The operations with the subset of pairs {m,l} are exactly the
same as the operations with the integers.
This example shows that no contradiction will be introduced
by adding a further axiom to the effect that besides the integers
there are numbers, called fractions, such that in the extended
system division is possible. Such an axiom is added and the
order relations among the fractions are defined as follows:
p m ..
-<— if pn<qm.
an
t The details needed to show that these integer pairs satisfy the alge-
braic laws of operation are to be found in Chapter 1, pages 5-12, of Pier-
pont's Theory of Real Functions. Pierpont's exposition differs from that
indicated above, in that he says that the integer pairs actuaUy are the frac-
tions.
12 INFINITESIMAL ANALYSIS.
By an analogous example t the possibility of negative num-
bers is shown and an axiom assuming their existence is justi-
fied. This completes the rational nimiber system and brings
the discussion to the point where this book begins.
Our Axiom K, which completes the real number system,
assuming that every bounded set has a least upper bound,
should, as in the previous cases, be accompanied by an exam-
ple to show that no tontradiction with previous axioms is intro-
duced by Axiom K. Such an example is the set of all lower
segments, a lower segment, S, being defined as any Jsounded
set of rational numbers such that if a; is a number of S, every
rational number less than x is in S. For instance, the set of
all rational nimibers less than a rational number a is a lower
segment. Of two lower segments one is always a subset of the
other. We may denote that aS is a subset of S' by the symbol
S©S\
According to the order relation, @, every bounded set of
lower segments [/S] has a least upper bound, namely the lower
segment, consisting of every number in any S of [S\. If S and
T are lower segments whose least upper bounds are s and t, we
may define
and S(g)r
as those lower segments whose least upper bounds are s + t and
sXt respectively. It is now easy to see that the set of lower
segments contains a subset that satisfies the same conditiohs
as the rational numbers, and that the set as a whole satisfies
axiom K. The legitimacy of axiom K from the logical point
of view is thus established, since our example shows that it
cannot contradict any previous theorem of arithmetic.
Further axioms might now be added, if desired, to postulate
the existence of imaginary numbers, e.g. of a number x for
t Cf. PiERPONT, loc. cit., pages 12-19.
THE SYSTEM OF REAL NUMBERS. 13
each triad of real numbers a, b, c, such that oT^+bx+c^O.
These axioms are to be justified by an example to show that
they are not in contradiction with previous assumptions. The
theory of the complex variable is, however, beyond the scope of
this book.
§ 5- Axioms for the Real Number System.
A somewhat more summary way of deahng with the prob-
lem is to set down at the outset a set of postulates for the
system of real numbers as a whole without distinguishing
directly between the rational and the irrational number. Sev-
eral sets of postulates of this kind have been published by E. V.
Huntington m the 3d, 4th, and 5th volumes of the Transac-
tions of the American Mathematical Society. The following
set is due to HuNTiNGTON.f
The system of real numbers is a set of elements related to
ope another by the rules of addition (-I-), multiplication (x),
and magnitude or order (<) specified below.
Al. Every two elements a and 6 determine uniquely aa
element a+b called their sum.
A 2. {a+b)+c=a + ib + c).
A3. (a+6) = (6+a).
A 4. li a+x=a+y, then x = y.
A 5. There is an element z, such that z+z=z. (This ele-
ment z proves to be unique, and is called 0.)
A 6. For every element a there is an element a', such that
a+a'=0.
M 1. Every two elements a and b determine uniquely an
element ab called their product; and if a?^0 and 6?^0, then
M2. iab)c=a{bc).
M 3. ab = ba.
M4. If ax = ay, and a^^O, then x=y.
t Bulletin of the American Mathematical Society, Vol. XII, page 228.
, J The latter part of M 1 may be omitted from the list of axioms, since
it can be proved as a theorem from A 4 and AMI.
14 INFINITESIMAL ANALYSIS.
M 5. There is an element u, different from 0, such that
wu = w. This element proves to be uniquely determined, and
is called 1.
M 6. For every element a, not 0, there is an element a",
such that aa" = l.
AMI. a{h+c)=ab + ac.
1. If a^b, then either a<b or b<a.
2. If a<6, then a5^6.
3. If a<b and b<c, then a<c.
O 4. (Continuity.) If [x] is any set of elements such that for
B certain element b and every x,x<b, then there exists. an ele-
ment B such that —
(1) For every x of [x], x<B;
(2) li y<B, then there is an zi of a; such that y<xi.
A 1. If x<y, then a+x<a + y.
M 1. If a>0 and b>0, then ab>0.
These postulates may be regarded as summarizing the prop-
erties of the real number system. Every theorem of real
analysis is a logical consequence of them. For convenience of
reference later on we summarize also the rules of operation
with the symbol |a;|, which indicates the "numerical " or "abso-
lute" value of x. That is, if a; is positive, la;|=a;, and if x is
negative, |x|= —x.
\x\ + \ymx + y\ (1)
.-. I\xk\^\lxk\, (2)
k-l k=l
n
where Ixk=xi+X2+ . . .+x„.
|N-M|^k-2/| = l2/-a;|^W + |j/| (3)
i2;-2/| = N-l2/l (4)
(5)
\X\ X
\y\ ~ y
If lx-t/|<ei, |2/-zl<e2, then |x-2|<ci+e2 (6)
THE SYSTEM OF REAL NUMBERS. 15
If [x] is any bounded set,
5[x]-5[x]=F[|xi-X2|] (7)
§ 6. The Number e.
In the theory of the exponential and logarithmic functions
(see page 97) the irrational number e plays an important r61e.
This number may be defined as follows :
e=B[Er,], (1)
where "^ T\'^2\'^ ' ' '"^n!'
where [n] is the set of all positive integers, and
n! = l-2-3...n.
It is obvious that (1) defines a finite number and not infin-
ity, since
J?„ = l+jj+2j+ . . •+^<l+l + 2 + 2? + - • • + 2^ = ^~2^-
The nimiber e may very easily be computed to any number of
decimal places, as follows:
^^0 =
= 1
1
1!"
= 1
1
2!"
= .5
1
3!"
= .166666 +
16
INFINITESIMAL ANALYSIS.
1
4!~
.041666 +
1
5!~
.008333 +
1
6!~
.001388+
1
7!~
.000198+
1
8!~
.000024+
1
9!"
.000002 +
E,=
2.7182
1
Lemma. — ^If k>e, then Ek>e--,y
Proof .—From the definitions of e and f?„ it follows that
'"■^*='^[(ATi)i+(Fr2)i+ •• •+(&+])!]'
where [Z] is the set of all positive integers. Hence
'^ik+2)...ik+t)]'
or e-E,<-g^^^'e.
If A>e, this gives
Ek>e-^^.
THE SYSTEM OF REAL NUMBERS. 17
Theorem 7. e=fil (iH — ) J,
where [n] is the set of all positive integers.
Proof. — By the binomial theorem for positive integers
Hence E„-(l+-) = ^ (^ '-j^, )
« n*-n(n-l) . .. (n-A + 1) ,,
k-2 kin''
»-, 7i*-(n-^ + l)fc
•^ Tim •
^iZi kin"
Hence by factoring
_ / 1\" ^(A;-l)(n*-^+n*-'(n-A;+l)
1^ "^ ^(_A; - 1) (n*-i +n*-g(n-A;+l) + ... + (n-A; + l)*-^)
"(A;-l);kn*-»
£-2
kin"
1 ^ (A;-1)A;
<n ifz k\ '
^-('-y<-n('^a)<K• ••••<')
From (a) ^„> (1+-)" d)
and from (6) r+^ '^^""n' ^^^
18 INFINITESIMAL ANALYSIS.
whence by the lemma
(l + l)">e-V- (3)
\ nl n\ n
From (1) it follows that e is an upper bound of
1\"-
[(^-ri.
and from (3) it follows that no smaller number can be an upper
bound. Hence
(('-y]-'
§ 7. Algebraic and Transcendental Numbers.
The distinction between rational and irrational nimibers,
which is a feature of the discussion above, is related to that
between algebraic and transcendental numbers. A number is
algebraic if it may be the root of an algebraic equation,
aox"+0ix"-i + . . .+o„_ia;+a„=0,
where n and ao, ai, . . . , a„ are integers and w> 0. A number is
transcendental if not algebraic. Thus every rational number —
n
is algebraic because it is the root of the equation
nx—m=0,
while every transcendental number is irrational. Examples of
transcendental numbers are, e, the base of the system of natural
logarithms, and ;:, the ratio of the circumference of a circle to
its diameter.
The proof that these numbers are transcendental follows
on page 19, though it makes use of infinite series which will
THE SYSTEM OF REAL NUMBERS. 19
not be defined before page 71, and the function e', which is
defined on page 57.
The existence of transcendental numbers was first proved
by J. LiouviLLE, Comptes Rendus, 1844. There are in fact
an infinitude of transcendental numbers between any two num-
bers. Cf. H. Weber, Algebra, Vol. 2, p. 822. No particular
number was proved transcendental till, in 1873, C. Hermite
(Crelle's Journal, Vol. 76, p. 303) proved e to be transcendental.
In 1882 E. LiNDEMANN (Mathematische Annalen, Vol. 20, p. 213)
showed that k is also transcendental.
The latter result has perhaps its most interesting application
in geometry, since it shows the impossibihty of solving the
classical problem of constructing a square equal in area to a
given circle by means of the ruler and compass. This is because
any construction by ruler and compass corresponds, according
to analytic geometry, to the solution of a special type of alge-
braic equation. On this subject, see F. Klein, Famous Prob-
lems of Elementary Geometry (Ginn & Co., Boston), and Weber
and Wellstein, Encyclop'ddie der Ekmmtarrmiherruitik, Vol. 1,
pp. 418-432 (B. G. Teubner, Leipzig).
§ 8. The Transcendence of e.
Theorem 8. // c, Ci, Cz, C3, . . . , c„ are integers (or zero bvi
Cj^O), then
c+c.c+c,e2+ ..,+c„c»?^0 (1)
Proof. — ^The scheme of proof is to find a number such that
when it is multipUed into (1) the product becomes equal to
a whole number distinct from zero plus a number between +1
and —1, a sum which surely cannot be zero. To find this
number N, we study the series t for e*, where A; is an integer <n:
t Cf. pages 71 and 99.
20 INFINITESIMAL ANALYSIS.
Multiplying this series successively by the arbitrary factors
i\-bi, we obtain the following equations:
/ k k^ \
c*l!-6i =61 -ll+ftiAU + 2+273 + -. -j;
e*-2!-&2=&2-2!(l+|)+62-A;2(l+|+^+...);
(k Jfc2\ / k k^ \
e*-s!-6.=6,-s!(l + j-,+2| + . . ^+-^^1)]}
ft2
^+.-Ti+ (.+i)(.+2) +---)
(s+l)(s+2)
For the sake of convenience in notation the numbers
61 ... 6, may be regarded as the coefficients of an arbitrary
polynomial
<f>{x)=bo+bix+b23^+. . .+6a,
the successive derivatives of which are
<l>'(x)=bi+2-b2X+. . .+s-b,-x'-\
,i^\, XT . T (m+1)! , s!
^<">(a;)=b„.-m!+6m+i- ,, ■x+...+b.-7- r-.-a*-"
The diagonal in (2) from 6-1! to b,-s \. _^.^ is obviously
^'(A), the next lower diagonal is ^"(A), etc. Therefore by
adding equations (2) in this notation we obtain
THE SYSTEM OF REAL NUMBERS. 21
eHV.hi+2\b2 + . . .+s\b,) = <}>'ik) + <j>"ik) + . . .
+ <l><'Kk) + Jb„,-k'--Rkm, (3)
m-l
k ^
m + l'^(m + l)(m+2)
in which ftjfcm = l + zrTT + 7r-rTT7r— rirr+. • .
Remembering that (f>(x) is perfectly arbitrary, we note tnat
if it were so chosen that
<f>'(.k)=0, <f>"ik)=0,..., .^(''-i)(A:)=0,
for every A; (k = l, 2, 3, . . . , n) then equations (2) and (3)
could be written in the form
m-l
+bp-p\
+6p+i-(p+l)!(l+^)
+6..«!(l+i|+2-, + ... + (^3^). (4)
A choice of (j>{x) satisfying the required conditions is
^{x) = (ao+aiX+as^ + ...+an3^y-j^:Ziy^= (p^iyT"' ^^^
where f{x) = {x-l){x-2)(x-S) . . . (x-n).
22 INFINITESIMAL ANALYSIS
Every jfc (*=!, 2, . . . , n) is a p-tuple root of (5). Here p
is still perfectly arbitrary, but the degree s of ^(x) is np + p-1.
If 4)(x) is expanded and the result compared with
<j){x)=bo+biX + . . .+b,x',
it is plain that
bo=0, bi=0, ..., &p_2=0,
on account of the factor x^~^, and
"-^ (p-1)!' " (p-1)!' ■*■' (p-1)!'
where 7p, Ip+i, ...,/, are all integers. The coefficient of e*=
in the left-hand member of (4) is therefore
Whenever the arbitrary number p is prime and greater than
Co, Np is the sum of Oo", which cannot contain p as a factor,
plus other integers each of which does contain the factor p.
Np is therefore not zero and not divisible by p.
Further, since
(p + Q! (p + l)(p+2)...(p+0
(p-l)!-r! P r!
is an integer divisible by p when r^t, it follows that all the
coefficients of the last block of terms in (4) contain p as a
factor. Since k is also an integer, (4) evidently reduces to
Np-e''=^pWkp + Ib^-k'--Rkm,
THE SYSTEM OF REAL NUMBERS.
23
where W^p is an integer or zero, and this may be abbreviated
to the form
Np-e>'=pWkp+rkp.
(6)
Before completing our proof we need to show that by choosing
the arbitrary prime number p sufficiently large, rtp can be
made as small as we please. If a is a number greater than n,
\R
km]
1 +
-+;
F
<
m+1 (m + l)(?ra + 2)
2
1 + -T +
W
m+1 (m + l)(m + 2)
+ .
+ .
<e«
for all integral values of m and oik<n.
\rkp\
Ibm-k^-Rkm
m=l
1 I \h„\-k'^-\Rk,m\.
1=1
Since the number bm is the coefficient of x" in <f>{x) and
since each coefficient of <j>ix) is numerically less than or equal
to the correspanding coefficient of
XP
-1
(p-1)!
it follows that
(|ao| + |ai|a; + |a2|x2 + . . . + |a„|z»)i',
!'-*p|<e''-(^ri)!(N + l«il«+---+W«")''
<(P-1)!''
24 INFINITESIMAL ANALYSIS.
where
Q=a(|oo| + |ci|a + . . . + |a„|a:«)
Qp /
is a constant not dependent on p. The expression , _^, ^ is
the pth term of the series for Qe^, and therefore by choosing p
sufficiently large Vkp, may be made as small as we please.
If now p is chosen as a prime number, greater than a and
ao and so great that for every k,
where d is the greatest of the numbers
C, C\, C2, Cz, . . . , Cn,
the equations (6) evidently give
iV p(c + Cie + C2e= + . . . + c„e")
= NpC + p{CiWij, + C2W2p + . . .+cJV„j>)
+ Cirip+C2r2p + . . . + c„r„p,
=NpC+pW + R, (8)
where W is an integer or zero and R is numerically less than
unity. Since NpC is not divisible by p and is not zero, while
pW is divisible by p, this sum is numerically greater than or
€qual to zero. Hence
Np(c+Cie+C2^ + . . .+c„e»)?^0.
Hence
and e is a transcendental number.
THE SYSTEM OF REAL NUMBERS. 25
§ 9. The Transcendence of n.
The definition of the number r. is derived from Euler's
formula
e^"^" ^ = cos x+\/ — 1 sin x;
by replacing x by n,
e
rN/^=-l (1)
If TT is assumed to be an algebraic number, rV - 1 is also
an algebraic number and is the root of an irreducible algebraic
equation F{x)=0 whose coefficients are integers. If the roots
of this equation are denoted by 21, 22, 23, ■ • • , Zn, then, since
^v — 1 is one of the z's, it follows as a consequence of (1) that
(e^' + l)(e'= + l)(e^»+l) ...(e^" + l)=0. ... (2)
By expanding (2)
l + Ie'i + Ie'i+'i + Ie''+'i+h + . . .=0.
Among the exponents zero may occur a number of times
e.g., (c - 1) t-mes. If then
Zi, Zi + Zj, Z, + Zj + Zk, ...,
be designated by Xi, X2, X3, ■ ■ . , x„, the equation becomes
c + e^'+e^=+.. . + e'"=0, (3)
where c is a positive number at least unity and the numbers
Xi are algebraic. These numbers, by an argument for which
the reader is referred to Weber and Wellstein's Encyclopddie
der Elemmiarmathematik, p. 427 et seq., may be shown to be
the roots of an algebraic equation
/(x)=ao+aiX+a2x2 + . . .+a„x" = 0, . . . (3')
26 INFINITESIMAL ANALYSIS.
the coefBcients being integers and Oo 5^0 and a„ j-^O. The rest of
the argument consists in showing that equation (3) is impossi-
ble when xi, X2, . . . , Xn are roots of (3'). The process is
analogous to that in § 8.
e^>c.V.b,=bi-V.+hxu{l+^+~ + . . .) ,
e^..3!63 = 63-3!(l+ff+f)+63X.3(l+^+^; + ...),
(4)
e^..s!.6.=6..«!(l+ff+... + ^)
+,^,.(l+_£L+^^ + ...)
The numbers 6i, . . . , 6„ may be regarded as the coefficients
of an arbitrary polynomial
^(x) = 6o+M+fe2a;2 + . . .+b^,
for which
<l>^'"^(x)^bm-m\+b„+i-^—:rY^-x + ... + b.j r.'X^'".
J. »-i
The diagonal in equations (4) from 6i-l! to b,-s\ , , is
(s-1)
obviously (f>'{xk), and the next lower diagonal 4>"{Xk), etc.
Therefore, by adding equations (4),
e**(l!6i+2!62 + . . .+s!6.) = .^'(xjt) + <^"(a;*) + . . .
■^cj>^'Kxk) + h^-Xk^Rk„„ . (5)
i»»=i
THE SYSTEM OF REAL NUMBERS. 27
in which
7? - 1 I ■ ^^ I ^fc^ I
"*" "^m + l^(m + l)(m+2) + -'-
Remembering that (f>{x) is perfectly arbitrary, let it be so
chosen that
<t>'{xk)=0, <A"(x,) = 0, <A'"(a;A)=0, ..., </.("- i)(xt)=0
for every xj.
Equation (5) may then be written as follows :
cMl!&i+2!62 + ...+s!6.) = i6„-(zt)'".i24.„
m=l
+ bp-pl
+i>,«(p+l)!(l + j-')
A choice of ^(x) satisfying the required conditions is
Q^ np—i . ^p—1
4>ix) = "(2? -1)1 {ao+aix+CiX^ + . . . +a„a;")p
O np-l.-cp-l
- (p-1)! (^("))''-
of which every xj is a p-tuple root. If <^(x) is expanded and
the result compared with
^(x)=feo+6iX+. . .+b,x',
it is plain that feo = 0, &i=0, . . . , 6p_2=0, on account of the
factor xP~i ; and
Op-i- (p_l)! ' ''p- (p_i)! ^' (p--l)! '
28 INFINITESIMAL ANALYSIS.
where /p, . . . , /, are all integers. The coefficient of e'k in (6)
may now be written
If the arbitrary number p is chosen as a prime number
greater than a^ and a„, Np becomes the sum of aoPa„"P~i, which
carnot contain p as a factor, and a number of other integers
each of which is divisible by p. Np therefore is not zero and
not divisible by p.
ip + t)\
Further, since , _^, , — j- is an integer divisible by p when
7^', it follows that all of the coefficients of the last block of terms
in (6) contain p as a factor. If then (6) is added by columns,
Npe^=pan^p-^[Po+PiXk+P2Xk'' + . . .+Ps-pXk'-p]
8
+ Ib„-Xk"'-Rk„,. . (7)
m= 1
where Po, Pi, . . . , P^^p are integers.
It remains to show tha.tIb,n-Xk"'-Rk„, can be made small at
will by a suitable choice of the arbitrary p. As in the proof
of the transcendence of e, it follows that
' Op
|rifcp| = Ib^-Xk""- Rkm < 7 TT-j • e",
m=l \P — 1^!
where
Q = \arJ'\a(}ao\ + \ai\a + . . . + |a„|a),
and a is the largest of the absolute values of Xkik = l, . . . , n).
If now p is chosen as a prime number, greater than unity,
greater than ao . . . an and greater than c, and so great also
that |rip| <-, it follows directly from equation (7) that
THE SYSTEM OF REAL NUMBERS. 29
= Njfi+pan^p--^{PoSo+PiSr + . . .+P._pS._p) + iVfcp, (8)
where \rkp\
Ihrn-XlT-Jtlcm
<n'
So=n, and Si = xi^+X2^+X3^+. . .+Xn\ and therefore
_ ttn-l „ fln^-1 2a„_2 ,
^1= — ;r~' '^2=-—^ — -— , ...,t
and therefore it follows that a„"P-i/Si, a„"P~i(S2, . . . , are all
whole numbers or zero. The term
po„»p-i'iPiSi
t =
is therefore an integer divisible by p, while, on the contrary,
Np and c are not divisible by p. The sum of these terms is
n
therefore a whole number ^+1 or ^ — 1, and since Irkp<l,
the entire right-hand member of (8) is not zero, and hence (3)
is not zero. Therefore —
Theorem g. The number tz is transcendental.
t Cf. BuRNSiDE and Panton Theory of Equations, Chapter VIII, Vol. I.
CHAPTER II.
SETS OF POINTS AND OF SEGMENTS.
§ I. Correspondence of Numbers and Points.
The system of real numbers may be set into one-to-one cor-
respondence with the points of a straight line. That is, a
scheme may be devised by which every niomber corresponds
to one and only one point of the line and vice versa. The
point is chosen arbitrarily, and the points 1, 2, 3, 4, ... are
at regular intervals to the right of in the order 1, 2, 3, 4, . . .
from left to right, while the points -1, -2, -3, . . . foUow at
regular intervals in the order 0, -1, -2, -3, . . . from right
to left. The poiBts which correspond to fractional numbers
are at intermediate positions as follows : t
To fix our ideas we obtain a point corresponding to a par-
ticular decimal of a finite number of digits, say 1.32.
? ■'. ■? ■?! ■■? ■? ?
Fia. 1.
Divide t he se gment 1 2 into ten equal parts. Then divide the
segment 3 4 of this division into ten equal parts. The point
marked 2 by the last division is the point corresponding to 1.32.
If the decimal is not terminating, we simply obtain an
infinite sequence of points, such that any one is to the right
of all that precede it, in case of a positive number, or to the
t It is convenient to think of numbers in this case as simply a notation
for points. In view of the correspondence of points and numbers the num-
bers furnish a complete notation for all points.
30
SETS OF POINTS AND OF SEGMENTS. 31
left in case of a negative number. The first few points of the
sequence for the number a- are the points corresponding to the
numbers 3, 3.1, 3.14, 3.141. This, set of numbers is bounded,
4, for instance, being an upper bound. Hence the points cor-
responding to these numbers all lie to the left of the point
corresponding to the number 4. To show that there exists a
definite point corresponding to the least upper bound B of the
set of numbers 3, 3.1, 3.14, 3.141, etc., use is made of the foUow-
ing:
Postulate of Geometric Continuity.— 7/ o set [x] of points
of a line has a right bound, that is, if there exists a point B on
the line svch that no point of the set \x] is to the right of B, then
there exists a leftmost right bound B of the set [x]. If the set
has a left bound, it has a rightmost left bound.
The leftmost right bound of the set of points corresponding
to the numbers 3., 3.1, 3.14, etc., is the point which corresponds
to the number n. In the same manner it follows from the pos-
tulats that there is a definite point on the line corresponding to
any decimal with an infinitude of digits.!
Conversely, given any point on the line, e.g., a point P, to
the right of 0, there corresponds to it one and only one num-
ber. This is evident since, in dividing the line according to a
decimal scale, either the point in question is one of the division-
points, in which case the number corresponding to the point is
a terminating decimal, or in case it is not a division-point we
will have an infinite set of division points to the left of it, the
point in question being the leftmost right bound of the set. If
now we pick out the rightmost point of this left set in every
division and note the corresponding nimiber, we have a set
of niunbers whose least upper bound corresponds to the
point P.
•f- It is not implied here, of course, that it is possible to write a decimal
with an infinitude of digits, or to mark the corresponding points. What
is meant is that if an infinite sequence of digits is determined, a definite
number and a definite point are thereby determined. Thus V2 determines
an infinite sequence of digits, that is, it furnishes the law whereby the
sequence can be extended at will.
32 INFINITESIMAL ANALYSIS.
The ordinary analytic geometry furnishes a scheme for set-
ting all pairs of real numbers into correspondence with aJI
points of a plane, and all triples of real numbers into corre-
spondence with all points in space. Indeed, it is upon this
correspondence that the analytic geometry is based.
It should be noticed that the correspondence between num-
bers and points on the hne preserves order, that is, if we have
three nimibers, a, b, c, so that a<b<c, then the corresponding
points A, B, C are under the ordinary conventions so arranged
that B is to the right of A, and C to the right of B.
It will be observed that we have not put this matter of the
one-to-one correspondence between points and numbers into
the form of a theorem. Rather than aiming at a rigorous
demonstration from a body of sharply stated axioms, we have
attempted to place the subject-matter before the reader in
such a manner that he will imderstand on the one hand the
necessity, and on the other the grounds, for the hypothesis.
§ 2 Segments and Intervals. Theorem of Borel.
Definition.— A segment a b is the set of all numbers greater
than a and less than b. It does not include its end-points a
and b. An interval ab is the segment a b together with a and
b. For a segment plus its end point a we use the notation
I— — I
a b, and when a is absent and b present a b. All - these nota-
tions imply that a<b.-\ Sometimes we denote a segment or
interval by a single letter. This is done in case it is not im-
portant to designate a definite segment or interval.
The set of all numbers greater than a is the infinite segment
a 00, and the set of all numbers less than a is the infinite segment
- 00 a. The infinite segments a oo and — oo a, together with the
point a, are respectively the infinite intervals a <x and — oo a.
t The notation ab,ao,ab, etc., to denote the presence or absence of end-
points is du3 to G. Peano, Analisi Infinitisimali. Torino, 1893.
SETS OF POINTS AND OF SEGMENTS. 33
Unless otherwise specified the expressions segment and interval
will be understood to refer to segments and intervals whose end-
points are finite.
By means of the one-to-one correspondence of numbers and
points on a line we define the length of a segment as follows:
The length of a segment a b with respect to the unit segment
1 is the munber \a-b\. This definition applies equally to all
segments whether they are commensurable or incommensurable
with the unit segment.
Definition. — A set of segments or intervals [a] covers a
segment or interval t if every point of < is a point of some a.
On the interval - 1 1 consider the set of points ^ . The
-1 % 'A K 1
I I 1 I 1 I
Fig. 2.
I— I I— I I— I I 1
set of intervals -1 0, ^ 1, 4 2' • ■ • > 2^ 2^^> • ■ ■ covers
I— I I— I. .
the interval —1 1, because every point of —1 1 is a point
of one of the intervals. On the other hand a set of segments
-10,-^ 1, . . . , ^ o^^j ^*c-j does not cover the interval
because it does not include the points — 1, 1, n, . . . , ^^ , . . . , or 0.
In order to obtain a set of segments which does cover the inter-
val, it is necessary to adjoin a set of segments, no matter how
small, such that one includes -1, one includes 0, one includes 1,
2) 4l • • •
The segment including 0, no matter how small it is, must
include an infinitude of the points ^, and there are only a finite
number of them which do not lie on that segment. It therefore
follows that in this enlarged set there is a subset of segments,
34 INFINITESIMAL ANALYSIS.
I— I
finite in number, which includes all the points of -1 1. This
turns out to be a general theorem, namely, that if any set of seg-
ments covers an interval, there is a finite subset of it which also
covers the interval. The example we have just given shows that
such a theorem is not true of the covering of an interval by a set
of intervals; furthermore, it is not true of the covering of a seg-
ment either by a set of segments or by a set of intervals.
I— I
Theorem lo.t If an interval a b is covered by any set [a] of
segments, it is covered by a finite number of segments ai, . . . , (j„
of [o].
Proof. — It is evident that at least a part of a 6 is covered
by a finite number of <j's; for example, if ao is the a or one of
— I
the a's which include a and if b' is any point of a 6 which lies
I— I
in (To, then o 6' is covered by oo- Let [b'] be the set of
—I I— I .
all points of a b, such that a b' is covered by a finite number
of a's. By Theorem 4 [b'] has a least upper bound B. To com-
plete our pr oof we show (a) that B is in [b'], and (6) that B=h.
(a) Let a" b" be a segment of [a] including B. Since B
is the least upper bound of [b'], there is a point of [6'], b', between
<i" and B. But if <ti, <t2, • • ■ . <t« be the finite set of segments
I — I
covering the interval a b', this set together with a" b" will
I — I
cover a B, which proves that S is a point of [b'].
(b) If B^b, then B<b and the set ai, 02, . . . , a„ together
' . I— I
with a" b", would cover an interval o c, where c is a point
between B and b" ; c would therefore be a point of \b'], which
is contrary to the hypothesis that B is an upper bound of [6'].
Hence B=b and the theorem is proved.
t This theorem is due to E. Borel, Annales de I'Ecole Nonnale Su-
p^rieure, 3d series, Vol. 12 (1895), p. 51. It is frequently referred to as
the Heine-Borel theorem, because it is essentially involved in the proof
of the theorem of uniform continuity given by E. Heine, Die Elemente der
FuTuiionetdehre, Crelle's Journal, Vol. 74 (1872), page 188.
SETS OF POINTS AND OF SEGMENTS. 35
An immediate consequence of this theorem is the following,
which may be called the theorem of unifarmity.
I — I
Theorem ii. // an interval a b is covered by a set of seg-
I — I
ments [a], then a b may be divided into N equal intervals such
that each interval is entirely within a a.
I — I
Proof. — By Theorem 10 a 6 is covered by a finite set of a's,
<ji, 02, ... , On. The end points of these <t's, together with a
and b, are a finite set of points. Let d be the smallest distance
between any two distinct points of this set. Because of the
overlapping of the a's, any two points not in the same segment
are separated by at least two end points. Therefore any two
points whose distance apart is less than d must he on the same
segment of cri, 02, ... , t7„. Now let N be such that — =rp<d,
then each interval of length — r^ is contained in a a.
Fig. 3.
By this argument we have also proved the following:
I — I
Theorem 12. If an interval a b is covered by a set of seg-
ments, then there is a number d such that for any two numbers
Xi and X2 such that alxi<X2=& o/nd \x\ -I2I <d, there is a seg-
ment a of [<t] which contains both Xi and 12. In other words,
any interval of length d lies entirely vrithin some a.
The sense in which these are theorems of uniformity is the
I— I
following. Any point x of a b, bemg witmn a segment a,
can be regarded as the middle point of an interval ix of length
Zi which is entirely within some a. The length Z, is in general
<iifferent for different points, x. Our theorem states that a
value I can be found which is effective as an l^ for every x, i.e.,
36 INFINITESIMAL ANALYSIS.
I— I
uniformly over the interval a b. The distinction here drawn is
one of the most important in rigorous analysis. It was first
observed in connection with the theorem of uniform continuity ;
see page 89.
I— I
The presence of both end points of o 6 is essential, as is
shown by the following example. 1 is covered by the seg-
. ~^ ^. ^ i i
ments ^ 2, ^ 1, g 2' • ■ • , 2" 2^^' " ' ' ' ^^ ^^ ^
points nearer to 0, Ix becomes smaller with the lower bound 0,
— I
and no I can be found which is effective for all points of 1.
When the end points are absent it is possible, however, to
modify the notion of covering, so that our theorem remains
true. This is sufficiently indicated by the following theorem,
which is an immediate consequence of Theorem 10.
Theorem 13. If on a segment a b there exists any set [a] of
segments such that
(1) [a] includes a segment of which a is an end point and a
segment oj which b is an end point.
(2) Every point of the segment a b lies on one or more of
the segments of the set [o].
Then among the segments of the set [a] there exists a finite set
of segments <ti, <t2, . . . , ct„ which satisfies conditions (1) ajid (8).
The theorems which we have just proved can be generalized
to space of any number of dimensions. A planar generalization
of a segment is a parallelogram with sides parallel to the co-
ordinate axes, the boundary being excluded. The planar gen-
eralization of an interval is the same with the boundary included.
The theorem of Borel becomes :
Theorem 14. // every point of the interior or boundary of a
parallelogram P is interior to at least one parallelogram p of a set
of parallelograms [p], then every point of P is interior to at least
one parallelogram of a finite subset pi . . .p„ of [p].
Proof.— Let z = 0, x=a>0, y=0, y=b>0 determine the
boundary of P. Let O^yi^b. Upon the interval i of the line
SETS OF POINTS AND OF SEGMENTS.
37
y=yi, cut off by P, those parallelograms of [p] that include
points of i as interior points determine a set of segments [;r]
such that every point of t is an interior point of one of these seg-
ments ;:. There is by Theorem 10 a finite subset of [n], 7:1. . . r.n,
including every point of i, and therefore a finite subset pi . . . pn
of [p], including as interior points every point of i. Moreover,
since the number of pi . . . p„ is finite, they include in their
interior all the points of a definite strip, e.g., the points be-
tween the lines y=y\—e and y=yi-\-e.
y=b
1/1
V=o
Fig. 4.
Thus for every y\ {O^yi^b) we obtain a strip of the parallel-
ogram P such that every point of its interior is interior to one
of a finite number of the parallelograms [p]. These strips in-
tersect the j/-axis in a set of segments that include every point
of the interval b. There is therefore, by Theorem 10, a
finite set of strips which mcludes every point in P. Smce
each strip is included by a finite number of parallelograms p,
the whole parallelogram P is included by a finite subset of [p].
The generalization of Theorems 11 and 12 is left to the reader.
§ 3. Limit Points. Theorem of Weierstrass.
Definition.— A neighborhood or vicinity of a point a in a line
(or simply a line neighborhood of a) is a segment of this fine such
that a lies within the segment. We denote a line neighborhood
38
INFINITESIMAL ANALYSIS.
of a point a by V{a). The symbol V*(a) denotes the set of all
points of V{a) except a itself. The symbols F(oo) and 7*(oo)
are both used to denote infinite segments a + oo , and F( — co )
and 'F'*( — oo) to denote infinite segments — ooa.f
A neighborhood of a point in a plane (or a plane neighbor-
hood of a point) is the interior of a parallelogram within which
the point lies. A neighborhood of a point (a, b) is denoted by
V(a, b) if (a, 6) is included and by V*{a, b) if (a, b) is excluded.
Instead of the three linear vicinities V(a), V(<x>), and F( — oo)
we have the following nine in the case of the plaile :
V(-aj,aj)
V(-oo,6)
V (o, oo ,)
■
V(o, 6)
V (oofoo)
V (oo.fc)
V(— oo, 05)
V(a,- oo)
V(oo,— od)
Fig. 5,
t This notation is taken from Pibhpont's Theory of Functions of Real
Variables. It is used here, however, with a meaning slightly different from
that of PlERPONT.
SETS OF POINTS AND OF SEGMENTS. 39
It follows at once from a consideration of the scheme for
setting the points on the Une into correspondence with all
numbers that in every neighborhood of a point there is a point
whose corresponding number is rational.
Definition. — A point a is said to be a limit point of a set
if there are points of the set, other than a, in every neighbor-
hood of a. In case of a line neighborhood this says that there
are points of the set in every V*{a). In the planar case this is
equivalent to saying that (a, b) is a limit point of the set [x, y],
either if for every F*(a) and V{b) there is an (x, y) of which x is
in y*{a) and y in F(6), or if for every V(a) and 7* (6) there is
an {x, y) of which x is in 7(a) and y in V*(b).
Thus is a limit point of the set I ^ I, where k takes all
positive integral values. In this case the limit point is not
a point of the set. On the other hand, in the set 1, 1-J,
1 — 2^, . . . , 1 — oA' • • ■ > 1 is a limit point of the set and also a
point of the set. In this case 1 is the least upper bound of the
set. In case of the set 1, 2, 3, the number 3 is the least upper
bound without being a limit point. The fimdamental theorem
about limit points is the following (due to Weierstrass) :
Theorem 13. Every infinite hounded set [p] of points on a line
has at least one limit point.
Proof. — Since the set [p] is bounded, every one of its points
lies on a certain interval a b. li the set [p] has no limit point,
I— I
then about every point of the interval a b there is a segment
a which contains not more than one point of the set [p]. By
Theorem 10 there is a finite set of the segments [a] such that
every point of a 6 and hence of [p] belongs to at least one of
them, but each a contains at most one point of the set [p],
whence [p] is a finite set of points. Since this is contrary to
the hypothesis, the assimiption that there is no limit point is
not tenable.
40 INFINITESIMAL ANALYSIS.
It is customary to say that a set which has no finite upper
bound has the upper bound + oo , and that one which has no
finite lower bound has the lower bound — oo . In these cases,
since the set has a point in every F*( + oo ) or in every V*( — oo )
+ 00 and — 00 are also called limit points. With these con-
ventions the theorem may be stated as follows:
Theorem i6. Every infinite set of points has a limit point,
finite or infinite.
The theorem also generahzes in space of any number of
dimensions. In the planar case we have:
Theorem 17. An infinite set of points lying entirely within a
parallelogram has at least one limit point.
Theorem 17 is a corollary of the stronger theorem that fol-
lows:
Theorem 18. // [{x, y)] is any set of number pairs and if a
is a limit point of the numbers [x], there is a value of b, finite or
+ 00 or —00, sv£h that for every V*{a) and V{b) there is an
(x, y) of which x is in V*{a) and y is in V{b).
Proof. — Suppose there is no value b finite or +00 or — 00
such as is required by the theorem. Since neither +00 nor
- 00 possesses the property required of b, there is a 7* (a) and
a F(oo ) and a V{-<xi) such that for every pair (x, y) of [(x, y)]
whose x lies in V*{fl) y fails to he in either y(oo) or F( — 00).
This means that there exists a pair of numbers M and m such
that for every (x, y) whose x is in V*{a) the y satisfies the con-
dition m<y<M. Further, since there exists no b such as is
required by the theorem, there is for every number k on the
I 1
interval m M & V{k) and a Fa*(o), such that for no (x, y) is x in
Vi^{a) and y in V{k). This set of segments [F(fc)] covers the
I 1
interval m M, whence by Theorem 10 there is a finite subset of
[V{k)], ViQc), . . . , 7„(fc) which covers m M, and hence a finite
set of corresponding Vk*{ays. Let V*(a) be a vicinity of a con-
tained in every one of the finite set of 74*(a)'s and in V*{a).
Hence if the x of a pair (x, y) is in V*{a), its y cannot he in one
SETS OF POINTS AND OF SEGMENTS 41
Of the infinite segments W^ and -^, or in one of the finite
segments V^ik), ..., V„{k), i.e., no y, corresponds to this x
which IS contrary to the hypothesis. This argument covers the
cases when a is + oo and when a is - « .
We add the definitions of a few of the technical terms that
are used in pomt-set theory, f
Definition.— A set of points which includes aU its Umit
points is called a closed set.
A set of pomts every one of which is a Umit point of the
set is called dense in itself. t
A set of points which is both closed and dense in itself is
called perfect.
A set having no finite limit point is called discrete.
A segment not including its end points is an example of a
set dense in itself but not closed. If the end points are added,
the set is closed and therefore perfect. The set of rational num-
bers is another case of a set dense in itself but not closed. Any
set containing only a finite number of points is cbsed, accord-
ing to our definition.
If every point of an interval a 6 is a limit point of a set
[x], then [x] is everywhere dense on a b. Such a set has a point
between every two points of the interval. A set which is
ever3rwhere dense on no interval is called nowhere dense. All
rational numbers between and 1 form an everywhere dense set.
§ 4. Second Proof of Theorem 15.
To make the reader familiar with a style of argument which
is frequently used in proving theorems which in this book are
made to depend upon Theorems 10 and 14, we adjoin the fol-
lowing lemma and base upon it another proof of Theorem 15.
t For bibliography and an exposition in English see W. H. Young and
G. C. YoTJNG, The Theory of Sets of Points. Cambridge, The University Press,
t In German "in sich dicht."
