Mathematical analysis : functions, limits, series, continued fractions

407 Pages·1965·16.668 MB·English

by  Brown, Donald Edward, Lyusternik, Lazar' Aronovich, Spence, E., Yanpol'skii, Avraam Ruvimovich

MATHEMATICAL
ANALYSIS
Functions, Limits, Series,
Continued Fractions
EDITED BY
L. A. LYUSTERNIK AND
A. R. YANPOL'SKÏÏ
TRANSLATED BY
D.E. BROWN
TRANSLATION EDITED BY
E. SPENCE
PERGAMON PRESS
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Copyright © 1965
PEROAMON PRESS LTD.
First Edition 1965
Library of Congress Catalog Card Number 63-19330
This is an edited translation of the original Russian volume entitled
MameMamunecKau αηαΜΐ3—Φγηκμιΐίΐ, npedeAU,
pndbi, qeriHue dpoou
(Matematichekii analiz—Funksiyi, predeli, ryady, tsepnye droby),
published in 1961 by Fizmatgiz, Moscow
FOREWORD
THE PRESENT book, together with its companion volume devoted to
the differential and integral calculus, contains the fundamental part
of the material dealt with in the larger courses of mathematical
analysis. Included in this volume are general problems of the theory
of continuous functions of one and several variables (with the
geometrical basis of this theory), the theory of limiting values for
sequences of numbers and vectors, and also the theory of numerical
series and series of functions and other analogous infinite processes,
in particular, infinite continued fractions.
Chapter I, "The arithmetical linear continuum and functions
defined there" (authors: L. A. Lyusternik and Ye. K. Isakova),
is devoted to real numbers, the arithmetical linear continuum, li­
miting values, and to functions of one variable. The material of this
chapter is more or less that which is usually called the introduction
to mathematical analysis.
Chapter II, "«-dimensional spaces and functions defined there"
(L. A. Lyusternik), effects the transition from functions of one
variable to functions of « variables, which, geometrically, corresponds
to the transition from the arithmetical linear continuum to «-dimen­
sional space E , the fundamental theory of which is given. §1 is devoted
n
to the fundamentals of «-dimensional geometry and, in particular,
of the theory of orthogonal systems of vectors in E which serves
n9
as a simpler model for the theory (Chapter IV) or orthogonal systems
of functions. §2 is devoted to limiting values in E to continuous
n9
functions of « variables and their systems (transformations in E^).
In this chapter also §3 deals with a subject which plays an important
part in pure and applied mathematics, the theory of «-dimensional
convex bodies.
Chapter III, "Series" (authors G. S. Salekhov and V. L. Danilov),
consists of the theory of series and practical methods of summation.
The theory of numerical series is dealt with in §1 including ques-
xiii
XIV FOREWORD
tions relating to infinite products, double series and the summation
of convergent series. Side by side with the classical material the reader
will find new results about the general tests for the convergence of
series and estimations of the remainder.
The more important classes of series of functions are considered
in §2: power, trigonometrical, and also asymptotic power series,
and their convergence. At this point some methods for the general
summation of divergent series are added. In §3 are to be found various
devices useful in calculations connected with the theory of series.
Chapter IV "Orthogonal series and orthogonal systems" (authors
A. N. Ivanova and L. A. Lyusternik), contains the general problems
of the reduction of functions to orthogonal (and also biorthogonal)
series. Here, also, general orthogonal systems of polynomials and
the classical systems of Legendre, Chebyshev, Hermite Polynomials,
and others, are considered.
Chapter V "Continued fractions" (author A. N. Khovanskii),
deals with that branch of analysis which occupied the attention
of the greatest mathematicians of the eighteenth and nineteenth
centuries, but which was afterwards unjustly forgotten. Continued
fractions did not find a place in the contemporary larger courses of
analysis; on the other hand comparatively recently, some elements
of the theory of continued fractions were studied even in middle
school. In the past few years the interest in continued fractions has
revived in connection with their application in computation and
other topics in applied mathematics.
Chapter VI, "Some special constants and functions" (authors
L. A. Lyusternik, L. Ya. Tslaf and A. R. Yanpol'skii), has more of
the nature of a manual (in the narrow sense of the word). The mate­
rial here concerns various constants, the most important systems
of numbers, including Bernoulli and Euler numbers, some discon­
tinuous functions, and the simpler special functions (elliptic integrals,
integral functions, the gamma and beta functions, some Bessel
functions, etc.). These functions, together with orthogonal polyno­
mials, after the elementary ones, are the most widely used in appli­
cations of mathematics. We would like to mention that these special
functions will be dealt with more fully and in the complex domain
in one of the following issues.
CHAPTER I
THE ARITHMETICAL LINEAR
CONTINUUM AND FUNCTIONS
DEFINED THERE
§ 1. Real numbers and their representation
1· Real numbers
All real numbers can be split into two classes: rational and irra­
tional. All integers and fractions (positive, negative and zero) are
rational numbers, while the remaining numbers are irrational.
