The Fundamentals of Mathematical Analysis: International Series in Pure and Applied Mathematics, Volume 1

499 Pages·1965·19.59 MB·English

by  G. M. Fikhtengol'ts

THE FUNDAMENTALS
OF
MATHEMATICAL ANALYSIS
Volume I
G. M. FIKHTENGOL'TS
Translation edited by
IAN N. SNEDDON
SIMSON PROFESSOR OF MATHEMATICS
IN THE UNIVERSITY OF GLASGOW
PERGAMON PRESS
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Copyright © 1965 Pergamon Press Ltd.
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First edition 1965
Reprinted 1979
Library of Congress Catalog Card No. 63-22750
This is a translation from the original Russian
OcHoebi MameMamunecKozo auanu3a (Osnovy
matematicheskogo analiza), published in 1960
by Fizmatgiz, Moscow
Printed in Great Britain by A. Wheaton & Co. Ltd., Exet
ISBN 0 08 013473 4
FOREWORD
THIS book is planned as a textbook of analysis for first and second
year mathematics students at Russian universities and consequently
is divided into two volumes. In compiling the book I have made
extensive use of my three-volume Course of Differential and Integral
Calculus, revising and abridging it in order to adapt it to the official
mathematical analysis programme and to make it meet the require-
ments of a lecture course.
The tasks I set myself and the points by which I was guided are
as follows:
1. First and foremost to provide a systematic and, as far as pos-
sible, rigorous treatment of the fundamentals of mathematical
analysis. I consider it obligatory for the contents of a textbook to
be presented in a logical sequence, in order to achieve a clearly
defined and systematic presentation of the facts.
This does not, however, prevent the lecturer from deviating from
a strict systematic approach, but, perhaps, even helps him in this
respect. In my own lecture courses, for example, I usually put aside
for a while such difficult tasks for beginners as the theory of real
numbers, the principle of convergence or the properties of con-
tinuous functions.
2. To uphold my own opinion that a course of mathematical
analysis should not appear to students to be merely a long chain
of "definitions" and "theorems", but that it should also serve as
a guide to action. Students must be taught to apply the theorems
in practice in order to assist them in mastering the computational
apparatus of analysis. Although this can be achieved largely with
the help of exercises, I have also included some examples in my
treatment of the theoretical material. The total number of these
examples is, out of necessity, small, but they have been selected in such
a way as to prepare students for conscientious work on the exercises.
[xxiii]
xxiv FOREWORD
3. It is well known that mathematical analysis has diverse and
remarkable applications both in mathematics itself and in related
scientific fields. Whilst students will realize this more and more as
time passes, it is essential that they should learn and get used to the
relationship of mathematical analysis with other mathematical
sciences and with the requirements of practical work whilst study-
ing the fundamentals of analysis. For this very reason I have pro-
vided, wherever possible, examples of the application of analysis
not only to geometry, but also to mechanics, physics and engi-
neering.
4. The problem of completing analytic work up to numerical
results is of both theoretical and practical importance. Since an
"exact" or "closed form" solution of a problem in analysis is possible
in the simplest cases only, it is important to acquaint students with
the use of approximate methods. Some attention has been given
to this within the pages of this book.
5. By way of a brief explanation of my treatment of the subject
matter, I have first of all considered the concept of a limit which
plays the principal role among the fundamental concepts of analysis
and which crops up in diverse forms literally throughout the entire
course. Hence arises the problem of establishing a unified form of all
variations of the limit. This is not only important from the viewpoint
of principles but also vital from a practical standpoint, to obviate
the necessity of having to construct the theory of limits anew each
time it arises. There are two ways of achieving this aim: we can
either immediately give the general definition of the limit of "directed
variable" (following, for example, Shatunovskii and Moore, or
Smith), or we can reduce every limit to the simplest case of the limit
of a variable ranging over an enumerated sequence of values. The
first alternative is'difficult for beginners, and I have, therefore, chosen
the second method of approaching the problem. The definition
of each new limit is given first by means of the limit of a sequence
and only later on "in ε-δ language".
6. To indicate a second feature of my treatment of the subject
matter I have in Volume II, when speaking of curvilinear and
surface integrals, emphasized the difference between the curvilinear
and surface "integrals of first kind" (the exact counterparts of the
FOREWORD XXV
ordinary and double integral over unoriented domains) and similar
"integrals of second kind" (where the analogy partly vanishes).
Experience has convinced me that this distinction not only leads
to a better understanding of the material, but is also convenient in
applications.