42 INFINITESIMAL ANALYSIS.
Lemma. — ^H3rpothesis : On a straight line there is an infinite
I II 1 I 1 ... , ,„
set of intervals ai bi, O262, • • • , CLnOn, ■ • • condiiionea as follows: j
. I — I I — I I — I I — I
(1) Interval 02^2 lies on interval aibi, 0363 on 02^2, etc.
In general o„6„ lies on a„_i&„_i. (This does not exclude the case
ak=ak+\.)
(2) For every interval e>0, however small, there is some n,
say n«, such that \bn,—an,\ <e.
Conclusion: There is one and only one point b which lies
1 — I
upon every interval a„ b„.
Proof. — Since the set of points ci . . . a„ . . . is bounded, we
have at once, by the postulate of continuity, that this set has a
leftmost right bound Ba- Similarly, the set 61 . . . 6„ . . . has
a rightmost left boimd Bf It follows at once that Ba = Bi„
for if not, we get either an a point to the right of Ba, or
a b point to the left of Bi when n, is so chosen that
\bn.-a„,\<B,,-B^.
We now give another proof for Theorem 11. Divide the in-
• I— I
terval a b on which aU points of [p] lie into two equal intervals.
Then there is an infinite number of points [pj on at least one
of these intervals which we call Ci 61. Divide this interval
j" In particular the set of segments assumed in the hypothesis may be
obtained by dividing any given segment into a given number of equal seg-
ments, then one of these segments into the same number of equal segments
and BO on indefinitely. To show that the sequential division into a num-
ber of equal segments gives a set of segments satisfying the conditions
of the hypothesis we have merely to show that such division gives a
segment less than any assigned segment a^b^. This is equivalent to the
statement tltat for every number e there is an integer n, such that — <e
n
a direct consequence of Theorem 3. This involves the notion that no con-
stant infinitesimal exists. It may appear at first sight that a proof of this
statement is superfluous. The fact is, however, as was first proved by
Vebonesi:, that the non-existence of constant infinitesimals is not provable
without some axiom such as the continuity axiom or the so-called Archime-
dean Axiom.
SETS OF POINTS AND OF SEGMENTS. 43
into two equal parts and so on indefinitely, always selecting for
division an interval which contains an infinite number of points
of the set [p]. We thus obtain an infinite sequence of intervals
I 1 1 1 I 1 . .
tti bi, a2 b2, . . . , an b„ . . . which satisfies the hypothesis of the
lenrnia. There is therefore a point B which belongs to every
I — I i — r I — I
one of the intervals aibi, az b2, . . . , (in bn . . . , and therefore
there is a point of the set [p] in every neighborhood of B.
It should be noticed that the intervals in this sequence may
be such that all intervals after a certain one will have, say, the
right extremities in common. In this case the right extremity
is the point B. Such is the sequence, obtained by decimal
division, representing the number 2 = 1.99999. . . .
CHAPTER III.
FUNCTIONS IN GENERAL. SPECIAL CLASSES OF FUNCTIONS
§ I. Definition of a Function.
Definition. — A variable is a symbol which represents any
one of a set of numbers. A ccnstant is a special case of a vari-
able where the set consists of but one number.
Definition. — A variable y is said to be a single-valued
Junction of another variable x if to every value of x there cor-
responds one and only one value of y. The letter x is called
the independent vaiiable and y the dependent variable.!
Definition. — A variable y is said to be a many-valued
function or multiple-valued function of another variable x if
to every value of x there correspond one or more values of y.
The class of multiple-valued functions thus includes the class
of single-valued functions.!
t This definition of function is the culmination of a long development of
the use of the word. The idea of function arose in connection with coordi-
nate geometry, Rene Descartes using the word as early as 1637. From
this time to that of Leibnitz "function" was used synonymously with the
word "power, " such as x', x', etc.
G. W. Leibnitz regarded "function" as "any expression standing for
certain lengths connected with a curve, such as coordinates, tangents, radii
of curvature, normals, etc."
JoHANN Bernoulli (1718) defined "function" as "an expression made
up of one variable and any constants whatever."
Leonard Euler (1734) called the expression described by Bernoulli an
analytic function and introduced the notation fix) . Euler also distinguished
between algebraic and transcendental functions. He wrote the first treatise
on "The Theory of Functions."
The problem of vibrating strings led to the consideration of trigonometric
series. J. B. Fourier set the problem of determining what kind of relations
<;an be expressed by trigonometric series. The possibility then under con-
44
SPECIAL CLASSES OF FUNCTIONS. 45
It is sometimes convenient to think of special values taken
by these two variables as arranged in two tables, one table con-
taining values of the independent variable and the other contain-
ing the corresponding values of the dependent variable.
Independent Variable Dependent Variable
X2
2/1
2/2
If 2/ is a single- valued function of x, one and only one value
of y will appear in the table for each x. It is evident that
functionality is a reciprocal relation; that is, if ?/ is a function of
X, then z is a function of y. It does not follow, however, that
if 2/ is a single-valued function of x, then x is a single-valued
function of y, e.g., y=x^. It is also to be noticed that such
tables cannot exhibit the functional relation completely when
the independent variable takes all values of the continuum,
since no table contains all such values.
Definition. — That y is & function of x (and hence that x
is a function of y) is expressed by the equation y=j{x) or by
x=f~^{y). If y and x are connected by the equation y=f{x),
f~^(y) is called the inverse function of /(x).
Thus y = x^ has the inverse function x=^±\/y. In this
case, while the first function y=x^ is defined for all real values
of X, the inverse function x= ±\/y is defined only for positive
values of y.
The independent variable may or may not take all values
between any two of its values. Thus n! is a function of n
where n takes only integral values. >S„, the sum of the first
sideration that any relation might be so expressed led Lejedne Dirichlet
to state his celebrated definition, which is the one given above. See the
Encyclopadie der mathematischen Wissenschaften, II A 1, pp. 3-5; also
Ball's History of Mathematics, p. 378. *
46 INFINITESIMAL ANALYSIS.
n terms of a series, is a function of n where n takes only integral
values. Again, the amount of food consumed in a city is a
function of the number of people in the city, where the inde-
pendent variable takes on only integral values. Or the inde-
pendent variable may take on all values between any two of
its values, as in the formula for the distance fallen from rest
by a body in time t, s = -n-
'It follows from the correspondence between pairs of num-
bers and points in a plane that the functional relation between
two variables may be represented by a set of points in a plane.
The points are so taken that while one of the two numbers
which correspond to a point is a value of the independent
variable, the other number is the corresponding value, or one
of the corresponding values, of the dependent variable. Such
representations are called graphs of the function. Cases in
point where the function is single-valued are: the hyperbola
referred to its asymptotes as axes \y=-) ; a straight line not
parallel to the y axis {y=ax+b); or a broken line such that
no line parallel to the y axis contains more than one of its points.
In general, the graph of a single-valued function with a single-
valued inverse is a set of points [{x, y)] such that no two points
have the same x or the same y.
Following is a graph of a function where the independent
variable does not take all values between any two of its values.
Consider Sn, the sum of the first n terms as a function of w in
the series
„ , 1 1 1
The numbers on the x axis are the values taken by the
independent variable, while the functional relation is repre-
sented by the points within the small circles. Thus it is seen
that the graph of this function consists of a discrete set of
points. (Fig. 6.)
SPECIAL CLASSES OF FUNCTIONS.
47
The definition of a function here given is very general. It
will permit, for instance, a function such that for all rational
values of the independent variable the value of the function is
4 5
Fig. 6.
unity, and for irrational values of the independent variable the
value of the function is zero.
§ 2. Bounded Functions.
Since the definition of function is so general there are few
theorems that apply to all functions. If the restriction that
fix) shall be bounded is introduced, we have at once a very im-
portant theorem.
Definition. — ^A function, f(x), has an upper bound for a
set of valves [x] of the independent variable if there exists a
finite nimaber M such that f{x) <M for every value of x in the
set [x]. The function has a lower bound m if f{x)>m for every
value of X in {x\. A function which for a given set of values of
X has no finite upper bound is said to be unbounded on that
set, or to have an upper bound + oo on that set, and if it has
48 INFINITESIMAL ANALYSIS.
no lower bound on the set the function is said to have the lower
bound — 00 on the set.
I — 1
Theorem 19. If on an interval aba Junction has an upper
bound M, then it has a least upper bound B, and there is at least
I — I
one value of x, x\ on a b such that the least upper bound of the
function on every neighborhood of Xi contained in a b is B.
Proof. — (1) The set of values of the function f{x) form a
bounded set of numbers. By Theorem 4 the set has a least
upper bound B.
I — I
(2) Suppose there were no point xi on a b such that the
least upper bound on every neighborhood of x\ contained in
I — I — I— I
a—bisB. Then for every x of a 6 there would be a segment
Ox containing x such that the least upper bound of /(x) for
I— I _
values of x common to (Ji and a 6 is less than B. The set [tTj] is
infinite, but by Theorem 10 there exists a finite subset [<;„] of
I — 1
the set [ctJ covering a b. Therefore, since the upper bound of
fix) is less than B on that part of every one of these segments
of [on] which lies on a b, it follows that the least upper bound
I — I _
of /(x) on a 6 is less than B. Hence the hypothesis that no
point Xi exists is not tenable, and there is a point Xi such
that the least upper bound of the function on every one of its
I — I _
neighborhoods which lies in a 6 is B.
This argument applies to multiple-valued as well as to single-
valued functions.
As an exercise the reader may repeat the above argument
to prove the following:
I — I
Corollary. — If on an interval 06a function has an upper
bound + 00 , then there is at least one value of x, xi on a b such
that in every neighborhood of xi the upper bound of the func-
tion is -I- 00 ,
SPECIAL CLASSES OF FUNCTIONS.
49
§ 3. Monotonic Functions ; Inverse Functions.
Definitions.— If a single-valued function f{x) on an interval
a b is such that /(x,)</(x2) whenever a;, <X2, the function is
said to be monotonic increasing on that interval. If /(xi)>
/(X2) whenever xi <X2, the function is said to be monotonic de-
creasing, ^'^j
Fig. 7.
If there exist three values of x on the interval o b, xi, X2,
and X3 such that /(x2)>/(xi) and /(x2)>/(x3), while xt<X2<X3
or /(x2)</(xi) and /(x2)</(x3), while Xi<X2<X3, the function
is said to be oscillating on that interval. A function which is
not oscillating on an interval is called non-oscillating. It should
be noticed that a function is not necessarily oscillating even if
it is not monotonic. That is, it may be constant on some parts
of the interval.
The terms monotonic and oscillating are not convenient of
application to multiple-valued functions. Hence we restrict
their use to single-valued functions.
Definition. — ^A function /(x) is said to have a finite niun-
I — I
ber of oscillations on an interval a 6 if there exists a, finite
so
INFINITESIMAL ANALYSIS.
number of points a=xo, Xi, . . . , Xn=b, such that on each inter-
val Xk-i xjfe (A; = l, 2, 3, . . . , n) f(x) is non-oscillating. It is
evident that if a function has only a finite number of oscilla-
I — !
tions on an interval a b and if there is no subinterval of
I — I I — I
a 6 on which the function is constant, then the interval a b
may be subdivided into a finite set of intervals on each of
2/>sin
Fig. 8.
which the function is monotonic. Such a function may be
called partitively monotonic (Abteilungsweise monoton).
The function /(x) =sin -, for a; 5^0, and /(i) =0, for a; =0, is an
example of a function with an infinite number of oscillations on
SPECIAL CLASSES OF FUNCTIONS.
51
every neighborhood of a point. fix)=x sin -, for Xf^O, /(O) = 0,
and f{x)=x^ sin -, for Xf^O, /(0)=0 have the above property
-and also are contmuous (see page 61 for meaning of the term
continuous function).
There exist continuous functions which have an infinite
number of oscillations on every neighborhood of every point.
y — x sin i
Fig. 9-
The first function of this type is probably the one discovered by
Weierstrass,t which is continuous over an interval and does not
possess a derivative at any point on this interval (see page 150).
t According to F. Klein, this function was discovered by Weierstrass in
1851. See Klein, Anwendung der Differential- und Integralrechnung auf
Geometrie, p. 83 et seq. The function wa^ first published in a paper en-
titled Abhandlungen aus der FunctionerUehre, Du Bois Beyuond, CreU^s
.Journal, Vol. 79, p. 29 (1874).
52
INFINITESIMAL ANALYSIS.
Other functions of this type have been published by Peano,
Moore, and others.f
These latter investigators have obtained the function in
question in connection with space-filling curves.
Theorem 20. If y is a monotonic function of x on the interval
a b, with bounds A and B, then in turn xisa svngle-valued monotonic
I — I
function of y on A B, whose upper and lower bounds are b and a.
Proof. — It follows from the monotonic character of 1/ as a
function of x that for no two values of x does y have the same
Fig. 10.
I 1
value. Hence for every value of y on A JB there exists one and
t G. Peano, Sut une courbe, qm remplit toiUe une aire plane, Mathematische
Annalen, Vol. 36, pp. 157-160 (1890). Cesaro, Sur la representation analy-
tique des regions et des courbes qui les remplisent, Bulletin des Sciences Mathi-
mati/juss, 2d Ser., Vol. 21, pp. 257-267. E. H. Moore, On Certain Crinkly
Curves. Transactions of the American Mathematical Society, Vol. 1, pp. 73-90
(1899). See also Steikitz, Mathematische Annalen, Vol. 52, pp. 58-69 (1899).
SPECIAL CLASSES OF FUNCTIONS. 53
only one value of x. That is, a; is a single-valued function of y.f
Moreover, it is clear that for any three values of y, yi, 2/2, 2/3,
such that 2/2 is between ?/i and j/3, the corresponding values of
^, Xi, X2, X3, are such that x-^ is between Xi and Xa, i.e., x is
a monotonic function of y, which completes the proof of the
theorem.
CcroUary.— If a function /(x) has a finite number ^ of oscil-
lations and is constant on no interval, then its inverse is at most
(A + l)-valued. For example, the inverse of y = x^ is double-
valued.
§ 4. Rational, Exponential, and Logarithmic Functions.
Definitions. — The symbol a"", where wi is a positive integer
and a any real number whatever, means the product of m
factors a. This definition gives a meaning to the symbol
y=a„x'"+am-ix"'-^ + . . .+aix + ao,
•where oo . . . a^ are any real numbers and m any positive inte-
ger. In this case y is called a rational integral function of x
or a polynomial in x.t
In case
amX"'+am_iX"'~^+ . . . +ai-x+ao
^~ 6„x" + 6„_iX"-»+ . . . +6i-x+6o '
m and n being positive integers and a* {k=0, . . .m) and bi
(Z = 0, . . . n) being real numbers, y is called a rational function
of X.
If
yn+yfi-lR^(x)+y^-m2{x)+ ... +yRn.l{x) +Rnix) =0,
where Ri{x) . . . R„(x) are rational functions of x, then y is said to
t it is clear that the independent variable y of the inverse function may
not take on all values of a continuum even if x does take on all such values.
% The notion of polynomial finds its natural generalization in that of
a power series
y=c^ + c-x + C2-x'+ . . . +c„i"+ . . .
For conditions under which a series defines y as a. function of x see
Chapter IV, § 3.
64 INFINITESIMAL ANALYSIS
be an algebraic function of x. Any function which is not
algebraic is transcendental.
The symbol a', where a; = — , m and n being positive integers
and a any positive real number, is defined to be the nth root of
the mth power of a. By elementary algebra it is easily shown
that
o*i.a=^=o*'+*» and {a''^)'"=a''^'".
If y=a',
then 2/ is an exponential function of x. At present this function
is defined only for rational values of x.
Fia. 11.
Theorem 21. The function a' for x on the set \ — \ is a monotonic
increasing function if l<a, and a monotonic decreasing function
ifO<a<l.
Proof. — (a) For integral values of x the theorem is obvious.
(6) If xi = — and X2 =—, where — >— , then
«i ni Til Wi
SPECIAL CLASSES OF FUNCTIONS. 55
c*'<a*' if o>l and a*'>a*» if a<l. The proof of this follows
at once from case (a), since a"i = \a»ij (by definition and ele-
mentary algebra) and
an, = ^a»ij .
(c) If xi= — and X2= — , where — < — , we have
Til n-z' rii 712
o"> =a"i"2 and a"2 =o"«"i, where mi 712 >in2-ni, which reduces
case (c) to case (&).
This theorem makes it natural to define a", where a > 1 and x
is a positive irrational number, as the least upper bound of all
r "1 m
numbers of the form La" J, where — is the set of all posi-
tive rational nimibers less than x, i.e., a'^=Bta^J. It is,
however, equally natural to define a" as ^Lo^J, where I I is
the set of all rational numbers greater than x. We shall prove
that the two definitions are equivalent.
Lemma. — If [x] is the set of all positive raiional numbers, then
5[a^] = l ifa>l
and B[a='] = l ifa<l.
Proof. — We prove the lemma only for the case o>l, the
argument in the other case being similar. If x is any positive
7?l 1
rational number, — , then the number - is less than or equal
i_ ^ fl 1
to x, and since a' is a monotonic function, a'*<a^. But - I
is a subset of I -~ I- Hence
where [n] is the set of all positive integers.
56 INFINITESIMAL ANALYSIS.
If 5La»J were less than 1, then there would be a value,
ni, of n such that a»i<l. This implies that a<l, which is con-
trary to the hypothesis. On the other hand, if 5La" J > 1, there
is a number of the form 1+e, where e>0, such that l + e<a"
for every n. Hence {l+e)"<a for every n, but by the binomial
theorem for integral exponents
(l+e)''>H-ne,
and the latter expression is clearly greater than a if
a
n>-.
e
Since 5La"J cannot be either greater or less than 1,
Theorem 22. // x is any real number, and \~\ the set of all
rational numbers less than x, and \-\the set 0} all rational numbers
greater than x, then
5La"J = BLaO ifa>l,
Bla'^j^Bla'^J ifO<a<l.
Proof. — We give the detailed proof only in the case a>l,
—
2 n]
is zero,
_ ai-a„J=Blai[l-a«~ vJJ
is also zero. Now if
B[a'^]^B[af'],
SPECIAL CLASSES OF FU^CT10I^S. 57
Since as is always greater than a",
Bla^j-Bla^j = £>0.
But from this it would follow that
p m
a9 -a"
is at least as great as e, whereas we have proved that
_ a«-o"J=0.
Hence 5La»J = sLa9j ifa>l.
Definition. — In case x is a positive irrational number,
and I — I is the set of all rational numbers greater than x, and
[— is the set of all rational numbers less than x, then
a» = Bla^J - 5L0 » J if a > 1
and a*=5La«] = B[a"] ifO<o<l.
Further, if a; is any negative real number, then
a'= — ; and a° = l.
a' .
Theorem 23. The function a* is a monotonic increasing func-
tion ofxifa>l, and a monotonic decreasing function if 0<a<l.
In both cases its upper bound ts + <» and Us lower bound is zero,
the function taking all values between these bounds; further,
ax,.^x,=ai,+i2 and {d'^Y^ = a^'-''K
The proof of this theorem is left as an exercise for the reader.
The proof is partly contained in the preceding theorems and
58 INFINITESIMAL ANALYSIS.
involves the same kind of argument about upper and lower
bounds that is used in proving them.
Definition. — The logarithm of x (x>0) to the base a{a>0) is
a number y such that a^=x, or a^°^'==x. That is, the func-
tion logo X is the inverse of a''- The identity
gives at once logo xi+ logo X2= logo (xi ■ X2) ,
and (a*')^'=a''"^^ gives Xi-logoX2=logaX2'^.
By means of Theorem 20, the logarithm logaX, being the
inverse of a monotonic function, is also a monotonic function,
increasing if 1 < a and decreasing if 0< a < 1. Further, the func-
tion has the upper bound + 00 and the lower bound — 00 , and
takes on all real values as x varies from to +00 . Thus it
follows that f or i< a, 1 < b,
B(\ogb x) =log6 a=log6 (Bx).
By means of this relation it is easy to show that the function
x", (a;>0)
is monotonic increasing for all values of a, a>0, that its lower
bound is zero and its upper bound is + w , and that it takes on
all values between these bounds.
The proof of these statements is left to the reader. The
general type of the argument required is exemplified in the
following, by means of which we infer some of the properties
of the function x".
If xi < xz, then
l0g2 Xi < log2 X2,
and Xi • log2 xi < X2 • log2 X2,
and log2 xi *' < log2 X2*».
.". Xi*><X2*».
SPECIAL CLASSES OF FUNCTIONS. 59
Hence i*, (x>0) is a monotonic increasing function of x.
Since the upper bound of x ■ log2 x = log2 x^ is + <» , the upper
bound of X* is +00. The lower bound of x* is not negative,
since x>0, and must not be greater than the lower bound of
2=^, since if x<2, x''<2*; since the lower bound of 2* is zerot
the lower bound of x* must also be zero.
Further theorems about these functions are to be found on
pages 64, 81, 97, 123, and 160.
•f The lower bound of a' is zero by Theorem 23.
CHAPTER IV.
THEORY OF LIMITS.
§1. Definitions. Limits of Monotonic Functions.
Definition. — If a point a is a limit point of a set of values
taken by a variable x, the variable is said to approach a upon
the set; we denote this by the symbol x = a. a may be finite
or +00 or — 00 .
In particular the variable may approach a from the left or
from the right, or in the case where a is finite, the variable may
take values on eacl^ side of the limit point. Even when the
variable takes all values in some neighborhood on each side of
the limit point it may be important to consider it first as taking
the values on one side and then those on the other.
Definition. — A value b (6 may be + oo or - oo or a finite
number) is a value approached by f{x) as x approaches a if for
every V*{a) and V(b) there is at least one value of x such that
x is in V*{a) and f(x) in V(b). Under these conditions f(x) is
also said to approach 6 as x approaches a.
Definition. — If b is the only value approached as x ap-
proaches a, then b is called the limit of f(x) as x approaches a.
This is also indicated by the phrase "f{x) converges to a unique
limit b as X approaches a," or "f{x) approaches b as a limit,"
or by the notation
L fix) =b.
x—a
The function f{x) is sometimes referred to as the limitand.
The set of values taken by x is sometimes indicated by the sym-
bol for a limit, as, for example,
60
THEORY OF LIMITS. 61
L j(x)=b or L f{x)=b or L f{x)=b.
x>a x<a x\[x]
x^a x±a z±a
The first means that x approaches a from the right, the second
that X approaches a from the left, and the third indicates that
the approach is over some set [x] otherwise defined.
Definition. — If f{x) is single-valued and converges to a
finite limit as x approaches a and
L/(x)=/(a),
x-a
then /(x) is said to be continuous at x=a.
By reference to § 3, Chapter II, the reader will see that if
6 is a value approached by fix) as x approaches a, then (a, b)
is a limit point of the set of points (x, /(x)). Theorem 18
therefore translates into the following important statement:
Theorem 24. If /(x) is any function defined for any set [x] of
which a is a {finite or +« or — 00 ) limit point, then there is at least
one value (finite or +« or -00) approached by f{x) as x ap-
proaches a.
Corollary. — If fix) is a bounded function, the values ap-
proached by fix) are all finite.
In the light of this theorem we see that the existence of
L fix)
x=a
simply means that fix) approaches only one value, while the
non-existence of
Lfix)
means that fix) approaches at least two values as x approaches a.
In case fix) is mono tonic (and hence single-valued), or more
generally if fix) is a non-oscillating function, these ideas are
particularly simple. We have in fact the theorem:
Theorem 25. If fix) is a non-oscillating function for a set of
values [x]<a, a being a limit point of [x], then as x approaches a
62 INFINITESIMAL ANALYSIS.
from the left on the set [x], f{x) approaches one and only one
value b, and if f{x) is an increasing function,
b = Bf{x)
for X on [x], whereas if f{x) is a decreasing function,
h = B}{x)
for X on [x].
Proof. — Consider an increasing non-oscillating function
and let
h = Bf{x)
for X on [x].
In view of the preceding theorem we need to prove only
that no Value 6V6 can be a value approached. Suppose h'>h;
then since Bf{x) =b, there would be no value of fix) between
b and b', that is, there would be a V(b') which could contain
no value of fix) , whence b'>b 'm not a value approached. Sup-
pose b'<b. Then take b'<b"<b, and since Bf{x)=b, there
would be a value Xi of [x] such that fixi)>b". If xi<x<a,
then b" <fixi)^fix), because fix) cannot decrease as x in-
creases. This defines a V*(a) and a V{b') such that if x is in
y*(a), fix) cannot be in V{b'). Hence b' <b is not a value
approached. A like argument applies if fix) is a decreasing
function, and of course the same theorem holds if x approaches
a from the right.
It does not follow that
Lfix)=Lfix),
x<a x^a
x±a x—a
nor that either of these limits is equal to /(a) . A case in point
is the following: Let the temperature of a cooling body of
water be the independent variable, and the amount of heat
given out in cooling from a certain fixed temperature be the
dependent variable. When the water reaches the freezing-
THEORY OF LIMITS.
63
point a great amount of heat is given off without any change
in temperature. If the zero temperature is approached from
below, the function approaches a definite limit point k. and if
the temperature approaches zero from above, the function
Hea«
Temp.
FiQ. 12.
approaches an entirely different point A-'. This function, how-
ever, is multiple-valued at the zero point. A case where the
limit fails to exist is the following: The function y=anl x
(see Rg. S, page 50) approaches an infinite number of values
as J approaches zero. The value of the fimction will be alter-
D&tely 1 and —1, as j = -. ^, -r-, etc., and for all values of
X between any two of these the function will take all values
between 1 and —1. Clearly every value between 1 and -1
is a value approached as r approaches zero. In like manner
64
INFINITESIMAL ANALYSIS.
1/ =- sin - approaches all values between and including + » and
- 00 , cf. Fig. 13.
tslni
Fig. 13.
The functions a*, logo x, x° defined in § 4 of the last chapter
are all monotonic and all satisfy the condition that
L f{x)=fia)=L fix),
x>a x<a
z£a x—a
at all points where the functions are defined. These functions
are therefore all continuous.
THEORY OF LIMITS. 66
§ 2. The Existence of Limits.
Theorem 26. A necessary and sufficient condition ■\ that f(i)
shall converge to a unique limit h as x approaches a, i.e., that
L f{x)=b,
is that for every V(b) there shall exist a V*{a) such that for every
xin r*{a), fix) is in r(b).
Proof. — (1) The condition is necessary. It is to be proved that
if L fix) =b, then for every T'(6) there exists a F*(a) such that
for every x in V*(a) the corresponding f{x) is in T'(b). If this
conclusion did not follow, then for some V{b) every V*(a)
w-ould contain at least one x' such that /(.j-') is not in T'(6).
There is thus defined a set of points [x'] of which a is a limit
point. By Theorem 20 f{x) would approach at least one value
h' as X approaches a on the set [/]. But by the definition of
[/], b' is distinct from h. Hence the hypothesis would be con-
tradicted.
(2) The condition is sufficient. We need only to show that
if for every. V(b) there exists a V*{a) such that for every
X in V*ia) the corresponding fix) is in V(b). then f{x) can
approach no other value than b. If b'j^b, then there exists
a 1(6') and a Vib) which have no point in common. Now if
V*ia) is such that for every x of T'*(a), f{x) is in Vib), then
fThis means: (o) If L f(x) = b, then for e\-ery V(5) there exists a T"*(o),
as specified by the theorem.
(6) If for e^-ery r(6) there exists a T"*(a) as specified,
then L /i,j) = b.
A condition is necessary for a certain conclusion if it can be deduced from
that conclusion ; a condition sufficient for a conclusion is one from which the
conclusion can be deduced, A man sufficient for a task is a man -who can
perform the task, \riiile a man necessary for the task is such that the task
cannot be performed without him.
66 INFINITESIMAL ANALYSIS.
for no such x is f{x) in V{b') and hence b' is not a value
approached.
The reader should observe that this proof applies also to
multiple-valued functions, although worded to fit the single-
valued case. It is worthy of note that in case 6 is a finite num-
ber, our theorem becomes :
A necessary and sufficient condition that
Lf{x)=h
xiza
is that for every £>0 there exists a Vt*(a) such that for every x
mV*(a), \f{x)-b\<B.
In case a also is finite, the condition may be stated in a
form which is frequently used as the definition of a limit, namely :
L fix) =b means that for every e>0 there exists a 5, >0 sv^h
that if \x—a\<d, and Xy^a, then |/(x)— 6|<£.t
Theorem 27. A necessary and sufficient condition that f{x)
■shall converge to a finite limit as x approaches a is that for every
e>0 there shall exist a V*{a) such that if Xi and X2 are any two
values of x in V,*{a), then
|/fe)-/(x2)|<e.
Proof. — (1) The condition is necessary. If Lf{x)=b and h
is finite, then by the preceding theorem for every ^>0 there
■exists a V *(o) such that if xi and X2 are in V *{a), then
l/(^i)-6|<|
and \fix2)-b\<^,
from which it follows that
|/(Xl)-/fe)|<£.
tThe E subscript to ^. or to F.*(o) denotes that d, or V,*{a') is a func-
tion of e. It is to be noted tha' inasmuch as any number less than S is
effective as dt, dt is a multiple- valued function of e.
THEORY OF LIMITS. 67
(2) The condition is sufficient. If the condition is satisfied,
there exists a V*{a) upon which the function j{x) is bounded.
For let 7 be some fixed number. By hypothesis there exists a
'V*{a) such that if x and Xo are on y*(o), then
|/(x)-/(xo)|<i.
Taking Zo as a fixed number, we have that
/(xo)-£</(x)</(xo)+7
for every x on V*{a). Hence there is at least one finite value,
6, approached by /(x). Now for every e > there exists a y,*(a)
such that if xi and X2 are any two valves of x in V*{a)^
|/(xi) — /(X2)| < £. Hence by the definition of value approached
there is an x, of 7.* (a) for which
W.)-'b\<^ (a)
and |/(x,)-/(x)l<£ (6)
for every x of F.*(a). Hence, combimng (a) and (6), for every x
of 7 *(o) we have
l/(x)-6|<2£,
and hence by the preceding theorem we have
L/(x)=6.
x-a
In case a as well as 6 is finite. Theorem 27 becomes:
A necessary and sufficient condition thai
Lfix)
x±a
shaU exist and be finite is that for every e>0 there exists a d.>0
such that
\i{Xx)-KXi)\<^
68 INFINITESIMAL ANALYSIS.
for every Xi and X2 such that
XiT^a, XzT^a, \xi—a\<d., \x2-a\<8,.
In case a is + 00 the condition becomes :
For every e > there exists a N.>0 such that
\Kxi)-f{x2)\<e
for every xi and X2 such that Xi>N„ X2>N,.
The necessary and sufficient conditions just derived have
the following evident corollaries :
Corollary 1. The expression
Lfix)=b,
x~a
where b is finite, is equivalent to the expression
Lif{x)-b)=0,
and whether h is finite or infinite
L fix) =6 is equivalent to L (-fix)) = —6.
Corollary 2. The expressions
L/(a;)=0 and L |/(a;)|=0
are equivalent.
Corollary 3. The expression
Lfix)=b
is equivalent to
Lfiy+a)=b,
»=o
where y+a=x.
THEORY OF LIMITS.
Corollary 4. The expression
L/(i)=6
x<a
i = o
IS equivalent to
jjHh-
where 2=
x—a
The reader should verify these corollaries by writing down
the necessary and sufficient condition for the existence of each
limit. The following less obvious statement is proved in detail
for the case when b is finite, the case when b is + « or — «
being left to the reader.
Corollary 5. If
Lfix)=b,
then L|/(x)| = |b|.
Xza
Proof. — By the necessary condition of Theorem 26 for every
e there exists a 7.* (a) such that for every Xi of V*{a)
\f{x^)-b\<e.
If /(Ji) and b are of the same sign, then
||/(xi)|-|6|| = |/(xO-6|<«,
and if /(xi) and b are of opposite sign, then
|l/(xi)i-|b||< 1/(^1) -M<^.
Hence, by the sufficient condition of Theorem 26,
L \fix)\
x=a
exists and is equal to |6|.
70 INFINITESIMAL ANALYSIS.
Corollary 6. If a function /(x) is continuous at x=a, then
|/(x)| is continuous at a;=a.
It should be noticed that
L|/(x)| = |6|
is not equivalent to
L fix) =b.
Suppose j{x) = 4-1 for all rational values of x and f{x) = — 1 for
all irrational values of x. Then L |/(a;)| = +1, but L f{x) does
x=a x=a
not exist, since both + 1 and — 1 are values approached by f{x}
as X approaches any value whatever.
Definition. — Any set of numbers which may be written [i„],
where
n=0, 1,2, ...,K,
or n=0, 1, 2, ...,«,.. .,
is called a Sequence.
To the corollaries of this section may be added a corollary-
related to the definition of a limit.
Corollary 7. If for every sequence of numbers [i„] having a
as a limit point,
L f{x) = h, then L f{x)=b.
x|[ln] x-a
x= a
Proof. — In case two values b and 6i were approached by
f{x) as X approaches a, then, as in the first part of the proof
of Theorem 26, two sequences could be chosen upon one of
which fix) approached b and upon the other of which fix)
approached bi.
§ 3. Application to Infinite Series.
The theory of limits has important apphcations to infinite
series. An infinite series is defined as an expression of the form
THEORY OF LIMITS. 71
00
i'ot=ai +02+03+ . . . +a„+ ...
If Sn is defined as
n
Ol + ... +an = -^ dkt
k=l
n being any positive integer, then the sum of the series is defined
as
LSr,=S
no- 00
if this limit exists.
If the Umit exists and is finite, the series is said to be conver-
gent. If (S is infinite or if <S„ approaches more than one value
as n approaches infinity, then the series is divergent. For exam-
ple, S is infinite if
lafc = l + l + l + l...,
and Sn has more than one value approached if
ioi = l-l + l-l + l...
It is customary to write
Rn = S — Sn.
A necessary and sufficient condition for the convergence of
an infinite series is obtained from Theorem 27.
(1) Fm- every e>0 there exists an integer N. such thai if
n>N, and n'>N., then
\S„-Sn'\<e.
This condition immediately translates into the following
form:
72 INFINITESIMAL ANALYSIS.
(2) For every £>0 there exists an integer N. swh that if
n>N„ then for every k
|o„+o„+i + . . .+a„+fc|<e.
Corollary. — If 2 ot is a convergent series, then L 04=0.
k=l *=«
Definition. — A series
2'ai=Oo+ai + . . .+On+. • •
is said to be absolutely convergent if
|ap| + |ai I + . . . + |a„| + . . . is convergent.
Since
|a„ + 0„+i+. . .+0„+fc|<|a„| + l0n+l|+. . .Iffln+ltij
the above criteria give
Theorem 28, A series is convergent if tt is absolutely conver-
gent.
00
Theorem 29. If I bkis a convergent series all of whose terms
k=0
00
are positive and I ak is a series such thai for every k, \ak\S)k,
k=Q
then I o*
k-O
is absolutely convergent.
Proof. — By hypothesis
k=0 k-0
THEORY OF LIMITS. 73
n
Hence I \ak\
is bounded, and being an increasing function of n, the series
is convergent according to Theorem 25.
This theorem gives a useful method of determining the con-
vergence or divergence of a series, namely, by comparison with
a known series. Such a known series is the geometric series
a+ar+ar^+ . . . +ar^+ . . . ,
where 0<r<l and a>0. In this series
„ l_rn+l a
which shows that the series is convergent. Moreover, it can
easily be seen to have the sum :; — .
•^ l-r
If rf 1, the geometric series is evidently divergent. This result
can be used to prove the "ratio-test " for convergence.
Theorem 30. // there exists a number, r, 0<r<l, such that
an ^
a„_il
for every integral value of n, then the series
ai+a2+ . . . +an + (1)
is absolutely convergent. If
a„-i
>1 for every n, the series is
divergent.
Proof. — ^The series (1) may be written
02 02 03 . , az an ,n~.
Oi+Oi-+ai + ■•• +°'^:r ■•■n — . ■ • (2)
74
and if
On
Ctn-l
INFINITESIMAL ANALYSIS.
<r, this is numerically less term by term than
Oi+Oir+Oir^ . . .+air" + (3)
and therefore converges absolutely. If
dn
^l,a„^i for every
n; hence, by the corollary, page 72, (1) is divergent.
Nothing is said about the case when
<1, but L
an_
dn-l
It is evident that the ratio test need be applied only to terms
beyond some fixed term a„, since the sum of the first n terms
ai+a2 + . . .+o„
may be regarded as a finite number Sn and the whole series as
i.e., a finite number plus the infinite series
an+i+an+2 + . • •
§ 4. Infinitesimals. Computation of Limits.
Theorem 31. A necessary and sufficient condition thai
L}{x)=b
x±a
is that for the function e(x) defined by the equation f{x) =b + e(x)
L e{x)=0.
THEORY OF LIMITS. 75
Proof.— Take e{x)=f{x)-b and apply Theorem 26. A
special case of this theorem is: A necessary and sufficient am-
dition for the convergence of a series to a finite value b is that for
every e>0 there exists an integer N. such that if n>N., then
\Rn\<e.
Definition.— A function f{x) such that
Lf{x)=0
is called an infinitesimal as x approaches o.f
Theorem 32. The sum, difference, or product of two infinitesi-
mals is an infinitesimal.
Proof. — Let the two infinitesimals be /i(x) and fzix). For
every £, 1> £>0, there exists a Vi*{a) for every x of which
and a F2*(o) for every x of which
Ux)\<^.
Hence in any V*(a) common to Vi*{a) and V2*{a)
\hix)+f2ixMfi{x)\ + Ux)\<e,
\h(x)-f2(xmflix)\ + \f2ix)\<,,
\hix)-f2{x)\ = \fi(x)\-\f2{x)\<e.
From these inequalities and Theorem 26 the conclusion follows.
Theorem 33. // f{x) is bounded on a certain V*{a) and £(i)
is an infinitesimal as x approaches a, then £{x) •f{x) is also an
infinitesimal as x approaches a.
t No constant, however small if not zero, is an infinitesimal, the essence
of the latter being that it varies so as to approach zero as a limit. Cf.
Goursat, Cours d' Analyse, tome I, p. 21, etc.
76 INFINITESIMAL ANALYSIS.
Proof. — By hypothesis there are two numbers m and M,
such that M>f{x)>m for every x on V*{a). Let k be the
larger of |m| and \M\. Also by hypothesis there exists for every
e a V,*(a) within V*{a) such that if x is in V*ia), then
or A;|£(x)|<£.
But for such values of x
\f{x)-E{x)\<k-\E(x)\<e,
and hence for every e there is a V*ia) such that for x an y,*(o)
\Kx)-c{x)\<e.
Corollary. — If }{x) is an infinitesimal and c any constant,
then c-f{x) is an infinitesimal.
Theorem 34. 7/ L/i(x)=6i ond Lf2{x)=b2,
x~a x~a
bi and 62 6et7ig /inite, then
L\fi{x)±f2(x)\=bi±b2, ... (a)
L\h{x)-f2{x)}=bi-b2; (/?)
andx/6.^0, lJ^^J^^ (,) ^
Proof. — According to Theorem 31, we write
/i(a;)=6i + £i(z),
h{x)=b2 + B2ix),
THEORY OF LIMITS. 77
where €i(a;) and £2(2;) are infinitesimals. Hence
fi{x)+f2{x)=bi+b2 + siix) + s2ix), . . . (a')
fi{x)-J2{x)=bi-b2 + bi-£2{x)+b2-ei{x) + ei{x)-e2{x). . (/?')
But by the preceding theorem the terms of (a') and (/3') which
involve £1(2) .and e2{x) are infinitesimals, and hence the con-
clusions (a) and (/?) are established.
To establish (;-), observe that by Theorem 26 there exists
a V*{a) for every x of which 1/2(2) -&2I <|62| and hence upon
which /aCx) t-^O. Hence
fijx) ^ bi + sijx) J}i b2eiix) -biezjx)
hix) 62 +£2(2) 62 62162+ £2(X)} '
the second term of which is infinitesimal according to Theorems
32 and 33.
Some of the cases in which 61 and 62 are ± 00 are covered
CO
by the following theorems. The other cases (oo -oo, — , -,
etc.), are treated in Chapter VI.
Theorem 35. i/ ]2{x) has a lower bound on some V*{a),
and if
L/,(2)=+oo,
1=0
then L i/2(a;)+/i(x)} = + oo.
Proof.— Let M be the lower bound of /2(x). By hypothesis,
for every number E there exists a VE*{a) such that for x on
VE*ia)
U{x)>E-M.
Since hix)>M,
this gives hix)+J2{x)>E,
which means that /i(x) +J2{x) approaches the limit + 00 .