The set of all rational numbers is everywhere dense, i.e. between
any two distinct rational numbers a and b (a<6) there is at least
one further rational number c (a<c<b), i.e. in fact, an infinity of
rational numbers.
Examples of irrational numbers are:>/2= 1-41421356..., π =
= 3-14159..., e = 2-7182818... — the base of natural logarithms, andso
on. Irrational numbers consist of algebraic and transcendental numbers.
Algebraic irrational numbers are defined as all non-integral real
roots of the algebraic equation
xn + a xn-1 +... + a _ x + a = 0,
x n x n
where a (/= 1, 2,...,n) are integers; for example, the roots
x
x =l/To, x =V8* of the equations ΛΓ'-ΙΟ = 0, *5-8 = 0, the
roots of the equation x5 — 3x* — 2x* -f- x2 -f1 = 0, and so on. The remain­
ing irrational numbers are described as transcendental; examples
of these are π, e, é1, 2^2, log n (where n is any integer not equal to
10Ä) and so on.
1
2 MATHEMATICAL ANALYSIS
2. The numerical straight line
We choose on a straight line E an origin of measurement — the
l9
origin of coordinates 0, a scale the unit of length, and a direction —
the orientation. We associate with every real number x a point A(x)
on the Une E having the coordinate (abscissa), x, conversely, with
l9
every point A(x) on E we associate a real number x — its abscissa.
l9
E is called a numerical straight line or a one-dimensional coordinate
x
space (see Chapter II for w-dimensional coordinate spaces isj.
The number
— a for a < 0,
for a ^ 0
is called the absolute or numerical value of the number Û.
The following relationships hold:
|β+6|^|β| + |Η
| -ô|*ll*Hô||,
e
| .*ΙΗ*|.|Η
β
a I |α|
*"!" W
" '
The number [0 — i] is called the distance between the points a and
6 on the straight line E (see Chapter II, § 1, sec. 1).
1
3. p-adic systems
Every real number is expressible as a decimal fraction, i.e. has a
definite expansion in the decimal system of numeration. The decimal
system is a particular case of a positional system to the base p, where
any positive integer p > 1 can be taken as the base. The numbers
0, 1, 2, ..., />— 1 are called the digits of this system, while pk
(fc = 0, ± 1, ±2, ...) are units of the &th order in the system.
Every positive integer N is uniquely expressible in the form
n
Ν=α>ρΡ + α ρι + ... + α !Ρ = Σ^ρ\ (1.1)
ί 1 η
where ^ are digits. Equation (1.1) is written as
N = α α ^α ^ ... a a . (1.10
η η η 2 ± 0
THE ARITHMETICAL LINEAR CONTINUUM 3
Similarly, any positive real number 5, rational or irrational, is
expressible as a fraction to the base p:
S= Σ a p\ (1.2)
h
ft«-oo
which is written as
5 = a a ^ ... a a , a^a^ a_ (1.20
n n 1 x Q 2 z
If 5 is an irrational number, it is uniquely expressible by an infinite
non-periodic fraction to the base p of the form (1.2) (or (1.2') ). ·
If 5 is rational, it is expressible as an infinite periodic fraction to
the base p, e.g. the number 5 ·= 1/6 is written in the decimal system
as
5 = 0-1666... = 0-1(6).
In the binary system, 5 = 1/6 is expressed by the infinite fraction
s = o-ooioioi... = o-o (oi) = 1+-L + ....
Rational numbers to the base/? are numbers expressible as fractions
with denominator pk (k = 1, 2, —2·... ); each such number has two
forms in the system to the base p: one with 0 repeated, the other
with p — 1 repeated.
For example, the number 5 = — is written in the binary system as
5 = 0-1000 ... = 0-1 (0),
5 = 0-0111 ... = 0-0(1);
and in the decimal system as
5 = 0-5000... = 0-5(0),
5 = 0-4999... = 0-4(9).
Having selected one of these forms for rational numbers to the
base p, say the first — with 0 repeated, we obtain a unique form for
rational numbers to the base p as infinite periodic fractions to the
base p, and at the same time a unique form for every real number.
The elements of various /abased systems were to be found in antiquity in diffe­
rent nations, and traces have been preserved into modern times in certain
languages, e.g. of p = 12 (dozens and grosses), p = 20 (traces of this system have
4 MATHEMATICAL ANALYSIS
been preserved in French), p = 40 ("forty times forty" refers to number of chur­
ches in Moscow (Trans.)), etc. The system to the base 60 was well developed; it
originated in ancient Babylon (traces are retained in measurement of angles and
time). The 60-base system must have competed with the decimal in the Middle
Ages in the Near East and Central Asia. The decimal system originated in
India, was further developed in Central Asia, and passed from there into
Europe.
