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 OXFORD · LONDON · EDINBURGH · NEW YORK PARIS · FRANKFURT PERGAMON PRESS LTD. Headington Hill Hall, Oxford 4 and 5 Fitzroy Square, London W.l| PERGAMON PRESS (SCOTLAND) LTD. 2 and 3 Teviot Place, Edinburgh 1 PERGAMON PRESS INC. 122 East 55th Street, New York 22, N.Y. GAUTHIER-VILLARS ED. 55 Quai des Grands-Augustins, Paris 6 PERGAMON PRESS G.m.b.H. Kaiserstrasse 75, Frankfurt am Main * 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/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 xb, 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.

Mathematical Analysis : Functions, Limits, Series, Continued Fractions.

Similar Mathematical Analysis : Functions, Limits, Series, Continued Fractions.