78 INFINITESIMAL ANALYSIS.
Theorem 36. 7/ L/i(x) = + oo or —00, and if f2(x) is such
that for a F*(a) /2(x) has a lower bound greater than zero or an
upper boundless than zero, then L {/i(x) 72(2;) } is definitely infinite;
x±a
i.e., if /2(x) has a lower bound greater than zero and Lfi{x) = + co,
z = a
then L {fi{x) ■f2{x) ! = +<», etc.
x~za
Proof. — Suppose /2(x) has a lower bound greater than zero,
say M, and that L fi{x) = + qo . Then for every E there exists a
x~a
- E
Te*{o) within V*{a) such that for every Xi of F£*(a), fi{xi) > t^,
and therefore fi{xi)-f2{xi)jfi{xi)-M>E. Hence by the defini-
tion of Umit of a function Lj/i(x) ■/2(a;)! =+». If we consider
the case where /2(x) has an upper bound less than zero, we
have in the same manner L {/i(x) •/2(a;) t = — <» ■ Similar state-
x±a
ments hold for the cases in which L /i(x) = — 00 .
Corollary. — If /2(x) is positive and has a finite upper bound
.andL/i(x) = +oo, thea
iv ■ , ^ = + 00 .
X±af2{x)
Theorem 37. If L /(x) = + (», then L 77-: = 0, and there is' a
x~a x~aJ\X)
■vicinity V*{a) upon which /(x) >0. Conversely, if L /(x) =0 and
x=a
there is a V*{a) upon which /(x) > 0, then L jr— = + 00 .
Proof. — If L f{x) = + 00 , then for every e there exists a
x-±a
V*{(i) such that if x is in V*{a), then
/(^)>7
THEORY OF LIMITS. 79
and 77-T<«-
since both f{x) and 7^ are positive.
Again, if L /(x) =0, then for every e there is a 7.* (a) such
x=.a
that for X in V*{a), |/(x)|<e or — ->- (/(x) being positive).
Hence L 7^-r = + 00 .
Corollary 1. If /i(x) has finite upper and lower bounds on
some V*{a) and L /2(x) = + « or - 00 , then
Corollary 2. If /2(x) is positive and /i(x) has a positive
lower bound on some V*{a) and L f2{x)=0, then
x£o
£( . , . = +00.
x=a/2(a;)
Theorem 38 (change of variable). If
(1) L/i(j)=6i and L/2(2/)=&2
wfecn y takes all values of /i (x) corresponding to valves of x on
some V*{a), and if
(2) /i{j) f^bi for X on V*{a),
then L /•,(/, (x)) =62.
x-a
80 INFINITESIMAL ANALYSIS.
Proof.— (a) Since L /zCj/) =62, for every 7(62) there exists a
F*(6i) such that if y is in V*(bi), jiiy) is in V{hi). Since
L fi{x) =61, for every F(6i) there exists a 7*(a) in F*(a) such
xda
that if a; is in y*(o), /i(x) is in F(6i). But by (2) if x is in
V*{a), fi{x)^bi. Hence (/?) for every V*(bi) there exists a
V*{a) such that for every x in V*(a), /i(x) is in V*{bi).
Combining statements (a) and (/3) : for every 1^(62) there
exists a F*(a) such that for every x in V*{a) fi{x) is in F*(6i),
and hence fzifix)) is in F(62). This means, according to Theo-
rem 26, that
Lf2ih{x))^b2.
x~a
Theorem 39. If L /i(x) =6 and L f2(y) =hQ>)} where y takes
x±a yxb
all valves taken by /i(x) for x on some V*ia),
then L/2(/i(x))=/,(6).
x~a
Proof. — The proof of the theorem is similar to that of Theorem
38. In this case the notation /2(6) implies that 6 is a finite
number. Thus for every ej there exists a V,*{a) entirely
within V*{a) such that if x is in V,*{a),
|/i(x)-fe|<n.
Furthermore, for every £2 there exists a 5,, such that for
every 2/, j/7^6, \y-b\<dE2,
\f2{y)-f2{b)\<e2.
But since Ifziy) -f2{b)\=0 when y = b, this means that for all
values of y (equal or unequal to b) such that \y—b\<d,^,
1/2(2/) -/2(b)|<e2. Now let e, =5„; then, if i is in V.*{a), it
follows that 1/1 (z) -b\<d„ and therefore that
|/2(/l(x))-/2(6)|<S2.
Hence L f2(f,{x))=f2{b).
THEORY OF LIMITS 81
Corollary 1. If /^(x) is continuous a,t x=a, and f-ziy) is con-
tinuous at 2/ = /i(a), then /2(/i(x)) is continuous at x = o.
Corollary 2. If ^v 0, /(x)^0, and L /(x) = b, then
under the convention that oo'= = oo if k>0 and oo*:=0ifjfc<0.
Corollary 3. If OO and /(x)>0 and 6>0 and Lf{x)=b,
^'^ Llog,/(x)=log,6,
under the convention that log,. ( + w ) = + oo and log„ = - oo .
The conclusions of the last two corollaries may also be ex-
pressed by the equations
L(/(x))* = (L/(x))*
x—a ar^a
and log, L/(x)=L log, '(x).
x'-a x±a
Corollary 4. If L (/(x))* or L log /(x) fails to exist, then
L /(x) does not exist.
§ 5. Further Theorems on Limits.
Theorem 40. 7/ f{x):^b for all valves of a set [x] on a certain
V*{a), then every value approached by fix) as x approaches a is
less than or equal to b. Similarly if fix)^b for all values of a set
[x] on a certain F*(o), then every valve approached by /(x) as x
approaches a is greater than or equal to b.
Proof. — If f{x)%b on V*{a), then if b' is any value greater
than b, and V{b') any vicinity of b' which does not include b,
there is no value of x on V*(a) for which /(x) is in V{b'). Hence
V is not a value approached. A similar argument holds for
the case where f{x)^b.
82 INFINITESIMAL ANALYSIS.
Corollary 1. If /(x)^0 in the neighborhood of x=c, then if
L fix) exist, L /(x)^0.
Corollary 2. If fiixj^fzix) in the neighborhood of x=a,
then L /i(x)2 L U^)
x±a i£o
if both these Umits exist.
Proof.— Apply Corollary 1 to fi(x)-f2{x).
Corollary 3. If t\{x)'^J2{x) in the neighborhood of x=a,
then the largest value approached by /i(x) is greater than or
equal to the largest value approached by /2(x).
Corollary 4. If /i(x) and ^(x) are both positive in the
neighborhood of x=a, and if /i(x)^/2(x), then if L /i(x)=0,
it follows that
L/2(x)=0.
Theorem 41. // [ucf] is a subset of [x], a being a limit point of
[x'], and if L f{x) exists^ then L /(a/) exists and
x-±a x±a
L f{x) = L fix').^
x~a x'±a
Proof. — By hypothesis there exists for every V{b) a V*{a)
such that for every x of the set [x] which is in V*{a), f{x) is in
F(6). Since [x'] is a subset of [x], the same V*{a) is evidently
efficient for x on [x'].
In the statement of necessary and sufficient conditions for
the existence of a limit we have made use of a certain positive
multiple-valued function of e denoted by 8,. If a given value
is effective as a d„ then every positive value smaller than this
is also effective.
Theorem 42. For every e for which the set of valves of d,
has an upper bound there is a greatest d,.
t The notation /(i') is used to indicate that x takes the values of the
8et[x'].
THEORY OF LIMITS. 83
Proof. — ^Let B[d.] be the least upper bound of the set of
values oi d, for a particular e. If x is such that \x—a\ < B[8,],
then there is a d, such that |x-a|<^,. But if \x-a\<d„
|/(z) -b\ < e. Hence, if \x-a\<B[d.l \f{x) -b\ < e.
Theorem 43. The limit of the least upper bound of a function
fix) on a variable segment a x,a<x, as the end point apjrroaches
a, is the least upper bound of the valv£s approached by the function
as X approaches a from the right.
Proof.^Let / be the least upper bound of the values ap-
proached by the function as x approaches a from the right, and
let b{x) represent the upper bound of fix) for all values of x
on a X. Since Bfix) on the segment a Xi is not greater than
Bfix) on a segment a X2 if xi lies on a x^, bix) is a non-oscillat-
ing function decreasing as x decreases. Hence L bix) exists
by Theorem 21; and by Corollary 3, Theorem 40, L bix)^l. If
x=a
L bix)=k>l, then there are two vicinities of k, V]ik) contained
x=a
in V2ik) and V2ik) not containing I. By Theorem 26 a Vi*ia)
exists such that if x is in 7i*(a), bix) is in V^ik). Further-
more, by the definition of bix), if Xi is an arbitrary value of x
on Fi*(a), then there is a value of x in a Xj such that fix) is
in Vik). Hence k would be a value approached by fix) con-
trary to the hypothesis k>l.
§ 6. Bounds of Indetennination. Oscillation.
It is a corollary of Theorem 43 that in the approach to any
point a from the right or from the left the least upper bound
and the greatest lower bounds of the values approached by
fix) are themselves values approached by fix). The four num-
bers thus indicated may be denoted by
fia+0) = L fix) = L fix),
1=0+0 lia
84 INFINITESIMAL ANALYSIS.
the least upper bound of the values approached from the right:
/(a-0)=L/(x)=L/(x),
x=a — x—a
the least upper bound of the values approached from the left:
j{a+0)=L f(x)=L fix),
x'-a+O <—
the greatest lower bound of the values approached from the right :
/(o-0)=L/(x)=L/(x),
lio- — >
x±a
the greatest lower bound of the values approached from the left.
If all four of these values coincide, there is only one value
approached and L fix) exists. If /(a+0) and fia+0) coincide,
x=a ■
this value is denoted by /(a+0) and is the same as L fix).
x>a
x±a
Similarly if /(o-O) and /(a-0 ) coincide, their common value,
L fix), is denoted by /(a - 0) . The larger of /(o+O) and /(a-0)
x<a
is denoted by L f{x), and is called the upper limit of f{x) as
X approaches a. Similarly L f(x), the lower limit of /(x), is
x=a
the smaller of /(a+0 ) and fja-O ). L fix) and L fix) are
called the bounds of indetermination of fix) at x = a (Unbe-
stimmtheitsgrenzen) . See the Encyclopadie der mathematischen
Wissenschaften, II 41.
In order that a function shall be continuous at a point a it
is necessary and sufficient that
/(a)=/(a+0)= /(a+0 )=/(o-0)= /(a+0) . ... (a)
The difference between the greatest and the least of these
THEORY OF LIMITS. 85
values is called the oscillc.tion of the function at the point a.
It is denoted by 0(,/(x), and according to the theorem above is
equivalent to the lower bound of all values of 0/(x), where
0/(3;) =Bf{x)-Bf{x) for a segment V{a).
0^(1) is used for the oscillation of fix) on the segment a b. It
is sometimes also used for the oscillation of }{x) on the interval
a b. The word oscillation may also be applied to the difference
between the upper and lower bounds of the function on a V*{a).
Denote this by Ov*(a)fi^)- The lower bound of these values
may be denoted by Oa*f{x) and is the difference between the
greatest and the least of the four values /(a+0), /(o-O),
f(a + ), /(g-O )-
The reader wiU find it a useful exercise to construct exam-
ples and to enumerate the different ways in which a func-
tion may be discontinuous, according as /(o+O) or /(a— 0) exist
or do not exist, and according as /(o) does or does not coincide
with any of the values approached by fix). (Compare the
reference to the E. d. m. W. given above.) The principal
classification used is into discontinuities of the first kind, where
/(a+0) and /(a-0) both exist, and discontinuities of the second
kind, where not both /(a+0) and /(a-0) exist.
Theorem 44. If a is a limit point of [x], then a necessary and
sufficient condition that &2 and bi shall be the upper and lower
bounds of indetermination of fix), as x=a, is that for every set
of four numbers Oi, a2, Ci, C2, such that t
ai<6i<Ci<C2<62<ffl2,
there exists a V*ia) such thai for every x on V*ia)
ai</(x)<a2,
and for som£ x', x" on V*ia)
fix')>C2 and fix")<Cy.
t If 6, = - 00, o, = 6, replaces a,<bu If 6a = + =0 , Oj = b, replaces 6, < Oj.
86 INFINITESIMAL ANALYSIS.
Proof. — I. The condition is necessary. It is to be proved^
that if 62 and 61 are the upper and lower bounds of indeter-
mination of f{x), as x = a on [x], then for every four numbers
ai<bi<Ci<C2<b2<a2 there exists a V*{a) such that: —
(1) For all values of x on V*{a), a^<f{x)<a2. If this con-
clusion does not follow, then for a particular pair of numbers
Ci, a2, there are values of f{x) greater than 02 or less than Gj
for X on any V*{a), and by Theorems 24 and 40 there is at least
one value approached greater than 62 or less than 61. This
would contradict the hypothesis, and there is therefore a V*{a)
such that for all values of x on F*(a), ai<f{x) <a2.
(2) For some x', x" on F*(o), f{x')>C2 and /(x")<Ci. If
this conclusion should not follow, then for some V*{a) there
would be no x' such that f(x') >C2, or no x" such that f{x") <Cj,
and therefore bi and 62 could not both be values approached.
II. The condition is sufficient. It is to be proved that 62
and 61 are the upper and lower bounds of the values approached.
If the condition is satisfied, then for every four numbers O], 02,
Ci, C2, such that ai<bi<Ci<C2<b2<a2 there is a V*(a) such
that for all j's on V*{a) ai</(x)<a2, and for some x', x",
f{x')>C2 and f{x")<Ci. By Theorem 24 there are values ap-
proached, and hence we need only to show that 62 is the least
upper and 61 the greatest lower bound of the values approached.
Suppose some B>b2 is the least upper bound of the values
approached; 02 may then be so chosen that b2<a2<B, so that
by hypothesis for x on V*(,a) B cannot be a v^lue approached.
Again, suppose B<62 to be the least upper bound; c may then
be chosen so that B<C2, and hence for some value x' on each
F*(o), /(x')<C2. By the set of values f{x') there is at least
one value approached. This value is greater than C2>B.
Therefore B cannot be the least upper bound. Since the least
upper bound may not be either less than 62 or greater than
62, it must be equal to 62- A similar argument will prove bi to
be the greatest lower bound of the values approached.
CHAPTER V.
CONTINUOUS FUNCTIONS.
§ I. Contintiity at a Point.
The notion of continuous functions will in this chapter, as
in the definition on page 61, be confined to single-valued func-
tions. It has been shown in Theorem 34 that if /i(i) and fzix)
are continuous at a point x=a, then
/lU)±/2(l), /l(x)-/2(x), /i(x)//2(x), (/2(X)?^0)
are also continuous at this point. Corollary 1 of Theorem 39
states that a continuous function of a continuous function is
continuous.
The definition of continuity at a;=a, namely,
Lf{x)=f{a),
x-a
is by Theorem 26 equivalent to the following proposition :
For every £>0 </icre exists a S,>0 such that if \x-a\<S„
then \fix)-j{a)\<e.
It should be noted that the restriction Xt^u which appears
in the general form of Theorem 26 is of no significance here,
since for x=a, |/(i)— /(a)|=0<£. In other words, we may
deal with vicinities of the type V{a) instead of V*{a).
The difference of the least upper and the greatest lower
I — I
bound of a function on an interval a b has been called in
Chapter IV, page 85, the oscillation of /(i) on that interval,
and denoted by 0*(x). The definition of continuity and
Theorem 27, Chapter III, give the following necessary and
suflScient condition for the continuity of a function /(i) at the
87
88 INFINITESIMAL ANALYSIS.
point x=a: For every e>0 there exists a d,>0 such that if
\xi-a\<d., and |x2-a|<^„ then \f{xi)-f{x2)\<K- This means
that for all values of ii and xz on the segment {a — d,) (a + 5,)
B\fix,)-fix2)\<^<e,
and this means Bfix) —Bf{x) < e,
or 0:lt'Jix)<s.
Then we have
Theorem 45. If fix) is continvaus for x=a, then for every
e > there exists a V,{o-) such that on V,(o) the oscillation of f{x)
is less than e.
Theorem 46. // f{x) is continvaus at a point x=a and if
f(a) is positive, then there is a neighborhood of x=a upon which
the function is positive.
Proof. — If there were values of x, [x'] within every neighbor-
hood of x=a for which the function is equal to or less than zero,
then by Theorem 24 there would be a value approached by
/(a/) as a/ approaches a on the set [a/]. That is, by Theorem
40, there would be a negative or zero value approached by f{x),
which would contradict the hypothesis.
§ 2. Continuity of a Function on an Interval.
Definition. — ^A function is said to be continuous on an in-
terval a 6 if it is continuous at every point on the interval.
Theorem 47. If f{x) is cordinujous on a finite interval a b,
then for every e>0,a b can be divided into a finite number of equal
intervals upon each of which the oscillation of f{x) is less than e.f
t The importance of this theorem in proving the properties of continu-
ous functions seems first to have been recognized by Goursat. See his
Coura d' Analyse, Vol. 1, page 161.
CONTINUOUS FUNCTIONS. 89
Proof. — By Theorem 45 there is about every point oi a b a.
segment a upon which the oscillation is less than e. This set
of segments [o] covers a b, and by Theorem 11 a 6 can be
divided into a finite number of equal intervals each of which is
interior to a <i; this gives the conclusion of our theorem.
Theorem 48. {Unijorm continuity.) If a function is con-
I — I
tmuous on a finite interval a b, then for every £ > there exists a
d,>0 such that for any two values of x, Xi, and X2, on a b where
\Xi-X2\<d,, |/(Xi)-/(X2)|<£.
Proof. — ^This theorem may be inferred in an obvious way
from the preceding theorem, or it may be proved directly as
follows :
By Theorem 27, for every e there exists a neighborhood
V^(x') of every a/ of a b such that if Xi and X2 are on V (x'),
then |/(a;i)-/(x2)|<£. The V'{xfys constitute a set of seg-
I— I
ments which cover a b. Hence, by Theorem 12, there is a, d,
such that if \xi—X2\>S„ Xi and X2 are on the same F(x') and
consequently |/(xi)— /(x2)l<e.
The uniform continuity theorem is due to E. Heine.! The
proof given by him is essentially that given above.
In 1873 LiJROTH J gave another proof of the theorem which
is based on the following definition of continuity :
A single-valued function is continuous at a point x=a' if
for every positive e there exists a d, such that for every Xi and
X2 on the interval a-d, a + S„ |/(a;i)-/(x2)|<« (Theorem 45).
By Theorem 42 there exists a greatest d for a given point
and for a given e. Denote this by J,(x). If the function is con-
I — I
tinuous at every point of a b, then for every £ there will be
a value of J,{x) for every point of the interval, i.e., d,{x), for
any particular e, will be a single-valued function of x.
t E. Heine: Die Elemente der Functionenlehre, Crelle, Vol. 74 (1872), p. 188.
1 Luroth: Bemerkung iiber Gleichmdssige Stetigkeit, Mathematische Anna-
len, Vol. 6, p. 319.
90 INFINITESIMAL ANALYSIS.
The essential part of Luroth's proof consists in establishing
the following fact: If f(x) is continuous at every point of its
interval, then for any particular value of e the function i/x)
is also a continuous function of x. From this it follows by
Theorem 50 that the function J,{x) will actually reach its greatest
lower bound, that is, will have a minimum value; and this
minimum value, like all other values of d„ will be positive.f
This minimum value of J (x) on the interval under consideration
will be effective as a <?, independent of x.
The property of a continuous function exhibited above is
called uniform continuity, and Theorem 48 may be briefly
stated in the form: Every function amtimwus on an interval
is uniformly continuous on that interval.t
This theorem is used, for example, in proving the integrability
of continuous functions. See page 157.
I — I .
Theorem 49. // a function is continuous on an interval a b, it
is bounded on that interval.
Proof. — By Theorem 46 the interval a b can be divided into
a finite number of intervals, such that the oscillation on each
interval is less than a given positive number s. If the number
of intervals is n, then the oscillation on the interval a 6 is
less than ne. Since the function is defined at all points of the
interval, its value being f{xi) at some point Xi, it follows that
I — I
every value of f(x) on a 6 is less than f{xi) +ne and greater
than /(xi)-n£; which proves the theorem.
Theorem 50. // a function f{x) is continuous on an interval
*
t It is interesting to note that this proof will not hold if the condition
of Theorem 26 is used as a definition of continuity. On this point see N. J.
Lennes: The Annals of Mathematics, second series, Vol. 6, p. 86.
Jit should be noticed that this theorem does not hold if "segment"
is substituted for "interval," as is shown by the function — on the segment
1, which is continuous but not uniformly continuous. The function is
defined and continuous for every value of x on this segment, but not for every
value of X on the interval 1.
CONTINUOUS FUNCTIONS. 91
I — I
a b, then the function assumes as values its least upper and
its greatest lovoer hound.
Proof.— By the preceding theorem the function is bounded
and hence the least upper and greatest lower bounds are finite.
By Theorem 19 there is a point k on the interval a b such that
the least upper bound of the function on every neighborhood of
x=kis the same as the least upper bound on the interval a b.
Denote the least upper bound of f{x) on a 6 by B. It follows
from Theorem 43 that 5 is a value approached by fix) as x
approaches k. But since L f{x)=f{k), the function being con-
tinuous at x=-k, we have that f{k) = B. In the same manner
we can prove that the function reaches its greatest lower
bound.
Corollary. — If A; is a value not assumed by a continuous
, ■ . I — I
function on an interval a b, then f{x) -k or k-f{x) is a con-
tinuous function of x and assumes its least upper and greatest
lower bounds. That is, there is a definite number i which is
the least difference between k and the set of values of /(x) 'on
the interval a b.
I — I
Theorem 51. If a function is continuous on an interval a b,
then the function takes on all values between its least upper and
its greatest lower bound.
Proof. — If there is a value k between these bounds which
is not assumed by a continuous function f{x), then by the
corollary of the preceding theorem there is a value i such that
no values of fix) are between k—J and k+J. With e less than
J divide the interval a b into subintervals according to
Theorem 47, such that the oscillation on every interval is less
than £. No interval of this set can contain values of fix)
both greater and less than k, and no two consecutive intervals
can contain such values. Suppose the values of fix) on the
first interval of this set are all greater than k, then the same is
92 INFINITESIMAL ANALYSIS.
true of the second interval of the set, and so on. Hence it
I — I
follows that all values of f{x) on a b are either greater than or
less than k, which is contrary to the hypothesis that k Ues be-
tween the least upper and the greatest lower bounds of the
function on a b. Hence the hypothesis that /(x) does not
assume the value k is untenable.
By the aid of Theorem 51 we are enabled to prove the fol-
lowing :
Theorem 51a. // /i(x) is continuous at every point of an inier-
' — I
val a' b' except at a certain point a, and if
Lfi{x) = + cc and L/2(x)= — 00,
then for every b, finite or +00 or — 00 , there exist two sequences
of points, [xj] and [x/] (i=0, 1, 2, . . . ), each sequence having a
as a limit point, such that
L i/ife)+/2(x/)!=6.
Proof. — Let [x/] be any sequence whatever on a' b' having a
... . I — I
as a limit point, and let Xq be an arbitrary point of a' V. Since
/i(x) assumes all values between /i(xo) and +00, and since
^ f2{x) = — oo, it follows, in case b is finite, that for every i
greater than some fixed value there exists an Xi such that
fl{Xi)+f2ix/)=b.
In case 6 = -I- « , Xj is chosen so that
fl{Xi)+f2iXi')>i.
Corollary. — Whether /j(x) and /2(x) are continuous or not,
if L /i(x)=4-oo and L /2(x)=-oo, there exists a pair of
CONTINUOUS FUNCTIONS. 93
sequences [xj and [x/] such that
L {/l(Xi)+/2(x/)|
»=«
is + 00 or — 00 .
Theorem 52. // y is a function, f{x), of x, monotonic and con-
I — I
tinuous on an interval a b, then x=f~^{y) is a function of y
which is monotonic and continuous on the intenal f{a) f{b).
Proof. — By Theorem 20 the function f~'^{y) is monotonic and
has as upper and lower bounds a and h. By Theorems 50 and
51 the function is defined for every value of y between and
including /(a) and f{b) and for no other values. We prove the
I — I
function contmuous on the interval /(a) f{b) by showing that
it is continuous at any point y=yi on this interval. As y
I- — I
approaches j/i on the interval /(a) j/i, f~^(y) approaches a definite
limit g by Theorem 25, and by Theorem 40 a<glf-^{yi)±b. If
g<f~^{yi), then for values of x on the interval g /(j/j) there is
no corresponding value of y, contrary to the hypothesis that f{x)
I — 1
is defined at every point of the interval a b. Hence g=f~'^{yi),
and by similar reasoning we show that f~^{y) approaches /~H2/i)
I — —*]
as y approaches 2/1 on the interval, 2/1 / ^{b).
Theorem 53. // f{x) is single-valued and continuous with A, B
I — I
as lower and upper bounds, on an interval a b and has a single-
I — I . I— I
valued inverse on the interval, A B then f{x) u monotonic on a b.
Proof. — If f{x) is not monotonic, then there must be three
values of x,
Xi<X2<X3,
such that either f{xi)^f(x2)'^f{xz)
or /(Xi)^/(X2)/^(X3).
In either case, if one of the equality signs holds, the hypothesis
tnat /(x) has a single-valued inverse is contradicted. If there
94 INFINITESIMAL ANALYSIS.
are no equality signs, it follows by Theorem 51 that there are
two values of x, Xt and x^, such that
Xi<X4<X2<3;5<2:3,
and /(X4)=/(X5), in contradiction with the hypothesis that f{x)
has a single-valued inverse.
Corollary.— li f{x) is single-valued, continuous, and has a
single-valued inverse on an interval a b, then the inverse func-
I — I
tion is monotonic on A B.
§ 3. Functions Continuous on an Everywhere Dense Set.
Theorem 54. // the functions fiix) and /2(x) are continuous
on the interval a b, and if fi{x)=f2{x) on a set everywhere dense,
then /i(x) =/2(2) on the whole interval.^
I — I
Proof. — Let [a/] be the set everywhere dense on a 6 for
which, by hypothesis, /i(x) =fi{x). Let x" be any point of the
interval not of the set [a/]. By hypothesis x" is a limit point of
the set [x'], and further /i(x) and /2(x) are continuous at x=x".
Hence L /i(x) =/i(x")
and L /2(x) =/2(x").
X=l"
But by Theorem 41 L fi{x')=L fi (x) ,
x'=i" lil"
and by Theorem 41 L f2{xf) =L fzix).
x'il" x=x"
Therefore /i(x") =/2(x").
t I.e., if a function f(x), continuous on an interval a b, is known on an
everywhere dense set on that interval, it is known for every point on that
interval.
CONTINUOUS FUNCTIONS. 95
I — I
Definition. — On an interval o 6 a function /(x') is uniformly
continuous over a set [x'] if for every £>0 there exists a 5,>0
such that for any two values of a/, Xi, and X2' an a b, for which
\xi' -X2'\<d„ \}{x,').-f{x2')\<e.
Theorem 55. // o function f{x') is defined on a set everywhere
I— I
dense on the interval a b and is uniformly continuous over that
set, then there exists one and only one function f{x) defined on
I — I
the full interval a b such that:
(1) /(i) is identical with f{xf) where fix') is defined.
1 — I
(2) f{x) is continuous on the interval a b.
I — I
Proof. — Let x" be any point on the interval a b, but not
of the set [a/]. We first prove that
L/(x')
x' £1"
exists and is finite. By the definition of uniform continuity, for
every e there exists a d, such that for any two values of x', Xi,
and x/, where |xi'-i2'|<5., |/(xi') -/(a;2')l<^- Hence we
have for every pair of values Xi' and Xz where |xi'-x"|<y
and 1x2' -x"| <Y that |/(xiO -/(xz') | < £. By Theorem 23 this
is a sufficient condition that
L/(x')
x-^x"
shall exist and be finite.
Let fix) denote a function identical with fixf) on the set
[x'] and equal to
Lfix')
at all points x". This function is defined upon the continuum,
96 INFINITESIMAL ANALYSIS.
since all points x" on a b are limit points of the set [a/]. Hence
the function has the property that L /(x')=/(x) for every x
Xi -X
of a b.
We next prove that f{x) is continuous at every point on the
I — I
interval a b, in other words that f{x) cannot approach a value
b different from /(zi) as x approaches Xi. We already know
that /(z) approaches f{xi) on the set [x']. If 6 is another value
approached, then for every positive e and d there is an x^ such
that
\X.,-Xi\<d, \f{X.,)-b\<€ (1)
Since f{xa) =L f{x') we have that for every e>0 there exists a
d,>0 such that for every x' for which \x' -x^| <5„
\fix')-f{x.,)\<e (2)
From (1) and (2) we have
\m-b\<2e (3)
Since the d of (1) is any positive number, there is an x^ on
every neighborhood of xj, and hence by (2) and (3) an a/ on
every neighborhood of Xi such that |/(x') — ?>| <2£, e being arbi-
trary and b a constant different from /(xi"). But this is con-
trary to the fact proved above, that L j{x') exists and is equal
to /(xi). Hence the function is continuous at every point of the
I— I
interval a b. The uniqueness of the function follows directly
from Theorem 54. ,
This theorem can be applied, for example, to give an ele-
gant definition of the exponential function (see Chap. III). We
first show that the function a" is uniformly continuous on
the set of all rational values between xi and X2, and then define
CONTINUOUS FUNCTIONS. 97
a* on the continuum as that continuous function which coin-
— TO
cides with a" for the rational values — . The properties of the
function then follow very easily. It will be an excellent exer-
cise for the reader to carry out this development in detail.
§ 4. The Exponential Function.
Consider the function defined by the infinite series
x^ x^ x^
Applying the ratio test for the convergence of infinite series
we have
.71! ■ (n-1)! n
If n' is a fixed integer larger than x, this ratio is always less than
— < 1 The series (1) therefore converges absolutely for every
n'
value of X, and we may denote its sura by
e(x).
From Chap. I, page 17, we have that
Theorem 56.
.iMT-
where [n] is the set of att positive irUegers, exists and is equal to
e(x) for all values of x.
98
INFINITESIMAL ANALYSIS.
Proof.— Let E„(x) = 2' ri
fc=oA;!
(where 0! = 1).
Then, since
\ n) " ■^(n-l)!'n"^(n-2)!-2!W "^•"'^nlW '
it follows that
(x\ "I " / 1 t? ^ \
1+-) = I (t-.-t ,,,• , J x*
< " /I to(w-I) ■■■ (n-A: + l)
)m
< -^ ri~~fc F •
A;!n*
Now, since
n*-(n-A; + l)* = (A-l){n*-i+n*-2.(n-A + l)+...
+ (n-A; + l)*-M<(i-l)A;-n*-i,
it follows that
^„(x)-(l+3"
< 2'
.^•e{\x\)
~k=.2ik-2)l-n^ n
For a fixed value of x, therefore, we have
(l+|)"=E„(x) + ei(»),
where ei(n) is an infinitesimal as n= oo.
At the same time
e{x)=E„{x) + £2{n),
where e2{n) is an infinitesimal as n= w.
Hence
L (l+-Y=e(x).
CONTINUOUS FUNCTIONS. 99
Theorem 57. L (iH — ) ,
where [z] is the set of all real numbers, exists and is equal to e{x).
Proof. — If z is any number greater than 1, let nz be the
integer such that
ni^z<nz + l.
Hence,it«>0, l+£i l+|>l + j^j ■ . . . . (1)
Henc (i+|)"""Mi+j)'>('+S:Ti)""' ' ' <^'
« (-3(-3""=(-?)'>(-i:Vi)-*"-V- (3)
Since L (l+f ) =1, and L (l+r4l)=l.
and L (l+f) =e(x), and L (l+Z-Tj) =e(a;;,
the inequaUty (3), together with Corollary 3, Theorem 40, leads
to the result:
L (l+4)'=e(x).
The argument is similar if i<0.
Corollary. ^L^(l+-) =e{x),
■where [z] is any set of numbers with limit point + oo.
Theorem 58. The function e{x) is the same as e^ where
1 1 ,
100 INFINITESIMAL ANALYSIS.
Proof. — By the continuity of 3^ as a function of 2 (see Corol-
lary 2 of Theorem 39), it follows that, since
L (1+-)" =e,
L (1+-) =e'.
where 2=na;. Hence by Theorem 39
&=L (1+-)'
and by the corollary of Theorem 57 the latter expression is equal
to e(x) , Hence we have
ex = i+a;+|+|^+ (1)
(1) is frequently used as the definition of e*, a' being defined
as e^logeO.
CHAPTER VI.
INFINITESIMALS AND INFINITES.
§ I. The Order of a Function at a Point.
An infinitesimal has been defined (page 75) as a function f{x)
such that
L fix) =0.
A function which is unbounded in every vicinity of a; = a
is said to have an infinity at a, to be or become infinite at
x = a, or to have an infinite singularity at x = a.t The recipro-
cal of an infinitesimal at x=a is infinite at this point.
A function may be infinite at a point in a variety of ways :
(a) It may be monotonic and approach +00 or - 00 as
x=a; for example, - as a; approaches zero from the positive
side.
(5) It may oscillate on every neighborhood of x=a and
still approach + 00 or — 00 as a unique limit; for example,
• 1 ^
sm--l-2
X
as X approaches zero.
t It is perfectly compatible with the.se statements to say that while
fix) has an infinite singularity at x=a, /{o)=0 or any other finite number.
For example, a function which is — for all values of x except x = is left
undefined for i=0 and hence at this point the function may be defined
as zero or any other number. This function illustrates very well how a
function which has a finite value at every point may nevertheless have infinite
smgularities.
101
102 INFINITESIMAL ANALYSIS.
(c) It may approach any set of real numbers or the set of
all real numbers; an example of the latter is
. 1
sm-
X
as X approaches zero. See Fig. 13, page 64.
(d) + 00 and — » may both be approached while no other
number is approached; for example, - as a; approaches zero from
both sides.
Definition of Order. — If /(x) and 0(x) are two functions
such that in some neighborhood V*{a) neither of them changes
sign or is zero, and if
where k is finite and not zero, then f{x) and <f>{x) are said to be
oi the same order aX- x= a. If
x=a<{>ix) '
then f{x) is said to be infinitesimal with respect to 4>{x), and
<fi{x) is said to be infinite with respect to f(x). If
L -rr^ = + 00 or — 00
then, by Theorem 37, <f>ix) is infinitesimal with respect to f(x),
and fix) infinite with respect to <f){x). If f{x) and <p{x) are both
infinitesimal at x=a, and f(x) is infinitesimal with respect to
<l){x), then fix) is infinitesimal of a higher order than <^(x), and
^ix) of lower order than fix). If ^(x) and fix) are both infinite
at x=o, and fix) is infinite with respect to ^(x), then /(x) is
INFINITESIMALS AND INFINITES 103
infinite of higher order than <i>{x), and 0(x) is infinite of lower
order than /(x).t
The independent variable x is usually said to be an infini-
tesimal of the first order as x approaches zero, x^ of the second
order, etc. Any constant j^O is said to be infinite of zero order,
- is of the first order, ^ of the second order, etc. This usage,
however, is best confined to analytic functions. In the general
case there are no two infinitesimals of consecutive order. Evi-
dently there are as many different orders of infinitesimals be-
tween X and a;2 as there are numbers between 1 and 2; i.e., xi+*
is of higher order than x for every positive value of k.
c,. -r /l(x) 1 , ^ foix)
oince L j-T-T^T whenever L '-rT~=k, we have
x-aj2\X) K x=a hW
Theorem 59. If /i(x) is of the same order as fzix), then fzix)
is of the same order as f\{x).
Theorem 60. The function cf{x) is of the same order as f{x),
c being any constant not zero.
Proof.— By Theorem 34, L ^^=c.
Theorem 61. If fiix) is of the same order as fiix), and /2(x)
is of the savfie order as fsix), then fiix) and fsix) are of the same
order.
t This definition of order is by no means as general as it might possibly
be made. The restriction to functions which are not zero and do not change
sign may be partly removed. The existence of
x=a't>(.x)
is dispensed with for some cases in § 4 on Rank of Infinitesimals and In-
finites. For an account of still further generalizations (due mainly to
Cauchy) see E. Borel, Series a Termes Positifs, Chapters III and IV, Paris,
1902. An excellent treatment of the material of this section together with
extensions of the concept of order of infinity is due to E. Borlotti, CaJr
colo degli Infinitesimi, Modena, 1905 (62 pages).
104 INFINITESIMAL ANALYSIS.
Proof. — By hypothesis L 7^,— - = fci and L r-r-r=k2.
By Theorem 34, ^L ^-^ ;£/^=,£ WV
(By definition, /2(a;) 5^0 and jz{x) y^O for some neighborhood of
x=a.) Hence
7- A^^) 7 7
x=al3W
Theorem 62, 7/ /i(x) and /2(x) are infinitesimal (infinite)
and neither is zero or changes sign on some V*{a), then f\{x) ■f2{x)
is infinitesimal (infinite) of a higher order than either.
Proof. L^-^fy^^=L/i(x)=0. (±00.)
xia /2W xia
Theorem 63. If fi(x), . . . , fn(x) have the same sign on some
V*(a) and if f2(x), . . . , fn(x) are infinitesimal (infinite) of the
same or higher (lower) order than fi(x), then
fl(x)+f2(x)+f3(x)+...+fn(x)
is of the same order as fi(x), and if /2(x), faix), . . . , f„(x) are of
higher (lower) order than fi(x), then fi(x)±f2(x)±f3(x)±. . .+fn(x)
is of the same order as /i (x) .
Proof. — ^We are to show that
J fl(x)+f2(x)+...+f„(x)
i=a fl(x)
By hypothesis,
xi fAx) -"" xta fl(x) -"" •'•' xia M^^^"'
A T A(^) 1
INFINITESIMALS AND INFINITES. 105
Hence, by Theorem 30,
r [hix) f2ix) hix) Ux)\
since all the ^'s are positive or zero.
Similarly, under the second hypothesis.
^ Mx)±f2(x)±...±Ux) ^ j^
/iJ^ + M^li +/"W ]
\h(x) hix) ••• /i(x)i
= 1+0 + . ..+0 = 1.
Theorem 64. — // /3(x) and fi{x) are infinitesimals with re-
spect to fi{x) and /aCx), then
J \h{x)+fz{x)\-\h{x)+j^{x)]
^ , J l/i(a:)+/3(x}-l/2(x)+/4(x)|
Froot. L ,/,.,, ,
^ ^ h(x) -hix) +fx(x) -Uix) +/3(X) ■f2{x) +f3ix) -f.jx )
xj,a h(x)-f2{x)
J /l(x)-/2fa) , J /1(X)-/4 (X) J hix)-f2(.x) f3{x)-U(x)
xLhix) •/2(X) xl'a hix) ■f2{x) JlafM ■ h{x) \Lh{x) -^ix) ^•
§ 2. The Limit of a Quotient.
Theorem 65. If as x = a, £i(x) is an infinitesimal vnth respect
to /i(x) and szix) with respect to fzix), then the valv£s approached by
/l(x) + £l(x) ^^^ M£)
/2(X)+«2(X) ^^ /2(X)
•as X approaches a are identical.
106 INFINITESIMAL ANALYSIS.
Proof.— This follows from the identity
^i(x)\
/l(x)+£i(x) _ /i(x)
f2{x)+e2{x) fiix)' /i ,i2_(i)
1 +
f2{x)'
Since tt4 and 4^,-^ are infinitesimal.
flix) }2{X)
Corollary. — If /i(x) and /2(x) are infinite at a;=a, then
fiix)+c /i(x)
f,{x)+d ^""^ /2(X)
approach the same values.
Theorem 66. // L ^,= L ^,^k, and if L^,=l
• ^ V ,. i. T hi. x)+h{x) J hix)
IS finite, then lc= L -,—, ^ , , , ^ = Li , , . ,
' ' x^a^\{x)-V^2{x) xUaA-^^^y
provided l^ -1 if k is finite, and provided l>0 if k is infinite.
fi{x)+f2{x) f2{x) fi(x)<j>2{x)-f2{x)Mx)
Proof.
4>iix)+Mx) Mx) Mx){<f>i{x)+Mx)) '
/l(x)+/2(x) _ Mx) //i(x) /2(X)\ 7 1 \
4>l{x)+(j>2{x) 4>2{X) \(/)i(x) (f>2{x)l "I ^2{X) ).
In case k is finite, the second term of the right-hand member
is evidently infinitesimal if Ij^—l and the theorem is proved.