7. As a short appendix to the book I have included a brief account
of elliptic integrals and in several cases I have presented problems
with solutions involving elliptic integrals. This may help to destroy
the harmful illusion, acquired by merely solving simple problems,
that the results of analytic calculations must necessarily be "elemen-
tary".
8. In various places throughout the book the reader will come
across remarks of an historical nature. Moreover, Volume I ends
with a chapter entitled, "Historical survey of the development of
the fundamental concepts of mathematical analysis" and Volume II
concludes with "An outline of further developments in mathema-
tical analysis". However, neither of these two "surveys" has been
introduced to serve as a substitute for a complete history of mathe-
matical analysis, which students meet with later in general courses
on "the history of mathematics". The first survey touches upon
the origin of the concepts, whilst the final chapter in Volume II
aims at providing the reader with at least a general idea of the chrono-
logy of the most important events in the history of analysis.
At this point, and in connection with the preceding paragraph,
I should like to give a warning to potential readers of this book.
The sequence in which I have treated various topics is closely con-
nected with modern demands for strict mathematical rigour—demands
which have become more and more acute over the years.
Historically speaking, therefore, the development of mathematical
analysis has not been followed as closely as it might have been.
Thus, Chapter 1 is devoted to "real numbers", Chapter 3 to
the "theory of limits", and it is not until Chapter 5 that I have
commenced to give a systematic account of the differential and
integral calculus. The historical sequence of events was, of course,
the complete reverse. The differential and integral calculus were
founded in the seventeenth century and developed in the eighteenth
century, being applied to numerous important problems; the theory
xxvi FOREWORD
of limits became the foundation-stone of mathematical analysis at the
beginning of the nineteenth century and only in the second half of
the nineteenth century did a clearly defined concept of real numbers
come into being, which justified the most refined propositions of
the theory of limits.
This book summarizes many years of experience in lecturing on
mathematical analysis in Leningrad University.
G. M. FIKHTENGOL'TS
CHAPTER 1
REAL NUMBERS
§ 1. The set of real numbers and its ordering
1. Introductory remarks. The reader is familiar, from school
courses of mathematics, with the rational numbers and their prop-
erties. However, already the demands of elementary mathematics
result in a need for the extension of this number domain. In fact,
among the rational numbers there frequently do not exist the roots
of positive integers, for instance γ2, i.e. there is no rational fraction
p/q where p and q are positive integers, the square of which is equal
to 2.
To prove this assertion assume the converse: let there exist a frac-
tion p/q such that (p/q)2 = 2. We may regard this fraction as irre-
ducible, i.e. p and q have no common factors. Since p = 2q2, p is
an even number, p = 2r (r is an integer) and, consequently, q is odd,
Substituting for p its expression we find that q2 — 2r2 which implies
that q is an even number. This contradiction proves our assertion.
Moreover, if we remain in the domain of rational numbers only,
it is clear that in geometry not all segments may be provided with
lengths. In fact, consider a square with side equal to the unit of
length. Its diagonal cannot have a rational length p/q, since if this
were the case, according to the Pythagoras theorem the square of
its length would be 2, which we know to be impossible.
In the present chapter we intend to extend the domain of rational
numbers by connecting with them numbers of a new kind—the
irrational numbers.
The irrational numbers appear in mathematics—in the form of expressions
containing roots—in medieval papers, but they were not regarded as genuine
numbers. In the seventeenth century the coordinate method created by Descartest
t René Descartes (1596-1650)—a celebrated French philosopher and scientist.
1 [1]
2 1. REAL NUMBERS
again raised the problem of the numerical description of geometric quantities.
This induced a gradual growth of the concept of the common nature of irrational
and rational numbers ; it was finally formulated in the definition of a (positive)
number given by Newtont in his General Arithmetic (1707): "By a number we
understand not so much the set of unities as an abstract ratio of a quantity to
another quantity of the same kind assumed to be unity.'* The integers and fractions
express numbers commensurable with unity while the irrational numbers express
those incommensurable with unity.
The mathematical analysis created in the seventeenth century and extensively
developed throughout the whole of the eighteenth century was for a long time
satisfied with this definition although it was alien to arithmetic and kept in the
background the most important property of the extended number domain—its
continuity (see Sec. 5 below). The critical trend in mathematics which arose at
the end of the eighteenth and the beginning of the nineteenth century advanced the
demand for a precise definition of the fundamental concepts of analysis and an
exact proof of its basic statements. This in turn soon made it necessary to construct
a logically sound theory of irrational numbers on the basis of a purely arithmetical
definition. In the seventies of the nineteenth century a number of such theories were
developed, superficially different in form but essentially equivalent. All these
theories define an irrational number by connecting it with some infinite set of
rational numbers.