At the present time the binary system is widely used in computers
(together with the related systems having powers of two as base:
p = 2k, k > 1 an integer). A system of numeration to the base three
is used in the Moscow State University "Setun" computer. A set of
numbers different from 0, 1, 2, ..., p -1 is occasionally used for
the p digits of the system to the base p. For instance, a convenient
choice of digits in the system to the base 3 is -1, 0, 1. The digits
-1 and 1 can be used in the binary system.
Non-homogeneous positional systems are more general; here, the ratios of
units of different orders are different numbers. Such systems were used (before
the introduction of the metric system) for representing "denominate" numbers,
i.e. for representing magnitudes such as length, weight, etc. For example, the
following system of units was used for measuring weight in pre-revolutionary
Russia: 1 pud (« 16 kg) — 40 funtov, 1 funt («400 g ) « 32 lota, etc.
4. Sets of real numbers
We shall discuss various sets of real numbers — for example, the
set of natural numbers: 1, 2, 3, 4, ..., n, ..., the set of all proper
fractions, the set of all rational numbers, the set of all real numbers
between 0 and 1, etc.
The numbers are called the elements of the set in question.
One can consider sets of elements of any kind, and not merely sets of real
numbers. For instance, the set of points of a plane, the set of trees in a district,
etc. The elements of these sets are respectively points of a plane, trees, etc.
Sets are denoted in this book by capital letters: M, N, A, B, X,
Y, etc., or by the symbol {x }, where x are the elements of the set
n n
(countable sets). The set of numbers satisfying the inequalities
a< x< ò (a, b are numbers) is called an interval and is written as (a, b).
The sets of numbers satisfying the inequalities x<a,x>b, are called
THE ARITHMETICAL LINEAR CONTINUUM 5
infinite intervals and are written as (— «>, a) and (è, + <»)
respectively. The set of numbers* satisfying the inequalities a ^ x ^b
is called a segment (or c/aserf interval), and is written as [a, b ].
The sets of points x satisfying the inequalities
a ^ x < è, a < x ^ 6,
are called semi-intervals and are written respectively as [a, è), (a, b ].
The infinite semi-intervals ( — ©o, a] and [6, + <») are similarly
defined.
The interval (x — ε, x+ ε), (ε > 0) is called an ^-neighbourhood of
the point x.
If an element x belongs {does not belong) to the set X, this is written
symbolically as χζ Χ(χζ X or x $ X).
If all the elements of a set X are simultaneously elements of a set
7, X is said to be a MOsei of the set 7, and we write symbolically:
X cY. Otherwise, X is not a subset of 7, and we write this symbol­
ically as X c 7(or ^ φ 7). For example,
y 6(0, 1), a €[«,*), β?(β, 6), *€[*,*);
(0, 1) e [0, 1), [1, 2] e (0, 1), (a, b) e [a, 0).
The set M of all the elements that belong both to a set A and a
set 5 is called the intersection or product of the sets /I and 5, and is
written symbolically as
M=A(]B (M= ΑχΒ = Α.Β = ΑΒ).
For example,
(0, l] = T-y, lln(0, 2), 6 = (a, 6]n[é, c)
and so on.
The set M consisting of all the elements that belong either to a set
A or to a set B is called the union or sum of sets ^4 and B and is denoted
9
symbolically by
M = A{jB (M= A + B).
For example,
(0, 2)uHj, +-)=(0, +oo), (-3, 7]U(5, 8] = (-3, 8].
6 MATHEMATICAL ANALYSIS
The set M consisting of the elements of a set B that do not belong
to a set A is called the complement of the set A with respect to the set
B or the difference between the sets A and B, and is written symboli­
cally as
M = B\A (M = B-A).
For example,
(7, 8] = (5, 8]\(-3, 7], (0, 2)\|~0, ±\ = Γ-ί, 2^ etc.
The notation B\A is employed in more general cases.
5. Bounded sets, upper and lower bounds
A set X of real numbers is said to be bounded from above (below)
if there exists a number M (m), not less (not greater) than all the
numbers JCÇI The number M (m) is called an upper (lower) bound
of the set X.
A set A" is said to be bounded if it is bounded from above and below.
For example, the set ( — », 0) is bounded from above, the set
(0, + oo) bounded from below, while (0, 1) is a bounded set.
The least (greatest) of all the upper (lower) bounds of a set X is
called the strict upper (lower) bound M* (m*) of the set X and is
written symbolically as
M* = sup* (m* = inf x).
The numbers M* and m* possess the following properties:
(1) The inequalities hold for all x € X:
M* ^ x, m* -^ x.
(2) Whatever the number ε > 0, a number x £ X can be found
0
for which, respectively,
x ^ M* — ε, ;c ^ m* + e.
0 0
For example,
sup x = 0, inf x = 0.
«€(-«>, 0) *€(0,*)
(3) If the set X = {*} is bounded from above (below), it has a
strict upper (lower) bound.