In the case where k is infinite we write the above identity in
the following form:
/1(X)+/2(X) /l(x) 1__.^,/2(X) 1
4>i{x)-\-<f>2(x) 0i(x) ^2(3:) (f>2{x) 9!>i(x) '
«^i.(x) %2(a;)
INFINITESIMALS AND INFINITES. 107
Both terms of the second member approach + oo or both - oo
if Z>0.
Corollary. ~li <f>i{x) and 962(3;) are both positive for some
VHa), and if A= L ilM= L ^4^, then L i#tM^ =k
whenever k is finite. If k is infinite, the condition must be added
that , , - has a finite upper and a non-zero lower bound.
Theorem 67. // /i(x) and /2(x) are both infinitesimals asx = a,
then a necessary and sufficient condition that
L , , - =k {k finite and not zero)
x = a /2W
is that in the equation fi(x)=k-f2{x) + s(x), e{x) is an infini-
tesimal of higher order than /i(x) or fzix).
Proof. — {lyThe condition is necessary. — Since L r~-( =/fc,
X4a/2(X)
}2{X) '' + '^^'^^'
or /i(x)=/2(2;)-A;+/2(x)-£'(a;), where L £'(x)=0 (Theorem 31).
x~a
By Theorems 60 and 61, fi(x) and /2(x) -k are of the same order,
since kf^O, while by Theorem 62 s'{x)-f2{x) is of higher order
than either /i(x) or f2(x). Hence the function s{x) = s'(x) •/2(x) is
infinitesimal.
(2) The condition is sufficient. — By hypothesis /i(x) =
f2(,x)-k + £(x), where /i(x) and /2(x) are of the same order as
£(t)
x=a, while e(x) is of higher order than these. Let e'{x) =ryT,
/2W
/ (x)
which by hypothesis is an infinitesimal. We then have j-j—
t / \
= k + e'{x). Hence, by Theorem 31, L j-j-r = k.
108 INFINITESIMAL ANALYSIS.
§ 3. Indeterminate Forms.t
a
Lemma. — // j- and -y are any two fractions such that b and d
are both positive or both negative, then the value of
a + c
b+d
I — I
,. , . . a c
lies on the interval j- -7.
Proof. — Suppose b and d both positive and
a ^a+c
b^b+d'
then db+ad^ab+ be.
.'. ad^ be;
.'. cd+ad^ cd+bc;
a+c ^c
• ■ 6+d = d*
The other cases follow similarly.
Theorem 68. // /(x) and (f>{x), defined on some F(+qo), are
'both infinitesimal as x approaches + 00 , and if for some positive
number h, (f>{x+h) is always less than <f>{x) and
, f{x+h)-f{x) ^^
then
„4>{x+h)—4>{x)
exists and is equal to k.X
f The theorems of this section are to be used in § 6 of Chap. VII.
% This and the following theorem are due to O. Stolz, who generalized
them from the special oases (stated in our corollaries) due to Cauchy. See
INFINITESIMALS AND INFINITES. 109
Proof.— Let Vi(k) and Vzik) be a pair of vicinities of k such
that Vzik) is entirely within T^(^-). By hypothesis there exists
an h and an A'o such that if j > A'2,
f{x + h)-fix)
(l>{x+h)-i>{x) W
is in V2{k). Since this is true for every x>X2,
f{x + 2h)-fix + h)
<i>ix+2h)-<j,{x+h) (2)
is also in V2{k). From this it follows by means of the lemma
fV,of f(x+2h)-fix)
^^^^ <l>{x+2h)-<pixy (3)
whose value is between the values of (1) and (2), is also in V2{k).
By repeating this argimient we have that for every integral
value of n, and for every x>X2,
jix+nh)-f{x)
(j)ix+nh)—<p{x)
is in F2(*)-
By Theorem 65, for any x
^ fjx+m-fix) _ /(i)
n=<x><f>{x+nh)-^{x) <j>{xy
Hence for every a; and for every e there exists a value of n, Nx„
such that if n> Nxi,
f{x+nh)-fix) fix)
<j>{x+nh)—(j)(x) <f>{x)
<£.
Taking e less than the distance between the nearest end-points
fix)
cf Viik) and V2ik) it is plain that for every i>A'2, ttt is
(pyX)
Stolz und Gmeiner, Functionentheorie, Vol. 1, p. 31. See also the referenca
to BoBTOLOTn given on page 103.
110 INFINITESIMAL ANALYSIS.
on Vi{k), which, according to Theorem 26, proves that
Corollary. — If [n] is the set of all positive integers and
^(n + 1) < <pin) and f(n) and ^(n) are both infinitesimal as w = qo ,
then if
/(n + l)-/(n) _
it follows that L -rr^ exists and is equal to k.
Theorem 69. If }{x) is bounded on every finite interval of a
certain y( + oo), and if ^{x) is monotonic on the same F( + oo)
and L ^(x) = +00 , and if for some positive number h
fix+h)-f{x)
,^^<j>{x+h)-<}>(x) "'
then L ~rl
exists and is equal to k.
Proof. — By hypothesis, for every pair of vicinities Vi{k) and
y2{k), V2(k) entirely within Vi(k), there exists an X2 such that
if x>X2, then
f{x+h)-f{x)
4{x+h)-<t>{x)
is in Viik). From this it follows as in the last theorem that
f{ x+nh)-f{x)
<j>(x+nh) —<f>{x)
is in Vzik). Now make use of the identity
fjx+nh) _ f{x+nh)-f{x) fix)
4>{x+nh) (j>{x+nh) ^{x+nh)
INFINITESIMALS AND INFINITES. Ill
f{x+nh)-fix) / <f>{x) \ fix)
'4>{x+nh)-<j>{x)\ <j>ix+nh)/'^<f>{x+nhy ' ' ^^
Let [a/] be the set of all points on the interval X2 X2+h, and
for this interval let A2 be an upper bound of \f{x^) | and B2 an
upper bound of <j>{x'). Then
cl>{x') _ B2
4>{xf +nh) ~ ''^^' "^ ^ <j>{X2+nh)
and JL/!./ , An = «2(a/, n) < — - —
<^(i' +n;i) ~ ''^ ' '"^ ^ <]>{X2 +nhy
Hence for every £ there exists a value of n, N^y, such that if
n>N£y^
£1(2/, n)<e and £2(2^, «)<« .... (2)
I — I
independently of xf so long as a/ is on X2 X2+h.
There are then three cases to discuss:
(1) A finite. (2) k= + <». (3) 4= -00.
(1) k finite. By the preceding argument, for x>Z2,
f{x+nh)-f{x)
<j>{x+nh)-<j){x)
is in V2(k), and hence
\fixf+nh)-f{xf)\ ^^, .
where ey, is the length of the mterval FaCA) and K the
absolute value of k.
Then, in view of (1),
fi^+r^ _ fjif +nh) -fix')
4>ix!+'nh) <l>ix'+nh)-4>ixf)
<iK+ev^€iix/,n) + e2ix',n).
112 INFINITESIMAL ANALYSIS.
Now take £v smaller in absolute value than the length of the
interval between the closer end-points of Vi(k) and V2ik). By
(2) there exists a value of n, N^y, such that if n>N£y,
-'i(^»<2(Ztt;;:)
and e2(x',n)<-^
for all values of 2/ on X2 X2 +h.
Hence for n> AT J J,
f(x'+nh) fix' + nh) -f{x')
4){x'+nh) ^{x'+nh)-4>{x')
«7 , «v
<^^+^^.)2(:^TT7;+T^'^
and since for a;> Z2 +Neyh there is an n> iVjj, and an a/ between
X2 and X2+h such that
3/+nh=x,
it follows that if a;> X2 +Ney,
]Kx) f(x'+nh)-fix') l
|^(x) 4>(.^+nh)-<f>ix)\'^^^'
fix)
and therefore, -j-p-!- is on Fi(A).
This means, according to Theorem 26, that
(2) A=+oo.
If the numbers wii and ma are the lower end pomts of
7i(A;) and Y^ih), then
<^(x'+nA)-,^(x')>"'2 forx'>Z2.
INFINITESIMALS AND INFINITES. 113
If ev is then chosen less than m2-mi, there will exist a value
of A'^g^ such that
for all values of n>Ngy independently of x' so long as a/ is in
I
X2 X2 + h. Then, in view of (1),
f{x'+nh) ii ^v \ ey ev 1 1 \
Since there is no loss of generality if m2> +1, this proves that
for X >X2 +Nsyn,
fix)
>m2 — £v>mi,
<P{x)
and hence -rrr is on VAk).
4>{x)
(3) A; = — 00 is treated in an analogous manner.
Corollary 1. If [n] is the set of all positive integers and if
<j>{n-\-\)> ^(n) and L ^(n) = 00 ,
11 = 00
. .f r /(n + l)-/(n)
then If „f.^(n + l)-^(n)-*' .
it follows that L -yt-x exists and is equal to k
„-=o?>W
I — I
Corollary 2. If f{x) is bounded on every interval, x (x + l),
and if L f{x + l)-}{x)=k,
I— 00
m
then L
XS3 00
x
exists and is equal to k.
114
INFINITESIMAL ANALYSIS.
§ 4. Rank of Infinitesimals and Infinites.
Definition. — If on some V*{a) neither /i(a;) nor /2(x) vanishes,
and
fi(x)
f2ix)
and
Ihix)
are both bounded as x approaches a,
f (x)
then fi{x) and /2(x) are of the same rank whether L j-i—. exists
or not.t
The following theorem is obvious.
Theorem 70. // /i(x) and /2(x) are of the same order, they
are of the same rank, and if fi{x) and f2{x) are of different orders,
they are not of the same rank. If fi{x) and ^(a;) are of the same
rank, they may or may not be of the same order.
Theorem 71. // fi{x) and fiix) are of the same rank as x
approaches a, then c-fi{x) and /2(x) are of the same rank, c being
any constant not zero.
Proof. — By hypothesis for some positive number M,
AW
hence
/2(X)
c-fiix)
<M and
/2(X)
hix)
<M,
/2(X)
<M-|c| and
f2{x)
c-fi{x)
<
M
Theorem 72. // /i(x) and /2(x) are of the same rank and
fiix) and fz{x) are of the same rank as x approaches a, then fi{x)
and fz{x) are of the same rank as x approaches a.
Proof. — By hypothesis,
fi{x)
/2(X)
<Mi and
/2(X)
hix)
<M2
in some neighborhood of 2;= a. Therefore
fiix)
f2{x)
/2(X)
/3(X)
<Mi-M2 or
/i(x)
fsix)
<Mi-M2.
1 1 and I -(sin — 1-2) are of the same rank but not of the same order
as I approaches zero.
INFINITESIMALS AND INFINITES.
In the same maimer
115
/2(X)
hix)
<Mi and
fsix)
f2{x)
<M2, whence
hix)
hix)
<Mi-M2.
Theorem 73. // /i(x) is infinitesimal {infinite) and does not
vanish on some V*{a), and if fiix) and fz{x) are infinitesimal
{infinite) of the sam£ rank as x approaches a, then fi{x) ■f2{x) is
0/ higher order than /aCx), and f\{x) ■fz{x) is of higher order than
}2{x). Conversely, if for every function, fi{x), infinitesimal {infi-
nite) at a, fi{x) ■f2{x) is of higher order than f3{x), and fi{x) ■fz{x)
is of higher order than f2{x), then f2{x) and /3(x) are of the same
rank.
fiix)
Proof. — Since
fsix)
by Theorem 33 that
is bounded as x approaches a, it follows
^ fi{x)-f2{x) _^
fsix)
which proves the first part of the theorem.
fsix)
Since Ukewise
/2(X)
is bounded, we have that
flix)-f3{x)
L
f2ix)
Suppose that for every fiix)
flix)-f2ix)
=0.
L
1=0
fsix)
=0 and L^^^W^^O,
and that /2(x) and fsix) are not of the same rank. Then, on
a certam subset [x J, L
r%fs{x')
f2{x)
same
0, or on some other subset
[x"l L ^=0. Let /i(x)=(7§ on the set [x'] for which
x'±al2iX ) /3W
^ M£)^Q j^j^d x-a on the other points of the continuum;
x=.Js{x) '
116 INFINITESIMAL ANALYSIS.
then /i(x) is an infinitesimal as x approaches a, while for the
set [x']
h{xf)-k{xf) _ Mxo U^),
which contradicts the hypothesis that
J h(x)-f3ix)
Li T~7~\ ~^'
x^a I2(X)
Similarly if on a certain subset L r-r^=0, we obtain a con-
t / \
tradiction by putting /i(x) =rT~:'
CHAPTER VII.
DERIVATIVES AND DIFFERENTIALS.
§ I. Definition and Illustration of Derivatives.
Definition.— If the ratio Z^_^ approaches a definite
limit, finite or infinite, as x approaches Xi, the derivative of
f{x) at the point Xi is the limit
J- fix) -/fa) _
x=xi X Xi
It is implied that the function }{x) is a single-valued function
Oh
Fig. 14.
of X. x—Xi is sometimes denoted by Jxi, and f{x)—f{xi) by
J/(xi), or, if 2/=/(x), by Ayi.
An obvious illustration of a derivative occurs in Cartesian
geometry when the fimction is represented by a graph (Fig. 14).
117
118
INFINITESIMAL ANALYSIS.
Here ^^^^ ^^^^^ is the slope of the line AB. If we sup-
X — Xi
pose that the line AB approaches a fixed du-ection (which
in this figure would obviously be the case) as x approaches Xi,
f(x) —f(xj)
then L -^^ — -^"-^ will exist and will be equal to the slope of
the limiting position of A B.
If the point x were taken only on one side of x\, we should
have two similar limiting processes. It is quite conceivable, how-
ever, that limits should exist on each side, but that they should
differ. That case occurs if the graph has a cusp as in Fig. 15.
Fig. 15.
These two cases are distinguished by the terms progressive
and regressive derivatives. WTien the independent variable
approaches its limit from below we speak of the progressive
derivative, and when from above we speak of the regressive
derivative. It follows from the definition of derivative that,
except in one singular case, it exists only when both these
limits exist and are equal. The exception is the case of a
derivative of a function at an end-point of an interval upon
which the function is defined. Obviously both the progressive
and the regressive derivative cannot exist at such a point. In
DERIVATIVES AND DIFFERENTIALS. 119
this case we say the derivative exists if either the progressive
or the regressive derivative exists.
Whether the progressive and regressive derivatives exist or
not, there exist always four so-called derived numbers (which
may be± w), namely, the upper and lower bounds of indeter-
mination of
/(x)-/(xi)
X—Xi '
a? x = Xi from the right or from the' left. (Compare page 84,
Chapter IV.) The derived numbers are denoted by the sjonbols
D, D, D, D,
analogous to the symbols on page £4. Of cotirse, in every cEise^
D^D and D^D.
— > < —
If we consider the curve representing the function
. 1
" X
at the point 2;=0, it is apparent that the Umiting position of
A B does not exist, although the function is continuous at the
point x=0 if defined as zero for x=0. For at every maxi-
mum and minimum of the curve sin—, a; -sin = ±x, and the
X X
curve touches the lines x=y and x=—y. That is, • — — - — —
^ ' X—Xi
approaches every value between 1 and —1 inclusive, as x
approaches zero.
The notion derivative is fimdamental in physics as well as
in geometry. If, for instance, we consider the motion of a
body, we may represent its distance from a fixed point as a
function of time, /(<). At a certain instant of time h its dis-
tance from the fixed point is /(ii), and at another instant ^2
it is f(t2) ; then
ti—tz
course indicated by the sign of the expression _
120 INFINITESIMAL ANALYSIS.
is the average velocity of the body during the interval of time
ti-t2 in a direction from or toward the assumed fixed point.
Whether the motion be from or toward the fixed point is of
f{t l}-f(t2) .
If we consider this ratio as the time interval is taken shorter
and shorter, that is, as <2 approaches h, it will in ordinary
physical motion approach a perfectly definite limit. This limit
is spoken of as the velocity. of the body at the instant ti.
Definition. — ^The derivative of a fimction y=f{,x) is denoted
df(x) diV
by fix) or by Di/(x) or -~^ '^^ T" /'(^) ^ ^^^° referred to as
the derived Junction of /(x) .
§ 2. Formulas of Differentiation.
Theorem 74. The derivative of a constant is zero. More
precisely: If there exists a neighborhood of Xi such that for every
valus of X on this neighborhood fix) =/(ii), then fixi) =0.
Proof. — In the neighborhood specified =0 for
every value of x.
Corollary. — If f(xi) exists and if in every F*(xi) there is a
value of X such that fix) =fixi), then fixi) =0.
Theorem 75. When for two functions /i(x) and fzix) the derived
functions fi'ix) and fz'ix) exist at xi it follows that, except in the
indeterminate case 00 — 00 ,
(a) // fsix) =/i(x) +/2(x), then fzix) has a derivative at Xi and
/3'(Xl)=/l'(Xi)+/2'(Xi).
(b) If fsix) =/i(x) •f2{x), then fsix) has a derivative at Xi and
fz'ix{) =/i'(xi) -/aCxi) +/i(xi) -//(xi).
(c) // fzix) =-7-r\, then, provided there is a F(xi) wpon which
./2(x) 7^0, fzix) has a derivative and
,,, . //(Xl)-/2(Xl)-/l(Xi)-//(Xi)
DERIVATIVES AND DIFFERENTIALS.
121
Proof. — By definition and the theorems of Chapter IV
(which exclude the case oo — oo),
(a) //(xO +h'{x.) = L /-iMzM^) + L /?M^MEl) (1)
x=Zi X X] x^x\ X — Xi
^ ^ I h(x)-h{x^) ^ Hx)-U{x{) I
^ ^ /i(x)+/2(x)-/ifa)-/2(xO
x-^x
I
X— a;i
= L
hix)-
X-
-/3(Xl)
-a;i
But
by definition,
/3'(Xl) =
__^Hx)-
-/3(a;i)
-Xi
(4)
Henoe /3'(a;i) exists, and /a'Cxi) =/i'(xi) +/2'(xi).
(&) h{x)=h(x)-Ux).
Whenever Xt^xi we have the identity
hix) -hjxi) _ /i(x) -Ux) -/i(xi) -/aCxi)
X — Xi X — Xi
_ /i(x) -/zCx) -/i(xi) -/aCx) +A(xi) -hjx) -/i(xi) -/zCxQ
X — Xj
=/2(a:)
f/ i(x)-/i(xi)
X— Xi
1 +,,(,) IteWa^l.
"' t X-Xi J
But the limit of the last expression exists as x=Xi (except
perhaps in the case 00 — 00 ) and is equal to
f2{x{)-U\x{)+h{x{)-U'{x{).
122 INFINITESIMAL ANALYSIS.
Hence L ^;^
exists and fs'ixi) =h{xi) -h'ixi) +/2'(xi) -fiixi).
(0 /^(^)=M^)-
The argument is based on the identity
/i(x)_/ifa)
/2(X) /2(Xl) _ /l(x)-/2fa)-/2(x)-/lfa) ,
x-ii /2(a;) 72(2:1) -(x-xi)
which holds when a; 7^X1 and when f2{x) 7^0.
3^^ fi(x)-f2(x^)-f2{x)-h(xr)
f2{x)-f2{Xl)ix-Xi)
Jljx) ■J2{x{)-h{x{) ■f2{x{)+h{Xi) ■U(Xl)-f2{x) -hJXi)
U{x)-J2{Xi){x-Xi)
/zfa) \h{x) -hjxi) I -/i(xi) {/2(x) -/aCzi) i
f2{x)-f2iXl)ix-Xl)
As before (excluding the case 00 — 00) we have
, ,. , /2(Xl)-/l^fa)-/2^fa)'-/lfa)
Corollary. — It follows from Theorems 74 and 75 of this
chapter that if ]2{x)=a-fi{x) where /i'(x) exists, then
/2'(x)=a-/i'(^).
Theorem 76. Ifx>0, then -^x^ = A; • x*"i.
DERIVATIVES AND DIFFERENTIALS. 123
(o) If A; is a positive integer, we have
L — ^ — = L 1 a;*- 1 + x*^2. 3.J _|_ .,. 2.4.3.^4-2 +2.jjr-ij
x=ii ■'' ^1 1=1,
171
{b) If A; is a positive rational fraction — , we have
x"— Xi" \x"/ — XXi"/
L/ = Li
JT ( J-'\"~l / J-^^n^Z ( 1\ ( l\n-l 2 i
~'\X^) + \X"/ • Vli"/ +. . .+ Ui"/ X"-Xi"
1 ( 1\'"-1
Tw:i-m\xi»/ ,
by the preceding case.
1 / J.\ "•""1 fn ——I
But —T-^:^^-m\xi") =^^^1" =A;-Xi*-i.
(c) If A; is a negative rational number and equal to — m,
then, by the two preceding cases,
^3,^, X-Xi arix, a;"-!!™ X-Xi Xi^"
= — Tnxi""*"^
But — TOXi~"'~^ = A;-x*"^
(d) If A; is a positive irrational number, we proceed as
follows :
124 INFINITESIMAL ANALYSIS
Consider values of x greater than or equal to unity. Let x
approach Xi so that x>xi. Since, by Theorem 23, x'' is a
monotonic increasing function of k for x>l, it follows that
t^^^^^u.}^ y^.u'.SL
X — Xi X — Xi X — Xi
for all values of k' less than k, and all values of x greater than x^.
If k' is a rational number, we have by the preceding cases that
xY'
KxJ -1
lii, X—Xi
Since Xi*~^ is a continuous function of k, it follows that for
every number N less than A;xi*~i there exists a rational num-
ber ki' loss than k such that
jV<A;i'-Xi*''-i<A;-2i*-i.
Hence, by Theorem 40,
xi«=-
X — Xi
cannot approach a value A^ less than A;xi*~i as x approaches Xi.
By a precisely similar argument we show that a number
greater than kxi''~'^ cannot be a value approached. Since there
is always at least one value approached, we have that
x'^-xi"
If x<Xi as X approaches Xi, we write
X*^ — Xi«
-=x''
X — Xi X\—X
and proceed as before. If A; is a negative number we proceed
DERIVATIVES AND DIFFERENTIALS. 125
as under (c). The case in which a;i < 1 is treated similarly. For
another proof see page 127.
Theorem 77. ^ log„ x=-- log„ e.
Proof. ^.2i?i^±^^)j:iog«_5=lw ^±i5
ix Jx ^ X
=i..o.(l.f)r..
But, by Theorem 57,
Therefore L — ^ — ^ ^2_ =, _ . i^g^ g_
Corollary. — log„ x = -.
Theorem 78. // }i'(x) exists and if there is a V{xi) upon which
fi(x) is continuous and possesses a single-valued inverse x = f2{y),
then J2{y) is differentiable and
If f'i^) is or +<x> or — co the convention
+ 00 — 00
is understood. Cf. Theorem 37.
Proof. — To prove this theorem we observe that
X = Xi ■^ ^l *=Ii ^ •''1
/l(x)-/i(Xi)
By the definition of single-valued inverse (p. 45),
x-X'^ _ hiy)-f2{yi)
/i(x)-/(xi) y-yi
t Theorem 78 gives a sufficient condition for the equality
dx dx'
dy
126 INFINITESIMAL ANALYSIS.
Hence, by Theorems 38 and 34 and 37,
x-x. x-xi u=m f2{y)-f2{yi) U'iy)'
]i.x)-f{xx) y-yi
Theorem 79. // (1) /i'(x) exists and is finite for x^Xi, and
fi(x) is continuous at x=Xi,
(2) f2'iy) exists and is. finite for 2/1 =fi{xi),
then ^/2{/i(xi)l=/2'(2/i)-/i'(2;i).t
Proof. — We prove this theorem first for the case when there
is a V*{xi) upon which fi(x) 7^/i(xi). In this case the following
is an identity in x :
/2{/l(x)i-/2l/lfa)i _ /2i/l(x)}-/2|/l(Xi)} /i(x)-/i(3:i)
X-Xi fi(x)-fiixi) ' X-Xi
By hypothesis (2) and Theorem 38,
' y^m y-yi xix. h{x)-fi{Xi)
By hypothesis (1),
Hence, by equation (1) and Theorem 34, we have the existence of
^fffC-rM r /2{/l(x)}-/2J/l(Xi)} f,..,,, .
^2|/i(a;)( = L^ ^3^^^ ■■f2{yi)-fi'(xi).
If /i(x)=/i(xi) for values of x on every neighborhood of
T=Xi, then, by hypothesis (1) and the corollary of Theorem 74,
/'(xi)=0.
t Theorem 79 gives a sufficient condition for the equality
dz _ dz dy
dx dydx'
DERIVATIVES AND DIFFERENTIALS. 127
Let [x'] be the set of points upon which /i(a;)7^/i(xi). (There
is such a set unless f(x) is constant in the neighborhood of
x = xi.) Then, by the same argument as in the first case, we
have
d
^,/2l/i(a;i)! =/2'(j/i)-/i'(xi)=0 for X on the set [x'].
Let [x"] be the set of values of x not included in [x']. Then
a^Mh (xi) } - ,,L , —:^r—^ = 0'
since the limitand function is zero. Hence both for the set [x']
and for the set [x"] the conclusion of our theorem is that the
derivative required is zero.
Theorem 80.
d
^^a^- a- log a.
Proof. — Let
2/ = a^,
therefore
log2/=x-logo
dy
and, by Theorem 77,
dx .
whence
dy
dx'
= j/-log a = a* logo.
This method also
affords an elegant proof of Theorem 76,
VIZ.,
d
;"=n2;"~'.
Let J/=x",
logj/=nlogx,
dy
dx _n
y 'x'
-r=n — =n-x"~i.
dx X
128
INFINITESIMAL ANALYSIS.
If
§ 3. Differential Notations.
y = f{x) and L
K,
x=a ^ •*'!
we denote f{x)-f{x{) hy Ay, and x-Xi by dx. Then, by
Theorem 31,
Mj^Ax-K+Ax- £{x),
where Ax- s{x) is an infinitesimal with respect to Ay and Ax for
x = a. This fact is expressed by the equation
dy=K-dx, where K=j'{x).
Here dy and dx are any numbers that satisfy this equation.
There is no condition as to their being small, either expressed
or implied, and dx and dy may be regarded as variable or
Fig. 16.
constant, large or small, as may be found convenient. When
either dx or dy is once chosen, the other is, of course, determined.
The numbers dx and dy are called the differentials of x and y
respectively.
DERIVATIVES AND DIFFERENTIALS 12&
In Fig. 16, f{xi) is the tangent of the angle CAB, dx is the
length of any segment AB with one extremity at A and parallel to
the j-axis, and dy is the length of the segment BC. If x is
regarded as approaching xi, then AB' is the infinitesimal ix,
WD' is Jy, while Wc' is e{x) -Ax. Hence, by Theorem 73, 'WC'
is an infimtesimal of higher order than Ax or Ay.
We thus obtain a complete correspondence between deriva-
tives and the ratios of differentials. Accordingly, for any for-
mula in derivatives there is a corresponding formula in differ-
entials. Thus corresponding to Theorem 75 we have :
Theorem 8i. When for two junctions /i(x) and /zCx)
dfi{x)=fi'{x)-dx and df2{x)=J2ix)-dx at Xi, it follows
that
(a) ///3(x)=/i(x)+/2(x),
(km dfsixi) = |/i'(xi) +/2'(xi) \dx
=d/i(xi)+d/2(xi).
ib) ///3(x)=/i(x)-/2(x),
then dU{x,) = \h'{x^)-U'ix{)\dx
= d/i(xi)-d/2(xi).
(c) // h{x)=h{x)-f2{x),
then d/3(xi) = {/i(xi)-/2'(xi)+/2(xi)+/i'(xi)! -dx
= /i(a:i)-d/2(xi)+/2(xi)-d/i(xi).
, ^, , , \f2(x,)-h'{x,)-h(xi)-f2'ixr)\-dx
then d/3(xi)= {f2{xi)\^
/2(x,)-d/i(x,)-/i(x,)d/2(xi)
i/2(Xl)P
The rule obtained on page 123 et seq. that the derivative of x* is
Jfc • X*- 1 corresponds to the equation dx* = A; • x*- ^ • dx. If , in the
130 INFINITESIMAL ANALYSIS.
equation dy=f{x)dx, dx is regarded as a constant while x
varies, then dyisa. function of x. We then obtain a differential
d2(dy) = {f'{x)-dx\d2X in precisely the same manner that we
obtain dy=f{x)-dx. Since d2X may be chosen arbitrarily, we
choose it equal to dx. Hence d{dy) =f"{x)dx^. We write this
d'^y=f'(,x)-dx'^.
The differential coefficient f"{x) is clearly identical with the de-
rivative of fix). In this manner we obtain successively
d^y=f^^\x)-dx^, etc.
We may write these results,
^^fix) ^=f"(x) ^-n-Hx)
dx '^^>' dx' ^ ^^^'•••-■dx"~' ^^''•
Evidently the existence of the differential coefficient is coexten-
sive with the existence of the derivative.
§ 4. Mean-value Theorems.
Theorem 82. // /(x) has a unique and finite derivative at
x = xi, then f(x) is continuous at x\.
Proof.— The proof depends upon the evident fact that if
f(x)-f{xi) approach anything but zero as x approaches Xi, then
one of the values approached by
m-f{x,)
X — Xi
is +00 or — 00 .
Definition.— The function f(x) is said to have a maximum
at x=xi if there exists a neighborhood V(xi) such that
(1) No value of fix) in 7(xi) is greater than /(xi).
(2) There is a value of x, X2, in 7(xi) such that X2<xi
a.nd/(x2)</(ii).
DERIVATIVES AND DIFFERENTIALS. 131
(3) There is a value of x, 13, in V{xi) such that X3>a;i and
/(X3)</(Xi).
Similarly we define a minimum of a function.
This definition allows any point of a constant stretch like
a, Fig. 17, to be a maximum, but does not allow any point of
b to be either a maximum or a minimum.
Fig. 17.
Theorem 83. // fixi) exists and if /(i) has a maximum or a
minimum at x=xi, then f{xi) =0.
Proof. — In case of a maximum at xi, it follows directly from
the hypothesis that
rix. X-Xi < xix, X-Xl
X>Xi *<^1
Since f(xi) exists these limits are equal, that is, the derivative
is equal to zero. Similarly in case of a minimum.
Theorem 84. // /(xi)=/(x2), fix) being corUimwus on th£
132
INFINITESIMAL ANALYSIS.
interval Xi X2, and if the derivative exists t at every point between
Xi and X2, then there is a value f between Xi and X2 such that /'(?)= 0.
The derivative need not exist at Xi and X2-
Proof. — (a) The function may be a constant between Xi and
X2, in which case f{x) = for all values of x between Xi and
X2 by Theorem 74.
(6) There may be values of the function between Xi and X2
which are greater than f{xi) and f{x2). Since the function is
continuous on the interval Xi X2, it reaches a least upper boimd
on this interval at some point X3 (different from Xi and '0:2).
By Theorem 83,
/'(X3)=0.
(c) In case there are values of the function on the interval
I — I
Xi X2 less than /(xi), the derivative is zero at the minimum
point in precisely the same manner as under case (b).
Fig. 18.
This theorem is called Rolle's Theorem. The restriction
that /(x) shall be continuous is unnecessary if the derivative
t Not necessarily finite.
DERIVATIVES AND DIFFERENTIALS. 133
exists, but simplifies the argument. The proof without this
restriction is suggested as an exercise for the reader.
The geometric interpretation is that any curve representing
a continuous fimction, }{x), such that f{xi) =f{x2), and having a
tangent at every point betweeen xi and X2 has a horizontal
tangent at some point between them. An immediate gener-
alization of this is that between any two points A and B on
a curve which satisfies the hypothesis of this theorem there
is a tangent to the curve which is parallel to the Une AB.
The following theorem is a corresponding analytical generali-
zation :
I — I
Theorem 85. // /(x) is continuous on the interval. Xi X2, and
if the derivative exists at every point between X\ and X2, then there
is a value of x, x = $, between xi and X2 such that
' X1-X2
Proof. — Consider a function /i(z) such that
fii.x)=f{x)-{x-x2) — ^^_^^ ;
then fi{xi)=fix2) and /i(x2)=/(x2).
Therefore /i(a;i)=/i(x2).
Hence, by Theorem 84, there is an x, x = $ on the segment xi X2
such that /i' (0=0.
That is, />'©=/'(« -'-^^^-0.
Therefore f(f)-^^^^^.
This is the "mean-value theorem." Its content may also be
■expressed by the equation
f{X2)=f{Xl) + {X2-Xl)n^).
134 INFINITESIMAL ANALYSIS.
Denoting xi -x by dx and f by x + Odx, where 0<5<1, it takes
the form
/(xi +dx) =/(xi) +f (xi + ddx)dx.
Theorem 86. // /i(x) and /2(x) are continiious on an interval
a b, and if /i'(x) and /a'Cx) exist between a and b, fz'ix) 5^ ± oo ,
and /2'(x)7^0, /2(a) 5^/2 (&), ^^len there is a value of x, x=f
between a and b such that
fi(a)-fi( b)_fi'{?)
/2(a) -/2(6) /2'(f)-
Proof. — Consider a function
Since /3(a) =0 and fsib) =0, we have as before /a'Cf) = 0.
Therefore AMzMl^M)
inereiore /2(a) -/2(6) /a'Ce)"
This is called the second mean-value theorem. The first
mean-value theorem has a very important extension to "Taylor's
series with a remainder," which follows as Theorem 87.
§ 5. Taylor's Series.
The derivative of f{x) is denoted by /"(x) and is called the
second derviative of /(x). In general the nth derivative is the
derivative of the n - 1st derivative and is denoted by /^"'(x) .
Theorem 87. // the first n derivatives of the function f{x)
l-l
exist and are finite upon the interval a b, there is a value of x, x„
l-l
on the interval a b such that
DERIVATIVES AND DIFFERENTIALS. 135
/(&)=/(a)+^/'(a)+^V(a) + .
^ (n-1)! ' ^'^^+ „, ; {Xn).
Proof. — Let i?„ be a constant such that
Fix) =f{x) -fia) - {x-a)na) J-^^f"{a) -...
_ (x-a)"-^ ,„_.) (.T-g)"
(n-1)! ' '■'^^ n! ""
is equal to zero for x=b. Since F{x)=0 for x = a, there is, by
Theorem 84, some value of x, Xi,a<xi<b such that i^'(xi)=0.
That is,
F'ix)=nx)-na)-{x-a)r{a)-. . .
_ (x-a)"-' , (x-a)"->
(n-2)! ^ "^"^ („_i), «n
is equal to zero for x=a;i. Since also F'{a) =0, there is a value
of X, X2, a<X2<Xi such that F"{x2)=0. Proceeding in this
manner we obtain a value of x, x„, a<x„<Xn-i such that
F^^KXn)=0.
But F">(x„)=p(a;n)-ii;n=0.
Therefore /i!„ = f"H2:»),
whence the theorem.
Corollary— In Theorem 87, f^'^Kx) need be supposed to exist
only on ab.
Definition. — The expression
n! n- *"=o "'•
is called the remainder, and the infinite series
t-o «•
is called Taylor's Series.
136 INFINITESIMAL ANALYSIS.
a constant different from zero,
» /W.(g)(b-g)n
then 2 —^
n=0 "•
is convergent but not equal to j(b), i.e.,
If L^-(&-a)»
fails to exist and be finite, then
00 /('')(a)
„=o «■!
(6 -a)"
is a divergent series.
Hence an obvious necessary and sufficient condition that
for a function /(x) all of whose derivatives exist for the values
of I, o<x<6,
n=0 "•
is that L '— V^ (6 -a)» = O.f
This leads at once, by Theorem 33, to the following sufficient
condition :
n—00 ni
for every value of i on o b is not sufficient, since in depends upon n.
DERIVATIVES AND DIFFERENTIALS. 137
Theorem 88. // /(")(x) exists and |/(")(x)| is less than a fixed
I — I
quantity M for every x on the interval a b and for every n
{n = l,2, . . .), then
m=m + ^^f'ia) + . . . + ^i^;(n)(a) + . . .
Functions are well known all of whose derivatives exist at
I — I
every point on an interval a b, but such that for some point
on this interval
n = "''
where fi is a function of x not identically zero. Other func-
tions are known for which the series is divergent. The classical
example of the former is that given by Cauchy,t e~? at the
point x = 0. If this function is defined to be zero for 2 = 0, all
its derivatives are zero for x=0, whence Taylor's development
gives a function which is zero for all values of x.
Pringsheim X has given a set of necessary and sufficient
conditions that a function shall be representable for the values
oi h, 0<h<R,hy means of the series
l-,./W(0)-A".
It was remarked above, p. 131, that a necessary condition for
f(x) to be a maximum at a; = a is f (a) =0 if the derivative exists.
Taylor's series permits us to extend this as follows :
Theorem 89. // on some V{a) the first n derivatives of f{x)
exist and are finite and on V*{a) /'"■•■ '^(x) exists and is bounded,^
and if
■|- Cauchy, Collected Works, 2d series, Vol. 4, p. 250.
j A. Pringsheim, Mathematische Annalen, Vol. 44 (1893), p. 52, 53.
See also Koxig, Mathematische Annalen, Vol. 23, p. 450.
§ Instead of assuming the existence of /<»+')(x) we might have assumed
A") (a:) continuous without essentially changing the proof.
138 INFINITESIMAL ANALYSIS.
0=/'(a)=/"(a) = ..;=/(»-i>(a),
then : (1) If n is odd, f{x) has neither a maximum nor a mini-
mum at a;
(2) // n is even, j{x) has a maximum or a minimum according
as f'^aXQ or f''\a)>Q.
Proof. — By Taylor's theorem, for every x in the vicinity of a
fix) =/(a) + (x-o)«/(")(o) + (a;-a)"+i-/(»+»(f«),
where ^x is between x and a.
Hence j{x) -f{a) = (x- a) "{/(») (a) + (i-a)/("+i)(f ^) } .
But since /^""""^'(fi) is bounded and x—a is infinitesimal, there
exists a 'V*{a) such that if x is in 7* (a),
Kx)-m
is positive or negative according as
(x-a)"-/(")(o)
is positive or negative.
(1) If n is odd,(a;— a)"is of the same sign as x—a, and hence
for /("'(a) >0
j{x)-f{a)>0 '\ix>a,
f{x)-J{a)<0 iix<a;
while for f"Ka)<0
/(x)-/(a)>0 ifa;<a,
/(x)-/(a)<0 ifx>a.
(2) If n is even, (x—a)" is always positive, and
hence if f"'(a)>0.
/(x)-/(o)>0 ifa;>a,
/(x)-/(a)>0 ifx<a;
then /(a) is a maximum.
DERIVATIVES AND DIFFERENTIALS. 139
If /(">(a)<0.
/(x)-/(a)<0 if a;>a, 1
tr \ t/ \ ^n -t ^ f then /(a) is a mimmum.
/(x)-/(a)<0 if i<a; J '^ -^
§ 6. Indeterminate Forms.
The mean-value theorems have an important application in
the derivation of l'Hospital's rule for calculating "indeter-
minate forms." There are seven cases.
fix)
(1) -, i.e., to compute L -tt-t if L /(x)=Oand L <}>{x)=0.
(2) — , i.e., to compute L -rr^ if L f{x) = ± oo and
L 0(1) = ±00.
(3) 00 —CO, i.e., to compute L j/(x)— ^(i)| if L f(x)= ±00
x=a x^a
and L <f>{x)= ±00.
(4) O-oo, i.e., to compute L /(x)-0(i) if L /(x)=0 and
Z/ <f>ix)= ±00.
(5) 1°°, i.e., to compute L /(x)*(^> if L /(x) = l and
L 4>(x) = ± 00 .
(6) 0°, i.e., to compute L /(x)*^^) if L /(x) = and L ^(x) =0.
x=a x=a x^a
(7) 00 0, i.e., to compute L f{xY^''^ if L/(x)=±oo and
L 0(x)=O.
These problems may all be reduced to one or the other of
the first two. The third may be written (since /(x) t-^O on some
V*{a))
/(i)-<^(x)=-^ ^(x) = — ^.
W) Jixj
which is either determinate or of type (1).
140 INFINITESIMAL ANALYSIS.
To the cases (5), (6), and (7) we may apply the corollaries of
Theorem 39 of Chapter IV, from which it follows (provided
fix) 7^0 on some F*(o)), that
exists if and only if
log L /(x)*(^>= L log/(a;)*(^>= L 9&(a;) log/(x) exists.
x=a x=a x^a
The evaluation of L -~
^)
comes under case (1) or case (2).