2. Definition of irrational number. We shall give the theory
of irrational numbers in the form due to Dedekind*. This theory
is based on the concept of a cut in the domain of rational numbers
We consider the division of the set of all rational numbers into two
non-empty (i.e. containing at least one number) sets A, A'; in other
words we assume that
(1) every rational number bolongs to one and only one set A or Ä.
We call such a division a cut if one more condition is satisfied,
namely:
(2) every number a of the set A is smaller than every number a' of
the set A'.
Set A is called the lower class, and set A' the upper class. The
cut will be denoted by A\A'.
The definition implies that any rational number smaller than
a number a of the lower class also belongs to this class. Similarly,
any rational number greater than a number a' of the upper class
belongs to the upper class.
t Isaac Newton (1642-1727)—great English physicist and mathematician.
t Richard Dedekind (1831-1916)—a German mathematician.
§ 1. SET OF REAL NUMBERS 3
Examples. (1) Define A as the set of all rational numbers a satis-
fying the inequality a<l, while set A' contains all numbers a'
such that a' > 1.
It can easily be verified that we have in fact obtained a cut. The
number unity belongs to class A' and obviously it is the smallest
number of this set. On the other hand there is no greatest number
in class A, since for any number a from A we can always indicate
a rational number a located between a and unity and, consequently,
x
greater than a and also belonging to class A.
(2) The lower class A contains all rational numbers a smaller
or equal to unity, a < 1, while the upper class contains all rational
numbers a' greater than unity, a' > 1.
This is also a cut, and now the upper class has no smallest number
whereas the lower does have the greatest (namely—unity).
(3) Class A contains all positive rational numbers a for which
a2 < 2, the number zero and all negative rational numbers, while
class A' contains all positive rational numbers a' such that a' 2>2.
It is easily seen that we again have a cut. Now class A has no
greatest number and class A' no smallest number. Let us, for instance,
prove the first assertion (the second can be proved in an analogous
way). Let a be an arbitrary positive number of class A; hence a2 < 2.
We prove that we can select a positive integer n such that
KÏ
<2,
so that the number a + (IIn) also belongs to class A.
This inequality is equivalent to the following two :
n nz
n ηΔ
The last inequality is certainly satisfied if« is such that (2a + l)/n <
2 —a2 for which it is sufficient to take
2a+l
n >-= =-.
4 1. REAL NUMBERS
Thus, regardless of the value of the positive number a of class A,
in the same class A there is always a greater number; since, for the
numbers a < 0 this assertion is obvious, no number of class A is
the greatest a in A.
Clearly, there cannot exist a cut such that there is simultaneously
a greatest number a in the lower, class and a smallest number a'
0 0
in the upper class.. In fact, assume that such a cut does exist. Then
we take an arbitrary rational number c which lies between a and
0
a' a <^c<aQ. The number c cannot belong to class A, since then
0i 0
a would not be the smallest number in this class; for an analogous
0
reason c cannot belong to class A' and this contradicts the property
(1) of the cut, the latter property being a part of the definition of
this concept.
Thus, cuts can be of three kinds illustrated in turn by Examples
(1), (2) and (3), either:
(1) in the lower class A there is no greatest number and the upper
class Ä contains a smallest number r, or
(2) there is a greatest number r in the lower class A while the
upper class A' has no smallest number, or, finally
(3) neither the lower class has a greatest number, nor the upper
class a smallest number.
It is said in the first two cases that the cut is made by the rational
number r (which is the boundary number between classes A and A')
or that this cut defines the rational number r. In Examples (1) and
(2) the number r was unity. In the third case a boundary number
does not exist and the cut does not define any rational number.
We now introduce new elements—the irrational numbers, by stating
that every cut of the form (3) defines an irrational number a. This
number a replaces the lacking boundary number; it seems to be
introduced between all numbers a of class A and all numbers a!
of class A'. In Example (3) this newly created number is evidently γ2.
Without introducing any unified notation* for irrational numbers
we shall always connect the irrational number a with the cut A\A'
in the domain of rational numbers, which defines it.
t We mean finite notation; the reader will become acquainted with a kind
of infinite notation in §1.4. Irrational numbers are usually denoted by forms
depending on their origin and role, e.g. \/2, log 5, sin 10°, etc.