The evaluation of cases (1) and (2) is effected by the follow-
ing theorems:
Theorem 90. // /(x) and 4>{x) are continuous and differentiable
and <^(x) is monotonic and (f>'{x) ^0 and <j>'(x) j^ 00 and
(1) i] L f{x)=0 and L <j>ix)=0 or
(2) if L ,^(x)=±oo,t
XwOO
0(x)
exists and is equal to K.
Proof .—For every positive h we have, by the second mean-
value theorem,
fix +h)- fix) _f(e.)
<t>ix+h)-<}>ix) <f>'i$^y
where ^^ lies between x and x + h. But since f ,, takes on values
which are a subset of the values of x, and since L f = 00
t It is not necessary that Lf{x)=aa ■ cf. Theorem 69.
DERIVATIVES AND DIFFERENTIALS. 141
which in turn implies L 77 r{ — -~rT = K
x^oo<p{x + h)-<j>{x) '
and this, according to Theorems 68 and 69, gives
Corollary. — If f{x) is continuous and differentiable, then
L — L fix).
The theorem above can be extended by the substitution
1
z = -
x-a
to the case where x approaches a finite value o. The approach
must of course be one-sided.
Theorem 91. // fix) and (f>{x) are continuous and differen-
tiable on same V*(a) and f{x) is hounded on every finite interval,
while (f>{x) is monotonic and
(1) L /(x)=0, L 0(1) =0 or
x=a x—a
(2) L ^(x) =+ 00 or - 00 :
thenxf xt¥(i) '
it follows that L 77-r
exists and is equal to K.
142 INFINITESIMAL ANALYSIS.
fix)
Proof. — If L ,,, . exists, the limit exists when the approach
x=a V \-^)
is only on values of x>a. Consider only such values of x.
Then if
^=^' /(^)=/(«+7)=-P'(2)
and 4>{x)='^{a+-)=0{z),
by hypothesis and Theorem 79, F'iz) and 0'iz) exist and
i^'(3)=/'(x)g.
Hence If LVi^r^'
\ hen, according to Theorem 38,
r F'iz)
exists and is equal to K.
Hence, by Theorem 90,
exists and is equal to K.
T ZM
.t^<I>iz)
fix)
Hence, by Theorem 38, L ., .
exists and is equal to K.
We have now derived conditions under which we can state
a general rule for computing an indeterminate form.
Provided fix) is not zero on every V*ia), any of the forms
<3) to (7) can be reduced to
Fix)
(Pix) W
DERIVATIVES AND DIFFERENTIALS. 143
where this is of type (1) or (2). Provided Fix) and 0{x) satisfy
the conditions of Theorem 91, the existence of the limit of (a)
depends on the existence of the Umit of
F'ix)
¥{xj- (^)
If (6) is indeterminate, and F'{x) and 0'{x) satisfy the condi-
tions of Theorem 91, the limit of (6) depends on the Umit of
F"{x)
'(X)
'ft-y\> 'W
and so on in general. If at each step the conditions of Theorem
91 are satisfied and the form is still indeterminate, the Umit of
i?'(n+l)(x)
depends on the limit of ^u+iv^ ('i+l)
If (n) is indeterminate for all values of n, this rule leads to no
result. If for some value of n
then all the preceding Umits exist and are equal to K, and so
x=o<P(a;)
The original expression is equal to X or e^ according to the case
under consideration.
144 INFINITESIMAL ANALYSIS.
§ 7. General Theorems on Derivatives.
Theorem 92. // /(x) is coniiniious and f(x) exists for every x
on an interval a b, then f'{x) takes on every value between any
two of its values.
Proof. — Consider any two values of f'(x), f'{x\), and f'ixi)
on the interval a b. Consider, further, the function
X — Xi
on the interval between Xi and X2. Since is a con-
X — Xi
tinuous function of x on this interval, it takes on every value
between and /'(xi), which is its limiting value as x
approaches xi. Hence, by Theorem 85, /'(x) takes on all values
between and including f{xi), and — — for values of x
I — I
on the interval Xi X2. By considering in a similar manner the
/(X2) -fix) i — I
function on the interval Xi X2, we show that f'{x)
takes on all values between - — ^-— ^ — — and /'(X2). Hence /'(x)
takes on all values between f(xi) and /'(X2).
Theorem 93. // the derivative exists at every point on an
interval, including its end-points, it does not follow that the de-
rivative is continuous or that it takes on its upper and lower
bounds.
Proof. — ^This is shown by the following example.
The curve shall he between the x-axis and the parabola
2" —1
j/ = ix2. The straight lines of slopes 1, IJ, If, . . . , 1 h — - — . . .
2"
through the points (i, 0), (i, 0), . . . , [^ifi, o) , . . ., respectively,
meet the parabola in points Ai, A2, A3, ... , A„, . . . The
broken line ^1 (J, 0) A2 (i, 0) A3 . . .An {^,0j . . . oo , has an
DERIVATIVES AXD DIFFERENTIALS. 145
infinitude of vertices. In each angle of the broken line con-
sider an arc of circle tangent to and terminated by the sides of
Fig. 19.
the angle, the points of tangency being one fourth of the distance
to the nearest vertex. The function whose graph consists of
these circular arcs and the portions of the broken line between
IH
them is continuous and differentiable on the interval 1.
Its derivative is discontinuous at j=0 and has the least upper
bound 2, which is never reached.
Theorem 94. // /'(x) exists and is equal to zero for every valiie
l-l , . ,
of X on the interval a h, then f{x) is a constant on thai interval.
Proof. — By Theorem 82, fix) is continuous. Suppose /(i)
not a constant, so that for two values of x, ii, and X2,
f{xi)7if{x2), then, by Theorem 85, there is a value of x,
x=$ between xi and X2 such that
^^^^' X2-X, '
146 INFINITESIMAL ANALYSIS.
which is different from zero, whence f'(x) is not zero for every
1-1
value of X on the interval a b. Hence f(x) is a constant on
l-l
a b.
Corollary. — If /i'(x) ^fzix) and is finite for every value of x
on an interval a b, then fi{x) —J2(x) is a constant on this interval.
Theorem 95. // j'ix) exists and is positive for every value
l-l
of x on the interval a b, then f(x) is monotonic increasing on this
interval. If f'{x) is negative for every value of x on this interval,
then f{x) is monotonic decreasing.
Proof, — If f{x) is positive for every value of x, then it fol-
lows from Theorem 85, provided that f{x) is continuous, that
the function is monotonic increasing, for if there were two
values of x, Xi and X2, such that /(xi) t /(X2) while Xi <X2, then
there would be a value of x, x = f , between xi and X2 such that
/'(f) _ /(^2)~/fa) <o
X2-X1
In case fix) is not supposed continuous, the argument can
be made as follows: If f{xi)>0, then, by Theorem 23, there
exists about the point xi a segment (xi— iJ), (xi-f-<J), upon
which
/(£W(£l)>0,
X — Xi
and hence, if x>Xi, /(x)>/(xi) and if x<X], /(x)</(xi). Now
about every point of the segment a b there is such a segment.
Let xf and x" be any two points of a 6 such that x'<x". By
Theorem 10, there is a finite set of these segments of lengths
Si . . . dn which include within them every point of the interval
I — I
x' x". We thus have a finite set of points, namely, the mid-
point and points on the overlapping parts of the segments
x'<Xi<X2<... <Xft<x", such that
DERIVATIVES AND DIFFERENTIALS. 147
/(a/X/CxiX/feX. . .</(xfc)</(x").
Hence /(x')</(x"). In a similar manner we prove that the
function is monotonic decreasing in case f{x) is negative.
Theorem 96. // a function f{x) is monotonic increasing on
l-l
an interval a b, and if f{x) exists for every value of x on this
interval, then there is no point on the interval for which f{x) is
negative. That is, f{x) is either positive or zero for every point
I — I
of a b.
Proof. — If /'(x) is negative for some value of x, say Xi,
then L = C, a negative number,
x=xi X — Xi
whence there is a neighborhood of Xi on which /(x) >/(xi), while
x<xi, or /(xi)>/(x), while x>Xi, \Yliicli is contrary to the
hypothesis that the function is monotonic increasing in the
neighborhood of x = ii. In the same manner we prove that if
the function is monotonic decreasing, and if the derivative
exists, then fix) cannot be positive.
The following theorem states necessary and sufficient condi-
tions for the existence of the progressive and regressive deriva-
tives. Conditions for the existence of a derivative proper are
obtained by adding the condition that the progressive and
regressive derivatives are equal.
Theorem 97. // /(x), x<Xi, is continuous in some neighbor-
hood of x=Xi, then a necessary and sufficient condition that f{xi)
shall exist and be finite is that there exists not more than one
linear function of x, ax+c, such that /(x) +ax+c vanishes on every
neighborhood 0/ x = xi.
Proof__(l) The condition is necessary. We prove that if /'(x)
exists and is finite, then not more than one function of the form
ax + c exists such that /(x) +ax+c vanishes on every neighbor-
hood of x=Xi. If no such function exists, the theorem is veri-
fied. If there is one such function, the following argument wUl
show that there is only one. Since, by hypothesis,
148 INFINITESIMAL ANALYSIS.
exists, we have, by Theorem 75, that
f{x)+ax+c-j{xi)-axi-c
exists. Let [a/] be the subset of the set of values of x on any
neighborhood of x=xi such that j{x')+ax!+c vanishes on the
set [a/]. By Theorem 41,
j{xf)+axf +c-f(xi) -axi -c
^ ^f(x)+ax+c-f(xr)-ax,-c ^^^^^^_^^^
X=Xi ■^ -^1
Since /'(xi) and a are both finite,
f{x')+ax^+(/-f{xi)-axi-c
x'^xi x! -Xi
is finite. But the numerator of this fraction is a constant,
f{x)+ax+c being zero on the set [x']. Hence
^ /(x)+ax+c-/(x0 .3ax^c^ or /'(xO+a=0,
xix, X-Xi
and, being continuous, /(xi)+axi+c=0.
The numbers o and c are miiquely determined by the equations
i/'(xi)+a=0,
(/(xi)+axi+c=0.
(2) The condition is sufficient. We are to show that
DERIVATIVES AND DIFFERENTIALS. 149
j- /W-/(Xi)
can fail to exist only when there are at least two functions of
the form ax+c such that /(x)+ax + c vanishes on every neigh-
borhood of i=xi. If L Mz/(£l)
x-xi X — Xi
does not exist, then ^S^^-J.^^
X — Xi
approaches at least two distinct values Ki and K2. Let K2<Ki.
Let A and B be two finite values such that K2<A<B<Ki.
On every neighborhood of x = xi there are values of x for which
X-Xi
is greater than B, and also values of x for which
f(x)-f{xi)
X-Xi
is less than A. Hence, since
/(x)-/(xi)
X-Xi
is continuous at every point except possibly xi, in a certain
neighborhood of Xi there are values of a; in every neighborhood
of xi for which =A,
X — Xi
or fix)-}{xi)=A(x-Xi),
which gives -f(xi) -A{x-xi)
as one function of the form ax+c.
150 INFINITESIMAL ANALYSIS.
In the same manner we show that —f(xi)—B(x—Xi) is
another function ax+c, which makes f{x)+ax+c vanish on
every neighborhood of x=^xi.
The geometric meaning of this theorem is obvious. If P is
a point on the curve representing f(x), then a necessary and
sufficient condition that this curve shall have a tangent at P is
that there exists not more than one line through P which inter-
sects the curve an infinite number of times on any neighborhood
of P- Compare the fvmctions x sin — and x^ sin — on page 51.
The earlier mathematicians supposed that every continuous
function must have a derivative except at particular points.
The first example of a fimctipn which has no derivative at any
point is due to Weiers iwadc . 7^
The function is
00
/(x)= I 6" cos (a'^nx),
where a is an odd integer, < 6 < 1 and ai>>l + |?r.
t For references and remarks see page 51 .
CHAPTER VIII.
DEFINITE INTEGRALS.
§ I. Definition of the Definite Integral.
The area of a rectangle the lengths of whose sides are exact
multiples of the length of the side of a unit square, is the num-
ber of squares equal to the unit square contained within the
rectangle, and is easily seen to be equal to the product of the
lengths of its base and altitude.t
In case the sides of the rectangle and the side of the unit
square are commensurable, the sides of the rectangle not being
exact multiples of the side of the square, the rectangle and the
square are divided into a set ..of equal squares. The area of the
rectangle is then defined as the ratio of the niunber of squares
in the rectangle to be measxared to the number of squares in the
unit square. Again, the area is equal to the product of the
base and altitude.
Any figure so related to the unit square that both figures can
be divided into a finite set of equal squares is said to be com-
mensurable with the miit.
The area of a rectangle incommensurable with the unit is
defined as the least upper boimd of the areas of all commensur-
able rectangles contained within it.
It foUows directly from the definition of the product of
irrational nvunbers that this process gives the area as the prod-
uct of the base and altitude. J
t Of course the units are not necessarily squares; they may be triangles,
parallelograms, etc.
X For the meaning of the length of a segment mcommensurable with the
unit segment, compare Chapter II, page 33.
151
152 INFINITESIMAL ANALYSIS.
Turning to the figure bounded by the segment a b (which
we take on the x axis in a system of rectangular coordinates)
the graph of a function y^fix) and the ordinates a;=a and x=6,
Fig. 20.
we obtain as follows an approximation to the common notion
of the area of such figures.
Let xo=a, xi, X2, . . . , Xn = b he a. set of points lying in order
from a to 6. Such a set of points is called a partition of a h,
and is denoted by n. The intervals xq xi, xi Xz, . . . , a;„_ii„
are intervals of ;:.
Let xi— zo=.^ia;, X2—xi=J2X, . . . , x„-Xn-i=^„x,
and let fi, $2, . . ■ , $n
be a set of points such that $1 is on the interval xq Xi, $2 is on
I 1 . I 1
Xi X2 . • • , and $n IS on i„_ix„.
Then /(fi), fiU), ..., /(f„)
are the altitudes of a set of rectangles whose combined area is
a more or less close approximation of the area of our figure.
Denote this approximate area by S.
Then S=m)Jix+m)J2X + . . .+/(f„)i„x= I fih)^kX.
k-l
As the greatest Ji^ is taken smaller and smaller, the figure
DEFINITE INTEGRALS. 153
composed of the rectangles comes nearer to the figure bounded
by the curve.
In consequence of these geometrical notions we define the
area of the figure as the limit of S as the J^x's decrease in-
definitely. The area S is the definite integral of /(x) from a
to 6. It has been tacitly assumed that the graph of y=f{x) is
continuous, since we do not usually speak of an area being
enclosed by a discontinuous curve. The definition of the defi-
nite integral when stated in its general form admits, however,
of functions which are discontinuous in a great variety of ways.
A more general definition of the definite integral is as follows :
1 — I I — I
Let a h {or h a) he an interval upon which a function f{x) is
defined, single-valued and bounded. Let n, stand for any par-
I — I I — I
tilicn of a b or b a by the points a = Xo, Xi, X2, . . . , x„ = 6 such
that the numbers A\X = Xi—a, A2X = X2 — X\,..., J„x~6— x„_i
are each numerically less than or equal to d.
fl, f2, ■ • • , fn
be a set of points on the intervals
I 1 I — I I 1 , ., ^ I 1 I 1 I 1
Xq-Xi, Xi X2, ■ . . , Xn-lXn (pr if b<a, Xi Xo, X2 Xi, X3 X2,
I 1
. . . , x„ x„_i) respectively, and let
// the many-valued function of d, S„ approaches a single limiting
value as d approaches zero, then
L S,= f){x)dx.
When we desire to indicate the interval' of integration we
write \Si and *?:, instead of Ss and n,. a and b are called
the limits of integration.
The details of this definition should be carefully noted.
154 INFINITESIMAL ANALYSIS.
For every d there is an infinite number of different partitions
TTg, and for every partition there is an infinite set of different
sets of f A, so that for every d the function Sg has an infinite
set of values. The graph of the function S» is of the type
shown in Fig. 21. Every value of S» for one d is assumed by
*S for every larger d. For any particular value of d the values
Fig. 21.
of Si lie on a definite interval BS3 BS,, whose length never in-
creases as d decreases. If this interval approaches as 5 ap-
proaches 0, the required hmit exists.
It is to be noticed that the set of ;r's, [:r*] includes every
possible n whose largest JkX is less than d. Thus we carmot
obtain the set of all ;r's by sequential repartitioning of any
given n, since there are partitions of the set [ng] which have
no partition points in common with any given partition. Inat-
tention to this point is perhaps the greatest source of error in
the development of the notion of a definite integral.
§ 2. Integrability of Functions.
The class of integrable functions is very large, including
nearly all the bounded functions studied in mathematics and
DEFINITE INTEGRALS. 155
physics. Even such an arbitrary function as
y=Qii X irrational,
y=^\/n^ if x=m/n,
is integrable. (See page 182, Theorem 127.)
Examples of non-integrable functions are y=\/x on the
interval 1 (where it is not bounded, see page 191), and the
function,
2/ =0 if a; is irrational and
y=\\i x\s rational.
To determine the conditions of integrability we introduce
the concept of integral oscillation. On any interval a b, f(x)
has a least upper bound A and a greatest lower bound B, be-
tween which the function varies. If A—B=Jy=''Ofix) is mul-
tiplied by the length of the interval, Jx = \b-a\, it gives the
area of a rectangle, including the graph of f{x). If the interval
is subdivided by a partition ;:, the sum of the products Jx-Jy
on the intervals of the partition is called the integral oscilla-
tion of fix) for the partition tt and is denoted by 0^. If we call
A]cy the difference between the upper and lower bounds of /(x)
I 1
on the intervals Xk-iXk, we have
n
Q^ = \Aix\-Aiy + \A2x\-A2y-\-... + \Anx\Jny-=I\Akx\-dky.
k= 1
Greometrically 0,, represents the areas of the rectangles Fi,...,F^
(Fig. 22), and so we exp)ect to find that if the lower bound
of 0, is zero, fix) is integrable. This proposition, which requires
some rather deUcate argument for its proof, will be taken up
in § 7. At present we shall show in a simple manner that every
continuous and every mono tonic function is integrable.
Lemma i. If S„ and SJ are two sums (formed by using differ-
ent $k's) on the same partition, then
156
INFINITESIMAL ANALYSIS.
Proof.
l'S,--S,'| =
I{f{^k)'f{^k')\JkX
4=1
^ ^ i/(fft)-/(.V)|-Mjfcx|.
A=l
But |/(f jt) -/(f*')l^ifc2/ by the definition of Jty.
Therefore |5„-S/|< I \Jkx\-dky (4)
k=l
(
f^^\
A,!/
V
;
T
h
a
\
1
Fry/
-\-^
Fio. 22.
A repartition of a partition ;r is formed by introducing new
points in t:.
Lemma 2. // tti is a repartition of n,
Proof. — Any interval Jita; of ;: is composed of one or more
DEFINITE INTEGRALS. 157
intervals Jk'x, dk"x, etc., of n^, and these contribute io S„
the terms
K^k')dk'x+f{^n^k"x+ (1)
The corresponding term of *S, is
/(ftM*x=/(fO^/a;+/(fi)J/'x + (2>
But since |/(fft)-/(f«:') 1=^*3/, the difference between (1) and
(2) is less than or equal to
Aky-\dk'x^Ak"x + . . .\^Aky-\dkx\
and hence \S:,-SA< I Aky-U,^\=0^.
' k=\ I
Theorem 98. Every Junction continuous on a b is integrable
I — I
on a b.
Proof. — ^We have to investigate the existence of the limit
LS} of the many-valued function Ss as d~0. Since S, ap-
proaches at least one value as d approaches zero (see Theorem
24), we need only to prove that it cannot have more than one
value approached. Suppose there were two such values, B
R — C
and C, B>C. Let £ = — —. By the definition of value
approached, for every 8 there must exist an S (which we call
Sb) such that
\SB-B\<e, (1)
and such that the corresponding ttb has its largest AkX<8.
Similarly there must be an Sc such that
\Sc-C\<^, (2)
and such that the corresponding nc has its largest JkPO<d. Let
TT be a partition made up of the points both of tcb and nc, and
let S be one of the corresponding sums. ;r is a repartition both
of TZB and nc.
158 INFINITESIMAL ANALYSIS.
Therefore |5-<Scl<0;rp (3)
and |S-5B|<ar^ (4)
But since f(x) is continuous, by the theorem of uniform conti-
nuity, d can be so chosen that if any two values of x differ by
less than d, the corresponding values of /(x) differ by less than
and hence on the partitions ttb and nc, whose J^x's are all
£
\b-a\
less than 8, the corresponding i^j/'s are all less than ,, _ , . So
n
we have (since I AkX = h—a)
On = i" \JkX\-Jky< - |itx|-|T-— T
= £•
Hence 0„„<£ and On„<s.
B—C
So we have, since e = — j— and d is so chosen that whenever
\^-x"\<d,\f{x')-fixf')\<~^^
\Sb-B\<s,
\Sc-C\<e,
\Sb-S\<b,
\Sc-S\<e.
From these inequalities it follows that \B—C\<4e, which con-
tradicts the statement that «= — j—. Hence the hypothesis
that fix) is not integrable is untenable.
Theorem 99. Every non-oscillating hounded function is
integrable.
Proof. — The proof runs, as in the preceding theorem, to the
DEFINITE INTEGRALS. 159
paragraph following (4). Let D and d be the upper and lower
bounds of /(z) . d, being arbitrary, can be so chosen that 8 = jr-^.
D—a
Then 0„^= 3 Jky-\^kx\< I /t^y-S,
° k-l A=l
and since f{x) is non-oscillating,
idky=D-d.
Therefore 0„ „ < (D - d) 5 = e.
Similarly Ojt < e. Hence again we have
\SB-B\<e,
\Sc-C\<^,
\Sb-S\<^,
\Sc-S\<^,
and therefore |5— C|<4e, whereas e was assumed equal to
R — C
— -7 — . Thus the hypothesis of a non-integrable non-oscillating
fimction is imtenable.
§ 3. Computation of Definite Integrals.
In computing definite integrals it is important to observe
that whfn the integral is known to exist the Umit can be cal-
culated on any properly chosen subset of the Si's. (See Theorem
41.) So we have that if <S,„ S)^, ... is any sequence of sums
such that L 5„=0, then
L Sin
ttsX
One case of this kind occurs when f * is taken as an end-
160 INFINITESIMAL ANALYSIS.
point of the interval Xk-i Xk and all the i^x's are equal. Then
we have
rb n I
/ f{x)dx= L I f{a+kJx)Jx, -where Ax = -
6— a
A simple example of this principle is the proof of the following
theorem .
Theorem loo. // ]{x) is a constant, C, then
rCdx=CQ)-a).
Proof. — The function f{x) = C is integrable either according
to Theorem 98 or Theorem 99. Hence
rCdx= L I C^—= L n-C^-^^=CQ)-a).
A few other examples follow. In each case the function is
known to be integrable by the theorems of § 2.
/}''
Theorem loi. / e^dx=e''— e".
Proof. —
=e^-Jx-—7- — 7-'- — 7^ — r^'^-^x .
, t. •. ^^
Ax
Whence the result follows since L -j- — :=1. (Differentiate
JiioC — i
mmierator and denominator with respect to Jx according to
Theorem 90.
DEFINITE INTEGRALS. 161
Instead of arranging the partition-points in an arithmetical
progression as in the cases above, we may put them in a geo-
metrical progression, that is, we let
6\" 6
aj ==«' a =3"'
Aix = aq — a, J2X = aq^—aq, . . . , J„x = ag"— 05""^,
fi=a, 62=03, . . . , f„=a5"-i,
and obtain the formula
rfix)dx= L a{q-lMa)+qfiaq)+... +q'^-'f(.aq--^)]
J <• « = i
= L a{q-l)'l qi^fiaq^).
jii 4=0
We apply this scheme to the following.
Theorem 102. In all cases where m is a whole number?^ -1,
and if a>0, b>0 for every value of m?^ -1,
/
6 ^m+1 (j'n+1
x'^dx = ■
m + 1
X^dx= L a{q-l) I q''{aq'')'"
a 9=1 i-0
=a'"+i L (3-l)[l + (r+i) + (r+^)' + - ■ . + ir^')^-n (1)
(om+l)n_l
= La'"+M(3")"'^'-i!jFrri
9 = 1 ^
162 INFINITESIMAL ANALYSIS.
Hence / x^dx = -
m + 1 '
T g-1 1
Theorem 103. / -dx = log b—loga, {0<a<b).
Proof. By equation (1) in the last theorem, since g™+i = g° = 1 ,
/ -dx= L niq-1);
>°^ a)
but n =— i , hence
log 3
£V^= ,il^- ^°S (^)=log (|)=log&-loga,
since (§6, Chapter VII) l'Hospital's rule gives
8=1 log g
The following theorem is of frequent use in computing both
derivatives and integrals.
I — I
Theorem 104. // on an interval a b two functions /(x) and
Fix) have the ■property that for every two valries of x, xi and X2,
■where a<xi<X2<b,
fiXl){X2-Xi) < F{X2) -F(Xi) < /(X2)(X2-Xi);
or if /(Xi)(X2-Xi) > i^(X2) -2^(Xi) > /(X2)(X2-Xi),
then (1), if f{x) is coniiniums,
DEFINITE INTEGRALS. 163
and (2) whether f{x) is continuous or not,
I }{x)dx exists and is equal to F{h) —F{a).
Proof. — We consider first the case
fiXi)iX2-Xi)<F{X2)-F{Xi)<fiX2){X2-Xi).
This gives /(xi) < — - — < /(X2).
X2 — Xi
Since f{x) is continuous at x = xi, L /(x2)=/(xi). Hence, by
Theorem 40 (Corollary 2),
F(x2)-Fixx)
^ — 77^:7, — =/(a;i),
zj-l, X2— Xl
which proves (1).
To prove (2) we observe that f{x) is non-oscillating and
therefore integrable according to Theorem 99. On any parti-
tion 7t whose dividing points are xi, X2, . . . , x„_i we have
/(o)(xi-a) <F{xi)-Fia) </(xi)(xi-a),
/(Xi)(X2-Xi) <F{X2)-FiXi) </(X2)(X2-Xi),
/(Xn_i)(5-x„_i) <F(b) -F(x„_i)</(6)(6-x„_i).
Adding, we get
/(o)(xi-a)+/(xi)(x2-xi) + . . .+/(x„_i)(6-x„_i)<F(6)-J!?'(o)
^/(a;i)(2;i-a)+/(x2)(x2-xi) + . . .+f(jb)ib-Xn-l).
But /(o)(xi-a) +. . .+/(x„_i)(6-x„_i)>BS,
and /(xi)(xi-a)+. . .+/(b)(&-x„_i)<B<S,.
164 INFINITESIMAL ANALYSIS.
Since this holds for every n, we have by Theorem 40 that as
(Theorem 99) / f{x)dx exists,
/^<'
x)dx=Fib)-F{a).
The proof in case fixi){x2-Xi)>F{x2) -Fixi)>f(x2){x2-xi}
is identical with the above when we write > instead of <.
§ 4. Elementary Properties of Definite Integrals.
Theorem 105. // a<b<c, and if a bounded function f{x) is
integrable from a to c, then it is integrable from a to b and from
b to c.
Proof. — Suppose f(x) not integrable from a to b, then .by
the definition of a limit (see Chap. II.) there must be a set of
values of IS3, [aS/], such that L 18,'= A, and another set [aS/']
such that L 183" = B, while A and B are distinct. Whether
}=0
£
f{x)dx exists or not, there must be a set of values of ISt^
[iSi'l such that the limit L IS/ =C. Now for every »S/ and
iS/ there exists a IS/ such that lS/=iS/+tS/. Therefore
A+C is a value approached by IS}. By similar reasoning,
5 + C is a value approached by IS}- Hence IS 3 has two values
approached, which is contrary to the hypothesis. Hence
/ {x)dx must exist. By similar reasoning / f{x)dx must exist.
Theorem 106. // a<b<c and if a bounded function f{x) is
integrable from a to b and from b to c, then f{x) is integrable from
f{x)dx= / f{x)dx+ I f{x)dx.
Proof. — Since / f{x)dx and / f{x)dx exist, we know by
Theorem 26 that for every e there exists a ^c' such that for
DEFINITE INTEGRALS.
every value of *5, where d < d/,
aSi- I f{x)di
<S'
and also a d." such that for every value of tSi where d<d/',
ts^-fj(
'x)dx
£
l-l
165
(1)
(2)
Now if the upper bound of /(x) on a c is M and its lower
bound is m, let oj" ■■
-, and let d, be smaller than the
smallest of 5/, dg" , d,'".
Consider any value of IS,. If the point b is one of the
points of the partition upon which ISa is computed, then tS,
is the s.im of one value of iS, and one value of IS,. If b is
not a point of this partition, let JbX be the length of the inter-
val of a-^i that contains b. Then for properly chosen ^Ss
and bS,
\%+lS,-'aSs\<AbX{M-m)<-^. ... (3)
So in evory case (whether or not 6 is a partition-point of „;:,)
by combining (1), (2), and (3) we obtain the result that for
every e there exists a d, such that for every aSi,
dx\<e.
IS,,- fj{x)dx- fj{x)dji
Therefore L 'aS,= / /(x)dx+ Imdx,
which proves the theorem.
Theorem 107. Provided both integrals exist, ■f and a<b,
r\m\dxt\ f){x)dx .
t That the first integral exists if Che second exists is shown in Theorem 135.
166 INFINITESIMAL ANALYSIS.
Proof. I\}i^k) MiX > I Ifi$k)^kX\.
Hence for every (Sj|/(x)| there is a smaller or equal Si fix), the
d's being the same. Hence by CoroUary 2, Theorem 40,
LS,\m\>\ LSem\.
Theorem io8. // / f{x)dx exists, then I f{x)dx exists and
f f(x)dx=- ffixjdx.
Proof. — ^This is a consequence of the theorem (Corollary 1
Theorem 27) that
L (-/(x)) = - Lfix),
a sum
cor-
for to every S used in defining / f{x)dx corresponds
equal to —S which is used in defining / f{x)dx.
Similarly to every S' used in defining / f{x)dx there
responds a sum -S' used in defining / fix)dx. Hence the
function Sg in the definition of / f{x)dx is the negative of the
function Sg used in the definition of / f{x)dx. Hence the
theorem follows from the theorem quoted.
We adjoin the following two theorems, the first of which is
an inmiediate consequence of the definition of an integral, and
the second a corollary of Theorems 105, 106, and 108.
DEFINITE INTEGRALS. 167
Theorem 109. J ^^^ f{x-h)dx exists and is equal to / f{x)dx,
provided the latter integral exists.\
Theorem no. // any two of the following integrals exist, so
does the third, and
£ f{x)dx+ Jjf{x)dx= r f{x)dx.
Theorem in. If C is any constant and if f{x) is integrable
I — I I — I
onab, then Cf{x) is integrable on ah and
£'cf{x)dx = cj'''fix)dx.
n
Proof.— Sa= I fi$k)^kX is an S) of the set which defines
i— 1
fb
J^ f{x)dx and 5/= I^ Cf{$k)J^ is the corresponding S» of the
set which defines / Cf{x)dx. Hence our theorem follows
immediately from Theorem 34, a special case of which is L Cf{x)
=CLf{x).
Theorem 112. // fi{x) and fzix) are any two functions each
I — I
integrable on the interval a b, then f{x) =f\{x) ±f2{x) is integra-
I — I
bU on ah and
rf(x)dx= rfi{x)dx± Pfzixjdx.
Proof. — ^The proof depends directly upon the theorem that
if L ^i(x)=6i, and L ^(x) =62; then L ^i(x)±^(x)=6i±62
x=a x=a x=a
(Theorem 34).
t First stated formally by H. Lebesode, Lemons smt VInUgration, Chapter
VU, page 98.
168 INFINITESIMAL ANALYSIS.
1-1
Theorem 113. // /i(x) and foix) are integrable onab and such
l-l
that for every value of x on a b fi(x)tf2{x), then
rf,(,x)dx> rf2{x)dx.
Proof. — Since Si is always greater than or equal to S2, then,
by Theorem 34, L Sit L S2, which proves the theorem.
1=0 oiO
Theorem 114. (Maximum-Minimum Theorem.)
If (1) the product fi{x)-f2{x) and the factor fi{x) are inte-
i-l
grable on a b,
I 1
(2) /i (x) is always positive or always negative on a b,
(.3)' M and m are the least upper and the greatest lower
I — I
bounds respectively of /2(x) on a b,
then m- / fi{x)dx< / fi{x)f2ix)dx<M ■ I fi{x)dx,
or m- f fi{x)dx > f)i{x) -fz^x >M- P fx{x)dx.
Proof. — By Theorem 111,
M ■ J fi{x)dx== rM-fi{x)dx
and m- / fi{x)dx= I m-fi{x)dx.
%/ a */o
But in case /i(x) is always positive,
m-fi{x)<fi{x)-f2{x)<M-fi{x).
Hence, by the preceding theorem.
J^m-fi{x)dx ^fjiix) ■f2{x)dx <f^M.f{x)dx,
DEFINITE INTEGRALS. 169
and therefore
m- rji(x)dx< rh{x)-f2ix)dx<M- P U{x)dx.
J a %J a J a
If/i(x) is always negative, it follows in the same manner that
m- rji{x)dxl /^fi{x)-h{x)dx>M- Ai(i)dx.
As an obvious corollary of this theorem we have the Mean-
value Theorem :
Theorem 115. Under the hypothesis of Theorem II4 there exists
a number K, m'S K^M, such that
rh{x)-f2(x)dx=K f''h{x)dx.
Corollary 1. In case f^^x) is continuous we have a value
I — I
f of X on a h such that
Ai(:c)-/2(x)dx=/2(f) f U^)dx.
J a 'J "■
In case /i(x)=l, / /i(x)dx = 6-a,
and the theorem reduces to this :
Theorem 116. // /(x) is any integrable function on the inter-
val a b, there exists a number M lying between the upper and lower
I — I
bounds of /(x) on a b such that
rfix)dx = M{b-a),
I — I
and if fix) is continuous, there is a value $ of x on a b swk
that fj{x)dx = fmh-a).
170 INFINITESIMAL ANALYSIS.
In many applications of the integral calculus the expression
Cmdx
—^ represents the notion of an average value of the
dependent variable y=f{x) as x varies from a to 6. An average
of an infinite set of values of /(x) is of course to be described
only by means of a limiting process. Consider a set of points
\-. — I
xj, X2, . . . , Xn-i,Xn=b ou thc interval a b such that
Xi—a = X2—Xi = X3—X2 = . ■ . = X„-i—X„^2=b—Xn-l.
Then M„=- I f{xk),
and we define the mean value of f{x), lM}(x)= L M„ if this
n=oo
,. . . T% b — a
linut exists. But Xk+i—Xk= =ix.
If the definite integral / f{x)dx exists, we may write
i/ a
/ f(x)dx= LS»,
where
Si= I f{xk)Ax= I f{xk)^^ = ^^^ 1 f{xk) = (b-a)M„.
k-i k"! n n jc^i
Therefore L Sa = ib-a) L M„.
} = n=ixi
We therefore have the theorem:
Theorem 117. In case the integral of f{x) exists on the interval
yf{x)dx
a
o b, iMfix) =
b — a
We note that \M is the same as the K which occvirs in the
mean-value theorem, and that the last theorem suggests a simple
DEFIXITE IXTEGRALS. 171
method of approximating 'the value of a definite integral by
multiplying the average of a finite nmnber of ordinates by b-a.
§ 5. The Definite Integral as a Function of the Limits of
Integration.
I 1
Theorem 118. // /(.r) i\>f integrahle on an intenal a b, and
I — I r^
if X i^ any poini of a b, I f{x)dx is a contimtous function of x.
Proof. — / /(,x)dr exists, b>- Theorem 105, and by the defini-
tion of a continuous function we need only to show that
L^(^fj{x)dx-fjKx)dj^ =0.
By the theorems of the preceding section,
A(j)dr - A(x) rf-r = rf{x)dx<\iB- (/ -x) \<\B- [x' -x)\,
%/a U a *J I
where {B stands for the least upper bound of f{x) on the inter-
I — 1 - I — I
val J x! , and B for the least upper bound of f{,x) on a b. Smce
5 is a constant, B\x' -x) approaches zero as x' approaches x.
and therefore by Theorem 40, Corollary 4, the conclusion of our
theorem follows.
Theorem 119. // /(x) Vs conXinuoM^ on an inten'ol a b,
/fix)dx (a<x<6) possesses a derivative icith respect to x such
that
Proof.— By the preceding theorem / /(x)dx is continuous.
172 INFINITESIMAL ANALYSIS.
To form the derivative we investigate the expression
f)(x)dx- f){x)dx f){x)dx
X' —X x' —X
as x' approaches x.
By Theorem 115 (the mean-value theorem),
(1)
f%)dx = mx!)){xf-x),
where f (x) is a value of x between x and x' and is a function
of x'. Hence (1) is equal to
m (2)
But as xf approaches x, f also approaches x and so, by Theorem
39, as x' approaches x, (2) approaches j{x). Therefore
/ l{x)dx- / j{x)dx ,
-=/(x)4/«x)i..
X —X
Following is a more general statement of Theorem 119.
Corollary. — ^If /(x) is continuous at a point Xi of a b ana
mtegrable on a b, then at a; = Xx
lf)(x)dx^Kx).
The proof is like that of Theorem 112 except that
y/(x) dx = {x- Xi)M(x) ,
and M(xi) is a value between the upper and lower bounds of
DEFINITE INTEGRALS. 173
fix) on xi X. But by the continuity of f{x) at i
L Mix) = fix,),
and hence the conclusion follows as in the theorem.
Theorem 120. // fix) is any continuous function on the inter-
I — I
val a b, and Fix) any function on this interval such that
im-fix),
then Fix) differs from I fix)dx at most by an additive constant.
Proof.— Let Fix) = j ^fix)dx + 4>ix).
Since Fix) and / fix)dx are both differentiable,
By the preceding theorem
d,
dx.
fjix)dx=fix).
Hence j-^ix) =0, whence, by Theorem 94, <^(a;) is a constant.
Theorem 121. // fix) is a continuous function on an interval
a b and Fix) is such that
then £fix)dx = Fib)-Fia).
/;
174 INFINITESIMAL ANALYSIS.
Proof. — By the last theorem,
f{x)dx=F{x)+c.
But 0= rf{x)dx=F{a)+c.
Therefore -F(a)=c.
Whence f''f{x)dx=F{b) +c=i?'(6) -F(a).
The symbol [F{x)fa or |S F{x) is frequently used for ^(6) -F{a).
In these terms the above theorem is expressed by the equation
){x)dx = \lF{x).
r
By this last theorem the theory of definite and indefinite
integrals is united as far as continuous functions are concerned,
and a table of derivatives gives a table of integrals. For dis-
continuous functions the correspondence does not in general
hold. That is, there are on the one hand integrable functions
](x) such that / \{x)dx is not differentiable with respect to x,
.and on the other hand differentiable functions ^(x) such that
■^'(x) is not integrable. t
§ 6. Integration by Parts and by Substitution.
The formulas for integration by parts and by substitution
are ordinarily written as follows:
/ udv = uv— / vd.u,
f M^y- J Kyy%dx.
t For a good discussion of this subject the reader is referred to H. Le-
BESGUE, L&;ons sur V Integration.
DEFINITE INTEGRALS. 175
The following theorems state sufficient conditions for their
validity.
Theorem 122. (Integration by parts.)
£ hix) ■h'{x)dx = \jM ■f2{x)^- fj2{x) ■h'{x)dx,
-provided //(x) and fz'ix) exist and are continuous on the interval
I — I
a b.
Proof. — By Theorem 75,
£{hix) -Mx)) =/i(x) -k'ix) +k(.x) ■h'i.x).
Therefore
X di^f^^""^ •/2W)dx = y^ /i(x) ■f2'{x)dx + £''j2{x) ■fi'ix)dx.
(The integral exists since it follows from the existence and
continuity of //(x) and ^'(a;) that /i(x) and /zCz) are continuous).
By Theorem 121,
£ii\h(x) •Hx)\dx=hQ>) ■f2ib) -hia) ./2(a).
Therefore
fjl(x) ■h'{x)dx = [hix) ■hix)l- fj2{x) ■ll'(x)dx.
Theorem 123. (Integration by substitution.) If y = cp(x) has a
I— I
continuous derivative at every point of ab and f(y) is continuous
for all values taken byy = (j)(x) as x varies from a to b,
where A = (j>(a), B = <f>(b).
176 INFINITESIMAL ANALYSIS.
Proof.— By Theorem 120 and by Theorem 79,
C being an arbitrary constant. C is determined by letting
X = a. Then if x = 6 we have*
Theorem
fjmy-£mt-d--
f(x)dx = J^ fi4>iy))-^dy,
where x — <j){y) and a = 4>{A), h = 4>{B); provided that both inte-
grals exist, and that <f>{y) is non-oscillating and has a finite
derivative.
Proof. f f{x)dx= L I f{$k)^kX (1)
whenever the least upper bound of J4X for each n approaches
„ , B-A
zero as n approaches + 00 . Now let jy= ,
yk=A+k-Jy,
4>(yk)-4>(.yk-i)-.dkX.
Hence, by Theorem 85, AkX = ^'(j)]^Ay,
where ij* lies between j/i and yk-i- Now if fj: = ^(ijA), it will
lie between <j){yk) and cj>{yk-i); moreover the Akx's are all of
the same sign or zero; and since the hypothesis makes ^(t/)
uniformly continuous, their least upper bound approaches zero
as n approaches + 00 .
Therefore f f{x)dx= L I f{^k)^k£
= L I f{4>irik))-<f>'{r,k)-Jy
nioo k=l
^fii>iyW(.y)dy,
DEFINITE INTEGRALS. 177
provided the latter integral exists.
Hence fji'')^^ = fj('f>iy)) '^dy.
Corollary. — The validity of this theorem remains if <f>(y)
has a finite number of oscillations.
Proof. — Suppose the maximum and minimum values of
<p{y) are
Oi, (h, 0,3, ... , o„,
corresponding to the values of y,
Ai, A2, A3, ... , An.
Then we have
ffix)dx= f''){x)dx+ f\x)dx + ...+ rf{x)dx
r^ dx
=//(*(x);5*
The form of this proposition given in Theorem 123 would per-
mit an infinitude of oscillations of ^(2/).
§ 7. General Conditions for Integrability.
The following lemmas are to be associated with those on
pages 155 and 156.
Lemma 3. If ttj is a repartition of n, then for any function
I — I
bounded on a b
0.,<0..
Proof. — Any interval iti of ^ is composed of one or more
intervals Ak'x, Ak'x, etc., of ni, and these contribute to 0,,
the terms
\Ak'x\Ak'y + \Ak"x\Ak"y + (1)
178 INFINITESIMAL ANALYSIS.
The corresponding term of 0^ is
\Jkx\Jky = \^k'x\J^y + \Jk"x\Jky+ (2)
Since each of JVy, ^k"y, etc., is less than or equal to Jky,
(1) is less than or equal to (2), and hence 0,. ^ 0».
I — 1
Lemma 4. If kq is any partition of the interval a b, and
eo any positive number, then for any bounded function there
exists a number do such that for every partition tc whose greatest
J is less than 80
Proof. — We prove the lemma by showing that if no has
N + 1 partition points Xo, Xi, X2, . . . , x„, an effective choice
of ^0 is
where R is the oscillation of the function on a b.
Of the intervals of tz there are at most JV — 1 which con-
tain as interior points, points oi xq, xi, . . . , xif. Denote the
lengths of these intervals of n by JpX, and denote by JgX the
lengths of the intervals of k which contain as interior points no
points of Xo, Xi, X2, . . . , x^. Then
V q
If ;:' is a repartition of no obtained by introducing the points
of n, then
Q
is a subset of the terms whose sum constitutes O^/. Hence, by
Lenuna 3,
i'|i^|.i,2/<0.-<0,„.
Since \A^\<s^ = J2-^
DEFIXITE INTEGRALS. 179
it follows that i'|Jpx| • Jp2/<«o.
p
Therefore 0,„ + £o^O,.
Lemma 5. If ;r is any partition, 0, is the least upper bound
of the expression
o» — »S, ,
where SJ and SJ' may be any two values of 5, corresponding
to different choices of the f 's.
Proof. — ^Without loss of generality we may assume every
Jtx positive.
Then BS^ -BS, = B\S: -S^"\.
But BS, = b\ I K^k)-A,A =• I {5(/et)|Jtx
It-i J *=i
and BS^ = B-
t=i J *=i -
Therefore BS^-BS,= -" [5/(ft)-J3/(ft)]Jtx
— *-i —
n
Therefore 5(5/ -5/') =0..
Theorem 125. A necessary and sufficient condition that a
function f{x), defined, single-valued, and bounded on an interval
I— I I— I
a b shaU be integrable on a b, is that the greatest lower bound of
On for this function shall be zero.
Proof. — ^We first show that if /(x) is integrable the lower
bound of 0, is zero. By hypothesis,
r^f{x)dx= L S)
■exists. By Theorem 27, Chapter IV, this implies that for every s
180 INFINITESIMAL ANALYSIS.
there exists a S, such that for every di<d. and 32<d,
Hence, if ;r be a partition whose intervals i^x are all less than
d„ we inust have
\SJ-S/'\<e
''•!
for every 5/ and S/'. By Lemma 3 this implies that 0„ < s.
But if for every £ there exists a tt such that 0^ < e, then
50. =0.
Secondly, we show that if the lower bound of 0. is zero,
S3 converges to a single value,
f){x)dx,
as d approaches zero. Given any positive quantity s there
exists a partition n, such that On,<-7. By Lemma 4 there
exists a S, such that for every ti whose intervals are numerically
less than d,
0,<0..+|<-|.
Now let Sn,' and Sn," be any two values of Sg^, and let
;:/" be the partition composed of the points of both ;r/ and
7:,". Then for any value of Sn^"' we have, by Lemma 2,
\S„; -Sn."'\ 20< <^,
\Sn,"-S„;"\<On;'<-^.
Therefore \S,t,' -Sn,"\<s.
DEFINITE INTEGRALS. 181
Hence for every e we have a d, such that for every two values
oiS,,d<d„
\S.,'-Sn,"\<e.
By Theorem 27, this imphes the existence of L Sg.
In case the definite integral does not exist it is sometimes
desirable to use the upper and lower bounds of indeterminate-
ness oi Ssas d approaches zero. These are denoted respectively
by the symbols / f{x)dx and / }{x)dx f
and are called the upper and lower definite integrals of /(x)
They are both equal to
f)ix)dx
if and only if the latter integral exists. They are usually
defined by the equations
BS.
f f{x)dx=R
-'it;
«/ a
where 5,= i" {Bfi^^)\Jia for all partitions of tc, and
/
fix)dx = BS,
where S,= 1 i^/Cc^) UhX for all partitions of n.
k=l
That f f{x)dx exists when the upper and lower integrals
are equal is evident under this definition, because
o.=s^-s^,
t For a more extended theory of these integrals, cf. Pierpont, page 337.
182 INFINITESIMAL ANALYSIS.
and thus B0^=0 if and only if
/* fix)dx= f fix)dx.
For every value of 5>0 there is an infinite set of partitions
n, for which the largest J^a; is less than d, and for each of these
there is a value of 0^. If Os stands for any such 0^, then
0) is a many-valued function of b.
Theorem 126. A necessary and sufficient condition that a
function f{x), defined, single-valued, and hounded on an interval
I — I
a b, is integrable is that
L O,=0.
Proof.T-r^e condition is necessary.
By Theorem 125 the integrability of f(x) implies B0„=0.
Hence for every e there exists a partition t: such that
By Lemma 4 there exists a 1?, such that for every s^ whose
greatest Jx is less than d,
0,'<0, + £<2s.
Hence L O' = 0.
The condition is sufficient.
Since L O*=0,
andO,>0, B0],=0.
Hence the function is integrable by Theorem 125.
Theorem 127. A necessary and sufficient condition that a
function, defined, single-valued, and bounded on an interval a b,
shall be integrable on that interval is that for every pair of positive
DEFINITE INTEGRALS. 183
numbers a and X there exists a partition n such that the sum of
the lengths of those intervals on which the oscillation of the function
is greater than a is less than A.
Proof. — The condition is necessary.
If for a given pair of positive numbers a and X there exists
DO ;r such as is required by the theorem, then 0^> a- A for every
n, which is contrary to the conclusion of Theorem 125. that
The condition is sufficient.
For a given positive e choose a and X so that
e £
a(b — a)<-^ and XR<-^,
where R is the oscillation of the function on a b. Let s^ be a
partition such that the sum of the lengths of those intervals on
which the oscillation of the function is greater than a is less
than A. Then the sum of the terms of 0^ which occur on these
intervals is less than
XR,
and the sum of the terms of 0^ on the remaining intervals is
less than
aQ) — a) .
Therefore On<X-R + aQ)-a)<£.
Hence BO, = 0,
whence by Theorem 125 the integral exists.
Defimtion.— The content of a set of points [x] on an interval
a 6 is a number C[x] defined as follows: Let w be any parti-
tion of a b, none of the partition points of which are points
of [x], and D, the sum 'of the lengths of those intervals of n
184 INFINITESIMAL ANALYSIS.
which contain points of [x] as interior points. Then
BD^ = C[x].
An important special case is where
C[x] = 0.
It is evident that if a set [x] has content zero, for every e
there exists a finite set of segments of lengths
«i, «2, £3, • • • , «n
which contain every point [x] and such that
n
1=1
It is also evident that if the sets [xi] and [2:2] are of content zero,
then the set of all Xi and X2 is of content zero.f
Theorem 128. A necessary and sufficient condition for the
integrability of a function f(x) on an interval a b is that for every
a>0 the set of points [x„] at which the oscillation of f{x) is greater
than or equal to a shall he of content zero.X
Proof. — If at every point of an interval c d the oscillation
of /(i) is less than a, then about each point of c d there is a
segment upon which the oscillation is less than a, and hence
by Theorem 11, Chapter II, there is a partition of c d upon
each interval of which the oscillation of f{x) is less than a.
Now to prove the condition sufficient we observe that if the
content of {x„] is zero, there exists for every X a partition TZi
such that the sum of the lengths of the intervals containing
points of [x,] is less than X. Moreover we have just seen
t For further discussion of the notion content see Pierpont, Real Func-
tions, Vol. I, p. 352, and Lebesque, Lemons sur I'lntigration.
X Compare the example on page 155.
DEFINITE INTEGRALS. 185
that the intervals which do not contain points on [x„] can be
repartitioned into intervals on which the oscillation is less than
a. Hence, by Theorem 127, the function is integrable.
To prove the condition necessary we note that on every
interval containing a point, x„ the oscillation of j{x) is greater
than or equal to or equal to a. Hence, if
C[xj>0,
the sum of the intervals upon which the oscillation is greater
than or equal to ct is greater than C[x<,].
Definition. — A set of points is said to be numerable if it is
capable of being set into one-to-one correspondence with the
positive integral numbers. If a set [x] is numerable, it can
always be indicated by the notation Xi, X2, X3, . . . , Xn, . . . , or
\xn\, but if it is not numerable, the notation \xn\ cannot be
apphed with the understanding that n is integral.
Theorem 129. A 'perfect set of points is not numerably infinite.^
Proof.— Suppose the theorem not true. Then there exists
a sequence of points \xn\ containing every point of a perfect set
[x]. Let Pi be any point of [x], and a^ bi a segment containing
Pi. Let x„, be the first of lx„! within oi 61. Since x„ is a
limit point of points of [x], there are points of the set other
than Pi and z„, on the segment ai bi. Let P2 be such a point,
and let 02 62 be a segment within ai 61 and containing P2
but neither Pi nor x„,. Let x„^ be the first point of {x„i
within 02 62. Proceeding in this maimer we obtain a sequence
of segments ja, bi\ such that every segment lies within the
preceding and such that every segment ai bi contains no point
^ni-ic of the sequence \xn\. By the lemma on page 42, Chapter
II, there is a point P on every segment of this set. Since there
are points of [x] on every segment a,- bi, P is a limit point of the
set [x]. Since [x] is a perfect set, P is a point of [x]. But if P
t For definition of perfect set see page 91.
186 INFINITESIMAL ANALYSIS.
were in the sequence jz„}, there would be only a finite number
of points of [x] preceding P, whereas by the construction there
is an infinitude of such points.
Theorem 130. A numerably infinite set of sets of points each
of content zero cannot contain every point of any interval.
Proof. — Let the set of sets be ordered into a sequence {[x]„] .
We show that on every segment a b there is at least one point
not of ![x]„). Since [x]i is of content zero, there is a segment
oi 61 contained in a 6 which contains no point of [x]i. Let
[x]„j be the first set of the sequence which contains a point of
Ci bi. Since [x]„, is of content zero, there is a segment 02 62
contained in ai 61 which contains no point of [x]„,. Continuing
in this manner we obtain a sequence of segments o b,ai 61, ... ,
a„ bn . . . such that every segment lies within the preceding,
and such that a„ 6„ contains no point of [x]i, . . . , [x]„. By
the lemma on page 42 there is at least one point P on all these
segments. Hence P is a point of a 6 and is not a point of any
set of |[x]„!.
Theorem 131. The points of discontinuity of an integrabk
function form at most a set consisting of a numerable set of sets,
each of content zero.
Proof. — Let <ti, 02, 0-3, .. .
be any set of numbers such that
and L a„ = 0.
n=oo
By Theorem 128 the set of points [x„J at which the oscillation
of f{x) is greater than or equal to On+i and less than «;„ is of
content zero. Since the set of sets j[x„Ji includes all the points
of discontinuity of fix) , this proves the theorem.
Theorem 132. // a function f{x) is integrable on an interval
I— I
o b, then it is continuous at a set of points which is everywhere
I— I
dense on a b.
DEFINITE INTEGRALS. 187
Proof. — ^If the theorem fails to hold, then there exists an
interval a 6 on which the function is discontinuous at every
point. By Theorem 131 an integrable function is discontinuous
at most on a numerably infinite set of sets each of content zero,
and by Theorem 130 such sets of sets fail to contain every
point of any interval.
Theorem 133. // / f{x)dx =
for every X of a b, then /(x)=0 on a set of points everywhere
I — I
dense on a b, and for every a>0 the points where \fix)\> a form
a set of content zero.
Proof. — ^At every point X where f{x) is continuous, accord-
ing to the corollary of Theorem 119,
smce
I f{x)dx IS a constant. The points of continuity of f{x)
J a
are everywhere dense, according to Theorem 132. Hence the
zero points of /(i) are everywhere dense. At a point of discon-
tinuity the oscillation of f{x) is greater than or equal to |/(x)|.
Hence the points where |/(x) | > a form a set of content zero.
Theorem 134. //
rx nx
I f{x)dx= I (f>{x)dx
I 1
for every X of a b, then f{x)=<f)(x) on a set of points everywhere
I — I
dense on a b, and for every <7>0 the points where \f{x) —<f){x)\>a
forms a set of content zero.
Proof. — Apply the theorem above to /(x) -<^(x).
Theorem 135. // /(x) is integrable from a to b, then |/(x)| is
integrable from a tob.'f
tThe converse theorem is not true; cf. example given on page 192.
188 INFINITESIMAL ANALYSIS.
Proof.—
Since 0<0,\Kx)\<OJ{x),
it follows that BO^f{x) = implies 5 0„|/(x)| = 0, and hence
the integrabilityof j{x) implies the integrability of |/(a;)|.
Theorem 136. // /(x) and <l>{x) are both integrabk on an inter-
I— I
vol a b, then
f{x)-4>{x) (1)
I— I ^ -
IS integrable on a b; and, provided there is a constant m > such
I — I
that |^(x)| — m>0 for x on a b, then
/(x)H-^(x) (2)
I— !
is integrable on a b.
Proof. — Since f{x) and ^(x) are both integrable on a 6, it
follows that for every pair of positive numbers a and A there
is a partition tti for fix) and a partition 712 for ^(x) such that
the sums of the lengths of the intervals on which the oscilla-
tions of f{x) and <f>(x) respectively are greater than a are
less than X. Let 7: be the partition consisting of the points
of both TTi and 712. Then the sum of the intervals of tz on which
the oscillation of either f{x) or (p{x) is greater than a is less
than 2A. Let M be the greater of B\f{x)\ and B\(f>{x)\ on a b.
Then on any interval of 71 on which the oscillations of /(x)
and (p{x) are both less than a the oscillation of /(x)-^(x)
is less than aM. Hence the sum of the intervals on which
the oscillation of /(x) ■ 0(x) is greater than aM is less than 2A.
Since a and A may be chosen so that 2 A and aM shall be any
pair of preassigned numbers, it follows by Theorem 127 that
1—1
fix) ■<f>ix) is integrable on a b.
In view of the argument above it is sufficient for the second
DEFINITE INTEGRALS. 189
part of the theorem to prove that -t-~. is integrable on a 6 if
</>(a;) is integrable and \(f>{x)\ >m. Consider a partition re such
that the sum of the intervals on which the oscillation of ^(x)
is greater than a is less than X. Since
1 1
^(Xi) <j>{X2)
.. \<f>(Xi)~(<j>X2)\
mXl)\-\<f>{X2)\
it follows that n is such that the sum of the intervals on which
the oscillation of -ri-^ is greater than —5 is less than X, and
T7-T is integrable according to Theorem 127.
A second proof may be made by comparing the integral
oscillations of /(x) and ^(x) with those of the functions (1) and
(2) and applying Theorem 125. j
Theorem 137. // /(x) is an integrable function on an interval
1—1
a b, and if <p{y) is a continuous function on an interval
Bf Bf, where Bf and Bf are the lower and upper bounds respecl-
~ ~~ I — I
ively of f(x) on a b, then (p\f{x)\ is an integrable function of x
I — I
on the interval a b.X
Proof. — By Theorem 48 there exists for every a>0 ad„ such
that for \yi-y2\<d„
\<f>(yi)-<f>(y2)\<<T (1)
Since f(x) is integrable on a 6 it follows by Theorem 127
that for every positive number X there is a partition ic such
t Cf. PlERPONT, Vol. 1, pp. 346, 347, 348.
X This theorem is due to Dtr Bois Retmond. It cannot be modified so
as to read " an integrable function of an integrable function is integrable."
Cf. E. H. Moore, Annals of Mathematics, new series, Vol. 2, 1901, p. 153.
190 • INFINITESIMAL ANALYSIS.
that the sum of the intervals on which the oscillation of f{x)
is greater than d„ is less than X. But by (1) this means that
the sum of the intervals on which the oscillation of ^{f{x)\
is greater than a is less than A. This, by Theorem 127, proves
that ^l/(x)l is integrable.
CHAPTER IX.
IMPROPER DEFINITE INTEGRALS.
§ I. The Improper Definite Integral on a Finite Interval.
If fix) is infinite at one or more points of the interval a b,
then, whatever may be the other properties of the function, the
definite integral of /(x) defined in Chapter VIII cannot exist on
I — I
the interval a b.
Definition. — ^If / f{x)dx exists for every x,a<x<b, and if f
L rfix)dx
x=aJ T.
exists and is finite, /(i) being unboimded on every neighbor-
hood of x = a, then this limit is the improper definite integral
I— I
on the interval a b. If f{x) is unbounded m every neigh-
borhood of x=a, and also in every neighborhood of x=b, but
bounded on some neighborhood of every other point of the
I— I . . I— I I— I
interval a b, we consider two intervals a c and c b where c
is any point a<c<b. If the improper definite integral exists
I— I 1—1 , . .
on a c and also on c b, then the sum of these mtegraJs is the
I— I
improper definite integral on a b.
t We will understand throughout this chapter that in the expression
(,x)dx
I 1
X approaches a on the interval o 6.
191
192 INFINITESIMAL ANALYSIS.
This definition can obviously be extended to the case where
the function is unbounded in the neighborhood of a finite num-
ber of points. Such points are then considered as partition
points, dividing the interval a b into a set of subintervals.
If the improper definite integral exists on each of these in-
I— I
tervals, their sum is the improper definite integral on a b.
Theorem 138. // / f{x)dx exists for every x, a<x<b, then
a necessary and sufficient condition that
L rf{x)dx
t=aJx
shall exist and be finite is that for every e there exists a V,*{a)
I — I
such that for every two values of x, Xi and xz, on the interval a b
and on V,*{a)
I rx,
\J f^'
|m/ Xi
x)dx
<s.
Proof. — This theorem is a special case of Theorem 27, since,
by Theorem 110,
f{x)dx= / f{x)dx- / f(x)dx.
Xi *J X\ \J X2
Theorem 139. // / f{x)dx exists for every x, a<x<6, and if
L r\f{x)\dx
x=a«/ X
is finite, then
L rf(x)dx
exists and is finite.^
t The first part of the hypothesis in this theorem is not redundant as
is shown by the following example. Let f{x) = x-i for positive rational values
IMPROPER DEFINITE INTEGRALS. 193
Proof.— By the necessary condition of Theorem 138 there
is a V*{a) corresponding to any preassigned e such that for
any two values of x, xi and Xz, which lie on the segment a~b
and on V*{a)
I r)fix)\d2
<£.
But, by Theorem 107,
\ r)f{x)\dx\>\ r){x)dx\,
since, by the hypothesis and Theorem 105, f^^f{x)dx exists.
Hence, by the sufficient condition of Theorem 138,
L f)ix)d2
exists and is finite.
Theorem 140
I. // / ]{x) dx exists for every x on the segment
a b, and if (x—a)''f(x) is bounded on V*(a) for some valve of
k, 0<k<l, then
L rf{x)dx
x=aJ X
exists and is finite.
Proof. — By hypothesis (i— a)*|/(a;)|<Af, i.e.,
of I and /(i)=— 2~J for positive irrational values of x. In this case
L I \f(x)\dx exists and is finite, while / j{x)dx does not exist for any
value of X on the interval o 5, and consequently L I f(x)dx has no mean-
xiavx
ing since the limitand does not exist.
194 INFINITESIMAL ANALYSIS.
where M may be taken greater than one. The proof of the
theorem consists in showing that for every e there exists a d,
such that if 0<xi-a<d„ 0<X2-a<d„ Xi<X2, then
I r^"'
I f{x)dx < e.
By Theorems 105 and 133,
=^^\ix2-ay-'-(xi-ay-''\.
That the last term of this series of inequalities is infinitesimal,
the reader may verify by choosing
This theorem may also be proved as a corollary of Theorem
Corollary. — If /(x) is integrable on x 6 for every x of a 6,
1 is of the same
oi k, 0<k<l, then
143
and is of the same or lower order than -. rr for some value
{x — a)"
L Prndz
exists and is finite.
Theorem 141. // for any positive number m and for any ktl
there exists a V*{a) on which f{x) does not change sign, and on
■which (x— o)* /(x) >m for every x, then
L rf{x)d2
^aJ X
cannot exist and he finite.
L
IMPROPER DEFINITE INTEGRALS. 195
Proof. — (1) In case
/jix)dx
fails to exist for some value of x between a and b,
L f''fix)dx
fails to exist because the limitand function does not exist.
(2) If fj^''^'^-
exists for every value of x between a and h, we proceed as follows :
Let 5< 1 be the length of a V*{a) on which f(x) does not change
sign, and on which {x — aYj{x)>m, and let Xi be the extrem-
ity of this neighborhood, which is greater than a. Then
l/(^)l>(^r^>(^^r^* ^^^ ^^^'■y ^ ""^ *^^ neighborhood.
Take xi so that (X2 - a) * = 2(a;2 - Xi) .
>fe?^^^^-^^) = ^-
Then / f{x)dx
Hence, by the necessary condition of Theorem 138,
L rf{x)dx
cannot exist and be finite.
Theorem 142. // L f{x)dx
cxiste and is finite and if fix) approaches infinity monotonicaliy
as x=a on some V*ia), then
L (x-a)-/(x)=0,
196
INFINITESIMAL ANALYSIS.
1
or in other words f{x) has an infinity of order lower than ^3^-t
Proof.— By means of Theorem 138 it follows from the hypoth-
esis that for every £ there exists a 7* (a) within V*{a) such
that for every Xi and X2 on a b, and also on V*ia),
/ fix)dx
<£.
Let X2 be any point of such a neighborhood and let Xi be so
chosen that Xi — a = X2 — Xi .
Since Xi and X2 are on V*{a),
/(Xi)>/(X2).
It follows from Theorem 116 that
But
/ fix)dx >|/(X2)|-(X2-Xi).
/(X2) • (X2-X1) = J/(X2) • (x2-a).
f L (i— o)-/(x) =0 is not a sufficient condition for the existence of
L I l(x)dx,
xia'' X
as is shown by the following example. Consider a set of points i,, I2
•Cs, . . . , i„, . . . such that Xn — a = 2(i„-|-i — a), Xi — a being unity.
2"
Define /(i,) = l, /(x2)=|, /(i,)=2, . . . , /(=r")=^:fri' Letthefunc-
tion be linear from f{xO to /(12), from /(xj) to /(xa), etc. Then
1/
Wx\
/(x)dx >i,
/(x)<ix
>J, etc.
Since these integrals are all of the same sign, their sum for any given num-
ber of terms is greater than the sum of the corresponding number of terms.
2
in the harmonic series. Also (x„ — o) ■ /(xr.) = — tT' ■'^l»e°ce L (x — a)/(x)=0.
IMPROPER DEFINITE INTEGRALS. 197
Hence for x = X2, \j{x) \-{x-a)<2e.
Since e is arbitrary, and since X2 is any point in V*{a), it follows
that L /(x)-(x-a)=0.
Corollary. — If / f(x)dx
exists for every x between a and h, and
l^ f)ix)dz
L
x=a\J X
exists and is finite, and if /(x) is entirely positive or entirely
negative, then zero is a value approached by {x — a)-f(x) as x
approaches a.
Proof. — Consider the case when the function is entirely
positive. Suppose zero is not a value approached. Then there
exists a pair of positive numbers £ and 8 such that for every
X, x — a<d,
{x—a)-f{x)> £.
I— I
On the interval, a a + d, consider the function
x—a
Since / ——zdx
Jx x—a
is a non-oscillating function of x, it follows from Theorem 25 that
L f ——dx
xLaJ T x-a
exists, and by Theorem 142 this limit must be infinite.
198 INFINITESIMAL ANALYSIS.
Since 1/(^)1 >^
on the neighborhood under consideration, it follows from Theo-
rem 107 and Corollary 2; Theorem 40, that
L f f{x)d2
L
x=a^ X
exists and is infinite, which is contrary to the hypothesis.
Theorem 143.! // (1) fii^) and /2(x) are of the same rank of
infinity at x = a, or if /i(x) is of lower order than fiix),
(2) / fi{x)dx and I f2ix)dx both exist
for every x on the segment a b,
(3) There is a neighborhood of x^a on
which /2(x) does not change sign,
P
(4) L I f2(.x)dx is finite, J
th£n it follows that L I f\{x)dx exists and is finite.
t This is what Professor Moore in his lectures calls the relative con-
vergence theorem. Theorems 143, 144, 151, 152 in this form are due to him.
t We notice that since under the hypothesis /zCi) does not change sign,
L I f2{x)dx
L I f2{x)dx
J X
cannot fail to exist either finite or infinite, for it follows from this hypothesis
that / fi{x)dx is a non-oscillating function of x and therefore, by Theorem 25
that the limit exists.
IMPROPER DEFINITE INTEGRALS. 199
Proof. — Since from the hypothesis
L I f2{x)dx
exists and is finite, we have by Theorem 138 that for every t
there exists a F*(a) such that for every xi and X2 on segment
a b and on V*{a)
IX
X2
J2{x)dx
xi
<e.
<M
Consider Xi and X2 on a neighborhood of a; = a for which ^^
)/2(x)
and for which /2(x) does not change sign. Then, by Theorem
113,
I f''h(x)dx <M-\ f'^f2(x)d2
<M-c.
Since M-e can be made small at will by making e small, it
follows by Theorem 138 that
£hix)d2
L
TJJ iU X
exists and is finite.
An important special case of this theorem is when /i(x)
is of the same or lower order of infinity than /2(x), i.e.,
L . , . =.K, a constant not zero.
x=a/2(2;)
The reader should verify for himself that Theorem 140 is a
corollary of Theorem 143. The other previous tests for the
existence of the improper definite integral can all be reduced to
special cases of Theorem 143. Cf., for example, the logarithmic
test on page 410 of Pierpont.
Theorem 144. // (1) /i(x) and /2(x) are of the same rank of
infinity at x=a, or if /i(x) is of higher order than f2{x),
(2) / f\{x)dx and I f2{x)dx both exist
I— I
for every x on the segment a b,
200 INFINITESIMAL ANALYSIS
(3) There is a neighborhood of x=a on
which /i(x) does not change sign,
(4) L I f2(x)dx is infinite (see note
under Theorem 143),
then L / fi (x)dx exists and is infinite or fails to exist.'f
x= a\J X
Proof. — This is a direct consequence of Theorem 143, since if
L / /i(x)dx,
x^aJ X
which exists by the foot-note of Theorem 143, were finite, then
L I f2{^)dx
x'=aJ X
would exist and be finite.
Theorem 145. // for a function /i(x) which does not change
sign in the neighborhood of x — a there exists a monotonic func-
tion f2{x) infinite of the same rank as fi(x) as x approaches a,
I fi{x)dx and / f2ix)dx both existing for every x on the seg-
ment a b, then a necessary condition that L I fi {x)dx shall exist
xLa\J X
and be finite is that
L (x-a)-/i(i)=0.
Proof. — By hypothesis
L rfx{x)dx
t This is what Professor Moore calls the relative divergence theorem.
IMPROPER DEFINITE INTEGRALS. 201
exists and is finite. Hence, by Theorem 143,
x'=.aJ X
exists and is finite. Therefore, by Theorem 142,
L (x-a)-/2(x)=0.
Since j^^ is bounded as x approaches a, i.e., |/i(2;)| <M- |/2(x)|,
we have (i-a) ■ \h{x)\<M-{x-a) ■ \U{x)\.
But L M.(x-a).|/2(a;)|=0.
Therefore, by Corollary 4, Theorem 40,
i'(x-a)-|/i(x)|=0,
or by Corollary 2, Theorem 27,
L ix-a)-fiix)=0.
x~a
§ 2. The Definite Integral on an Infinite Interval.
The integral over an infinite interval, viz..
L h{x)dx,
c=oo«/ a
has properties analogous to those of the improper definite inte-
gral on a finite interval discussed in the preceding section, and
is likewise called an improper definite integral.
The following theorems correspond to Theorems 138 to 145.
202 INFINITESIMAL ANALYSIS.
Theorem 146. // / f{x)dx
exists for every x, a<x, then a necessary and sufficient condition
that L f{x)dx
exists and is finite, is that for every e there exists a D, such that
for every two values of x, Xi and Xj, each greater than D„
I r'f(.x)d
\x\<t
Proof. — The theorem is a, direct consequence of Theorems 105
and 27.
Theorem 147. // / f{x)dx
exists for every x greater than a, and if
L r\m\dx
is finite,'\ then
L f\x)dx
exists and is finite.
Proof. — The proof is like that of Theorem 139.
Theorem 148. // / f{x)dz
exists for every x greater than a, and if {x—a)^-f{x) is bounded as
X approaches infinity for some k, k>\, then
L rf{x)di
exists and is finite.
t Note on page 192 shows that this hypothesis is not redundant.
IMPROPER DEFINITE INTEGRALS. 203
Proof. — ^If in the proof of Theorem 140 we write Z),i-* =
^ j^f instead of d.^-''= ' ~ \ and use Theorem 146 in-
stead of 138, the proof of Theorem 140 will apply to Theorem
148.
Theorem 149. // f{x) does not change sign for x greater than
some fixed number D, and if for some positive number m and
some number k^\ \{x-aY-f{x)\ >m for every x greater than D,
then
L f){x)dx
cannot exist and be finite.
Proof. — By making suitable changes in the proof of Theorem
141 so as to make xi and X2 approach infinity instead of a, that
proof applies to this theorem.
Theorem 150. // L / f{x)dx
1=00 »/(i
exists and is finite, and if f{x) is monotonic for all values of x
greater than some fixed number, then
L (x-a)-/(x)=0.
Proof. — By making slight modifications of the proof of
Theorem 142, that proof applies to this theorem.
Corollary. — If / f{x)dx
exists for every x greater than a, and
L f'f{x)dx
«
exists and is finite, and if /(x) does not change sign for x greater
204 INFINITESIMAL ANALYSIS.
than some fixed number, then zero is a value approached by
(x-a)/(x) as X approaches oo.
The proof is similar to that of the corollary of Theorem 142.
Theorem 151. //
(1) fi{x) and /2(x) are infinitesimals of the same rank as x
approaches 00, or if fi{x) is of higher order than f2{x),
(2) / fi{x)dx and I f2{x)dx both exist for every x, a<x,
(3) /2(2;) does not change sign for x greater than some fi^ed
number,
(4) L I f2{x)dx is finite,
then it follows that
L rfi{x)dx
x^ootJa
exists and is finite.^
Proof. — The proof is analogous to that of Theorem 143.
Theorem 152. //
(1) /i(x) and fzix) are infinitesimals of the same rank as x
approaches infinity, or if fi{x) is of lower order than fiix),
(2) / fi{x)dx and I f2{x)dx both exist for every x, a<x,
%J a J a
(3) /i(x) does not change sign for x greater than some fixed
number,
(4) L I f2{x)dx is infinite,
then
L I fi{x)dx
exists and is infinite or fails to exist.
Proof like that of Theorem 144.
Theorem 153. // for a function fi{x) which does not change
sign in the neighborhood of x=<xi there exists a monotonic func-
tion fzix) such that /i(x) and /2(x) are infinitesimals of the same
t See note under Theorem 143.
IMPROPER DEFINITE INTEGRALS. 205
rank as X approaches infinity, I fi(x)dx and / f2(x)dxbotheJ>-
isling for every x>a, then a necessary condition that
L rh{x)dx
shall exist and he finite is that
L (x-a)-/i(x) = 0.
The proof is like that of Theorem 145.
§ 3. Properties of the Simple Improper Definite Integral.
The following definition of the simple improper definite
integral is equivalent in substance to that given on page 192,
and in form is partly the definition of the general improper
definite integral given on page 210.
The definite integral of a function is said to exist properly at
a point Xi or in the neighborhood of this point, on the interval
l-i I — I . . . .
a 6 if there exists an interval on ai bi contaimngxi as anmterior
point (or as an end point in case Xi=a or Xi=b) ^vrh that the
proper definite integral of fix) exists on this : te val. The
integral is said to exist improperly at a point xi on the interval
aTb if f(x) has an infinite singularity at Xj and there exists an
interval ai bi on a b containing Xi as an interior point (or end
noint in case Xi=a or ii=6) such that the improper definite
I — I . I— I
integral exists on each of the intervals a, xi and xi 61.
If on an interval a b the definite integral exists properly
at every point except a finite number of points, and ex-
ists improperly at each of these points, then the improper
definite integral is said to exist simply on the interval
iTb, or the simple improper definite integral is said to exist on
206 INFINITESIMAL ANALYSIS.
1-1 . I-I
the interval a b. Let Xi, X2, . . . , Xn be the points of a 6 at
which the integral exists improperly. The simple improper
l-l
definite integral on a 6 is the sum of the improper definite
l-l I— I 1 1 I I — I
integrals on the intervals a Xi, xi X2, ■ . . , x„_i x„, x„ b.
We denote the simple improper definite integral of f{x) on
l-l
the interval o 6 by
rmdx
This symbol is used generically to include the proper as well as
the improper definite integral.
Theorem 154. // a<b<c, and if two of the three simple im-
proper definite integrals
I f{x)dx, I f(x)dx, and / f{x)dx
S^a SJb S«/ o
exist, then the third exists and
rf{x)dx+ rf{x)dx^ rf{x)dx.
5«/ o St/ b St/ o
Proof. — If 6 is a point at which the integral exists improperly,
and if
/ f{x)dx and / f{x)dx
sJa sJb
S'Ja SJb
both exist, then by the definition of
rf(x)dx
SJ a
the latter exists and is equal to the sum of the two former.
If one of the two integrals, say
/ Kx)dx,
sJ a
IMPROPER DEFINITE INTEGRALS. 207
exists, and if f" f{x)dx
Sja
exists, then / ^{x)dx
• sjb
exists since only in that case does
frndx
Sja
exist. The equation
/ f{x)dx+ /fix)dx= f'f{x)dx
Sja sJb sJ a
likewise holds.
If 6 is a point at which the integral exists properly, then the
theorem follows from the above argument and the definition on
page 205.
Theorem 155. // / f{x)dx
exists, then I f{x)dx
sJb
exists and I f{x)dx= — I f(x)dx.
sJ a sJb
Proof. — ^In case the integral exists improperly only at one point
of the interval, then the theorem is an immediate consequence
of Theorem 108 and CoroUary 1, Theorem 27. (If L fix) =K,
then L{ —fix) j = —K.) The theorem in the general case follows
directly from this case and the definition of the simple improper
definite integral.
Theorem 156. If c is a constant and if the simple improper
208 INFINITESIMAL ANALYSIS.
definite integral of j{x) exists on a b, then the simple improper
l-l
definite iniegral of c-f{x) exists on a b and
rf(x)dx=' rcf{x)dx.
J a Sj a
Proof. — ^The theorem is a direct consequence of Theorems
111 and 34.
Theorem 157. // the simple improper definite integrals of fi{x)
and fiix) both exist ona b, then the simple improper definite inte-
gral of /i(x) +/2(x) and of /i(x) —f^ix) both exist and
f\fl{x)±f2{x)\dx= rfl{x)dx± f)2{x)dx.
SJa St/o Sja
Proof. — ^The theorem is a direct consequence of Theorems 112
and 34.
Theorem 158. // the simple improper definite integrals of
/i(i) and fzix) both exist, and if fi{x) > fzix), then
rf,{x)dx> rf2{x)dx.
St/ a SJ a
Proof. — The theorem is a direct consequence of Theorem
113 and Corollary 2, Theorem 40.
Theorem 159. // / f{x)dx
sJa
exists, then I f{x)dx
St/a
is a continiums function of the limit of integration on the interval
a 0.
Proof. — If a; is a point at which the integral exists properly,
the theorem is the same as 118. If a; is a point at which
IMPROPER DEFINITE INTEGRALS. 209
the integral exists improperly, then the theorem follows from
Theorems 138 and 27.
Theorem i6o.
// frndx
sJa
eocists, it does not
follow that
fimidx
SJ a
exists.
Proof.— Let
Xl, X2, X3, . . . , Xn,
be an infinite sequence of points on 1 in the order indicated
from 1 towards such that
Jxn X n
Consider a function /(i) defined as follows:
fix) = - on xi 1, X3 xg, etc.
1 I— I I— I
fw=-- on i2 a;i, X4 xs, etc.
Obviously L I f(x)dx t
exists and is finite since the series J— J + i . . . is convergent,
while L C\f(x)\dx
x=oJx
is divergent since the harmonic series is divergent.
t That is a limit point of the sequence of points is obvious since in case
this sequence has a limit point greater than zero the proper definite integral
of the function — would fail to exist on some interval ah where 0<a<6,
X '
which is impossible.
210 INFINITESIMAL ANALYSIS.
§ 3. A More General Improper Integral.
The problem of defining and studying the properties of the
improper integral when the set of points of singularity is infinite
has been treated by many writers.! In this section we give a
few properties of improper integrals as defined by Harnack
and Moore.
I— I
Denote by Po any set of points of content zero on a b, and
by P the set of all points of a 6 not points of Po- P and Pq
I— I
are complementary sub-sets of a b. Denote by / any fimte set
I— I
of non-overlapping intervals of a 6 which contain no point of the
set Pq. The symbol m(7) stands for the sum of the lengths of the
intervals of I. Eor the sake of brevity D will be used for |a— 6|.
The following conditions are assumed to be satisfied :
(a) The definite integral of f{x) exists properly at every
point of P. The sum of the integrals of /(x) on the intervals of
/ is denoted by
Prndx.
'J a I
(b) For every positive e there exists a positive 8. such that
for any two sets, I' and I", of intervals none of which contain
any point of Pq and for which
|D-m(7')|<^. and \D-miI")\<8„
t A. Cauchy and B. Riemann studied the case of a fiflite number of sin-
gularities in papers which are to be found in these writers' collected works.
The infinite case has been treated by
A. Hahnack, Mathematische Annalen, Vols. 21 and 24 (1883-84).
O. Holder, Mathematische Annalen, Vol. 24 (1884).
C. Jordan, Cours d' Analyse, Vol.' 2 (1894, 2d ed.).
O. Stolz, Grundzuge der Differential- und Integralrechnung , Vol. 3.
A. ScHOENFLiES, JahresbeHcht der Deutschen Mathematiker-Vereinigung
Vol. 8 (1900).
Valleb-Potjssin, LiouvUle's Journal, Ser. 4, Vol. 8 (1892).
E. H. Moore, Transactions of the American Mathematical Society Vol 2
(1901).
J. PlERPONT, Theory of Functions of Real Variables (1906).
IMPROPER DEFINITE INTEGRALS. 211
/ f{x)dx— I f{x)dx <e.
\J al' JaV
It follows by Theorem 27 that
exists and is finite. This limit is denoted by
r }{x)dx
bJaP„
and is called the hroad improper definite integral with respect
to Po of the fmiction /(x) on the interval a b.
It is to be noticed that all the points of Po need not be on
1 — I . I— I
a b] those which are not on a 6 do not affect the existence of
/ f{x)dx.
W aPa
Therefore if f{x) is improperly integrable on some sub-interval
I — I I— I . . , ,
a' V of o h, its mtegral may be denoted by
. r fix)dx.
Wa'Pa
Theorem i6i. If a<b<c and if of the integrals
r f{x)dx, r f{x)dx, r fix)dx,
WaPo b^bPo b-^ aPo
either (a) / fix)dx and / fix)dx exist,
<y (^) / f{x)dx exists,
b'^aPo
212 INFINITESIMAL ANALYSIS,
then all three integrals exist and
/' fix)dx+ r f{x)dx= f f{x)dx. . . (1)
JaPa -JlPo b^aPo
I— I , , ,
Proof .—Every set I of intervals on o c may be regarded as
composed of a set Ton a h and a set 7 on 6 c, while, conversely,
every pair of sets / and / constitute a set I. Hence
r Hx)dx= f f{x)dx+ I -f{x)dx.
J al "J a'l '^ b I
(Note that both members of this equation are multiple-valued
functions of mil) and of m (7) and m(T)). The conclusion of
our theorem follows in case (a) from Theorem 34.
It remains to show that if / /(a;)dx exists, then / f{x)dx
bJ a Po bJa Po
and / j{x)dx exist, and in that case also equation (1) holds.
bJ bPo
Suppose that on some sequence of sets [/] one of the two expres-
sions / _ f{x)dx and / _ f{x)dx, say / _ f{x)dx, approaches
Jal J bl J al
two distinct values as m(/) approaches D. Since there is some
sequence of sets of intervals 1 7' | on which / _ /(x) approaches
'^ bl
only one value, it follows that on the sequence of sets of intervals
obtained by associating with each 7 an V and with each 7' an 7',
/ f{x)dx approaches two distinct values as m{I) = D, which
Jal
is contrary to hypothesis.
If / _f{x)dx approaches infinity, then clearly / f{x)dx
must approach infinity of the opposite sign. Hence, by the
corollary of Theorem 51a, / f{x)dx will approach both + oo
mJ a I
IMPROPER DEFINITE INTEGRALS. 213
and -00 as m{I)^D, which again contradicts the hypothesis
that / f{x)dx exists. The equaUty
I f{x)dx= f f{x)dx+ r f(x)dx
b'J n P„ b'J a Po b'J b Pa
now follows from the identity of the limitands
/ f{x)dx and / f{x)dx+ / _f{x)dx.
'J a I 'J a Y ^ bY
Theorem 162. // / f{x)dx exists, then I f(x)dx exists
W a. Pa b-J b P„
and
f f(x)dx=- f f{x)dx.
Wa Po b'Jb Po
Proof.— By Theorem 108» for every /
/ f{x)dx= - I f{x)dx,
J al «/ 6 /
whence / ]{x)dx=- I j{x)dx.
W aPo b^ b Po
Theorem 163. // / f{x)dx exists, then I c-f{x)dx exists
bJ a Pa b^ a Pa
end
I c-f(x)dx = c- I fix)dx.
Wa Pa b'^a Po
Proof. — This is a direct consequence of Theorems 111 and 34.
Theorem 164. // / fi{x)dx and I f2{x)dx both exist,
bJ a Po b^ o Pa
then / (fi{x)±f2{x))dx exists and
214 INFINITESIMAL ANALYSIS.
r hi^)dx± r J2{x)dx= f {h{x)dx±h{x))dx.
W a Pa h^ a Po b^ a Pq
Proof. — This is a direct consequence of Theorems 112 and 34.
Theorem 165. // h{x) ^fzix), then
r h{x)dx^ f f2{x)dx,
provided these integrals exist.
Proof. — By Theorems 113 and 40.
Theorem 166. // / fi{x)dx and I hix)dx both exist,
r hix)-f2{x)dx
b-^aPo
does not in general exist.
Proof. — Let fi{x)=J2{x) = -j=. In this case the hypothesis
of the theorem is verified but the product, - fails to be inte-
I — I
grable on the interval 1.
Theorem 167. / f{x)dx is a continuous function of x.
b^aPo
Proof. — If a; is a point at which the integral exists properly,
the continuity follows by Theorem 118. If a; is a point of the
set Po, then, by Theorem 26, we need to show that for every e
I — I
there is a d, such that for every interval a' h' containing Xi and
I n
of length less than d„ \ I f{x)dx <s. By definition there
|6>/o' Po
exists a d, such that for every I' and /'' for which lm(/') -D\<d,
md\mir)-D\<d.,
I r f(x)dx- f f{x)dx
|«/ a P J a V
<e.
IMPROPER DEFINITE INTEGRALS. 215
I— I
Let a' b' be an interval containing xi such that
|a'-fe'l<y-
Let /' be any set of intervals not containing any point of Pq and
. . . I — I
containing no point of a' h', and such that |m(/') - D| < d,. De-
note hyl^a'b') any set of non-overlapping intervals on a' V con-
taining no point of Pq, and let /" be the set of all intervals in
/' and 7(„-i,o- Then
\m(I")-D\<d.
and /_ t{x)dx= /_ /(x)dx+ r_f(x)dx
^al" ^aV 'J a! Ka'V)
\ pv (I Ph />b
and / _ /(x)dd= / _ j{x)dx- / _/(x)da
Hence / f{x)dx
\b^ a' Po
<e.
Corollary. — For Xi any point on a 6
L
x=xi b'^ Xj
f'f{x)dx=0.
Theorem i68. If f{x) is integrable with respect to Pq, and if
Pi is a set of points of content zero, then f{x) is integrable with
respect to the set P2 consisting of all points in Po and in Pi and
r f{x)dx= f f{x)dx.
Proof. — Obviously the set P2 is of content zero. Any set
of intervals / not containing a point of P2 is also a set 1 not
216 INFINITESIMAL ANALYSIS.
containing a point of Po- Hence any value approached by
/ J{x)dx as m(/) approaches D is a value approached by
'J al
yf{x)dx as m(7) approaches D. Hence / f{x)dx exists
and
/ f{x)dx= f f{x)dx.
Theorem 169. If fi{x) is integrable vnth respect to Pi and
J2{x) is integrable with respect to P2, then fi{x) ±f2{x) is integrable
with respect to the set, P3, of all points in Pi and P2 and
r f{x)dx± r f{x)dx= r (/iwi/acx))^.
i«/ a Pi 6*^0 Pj 6«^ a Pa
Proof. — By Theorem 168 each of the functions fi{x) and
fzix) is integrable with respect to P3, and
and
/ fi{x)dx= / fi{x)dx,
WaPi WaP}
/ f2(x)dx= / f2{x)dx,
Wa P, W a P,
and hence, by Theorem 164, fi{x)±f2{x) is integrable with
respect to P3 and
r fi{x)dx+ r f{x)dx= r (j(x)±f{x))dx.
W a Pi b^ a Pi W a Pi
The broad improper definite integral as here defined con-
tains as a special case the proper definite integral, the integral
in that case existing properly at every point of the interval a h.
It does not, however, contain as a special case the simple im-
proper definite integral considered in § 3. This may readily
be shown by means of the function used on page 209 to show
IMPROPER DEFINITE INTEGRALS. 217
that the simple improper definite integral is not absolutely
convergent. In the case of this function a sequence of sets
of intervals /„ may be so chosen that / f{x)dx shall ap-
proach any value whatever as m(/a) approaches D.
An improper integral which includes both the simple and
the broad improper integrals is obtained as follows; Every
set / is to be such that if /' is its complementary set of seg-
ments on a b, then every segment of I' contains at least one
point of Pq. The limit of / f{x)dx as m(7) approaches D, if
•-'a /
existent, is called the narrow improper definite integral and is
denoted by / f{x)dx.
It is evident that if the broad integral exists, then the narrow
integral also exists. The narrow integral includes the simple
improper definite integral of the preceding chapter. Hence it
follows that the broad and the narrow integrals are not equiva-
lent.! Theorems 161 to 167 hold of the narrow integral as well
as of the broad integral. The proofs are identical with the
above except that the sets 7 are limited as in the definition of the
narrow integral. It may be shown by examples that Theorems
168 and 169 do not hold in the case of the narrow integral.
To show that 168 does not hold consider the function defined
in the proof of Theorem 160, where Po consists of the point 0.
Let Pi be the [xj of that example. Then obviously the narrow
integral / ti^)dx, where P2 contains all the points of Pi
and P2, fails to exist. The same example shows that Theorem
169 does not hold of the narrow integral.
■f The narrow integral is so called because it has fewer properties than the
broad integral. It exi.sts for a wider class of functions.
218 INFINITESIMAL ANALYSIS.
§ 5. Special Theorems on the Criteria of Existence of the
Improper Definite Integral on a Finite Interval.
The examples of this section are intended to give an idea of
the possible singularities of improperly integrable functions, and
to indicate the difficulty of obtaining more general criteria of
the divergence or convergence of the simple improper integral
than those given in § § 1 and 2 of this chapter.
Lemjna. — 'For every function /i(x) which is unboimded in
every neighborhood of x=a there is a function /2(x) which is
infinitesimal as x approaches a, such that ji{x)-f2{x) is un-
bounded in every neighborhood of i = o, and such that
/2(X)
x—a
is monotonic increasing as x approaches o.
Proof. — Since /i(i) is imboimded in every neighborhood of
x = o, it follows that for every point Xi of the segment a b there
is a point X2 on the segment a xi such that
|/ife)|>2|/i(xi)l>2M,
and such that (xg —a) 5 J(xi — a).
Let Xi, X2, X3, . . . , x„, . . . be a sequence of points dense only
at a such that
|/i(a;„)|>2|/i(x„_i)|>2''-i-M,
and such that |x„— a| ^ i|x„_i -a\.
We define /2(x) as follows:
hM = - on the points Xi, X2, . . . , x„, . . . .
TV
IMPROPER DEFINITE INTEGRALS. 219
avd fzix) is linear between the points of the sequence Xi, X2, . . . ,
Xn,... Then there are values of x on a;„ x„_i such that
l/i(a;)|-/2(x)>|-M,
whence hi^)-hi^) is unbounded in the neighborhood of a.f
Ubviously ^— ^ IS monotomc increasing as x approaches a.
Theorem 170. For every function /i(x) which is unbounded
in every neighborhood of x = a there exists a non-oscUlating func-
tion f2{x) such that
L I f2(x)dx
exists and is finite, while
(.x-a)-fi{x)-f2{x)
is unbounded in the neighborhood of x=a.
Proof. — According to the lemma there exists a function
/3 (x) such that L fa (x) = 0,
while fz{x)-fi(x) is imbounded and the fimction
fsix)
U{x) =
x—a
is monotonic increasing as x approaches a. Since
ix-a)f4(x)-fiix)^f3ix)-f,ix),
t In case L f,(x)=<x, f,(x) ==== or ftipc)^- — -^ would satisfy the
/ (x)
requirements of the lemma except that they need not make -^^^ monotonic.
220 INFINITESIMAL ANALYSIS.
(x-a)-}i{x)-fi{x) is unbounded in the neighborhood of x=a.
I — I
Let xi, . . . , a;„, . . . be a sequence of points on a b whose only-
limit point is a, such that /3(x)-/i(x) is unbounded on this set.
In the sequence
(xi -o)/4(xi), (x2-a)/4(x2), ..., ix„-a)U{x)„ . (1)
L (x„-o)/4(x„)=0, since L {x-a)fi{x)=0.
71=00 x=a
Hence there is a value of n, rii, such that
l(xi-a)/4(xi)l>2l(x„.-a)/4(xOI,
and another value of n, Ui, such that
|(x„,-a)/4(x„J| >2|(x„,-a)/4(x„,)|, etc.,
rim+i being so chosen that
|(2„„-a)/4(x„JI > 2l(x„„^i -a)/4(a;n„+i)|.
In this manner we select from the sequence (1) a set of tenns
forming the convergent series
(xi-a)/4(xi) + (x„, -a)/4(x„,) + . . . + (x„^-a)/4(x„J+. . . (2)
We then obtain a function f^ix) as follows: For the set of values
of X
x„„^i<x<x„„, /2(X)=/4(X„„).
Then (1) /zC^;) is non-oscillating since
/4(a;"m)</4(a;nm+l).
(2) {x-a)J2(x)-fi{x) is unbounded on the set ij, x^,
Xr^, ■ ■ ■ , Xn„, ■■-, since on this set
/2(X)=/4(X).
IMPROPER DEFINITE INTEGRALS. 221
But the terms of this series are numerically smaller than the
corresponding terms of the convergent series (2). Hence
^ / }2(x)dx
exists and is finite.
Theorem 170 may be regarded as showing that
L (x-a)/2(a;)=0
is a strong necessary condition that, under the hypothesis of
Theorem 142,
^ / f2{x)dx
L
z— a*/ X
shall exist and be finite. For, according to Theorem 170, it is
impossible to modify the function (a; -a) by any factor /i(i)
which shall approach infinity so slowly that for every function
fzix) where
L
/ f2{x)dx
\U X
exists and is finite
L(x-a)A(z)-/2(a:)=0.t
x^a
Theorem 171. For every function fiix) defirMi on the interval
I — I
a b there exists a function fzix) such that
(1) fzix) is continuous and does not change sign on a certain
neighborhood of x = a.
t See Prinosheim, Mathematische Annilen, Vo!. 37, pp. 591-694 (1890).
222 INFINITESIMAL ANALYSIS.
(2) L I f2(x)dx exists and is finite.
(3) For X on a certain set [a/]
x=a h{x')
Proof. — Let Xi , x-i, . . . , x„', ... be a set of points of the
I— I
interval a b dense only at a. Let Bi, Bz, B3, . . . , Bn, ... be
a set of numbers such that
5„-n|/i(x'„)|>2-fi„+i(n + l)i/i(z'„+i)|. (n = l, 2, 3, . . .)
On the X axis lay off a set of segments [a„] such that a„ is of
length Bn and a;„ is its middle point. On the segments (t„ as
bases construct isosceles triangles on the positive side of the
X axis whose altitudes are n-|/i(a;)|. The measures of areas
of these triangles form a convergent series. Let fsix) be
any continuous, monotonic, unbounded function such that
L rf3(x)dx
exists and is finite. We then define /2(a;) as the function repre-
sented by the following curve :
(1) Those parts of the boundaries of the isosceles triangles
just described which lie above the curve defined by ]z{x).
(2) Those parts of the curve defined by ]z{x) which he out-
side the triangles or on their boundary. Obviously the func-
tion so "defined has the properties specified in the theorem, the
points xx, X2,..., Xn', . . . being the set [a/] specified by (3)
of the theorem.
Theorem 171 means that from the hypothesis that the
improper definite integral of f{x) exists on a 6 it is impossible
to obtain any conclusion whatever as to the order of infinity
or the rank of infinity of f{x) atx=a. This is what one would
IMPROPER DEFINITE INTEGRALS. 223
expect a -priori, since the definite integral is a function of two
parameters, while the necessary condition in terms of t ounded-
ness would be in terms of only one of these.
§ 6. Special Theorems on the Criteria of the Existence of the
Improper Definite Integral on the Infinite Interval.
Theorem 172. For every function fi{x) which is unbounded as
X approaches 00 there exists a non-osciUating function /2(x) such
that
L I f2{x)dz
z^oDc/a
exists and is finite, while (x — a)fi{x)-f2^x) is unbounded as x
approaches <x>.
Proof. — Obviously the lemma of Theorem 170 can be stated
so as to apply to the case where x approaches « instead of 0.
If then in the proof of Theorem 161 the set of points ii . . . x„ . . .
is so taken that
L Xn=CO
instead of a, the proof of Theorem 161 applies with the excep-
tion that fzix) is non-oscillating decreasing instead of non-oscil-
lating increasing.
Theorem 173. For every function f\ {x) defined on the interval
a 00 there exists a function /2(x) such that
(1) /zCx) is corUinuou^ and does not change sign for x greater
than a certain fixed number.
(2) L I f2(x)dx
exists and is finite.
224 INFINITESIMAL ANALYSIS.
(3) For X on a certain set [x^
Proof. — Such a function /aCx) may be defined in a manner
analogous to that of the proof of Theorem 171.
The remarks as to the meaning of Theorems 170 and 171
apply with obvious modifications to Theorems 172 and 173.
INDEX.
31
Absolute convergence
series, 72
Absolute value, 14
Algebraic functions, 53
" numbers, 18
Approach to a limit, 60
Axioms of continuity, 4,
' ' of the real number system, 13
Bounds of indetermination, 84
" upper and lower, 3, 47
Broad improper definite integral, 211
Change of variable, 79, 126, 175
Class, 3
Closed set, 41
Constant, 44
Content of a set of points, 183
Contin ity at a point, 61
' axioms of, 4, 31
" over an interval, 88
uniform, 89
Continuous function, 61
' ' real number system, 4
Continuum, linear, 4
Convergence of infinite series, 71
to a limit, 60
Covering of interval or segment, 33
Decreasing function, 49
Dedekind cut, 7
Definite integral, 153
Dense, 41
Dense in itself, 41
Dependent variable, 44
Derivative, 117
" progressive and regres-
sive, 118
Derived function, 120
Difference of irrational numbers, 8
Differential, 128
Differential coefficient, 130
Discontinuity, 62, 63
' ' of the first and second
kind, 85
The references are to pages,
of infinite Discrete set, 41
Divergence, 71
Everywhere dense, 41
Exponential function, 54
Function, 44
" algebraic, 51
continuity of, at a point.
61
" continuity of, over an in-
terval, 88
' ' exponential, 54
" graph of, 46
" mfinite at a point, 101
" inverse, 45
" limit of, 60
" monotonic decreasing, 49
" " increasing, 49
" non-oscillating, 49
" oscillating, 49
" partitively monotonic, 50
" rational, 53
" " integral, 53
' ' transcendental, 54
" unbounded, 47
" uniform continuity of, 89
' ' upper and lower liound of,
47
' ' value approached by, 60
Geometric series, 73
Grapn of a function, 46
Greatest lower bound, 4
Improper definite integral, 191
Improper definite integral, broad,
211
Improper definite integral, narrow,
217
Improper definite integral, simple,
205
Improper existence of the definite
integral, 205
225
226
INDEX.
Increasing function, 49
Independent variable, 44
Infinite, 101, 102
Infinite segment, 32
" series, 70
" " convergence and di-
vergence of, 71
Infinitesimals, 75
Infinity as a limit, 40, 47, 60
Integral, definite, 153
existing properly at a point,
205
Integral oscillation, 155
Interval, 32
Inverse function, 45
Irrational number, 1, 4
' ' numbers, difference of, 8
" product of, 8
" " quotient of, 9
" sum of, 8
Least upper bound, 4
L'Hospital's rule, 139
Limit, lower, 84
" of a function, 60
" of integration, 153
" point, 39
' ' upper, 84
Limitand function, 60
Linear continuum, 4
Logarithms, 58
Lower bound of a function, 47
" " ■" " set of numbers, 3
" integral, 181
" segment, 12
Many-valued function, 44
Maximum of a function, 130
Mean-value theorem of the differen-
tial calculus, 133
Mean-value theorem of the integral
calculus, 169
Minimum of a function, 131
Monotonic function, 49
Narrow improper definite integral,
217
Necessary and sufficient condition, 65
Neighborhood, 38
Non-differentiable function, 150
Non-integrable function, 155
Non-numerably infinite set, 185
Non-oscillating function, 49
Nowhere dense, 41
Number, 1
" algebraic, 18
" irrational, 1, 4
Number, system, 4
Numbers, transcendental, 18
' ' sequence of, 70
" sets of, 3
Numerably infinite set, 185
One-to-one correspondence, 30
Order of function, 102
Oscillation of a function, 49 "
" " function at a point, 85
Partitively monotonic, 50
Perfect set, 41
Polynomial, 53
Product of irrational numbers, 8
Progressive derivative, 118
Proper existence of the definite inte-
gral at a point, 205
Quotient of irrational mmibers, 9
Rank of infinitesimals and infinites,
114
Rational functions, 53
. ' ' integral functions, 53
" numbers, 1
Ratio test for convergence of infinite
series, 73
Real mmiber system, 4, 13
Regressive derivative, 118
RoUe's theorem, 132
Segment, 32
infinite. 32
" lower, 12
Sequence of numbers, 70
Series, infinite, 70
" convergence and divergence
of, 71
" geometric, 73
" Taylor's, 134, 135
Sets of numbers, 3
Simple improper definite integral, 205
Single-valued functions, 44
Singularity, 101
Sum of irrational numbers, 7
Taylor's series, 134
Theorem of uniformity, 35
Transcendental functions, 54
" numbers, 18
Unbounded function, 47
Uniform continuity, 89
Uniformity, 35
Upper bound of a function, 47
" "" set of numbers, 3
INDEX.
227
Upper integral, 181
" limit, 84
Value approached by a function, 60
«' " " the independ-
ent variable, 60
Variable, 44
' ' dependent, 44
" independent, 44
Vicinity, 38
V(a), 38
V*{a), 38
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Planning and Construction of High Office Buildings. Svo, 3 50
Skeleton Construction in Buildings Svo, 3 00
Brigg's Modem American School Buildings. Svo, 4 00
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Carpeater's Heating and Ventilating of Buildings 8vo,
Freitag's Architectural Engineering 8vo,
Fireproofing of Steel Buildings 8vo,
French and Ives's Stereotomy 8vo,
Gerhard's Guide to Sanitary House-inspection 26mo,
Theatre Fires and Panics x2mo,
*Greene's Structural Mechanics 8vo,
Holly's Carpenters* and Joiners' Handbook iSmo,
Johnson's Statics by Algebraic and Graphic Methods 8vo,
JCidder's Architects' and Builders' Pocket-book. Rewritten Edition. i6mo, mor.,
Uerrill's Stones for Building and Decoration 8vo,
Non-metallic Minerals: Their Occurrence and Uses 8vo,
Monckton's Stair-building 4to,
Patton's Practical Treatise on Foundations 8vo,
Peabody'B Naval Architecture 8vo,
Rice's Concrete-block Manufacture 8vo,
Richey's Handbook for Superintendents of Construction i6mo, mor.,
* Building Mechanics' Ready Reference Book. Carpenters' and Wood-
workers' Edition i6mo, morocco,
Sabin's Industrial and Artistic Technology of Paints and Varnish 8vo,
Siebert and Biggin's Modem Stone-cutting and Masonry 8vo,
Snow's Principal Species of Wood 8vo,
Sondericker's Graphic Statics with Applications to Trusses, Beams, and Arches.
8vo,
Towne's Locks and Builders' Hardware i8mo, morocco.
Wait's Engineering and Architectural Jurisprudence 8vo,
Sheep,
Law of Operations Preliminary to Construction in Engineering and Archi-
tecture 8vo,
Sheep,
Law of Contracts gvo.
Wood's Rustless Coatings: Corrosion and Electrolysis of Iron and Steel. .8vo,
Worcester and Atkinson's Small Hospitals, Establishment and Maintenance,
Suggestions for Hospital Architecture, with Plans for a Small Hospital
iimo.
The World's Columbian Exposition of 1893 Large 4to
ARMY AND HAVY.
Bemadou's Smokeless Powder, Nitro-cellulose, and the Theory of the Cellulose
M<>'«»'» umo, 2 so
• Bruff's Text-book Ordnance and Gunnery gvo 5 „„
Chase's Screw Propellers and Marine Propulsion gvo' 3 00
Cloke's Gunner's Examiner g^o'
Craig's Azimuth ...4to[ 3 50
Crehore and Squier's Polarizing Photo-cbronograph gyo 3 00
• Davis's Elements of Law g^^' ^
• Treatise on the Military Law of United States ' ' gvo,' 7 00
Sheep, 7 50
De Brack's Cavalry Outposts Duties. (Carr. ) 24mo, morocco, 2 00
Dietz's Soldier's First Aid Handbook i6mo, morocco, 1 2s
• Dudley's Military Law and the Procedure of Courts-martiaL . . Large i2mo, ., so
Durand's Resistance and Propulsion of Ships gvo' _ Xo
• Dyer's Handbook of Light ArtiUery ...lamo' 3
Eissler's Modem High Explosives g '
• Fiebeger's Text-book on Field Fortification Small gvo' 2 00
Hamilton's The Gunner's Catechism igmo'
• Hofi'B Elementary Naval Tactics gvo'
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Ingalls's Handbook of Problems in Direct Fire 8to,
* Ballistic Tables 8yo,
* Lyons'B Treatise on Electromagnetic Phenomena. Vob. L and II. .8vo, each,
* Mahan's Permanent Fortifications. (Mercur.) 8to, ball morocco*
Manual for Conrts-martiaL i6mo, morocco,
* Mercur's Attack of Fortified Places i2mo,
* Elements of the Art of War 8vo,
Metcalf's Cost of Manufactures — And the Administration of Workshops. .8vo,
* Ordnance and Gunnery. 2 vols i2mo,
Murray's Infantry Drill Regulations i8mo, paper,
Nixon's Adjutants' ManuaL 24mo,
Peabody's Naval Architecture 8vo,
* Phelps's Practical Marine Surveying 8to,
Powell's Army Officer's Examiner i2mo,
Sharpe's Art of Subsisting Armies in War iSmo, morocco,
* Tupes and Poole's Manual of Bayonet Exercises and Musketry Fencing.
24mo, leather,
* Walke's Lectures on Explosives 8vo,
Weaver's Military Explosives 8vo,
* Wheeler's Siege Operations and Military Mining 8vo,
Winthrop's Abridgment of Mihtary Law i2mo,
Woodhull's Notes on Military Hygiene i6mo,
Young's Simple Elements of Navigation i6mo, morocco,
ASSAYING.
Fletcher's Practical Instructions in Quantitative Assaying with the Blowpipe.
i2mo, morocco,
Furman's Manual of Practical Assaying 8vo,
Lodge's Notes on Assaying and Metallurgical Laboratory Experiments. . . .8vo,
Low's Technical Methods of Ore Analysis 8vo,
Miller's Manual of Assaying i2mo.
Cyanide Process i2mo,
Minet's Production of Aluminnm and its Industrial Uge. (Waldo.) i2mo,
O'DriscoU's Notes on the Treatment of Gold Ores 8vo,
Ricketts and Miller's Notes on Assaying 8vo,
Robine and Lenglen's Cyanide Industry. (Le Clerc.) 8vo,
nike's Modem Electrolytic Copper Refining 8vo,
Wilson's Cyanide Processes i2mo,
Chlorination Process i2mo,
ASTRONOMY.
Comstock's Field Astronomy for Engineers 8vo, 2 50
Craig's Azimuth 4to, 3 so
Crandall's Text-book on Geodesy and Least Squares 8vo, 3 00
Doolittle's Treatise on Practical Astronomy 8vo, 4 00
Gore's Elements of Geodesy ..8vo, 2 50
Hayford's Text-book of Geodetic Astronomy 8vo, 3 00
Merriman's Elements of Precise Surveying and Geodesy 8vo, 2 50
* Michie and Harlow's Practical Astronomy 8vo, 3 00
* White's Elements of Theoretical and Descriptive Astronomy i2mo 00
BOTANY.
Davenport's Statistical Methods, with Special Reference to Biological Variation.
i6mo, morocco, i 25
Thom^ and Bennett's Structural and Physiological Botany. i6mo, 2 25
Westermaier's Compendium of General Botany. (Schneider.) 8to, 2 00
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CHEMISTRY.
* Abegg's Theory of Electrolytic Dissociatioo. (Von Ende.) i2mo, i 25
Adriance's Laboratory Calculations and Specific Gravity Tables i2mo, i 25
Alezeyeff's General Principles of Organic Synthesis. (Matthews.) 8vo, 3 00
Allen's Tables for Iron Analysis 8vo, 3 00
Arnold's Compendium of Chemistry. (Mandel.) Small 8vo, 3 50
Austen's Notes for Chemical Students lamo, i 50
Bemadou's Smokeless Powder. — Nitre-cellulose, and Theory of the Cellulose
Molecule i2mo, 2 50
* Browning's Introduction to the Rarer Elements 8vo, i so
Bnuh and Penfield's Manual of Determinative Mineralogy 8vo, 4 00
* Claassen's Beet-sugar Manufacture. (Hall and Rolfe.) 8vo, 3 00
Classen's Quantitative Chemical Analysis by Electrolysis. (Boltwood.). .8vo, 300
Cohn's Indicators and Test-papers Z3mo, 2 00
Tests and Reagents 8vo, 300
Crafts's Short Course in Qualitative Chemical Analysis. (Schaeffer.). . . z3mo, i 50
* Danneel's Electrochemistry. (Merriam.) i2mo, 1 25
Dolezalek's Theory of the Lead Accumulator (Storage Battery). (Von
Ende.) i3mo, 2 50
Drechsel's Chemical Reactions. (MerrilL) i2mo, i 25
Duhem's Thermodynamics and Chemistry. (Burgess.) 8vo, 4 00
Eissler's Modem High Explosives 8vo, 4 00
Effront's Enzymes and their Applications. (Prescott.) 8to, 3 00
Erdmann's Introduction to Chemical Preparations. (Dunlap.) i2mo, i 25
Fletcher's Practical Instructions in Quantitative Assaying with the Blowpipe.
i2mo, morocco, i 50
Fowler's Sewage Works Analyses i2mo, 2 00
Fresenius's Manual of Qualitative Chemical Analysis. (Wells.) 8vo, 5 00
Manualof (Qualitative Chemical Analysis. Part I. Descriptive. (Wells.) 8vo, 3 00
System of Instruction in Quantitative Chemical Analysis. (Cohn.)
2 vols 8vo, 12 so
Fuertes's Water and Public Health i2mo, i so
Furman's Manual of Practical Assaying 8vo, 3 00
* Getman's Exercises in Physical Chemistry i2mo, 2 00
Gill's Gas and Fuel Analysis for Engineers izmo, i 25
* Gooch and Browning's Outlines of Qualitative Chemical Analysis. Small 8vo, 1 25
Grotenfelt's Principles of Modem Dairy Practice. (WolL) i2mo, 2 00
Groth's Introduction to Chemical Crystallography (Marshall) i2mo, i 25
Hammarsten's Text-book of Physiological Chemistry. (MandeL) Svo, 4 00
Helm's Principles of Mathematical Chemistry. (Morgan.) iimo, i so
Bering's Ready Reference Tables (Conversion Factors) i6mo, morocco, 2 so
Hind's Inorganic Chemistry 8vo, 3 00
* Laboratory Manual for Students i2mo, i 00
Holleman's Text-book of Inorganic Chemistry. (Cooper.) 8vo, 2 50
Text-book of Organic Chemistry. (Walker and Mott.) Svo, 2 50
* Laboratory Manual of Organic Chemistry. (Walker.) i2mo, i 00
Hopkins's Oil-chemists' Handbook Svo, 3 00
Iddings's Rock Minerals Svo, 5 00
Jackson's Directions for Laboratory Work in Physiological Chemistry . . Svo, 125
Keep's Cast Iron Svo, 2 50
Ladd's ISanual of Quantitative Chemical Analysis i2mo, z 00
Landauer's Spectrum Analysis. (Tingle.) Svo, 3 00
* Langworthy and Austen. The Occurrence of Aluminium in Vegetable
Products, Animal Products, and Natural Waters 8vo, 2 00
Lassar-Cohn's Application of Some General Reactions to Investigations in
Organic Chemistry. (Tizigle.) z2mo, z 00
Leach's The Inspection and Analysis of Food with Special Reference to State
ControL Svo, 7 50
Lob's Electrochetnistry of Organic Compounds. (Lorenz.) Svo, 3 oo
4
Lodge's Notes on Assaying and Metallurgical Laboratory Experiments. .. .8to,
Low's Technical Method of Ore Analysis 8vo,
Lunge's Techno-chemical Analysis. (Cohn.) i2mo
* McKay and Larsen's Principles and Practice of Butter-making 8to,
Mandel's Handbook for Bio-chemical Laboratory umo,
'^ Martin's Laboratory Guide to Qualitative Analysis with the Blowpipe . . x2mo.
Mason's Water-supply. (Considered Principally from a Sanitary Standpoint.)
3d Edition, Rewritten 8to,
Examination of Water. (Chemical and BacteriologicaL). X2mo,
Matthew's The Textile Fibres 8to,
Meyer's Determination of Radicles in Carbon Compounds. (Tingle.). . xamo.
Miller's Manual of Assaying lamo.
Cyanide Process i2mo,
Minet's Production of Aluminum and its Industrial Use. (Waldo.) . . . . i2mo,
Mixter's Elementary Text-book of Chemistry x2mo,
Morgan's An Outline of the Theory of Solutions and its Results. ...... i2mo,
Elements of Physical Chemistry i2mo,
* Physical Chemistry for Electrical Engineers i2mo,
Morse's Calculations used in Cane-sugar Factories z6mo, morocco,
* Muir's History of Chemical Theories and Laws 8vo,
Mulliken's General Method for the Identification of Pure Organic Compounds.
VoL I Large 8to,
O'Brine's Laboratory Guide in Chemical Analysis 8to,
O'Driscoll's Notes on the Treatment of Gold Ores 8vo.
Ostwald's Conversations on Chemistry. Part One. (Ramsey.) l2mo,
" " " " Part Two. (Turnbull.) i2mo,
* Pauli's Physical Chemistry in the Service of Medicine. C Fischer.) .... z2mo,
* Penfield's Notes on Determinative Mineralogy and Record of Mineral Tests.
8vo, paper,
Pictet's The Alkaloids and their Chemical Constitution. (Biddle.) 8vo,
Pinner's Introductian to Organic Chemistry. (Austen.) i2mo,
Poole's Calorific Power of Fuels 8vo,
Prescott and Winslow's Elements of Water Bacteriology, with Special Refer-
ence to Sanitary Water Analysts i2mo,
* Reisig's Guide to Piece-dyeing 8to,
Richards and Woodman's Air, Water, and Food from a Sanitary Standpoint. .8vo ,
Ricketts and Russell's Skeleton Notes upon Inorganic Chemistry. (Part I.
Non-metallic Elements.) 8vo, morocco,
Ricketts and Miller's Notes on Assaying 8vo,
Rideal's Sewage and the Bacterial Purification of Sewage 8vo,
Disinfection and the Preservation of Food 8vo,
Riggs's Elementary Manual for the Chemical Laboratory 8vo,
Robine and Lenglen's Cyanide Industry. (Le Clerc.) 8vo,
Ruddiman's Incompatibilities in Prescriptions 8vo,
* Whys in Pharmacy i2mo,
Sabin's Industrial and Artistic Technology of Paints and Varnish 8vo,
Salkowski's Physiological and Pathological Chemistry. (OrndorS.) 8vo,
Schimpf's Text-book of Volumetric Analysis i2mo.
Essentials of Volumetric Analysis i2mo,
* Qualitative Chemical Analysis 8vo,
Smith's Lecture Notes on Chemistry for Dental Students 8vo,
Spencer's Handbook for Chemists of Beet-sugar Bouses i6mo, morocco,
Handbook for Cane Sugar Manufacturers i6mo, morocco,
Stockbridge's Rocks and Soils 8vo,
* Tillman's Elementary Lessons in Heat 8vo,
* Descriptive General Chemistry 8vo,
Treadwell's QuaUtative Analysis. (HaU.) 8vo,
Quantitative Analysis. (Hall.) 8vo,
Tumeaure and Russell's Public Water-supplies 8vo,
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Van Deventer's Physical Chemistry for Beginners. (Boltwood.) x2mo, z 50
• Wa]ke*s Lectures on Explosives 8vo, 4 00
Ware's Beet-sugar Manufacture and Refining Small 8vo, cloth, 4 00
Washington's Manual of the Chemical Analysis of Rocks 8vo, 2 00
Weaver's Military Explosives 8vo, 3 00
Wehrenfennig*5 Analysis and Softening of Boiler Feed-Water 8vo, 4 00
Wells's Laboratory Guide in Qualitative Chemical Analysis 8vo, t 50
Short Course in Inorganic Qualitative Chemical Analysis for Engineering
Students iimo, i 50
Text-book of Chemical Arithmetic i2mo» i 25
Whipple's Microscopy of Drinking-water 8vo, 3 50
Wilson's Cyanide Processes i2mo» i 50
Chlorination Process z2mo, i 50
Winton's Microscopy of Vegetable Foods 8vo, 7 50
Wulling's Elementary Course in Inor^ai^ic, Pharmaceuticai, and Medical
Chemistry. z2mo, 3 00
CIVIL ENGINEERING.
BRIDGES AND ROOFS. HYDRAULICS. MATERIALS OF ENGINEERING.
RAILWAY ENGINEERING.
Baker's Engineers' Surveying Instruments z2mo, 3 00
Bixby's Graphical Computing Table Paper 19^X241 inches. 25
Breed and Hosmer's Principles and Practice of Surveying 8vo, 3 00
* Burr's Ancient and Modero Engineering and the Isthmian Canal .... Svo, 3 50
Comstock's Field Astronomy for Engineers 8vo» 2 50
Crandall's Text-book on Geodesy and Least Squares 8vo, 3 00
Davis's Elevation and Stadia Tables 8vo, i 00
Elliott's Engineering for Land Drainage 1200, i 50
Practical Farm Drainage i2mo, z 00
♦Fiebeger's Treatise on Civil Engineering 8vo,
Flemer's Phototopographic Methods and Instruments 8vo,
Folwell's Sewerage. (Designing and Maintenance.) 8vo,
Freitag's Architectural Engineering. 2d Edition, Rewritten 8vo,
French and Ives's Stereotomy 8vo,
Goodhue's Municipal Improvements i2mo,
Gore's Elements of Geodesy 8vo,
Hayford's Text-book of Geodetic Astronomy 8vo,
Bering's Ready Reference Tables (Conversion Factors') x6mo, morocco,
Howe's Retaining Walls for Earth i2mo,
* Ives's Adjustments of the Engineer's Transit and Level x6ino, Bds.
Ives and Hilts's Problems in Surveying x6mo, morocco,
Johnson's (J. B.) Theory and Practice of Surveying Small 8vo,
Johnson's (L. J.) Statics by Algebraic and Graphic Methods 8vo,
Laplace's Philosophical Essay on Probabilities. (Truscott and Emory.) . z2mo.
Mahan's Treatise on Civil Engineering. (1873.) (Wood.) 8vo,
* Descriptive Geometry 8vo,
Merriman's Elements of Precise Surveying and Geodesy 8vo,
Merriman and Brooks's Handbook for Surveyors z6mo, morocco,
Nugent's Plane Surveying 8vo,
Ogden's Sewer Design i2mo,
Parsons's Disposal of Municipal Refuse 8vo,
Patton's Treatise on Civil Engineering 8vo half lealher.
Reed's Topographical Drawing and Sketching 4to,
Rtdeal's Sewage and the Bacterial Purification of Sewage 8vo,
Siebert and Biggin's Modern Stone-cutting and Masonry 8vo,
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Smith's MbihibI of Topographical Drawing. (McMillan,) 8vo, 3 50
Sondericker's Graphic Statics, with Applications to 'irusses, Beams, and Arches.
8vo,
Taylor and Thompson's Treatise on Concrete, Plain and Reinforced 8to,
* Trautwine's Civil Engineer's Pocket-book i6mo, morocco,
Venable's Garbage Crematories in America 8to,
Wait's Engineering and Architectural Jurisprudence 8to
Sheep,
Law of Operations Preliminary to Construction in Engineering and Archi-
tecture 8vo,
Sheep,
Law of Contracts 8vo,
Warren's Stereotomy — Problems in Stone-cutting 8to,
Webb's Problems in the Use and Adjustment of Engineering Instruments.
xomo, morocco,
Wilson's Topographic Surveying 8vo,
BRIDGES AND ROOFS.
Boiler's Practical Treatise on the Construction of Iron Highway Bridges. .8vo, 2 00
• Thames River Bridge 4to, paper, 5 00
Burr's Course on the Stresses in Bridges and Roof Trueses, Arched Ribs, and
Suspension Bridges 8vo, 3 50
Burr and Falk's Influence Lines for Bridge and Roof Computations 8to, 3 00
Design and Construction of Metallic Bridges 8to 5 00
Du Bois's Mechanics of Engineering. Vol. II Small 4to, xo 00
Foster's Treatise on Wooden Trestle Bridges 4to, 5 00
Fowler's Ordinary Foundations 8vo, 3 50
Greene's Roof Trusses 8vo, i 25
Bridge Trusses 8to, 2 50
Arches in Wood, Iron, and Stone 8vo 2 50
Howe's Treatise on Arches 8vo, 4 00
Design of Simple Roof-trusses in Wood and Steel , 8vo, 2 00
Symmetrical Masonry Arches 8vo, 2 50
Johnson, Bryan, and Turneaure's Theory and Practice in the Designing of
Modem Framed Structures Small 4to, 10 00
Merriman and Jacoby's Teit-book on Roofs and Bridges:
Part I. Stresses in Simple Trusses 8vo, 2 so
Part n. Graphic Statics 8vo, 2 so
Part ni. Bridge Design 8vo, 2 so
Part IV. Higher Structures 8vo, 2 50
Morison's Memphis Bridge 4to, 10 00
Waddell's De Pontibus, a Pocket-book for Bridge Engineers . . i6mo, morocco, 2 00
* Specifications for Steel Bridges i2mo, 50
Wright's Designing of Draw-spans. Two parts in one vohime 8vo, 3 SO
HYDRAULICS.
Barnes's Ice Formation 8vo, 3 00
Bazin's Experiments upon the Contraction of the Liquid Vein Issmng from
an Orifice. (Trautwine.) 8vo. 2 00
Bovey's Treatise on Hydraulics ^°' 5 00
Church's Mechanics of Engineering 8vo, 6 co
Diagrams of Mean Velocity of Water in Open Channels paper, ' 50
Hydraulic Motors ^vo, 2 00
Coffin's Graphical Solution of Hydrr.uUc Problems i6mo, morocco, 2 50
Flather's Dynamometers, and the Measurement of Power "mo, 3 00
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Folwell's Water-supply Engineering 8vo, 4 co
Frizell'a Water-power 8vo, s 00
Fuertes's Water and Public Health ■ . izmo, i 50
Water-filtration Works lamo. 2 50
Ganguillet and Kutter's General Formula for tlie Uniform Flow of Water in
Rivers and Other Channels. (Hering and Trautwine.) 8vo» 4 00
Hazen's Filtration of Public Water-supply Svo* 3 00
Hazlehurst's Towers and Tanks for Water-works 8vo, 2 50
Herschel's 1x5 Experiments on the Carrying Capacity of Large* Riveted, Metal
Conduits 8vo, 2 00
Mason's Water-supply. (Considered Principally from a Sanitary Standpoint.)
8vo»
Uerriman's Treatise on Hydraulics. . Svo,
* Hichie's Elements of Analytical Mechanics 8vo,
Schuyler's Reservoirs for Irrigation, Water-power* and Domestic Water-
supply Large 8vo ,
■*■ Thomas and Watt's Improvement of Rivers 4to»
Turneaure and Russell's Public Water-supplies Svo,
Wegmann's Design and Construction of Dams 4to,
Water-supply of the City of New York from 1658 to 1895 4to,
Whipple's Value of Pure Water Large i2mo,
Williams and Hazen's Hydraulic Tables 8vo,
Wilson's Irrigation Engineering Smail Svo,
Wolff's Windmill as a Prime Mover Svo,
Wood's Turbines. Svo,
Elements of Analytical Mechanics Svo,
MATERIALS OF ENGINEERING.
Baker's Treatise on Masonry ConBtruction Svo, 5 00
Roads and Pavements Svo, s 00
Black's United States Public Works Oblong 4to, 5 00
♦ Bovey's Strength of Materials and Theory of Structures Svo, 7 50
Burr's Elasticity and Resistance of the Materials of Engineering Svo, 7 50
Byrne's Highway Construction Svo, 5 00
Inspection of the Materials and Workmanship Employed in Construction.
i6mo, 3 00
Church's Mechanics of Engineering. . Svo, 6 00
Du Bois's Mechanics-of Engineering. Vol. I Small 4to, 7 50
•Eckei's Cements, Limes, and Plasters Svo, 6 00
Johnson's Materials of Construction Large Svo, 6 00
Fowler's Ordinary Foundations Svo, 3 50
Graves's Forest Mensuration. Hvo, 4 co
* Greene's Structural Mechanics. . Svo, 2 50
Keep's Cast Iron Svo, 2 50
Lanza's Applied Mechanics Svo, 7 50
Marten's Handbook on Testing Materials. (Henning.) 2 vols. Svo, 7 50
Maurer's Technical Mechanics. , . Svo, 4 00
Merrill's Stones for Building and Decoration Svo, 5 00
Merriman's Mechanics of Materials Svo, 5 00
* Strength of Materials i2mo, i 00
Metcalf 8 Steel. A Manual for Steel-users i2mo, 2 00
Patton's Practical Treatise on Foundations Svo, 5 00
Richardson's Modern Asphalt Pavements Svo, 3 00
Richey's Handbook for Superintendents of Construction i6mo, mor., 4 00
• Ries's Clays: Their Occurrence, Properties, and Uses Svo, 5 00
Rockwell's Roads and Pavements in France i2mo i 25
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Sabin's Industrial and Artistic Technology of Paints acd Varnish. 8vo,
Smith's Materials of Machines i3mo,
Snow's Principal Species of Wood 8vo,
Spalding's Hydraulic Cement i3mo.
Text-book on Roads and Pavements xsmo,
Taylor and Thompson's Treatise on Concrete, Plain and Reinforced. 8vo,
Thurston's Materials of Engineering. 3 Parts 8vo,
Part I. Non-metallic Materials of Engineering and Metallurgy SyOp
Part n. Iron and Steel gvo,
Part m. A Treatise on Brasses, Bronzes, and Other Alloys and their
Constituents 8to,
Tillson's Street Pavements and Paving Materials 8vo,
Waddell's De Pontibus. (A Pocket-book for Bridge Engineers.). .i6mo, mor., 300
• Specifications for Steel Bridges i3mo, 50
Wood's (De V.) Treatise on the Resistance of Materials, and an Appendix on
the Preservation of Timber 8vo, 3 00
Wood's (De V.) Elements of Analytical Mechanics 8vo, 3 00
Wood's (H. P.) Rustless Coatings; Corrosion and Electrolysis of Iron and
SteeL 8vo, 4 00
RAILWAY ENGIWEERIHG.
Andrew's Handbook for Street Railway Engineers 3x5 inches, morocco, x 35
Berg's Buildings and Structures of American Railroads 4to, 5 00
Brook's Handbook of Street Railroad Location. i6mo, morocco, i 50
Butf s Civil Engineer's Field-book. x6mo, morocco, 2 50
Crandall's Transition Curve x6mo, morocco, x 50
Railway and Other Earthwork Tables 8vo, i 50
Dawson's "Engineering" and Electric Traction Pocket-book . . i6mo, morocco, 5 00
Dredge's History of the Pennsylvania Railroad: (1879) Paper, 5 00
Fisher's Table of Cubic Yards Cardboard, 33
Godwin's Railroad Engineers' Field-book and Explorers' Guide. . .i6mo, mor., 3 so
Hudson's Tables for Calculating the Cubic Contents of Excavations and Em-
bankments 8vo, X 00
Molitor and Beard's Manual for Resident Engineers x6mo, x 00
Nagle's Field M<^n^*^ for Railroad Engineers x6mo, morocco, 3 00
Phitbrick's Field Manual for Engineers i6mo, morocco, 3 00
Searles's Field Engineering x6mo, morocco, 3 00
Railroad SpiraL x6mo, morocco, i 50
Taylor's Prismoidal Formuls and Earthwork 8vo, x 50
* Trautwine's Method of Calculating the Cube Contents of Excavations and
Embankments by the Aid of Diagrams 8vo, 2 00
The Field Practice of Laying Out Circular Curves for Railroads.
i3mo, morocco, 3 50
Cross-section Sheet Paper, 3S
Webb's Railroad Construction x6mo, morocco, s 00
Economics of Railroad Construction Large xsmo, 3 50
Wellington's Economic Theory of the Location of Railways Small 8vo, S 00
DRAWHTG.
Barr's Kinematics of Machinery 8vo, 3 50
» Bartlett's Mechanical Drawing 8vo, 3 00
• " " " Abridged Ed 8vo, x so
Coolidge's Manual of Drawing 8vo, paper, i 00
9
Coolidge and Freeman's Elements ot General Drafting for Mechanical Engi-
neers Oblong 4to,
Durley's Kinematics of Hachioes .8to«
Emch's Introduction to Projective Geometry and its Applications 8to,
Hill's Text-book on Shades and Shadows, and Perspective 8vo,
Jamison's Elements of Mechanical Drawing 8vo,
Advanced Mechanical Drawing 8vo,
Jones's Machine Design :
Part I. Kinematics of Machinery 8vo,
Part n. Form, Strength, and Proportions of Parts 8vo,
MacCord's Elements of Descriptive Geometry 8vo,
Kinematics; or. Practical Mechanism 8vo,
Mechanical Drawing 4to,
Velocity Diagrams 8vo,
MacLeod's Descriptive Geometry.. Small Svo,
* Mahan's Descriptive Geometry and Stone-cutting 8vo,
Industrial Drawing. (Thompson.) 8vo,
Moyer's Descriptive Geometry 8vo,
Reed's Topographical Drawing and Sketching 4to,
Reid's Course in Mechanical Drawing Svo,
Text-book of Mechanical Drawing and Elementary Machine Design. Svo,
Robinson's Principles of Mechanism Svo,
Schwamb and Merrill's Elements of Mechanism Svo,
Smith's (R. S.) Manual of Topographical Drawing. (McMillan.) Svo,
Smith (A. W.) and Marx's Machine Design Svo,
* Titsworth's Elements of Mechanical Drawing Oblong Svo,
Warren's Elements of Plane and SoUd Free-hand Geometrical Drawing. i2mo.
Drafting Instruments and Operations X2mo,
Manual of Elementary Projection Drawing i2mo.
Manual of Elementary Problems in the Linear Perspective of Form and
Shadow z2mo.
Plane Problems in Elementary Geometry i2mo,
Primary Geometry. lamo.
Elements of Descriptive Geometry, Shadows, and Perspective 8vo,
General Problems of Shades and Shadows Svo,
Elements of Machine Construction and Drawing Svo,
Problems, Theorems, and Examples in Descriptive Geometry Svo,
Weisbach's Kinematics and Power of Transmission. (Hermann and
Klein.): Svo,
Whelpley's Practical Instruction in the Art of Letter Engraving i2mo.
Wilsoa's (H. M.) Topographic Surveying 8vo,
Wilson's (V. T.) Free-hand Perspective Svo.
Wilson's (V. T.) Free-hand Lettering , gvo,
Woolf's Elementary Course in Descriptive Geometry Large Svo,
ELECTRICITY AND PHYSICS.
* Abegg's Theory of Electrolytic Dissociation. (Von Ende.) i2ma, i 25
Anthony and Braclcett's Text-book of Physics. (Magie.) Small Svo 3 00
Anthony's Lecture-notes on the Theory of Electrical Measurements. .. .i3mo, 1 00
Benjamin's History of Electricity 8vo, 3 00
Voltaic Cell Svo, 3 00
Classen's Quantitative Chemical Analysis by Electrolysis. (Boltwood.).Svo, 3 00
* Collins's Manual of Wireless Telegraphy i2mo, i 50
Morocco, 2 00
Crehore and Squier's Polarizing Photo-chronograph Svo, 3 00
* Danneel's Electrochemistry. (Merriam.) i2mo, i 25
Dawson's "Engineering" and Electric Traction Pocket-book. i6mo, morocco, 5 00
10
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Dolezalek's Theory of the Lead Accumulator (Storage Battery). (Von
Ende.) i2ino, a jo
Duhem's Thermodynamics and Chemistry. (Burgess.) 8vo, 4 00
Flather's Dynamometers, and the Measurement of Power i2mo, 3 00
Gilbert's De Hagnete. (Hottelay.) 8vo, 2 50
Hanchett's Alternating Currents Explained. i2mo, i 00
Bering's Ready Reference Tables (Conversion Factors) i6mo morocco, 2 so
Holman's Precision of Measurements 8to, 2 00
Telescopic Hirror-scale Method, Adjustments, and Tests .... Large 8vo, 75
Kinzbrunner's Testing of Continuous-current Macliines 8to, 2 00
Landauer's Spectrum Analysis. (Tingle.) Svo,
Le Chatelier's High-temperature Measurements. (Boudouard — Burgess.) i2mo.
Lob's Electrochemistry of Organic Compounds. (Lorenz.) . . .8to,
* Lyons'] Treatise on Electromagnetic Phenomena. Vols. I. and IL 8vo, each,
* Michie's Elements of Wave Motion Relating to Sound and Light. Svo,
Niaudefs Elementary Treatise on Electric Batteries. (Fishback.) i2ma,
* Parshall and Hobart's Electric Machine Design 4to, half morocco,
Reagan's Locomotives: Simple, Compound, and Electric. iHew Edition.
Large i2mo, 3 So
* Rosenberg's Electrical Engineering. (Haldane Gee — Kinzbronner.). . Svo, 2 oo
Ryan, Norris, and Hoxie's Electrical Machinery, VoL L Svo, 2 50
Thurston's Stationary Steam-engines. Svo, 2 E<!
* Tillman's Elementary Lessons in Heat Svo, i 50
Tory and Pitcher's Manual of Laboratory Physics Small Svo, 2 00
Ulke's Modem Electrolytic Copper Refining. Svo, 3 00
LAW.
* Davis's Elements of Law Svo, 2 50
* Treatise on the Military Law of United States Svo, 7 oo
* Sheep, 7 JO
* Dudley's Military Law and the Procedure of Courts-martial . . . Large i2mo, 3 50
Mqni f l for Courts-martiaL i6mo, morocco, i 5®
Wait's Engineering and Architectural Jurisprudence Svo, 6 oo
Sheep, 6 so
Law of Operations Preliminary to Construction in Engineering and Archi-
tecture ...Svo 500
Sheep, 5 50
Law of Contracts. 8vo, 3 00
Winthrop's Abridgment of Military Law i2mo, 2 50
MAITOFACTURES.
Bemadou'g Smokeless Powder— Nitro-cellulose and Theory of the Cellulose
Molecule ""»»•
Bolland's Iron Founder i2mo,
The Iron Founder," Supplement 1 2mo,
Encyclopedia of Founding and Dictionary of Foundry Terms Used in the
Practice of Moulding i2mo,
• Claassen's Beet-sugar Manufacture. (Hall and RoKe.) Svo,
• Eckel's CemenU, Limes, and Plasters 8vo,
Eissler's Modem High Explosives 8vo,
Effront's Enzymes and their Applications. (Prescott) Svo,
Fitzgeraii's Boston Machinist i2mo.
Ford's Boiler Making for Boiler Makers. iSmo,
Hopkin's Oil-chemists' Handbook. 8™>
Keep's Cast Iron. ^'O'
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Leach's The Inspection and Aiutlysis of Food with Special Reference to State
Control Large 8vo, 7 so
* McKay and Larsen's Principles and Practice of Butter-making 8vo, i 50
Uatthewe's The Textile Fibres 8vo, 3 50
Metcalf's SteeL A Uanual for Steel-users: i2mo, 2 00
Hetcalfe'r Cost of Manufacttires — And the Administration of Workshops . 8to, 5 00
Heyer's Modem Locomotive Construction 4to, zo 00
Horse's Calculations used in Cane-sugar Factories i6mo, morocco, z 50
* Reisig's Guide to Piece-dyeing 8vo, 25 00
Rice's Concrete-block Manufacture 8vo, ^ 00
Sabin's Industrial and Artistic Technology of Paints and Vamislu 8vo, 3 00
Smith's PresE-workizig of Metals 8vo, 3 00
Spaldizig's HydrauUc Cement. z2mo, 2 00
Spencer's Hancfbook for Cheznists of Beet-sugar Houses. .... z6mo morocco* 3 00
Handbook for Cane Sugar Manufacturers i6mo morocco* 3 00
Taylor and Thoznpson's Treatise on Concrete, Plain and Reinforced 8vo, 5 00
Thurston's Manual of Steam-boilers, their Designs, Construction and Opera-
tiozi. 8to, s do
* Walke's Lectures on Explosives 8vo, 4 00
Ware's Beet-sugar Manufacture and Refining Small 8vo, 4 00
Weaver's Military Explosives 8vo,
West's American Foundry Practice z2mo,
Moulder's Text-book z2mo,
Wolff's Windmill as a Prime Mover 8vo,
3 00
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50
3
Wood's Rustless Coatings: Corrosion and Electrolysis of Iron and Steel. .8vo, 4 00
MATHEMATICS.
I so
50
7S
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z2mo, I 7S
Baker's Elliptic Functions. 8vo,
• Bass's Elements of Differential Calculus Z2mo, 4
Briggs's Elements of Plane Azzalytic Geometry Z2mo i 00
Compton's Moniul of Logarithznic Computations i2ino i 50
Davis's Introduction to the Logic of Algebra. gvo, i 50
• Dickson's CoUege Algebra Large Z2mo! i SO
• Introduction to the Theory of Algebraic Equations Large z2mo, z 25
Emch's Introduction to Projective Geometry and its Applications 8vo
Halsted's Elements of Geometry. ^vo
Elementary Synthetic Geometry gyo'
Rational Geometry z2mG
• Johnson's (J. B.) Thr:.e-place Logarithmic Tables: Vest-pocket size. paper! is
zoo copies for s 00
• Mounted on heavy cardboard, 8X10 inches, 25
zo copies for 2 00
Johnson's (W. W.) Elementary Treatise on Differential Calculus. .Small 8vo, 3 00
Elementary Treatise on the Integral Calculus SmalfSvo, z so
Johnson's (W. W.) Curve Tracing in Cartesian Co-ordinates z2mo, z 00
Jolmson's (W. W.) Treatise on Ordiziary and PartiaT Differential Equations.
SnzallSvo, 3 so
Johnson's (W. W.) Theory of Errors and the Method of Least Squares. z2mo, z 50
• Johnson's (W. W.) Theoretical Mechaziics z2mo, 3 00
Laplace's Philosophical Essay on ProbabiUties. (Truscott and Emory.) . z2mo, 2 00
• Ludlow and Bass. Elements of Trigonometry and Logarithmic and Other
Tables 8vo, 3 00
Trigonometry and Tables published separately Each, 2 oc
• Ludlow's Logarithiziic and Trigonometric Tables gvo t 00
Mann i n g's Irrational Numbers and their Representation by Sequences and Series
Z2mo, z 25
12
Hathenuitlcal Monocraphs. Edited by Mansfield Merriman and Robert
S. Woodward Octavo, each i oo
No. 1. History of Modern Mathematics, by David Eugene Smith.
No. J. Synthetic Projective Geometry, by George Bruce Balsted.
No. 3. Determinants, by Laenas OiSord Weld. No. 4. Hyper-
bolic Functions, by James McMahon. No. 5. Harmonic Func-
tions, by William E. Byerly. No. 6. Grassmann's Space Analysis,
by Edward W. Hyde. No. 7. ProbabiUty and Theory of Errors,
by Robert S. Woodward. No. 8. Vector Analysis and Quaternions,
by Alexander Macfarlaae. No. 9. Differential Equations, by
William Woolsey Johnson. No. 10. The Solution of Equations,
by Mansfield Merriman. No. 1 1. Functiotas of a Complex Variable,
by Thomas S. Fislcr.
Haurer's Technical Mechanics 8vo, 4 00
Merriman's Method of Least Squares 8vo a 00
Rice and Johnson's Elementary Treatise on the Differential Calculus. . Sm. 8vo, 3 00
Differential and Integral Calculus, a vols, in one Small 8vo, a 50
* Veblen and Lennes's Introduction to the Real Infinitesimal Analysis of One
Variable gvo, a 00
Wood's Elements of Co-ordinate Geometry gvo, a oo
Trigonometry: Analytical, Plane, and Spherical lamo, i 00
MECHANICAL ENGINEERmO.
MATERIALS OF ENGINEERING, STEAM-ENGINES AND BOILERS.
Bacon's Forge Practice lamo,
Baldwin's Steam Heating for Buildings lamo,
Barr's Kinematics of Machinery 8vo,
• Bartlett's Mechanical Drawing 8vo,
• " " " Abridged Ed 8vo,
Benjamin's Wrinkles and Recipes lamo.
Carpenter's Experimental Engineering 8vo,
Heating and Ventilating Buildings 8vo,
Clerk's Gas and Oil Engine Small 8vo,
Coolidge's Manual of Drawing 8vo, paper.
Coolidge and Freeman's Elements of General Drafting for Mechanical En-
gineers Oblong 4to,
Cromwell's Treatise on Toothed Gearing lamo.
Treatise on Belts and Pulleys lamo,
Durley's haematics of Machines 8vo,
Flather's Dynamometers and the Measurement of Power lamo,
Rope Driving -. lamo.
Gill's Gas and Fuel Analysis for Engineers lamo.
Hall's Car Lubrication lamo,
Bering's Ready Reference Tables (Conversion Factors) i6mo, morocco.
Button's The Gas Engine 8vo,
Jamison's Mechanical Drawing Svo,
Jones's Machine besign:
Part I. Kinematics of Machinery Svo,
Part II. Form, Strength, and Proportions of Parts Svo,
Kent's Mechanical Engineers' Pocket-book i6mo, morocco,
Kerr's Power and Power Transmission Svo,
Leonard's Machine Shop, Tools, and Methods Svo,
• Lorenz's Modem Refrigerating Machinery. (Pope, Haven, and Dean.) . Svo,
MacCord's Kinematics; or. Practical Mechanism Svo,
Mechanical Drawing 4to,
Velocity Diagrams Svo, i 50
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MacForland's Standard Reduction Factors for Gaaes 8vo, i 50
Uahan's Industrial Drawing. (Thompson.) 8vo 3 50
Pooies Calorific Power of Fuels 8to, 3 00
Rcid's Course in Uechanical Drawing 8to, 2 00
Text-book of Hecbanical Drawing and Elementary Machine Design. 8vo, 3 00
Richard's Compressed Air zsmo, i so
Robinson's Principles of Mechanism. 8to, 3 00
Schwaub and Merrill's Elements of Mechanism 8vo, 3 00
Smith's (0.) Press-working of Metals 8vo 3 00
Smith (A. W.) and Marx's Machine Design 8vo, 3 00
Thurston's Treatise on Friction and Lost Work in Machinery and Mill
Work 8vo, 3 00
Animal as a Machine and Prime Motor, and the Laws of Energetics. i2nio, 1 00
Tillson's Complete Automobile Instructor i6mo, i 50
Morocco, 2 00
Warren's Elements of Machine Construction and Drawing 8vo, 7 50
Weisbach's Kinematics and the Power of Transmission, (Herrmann —
Klein.) 8vo, 5 00
Machinery of Transmission and Governors. (Herrmann — Klein.). .Svo, s 00
Wolff's Windmill as a Prime Mover 8vo, 3 00
Wood's Turbines 8vo, 2 50
MATERIALS OP ElTGmEERING.
* Bovey's Strength of Materials and Theory of Structures 8vo, 7 50
Burr's Elasticity and Resistance of the Materials of Engineering. 6th Edition.
Reset 8to,
Church's Mechanics of Engineering 8vo,
* Greene's Structural Mechanics gvo,
Johnson's Materials of Construction 8vo,
Keep's Cast Iron gyo
Lanza's Applied Mechanics gvo
Martens's Handbook on Testing Materials. (Henning.) 8vo,
Haurer's Technical Mechanics gvo,
Heiriman's Mechanics of Materials gvo
* Strength of Materials i2mo
Metcalf's SteeL A Manual for Steel-users i2mo',
Sabin's Industrial and Artistic Technology of Paints and Varnish 8vo,
Smith's Materials of Machines l2mo'
Thurston's Materials of Engineering 3 vols., 8vo
Part n. Iron and SteeL gvo!
Part in. A Treatise on Brasses, Bronzes, and Other Alloys and their
Constituents gvo
Wood's (De V.) Treatise on the Resistance of Materials and an Appendix on
■ the Preservation of Timber gvo.
Elements of Analytical Mechanics gvo.
Wood's (M. P.) Rustless Coatings: Corrosion and Electrolysis of Iron and
Steel gvo.
STEAM-ENGINES AND BOILERS.
Berry's Temperature-entropy Diagram i2mo
Carnot's Reflections on the Motive Power of Heat (Thurston.) i2mo, i so
Dawson's "Engineering" and Electric Traction Pocket-book i6mo, mor., s 00
Ford's Boiler Making for Boiler Makers i8mo,
Goss's Locomotive Sparks gvo
Locomotive Performance gvo
Hemenway's Indicator Practice and Steam-engine Economy . ismo
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Button's Hechanical Engineering of Power Plants 8to, 5 00
Heat and Heat-engines 8to. s 00
Kent's Steam boiler Economy 8vo, 4 00
Kneass's Practice and Theory of the Injector 8to, i so
HacCord's Slide-valves 8vo, 2 00
Meyer's Modem Locomotive Construction 4to, 10 00
Peabody's Manual of the Steam-engine Indicator lamo. i 50
Tables of the Properties of Saturated Steam and Other Vapors 8vo, i oo
Thermodynamics of the Steam-engine and Other Heat-engines 8vo, 5 00
Valve-gears for Steam-engines 8vo, i 50
Peabody and Miller's Steam-boilers 8vo, 4 00
Pray's Twenty Years with the Indicator Large 8vo, 2 50
Pupin's Thermodynamics of Reversible Cycles in Gases and Saturated Vapors.
(Osterberg.) umo, i 15
Reagan's Locomotives: Simple, Compound, and Electric. New Edition.
Large z3mo, 3 50
Rontgen's Principles of Thermodynamics. (Du Bois.) 8vo, 5 oc
Sinclair's Locomotive Engine Running and Management xamo, 2 00
Smart's Handbook of Engineering Laboratory Practice lamo, a 50
Snow's Steam-boiler Practice 8vo, 3 00
Spangler's Valve-gears 8vo, 2 50
Notes on Thermodynamics i2mo, i 00
Spangler, Greene, and Marshall's Elements of Steam-engineering 8vo, 3 00
Thomas's Steam-turbines 8vo, 3 50
Thurston's Handy Tables 8vo, x so
Manual of the Steam-engine 2 vols., 8vo, 10 00
Part I. History, Structure, and Theory 8vo, 6 00
Part n. Design, Construction, and Operation 8vo, 6 00
Handbook of Engine and Boiler Trials, and the Use of the Indicator and
the Prony Brake 8vo, 5 00
Stationary Steam-engines 8vo, 2 go
Steam-boiler Explosions in Theory and in Practice i2mo, i 50
Manual of Steam-boilers, their Designs, Construction, and Operation . Svo, 5 00
Wehrenfenning'sAnalysisandSofteningof Boiler Feed-water (Patterson) 8vo, 400
Weisbach's Heat, Steam, and Steam-engines. (Du Bois.) Svo, 5 00
Whitham's Steam-engine Design 8vo, s 00
Wood's Thermodynamics, Heat Motors, and Refrigerating Machines. . .8vo, 4 00
MECHANICS AND MACHINERY.
Barr's Kinematics of Machinery Svo, 3 so
* Bovey's Strength of Materials and Theory of Structures Svo, 7 50
Chase's The Art of Pattern-making i2mo, 2 50
Church's Mechanics of Engineering Svo, 6 00
Notes and Examples in Mechanics Svo, 2 00
Compton's First Lessons in Metal-working i2mo, i so
Compton and De Groodt's The Speed Lathe i2mo, 1 ko
Cromwell's Treatise on Toothed Gearing i2mo, i 50
Treatise on Belts and Pulleys i2mo, i 50
Dana's Text-book of Elementary Mechanics for Colleges and Schools. . i2mo, i 50
Dingey's Machinery Pattern Making i2mo, 2 00
Dredge's Record of the Transportation Exhibits Building of the World's
Columbian Exposition of 1893 4*0 half morocco, s 00
Du Bob's Elementary Principles of Mechanics:
VoL I. Kinematics 8vo, 3 so
Vol n. Statics 8vo. 4 00
Mechanics of Engineering. VoL I SmaU4to, 7 SO
VoL n Small 4to, 10 00
Durley's Kinematics of Machines 8™- 4 00
15
Fitzgerald's Boston Machinist i6mo, i oo
Flather's Dynamometers, and the Measurement of Power iimo, 3 00
Rope Driving i2mo, 2 00
Goss's Locomotive Sparks 8vo, 2 00
Locomotive Performance 8vo, s 00
* Greene's Structural Mechanics 8vo, 2 so
Hall's Car Lubrication i2mo, i 00 .
Holly's Art of Saw Filins i8mo, 75
James's Kinematics of a Point and the Rational Mechanics of a Particle.
Small 8va, 2 00
* Johnson's (W. W.) Theoretical Mechanics i2mo, 3 00
Johnson's (L. J.) Statics by Graphic and Algebraic Methods Svo, 2 00
Jones's Machine Design:
Part I. Kinematics of Machinery Svo, i 50
Part n. Form, Strength, and Proportions of Parts Svo, 3 00
Kerr's Power and Power Transmission Svo, 2 00
Lanza's Applied Mechanics Svo, 7 50
Leonard's Machine Shop, Tools, and Methods Svo, 4 00
* Lorenz's Modem Refrigerating Machinery. (Pope, Haven, and Dean.). Svo, 400
MacCord's Kinematics; or, Practical Mechanism Svo, 5 00
Velocity Diagrams Svo, i 50
* Martin's Text Book on Mechanics, Vol. 1, Statics i2mo, 1 25
Maurer's Technical Mechanics. Svo, 4 00
Herriman's Mechanics of Materials Svo, s 00
* Elements of Mechanics i2mo, i 00
* Michie's Elements of Analytical Mechanics Svo, 4 00
*Parshall and Hobart's Electric Machine Design 4to, half morocco, 12 50
Reagan's Locomotives : Simple, Compound, and Electric. New Edition.
Large i2mo, 3 00
Reid's Course in Mechanical Drawing Svo, 2 00
Text-book of Mechanical Drawing and Elementary Machine Design. Svo, 3 00
Richards's Compressed Air i2mo, i 50
Robinson's Principles of Mechanism Svo, 3 00
Ryan, Norris, and Hoxie's Electrical Machinery. VoL I Svo, 2 50
Sanborn's Mechanics: Problems Large 12310, i 50
Schwamb and Merrill's Elements of Mechanism 8to, 3 00
Sinclair's Locomotive-engine Running and Management. i2mo, 2 00
Smith's (O.) Press-working of Metals Svo, 3 00
Smith's (A. W.) Materials of Machines i2mo, i 00
Smith (A. W.) and Marx's Machine Design Svo, 3 00
Spangler, Greene, and Marshall's Elements of Steam-engineering Svo, 3 00
Thurston's Treatise on Friction and Lost Work in Machinery and Mill
Work Svo, 3 00
Animal as a Machine and Prime Motor, and the Lawc of Energetics. i2mo, 1 00
Tillson's Complete Automobile Instructor i6mo, i 50
Morocco, 2 00
Warren's Elements of Machine Construction and Drawing Svo, 7 50
Weisbach's Kinematics and Power of Transmission. (Herrmann— Klein.). Svo, 500
Machinery of Transmission and Governors. (Herrmann — Klein.). Svo, 5 00
Wood's Elements of Analytical Mechanics Svo, 3 00
Principles of Elementary Mechanics i2mo, i 25
Turbines Svo, 2 50
The World's Columbian Exposition of 1893 4to, i 00
MEDICAL.
De Fursac's Manual of Psychiatry. (RosanoS and Collins.) Large i2mo, 2 50
Ehrlich's Collected Studies on Immunity. (Bolduan.) Svo, 6 00
Hammarsten's Text-book on Physiological Chemistry. (Mandel.) Svo, 4 00
16
Lassai-Cohn's Practical Urinary Analysis. (Lorenz.) iimo, i oo
* Pauli's Physical Chemistry in the Service of Hedicjae. (Fischer.). ■ ■ i2mo, i 25
* Pozzi-Escot's The Toxins and Venoms and their Antibodies. (Cohn.). izmo, i 03
Rostoski's Senun Diagnosis. (Bolduan.) izmo, i 00
Salkowski's Physiological and Pathological Chemistry. (OrndorCE.) &ro, 2 50
* Satterlee's Ovtlines of Human Embryology izmo, z 25
Steel's Treatise on the Diseases of the Dog 8vo, 3 50
Von Behring's Suppression of Tuberculosis. (Bolduan.) i2mo, i 00
Wassermann's Immune Sera ■ Hemolysis, Cytotoxins, and Precipitins. (Bol-
duan.) i2mo, cloth, I 00
Woodhull's Botes on Military Hygiene 26mo, i 50
* Personal Hygiene izmo, i 00
Vulling's An Elementary Course in Inorganic Pharmaceutical and Medical
Chem&try izmo, 2 00
METALLURGY.
Egleston's Metallurgy of Silver, Gold, and Mercury :
VoL L Silver 8vo,
VoL n. Gold and Mercury 8vo,
Goesel's Minerals and Metals: A Reference Book. i6mo, mor.
* Iles's Lead-smelting izmo,
Keep's Cast Iron 8vo,
Kunhardt's Practice of Ore Dressing in Europe 8vo,
Le Cbatelier's High-temperatuxe Measurements. (Boudouard — Burgess. )x2mo,
MetcalTs SteeL A Manual for Steel-users. izmo.
Miller's Cyanide Process. izmo,
Minet's Production of Afaiminom and its Industrial Use. (Waldo.). . . . izmo,
Robine and Lenglen's Cyanide Industry. (Le Clerc.) 8to,
Smith's Materials of Machines. izmo,
Thurston's Materials of Engineering. In Three Parts 8vo,
Part n. Iron and SteeL 8vo,
Part HL A Treatise on Brasses, Bronzes, and Other Alloys and their
Constituents 8™>
Ulke's Modem Electrolytic Copper Refining 8yo,
JimERALOGY.
Barringer's Description of Minerals of Commercial Value. Oblong, morocco, 2 50
Boyd's Resources of Southwest Virginia 8vo, 3 00
Map of Southwest Virignia Pocket-book form. 2 00
Brush's Manual of Determinative Mineralogy. (Penfield.) 8vo, 4 00
Chester's Catalogue of Minerals 8". paper, i 00
Cloth, I zs
Dictionary of the Hames of Minerals 8vo, 3 so
Dana's System of Mineralogy Large 8vo. half leather. .2 so
First Appendix to Dana's New " System of Mineralogy." Large 8vo, i 00
Text-book of Mineralogy 8vo, 4 00
Minerals and How to Study Them "•>«>• ' SO
Catalogue of American LocaUties of Minerals Large Bvo,
Manual of Mineralogy and Petrography >s™o
Douglas's Untechnical Addresses on Technical Subjects "mo, i 00
Eakle's Mineral Tables ™' ' \^
Egleston's Catalogue of Minerals and Synonyms ■• ■ »»<>, so
Goesel's Minerals and Metals: A Reference Book i6mo,mor. 300
Groth's Introduction to Chemical Crystallography (Marshall) "mo. i 25
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00
2
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I
00
2
so
4
oo
I
00
8
00
3
50
z
50
3
00
00
IddingB's Rock Minerals 8vo, s oo
HeiTill's Non-metallic Minerals: Their Occurrence and Uses 8to, 4 00
* Penfield's Notes on Determinative Mineralogy and Record of Mineral Tests.
8to, paper, 50
* Richards's Synopsis of Mineral Characters izmo, morocco, i 25
* Rles's Clays: Their Occurrence, Properties, and Uses 8vo, 5 00
Rosenbusch's Microscopical Physiography of the Rock- mak i ng Minerals.
(Iddings.) 8vo, s 00
* Tillman's Text-book of Important Minerals and Rocks 8vo, 2 00
Boyd's Resources of Southwest Virginia 8vo, 3 00
Map of Southwest Virginia Pocket-book form 2 00
Douglas's Untecbnical Addresses on Technical Subjects i2mo, I 00
EisSler's Modem High Eiplosives "'' 4 "o
Goeael's Minerals and Metals : A Reference Book . . i6mo, mor. 3 00
Goodyear's Coal-mines of the Western Coast of the United States i2mo, 2 50
Ihlseng's Manual of Mining 8vo, 5 00
• Iles's Lead-smelting i2mo, 2 so
Kunhardt's Practice of Ore Dressing In Europe 8vo, i 50
Miller's Cyanide Process i2mo, i 00
O'Driscoll's Notes on the Treatment of Gold Ores 8vo, 2 00
Robine and Lenglen's Cyanide Industry. (Le Clerc.) 8vo, 4 00
• Walke's Lectures on Explosives 8vo, 4 00
Weaver's Military Explosives 8vo, 3 00
Wilson's Cyanide Processes i2mo, i 50
Chlorination Process i2mo, 1 50
Hydraulic and Placer Mining i2mo, 2 00
Treatise on Practical and Theoretical Mine Ventilation. i2mo, 125
SANITARY SCIENCE.
Bashore's Sanitation of a Country House i2nio, 1 00
* Outlines of Practical Sanitation i2mo, i 25
FolweH's Sewerage. (Designing, Construction, and Maintenance. J 8vo, 3 00
Water-supply Engineering 8vo, 4 00
Fowler's Sewage Works Analyses I2m3, 2 00
Fuertes's Water and Public Health i2mo, x 50
Water-filtration Works i2mo, 2 50
Gerhard's Guide to Sanitary House-inspection x6mo, 1 00
Hazen's Filtration of Public Water-supplies 8vo, 3 00
Leach's The Inspection and Analysis of Food with Special Reference to State
Control 8vo, 7 so
Mason's Water-supply. (ConsideredprincipaUyfromaSanitaryStandpoint)8vo, 4 00
Examination of Water. (Chemical and Bacteriological) i2mo, i 25
* Merriman's Elements of Sanitary Engineering 8vo, 2 00
Ogden's Sewer Design i2mo, 2 00
Prescott and Winslow's Elements of Water Bacteriology, with Special Refer-
ence to Sanitary Water Analysis i2mo, i 25
* Price's Handbook on Sanitation i2mo, 1 so
Richards's Cost of Food. A Study in Dietaries i2mo, x 00
Cost of Living as Modified by Sanitary Science i2mo, x 00
Cost of Shelter .• i2mo, i 00
18
2
oo
I
so
4
oo
S
oo
I
oo
3
50
7
50
I
50
I
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Richards and Woodman's Air. Water, and Food from a Sanitary Stand-
point 8vo,
• Richards and Williams's The Dietary Computer 8vo.
Rideal's Sewage and Bacterial Purification of Sewage 8vo,
Tumeaure and Russell's Public Water-supplies 8vo,
Von Bebring's Suppression of Tuberculosis. (Bolduan.) i2mo,
Whipple's Microscopy of Drinking-water 8vo,
Winton's Microscopy of Vegetable Foods 8vo,
Woodhull's Notes on Military Hygiene i6mo.
♦ Personal Hygiene X2mo,
MISCELLANEOUS.
Emmons's Geological Guide-book of the Rocky Mountain Exctirsion of the
International Congress of Geologists Large t-vo, i 5-.
Ferrel's Popular Treatise en the Winds 8vo, 4 00
Gannett's Statistical Abstract of the World 24mOp 75
Haines's American Railway Management i2mo, 2 50
Ricketts's History of Rensselaer Polytechnic Institute 1824-1894. .Small 8vo, 3 00
Rotherham's Emphasized New Testament Large 8vo , 2 00
The World's Columbian I xposition of 1893 4to, i 00
Winslow's Elements of Applied Microscopy i2mo, x 50
HEBREW AND CHALDEE TEXT-BOOKS.
Green's Elementary Hebrew Grammar i2mo, i 25
Hebrew Chrestomathy 8vo, 2 00
Gesenius's Hebrew and Chaldee Lexicon to the Old Testament Scriptures.
(Tregelles.) Small 4to, half morocco, s 00
Letteris's Hebrew Bible 8vo, 2 as
19
oORNEaUNlVaSlTYllBfWRY
slUL U 1991
MATHEMATlCSUBflARY
■¥• ^ AfS, '2?! ■