Mathematical Analysis: Differentiation and Integration

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I. G. Aramanovich, R.S.Guter, L.A.Lyusternik, I.L. Raukhvarger, M. I. Skanavi, A. R.Yanpol'skii

MATHEMATICAL ANALYSIS Differentiation and Integration

MATHEMATICAL ANALYSIS Differentiation and Integration I. G. ARAMANOVICH • R.S .G U TER L.A.LYUSTERNIK • I.L. RAUKHVARGER M. I. SKANAVI • A. R.YANPOL’SKII

Translated by H. MOSS

English edition edited by I.N. S N E D D O N Simson Professor o f Mathematics University o f Glasgow

PERG AM ON PRESS O X FO R D • L O N D O N • E D IN B U R G H • N EW Y O RK PA R IS • F R A N K F U R T

Pergamon Press Ltd., Headington Hill Hall, Oxford 4 & 5 Fitzroy Square, London W. 1 Pergamon Press (Scotland) Ltd., 2& 3Teviot Place, Edinburgh 1 Pergamon Press Inc., 122 East 55th St., New York 22, N.Y. Pergamon Press GmbH, Kaiserstrasse 75, Frankfurt-am-Main

Copyright © 1965 P er g a m o n P ress L t d .

First English Edition 1965

Library of Congress Catalog Card No. 64-8051 This book is a translation of MameMamynecKuH anaAua (Matematicheskii analiz) in the series Spravochnaya Matematicheskaya Biblioteka under the editors!) ip of L. A. Ly usternik and A. R. Yanpol’skii, and published by Fizmalgiz, Moscow, 1961.

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FOREWORD T he present volume of the series in Pure and Applied Mathe­

matics is devoted to two basic operations of mathematical analysis — differentiation and integration. It discusses the complex of problems directly connected with the operations of differentiation and integra­ tion of functions of one or several variables, in the classical sense, and also elementary generalizations of these operations. Further generalizations will be given in subsequent volumes of the series, volumes devoted to the theory of functions of real variables and to functional analysis. Together with an earlier volume in the series, volume 69, L. A. Lyusternik and A. R. Yanpol’skii, Mathematical Analysis (Functions, Limits, Series, Continued Fractions), the present one includes material for a course of mathematical analysis, which is treated in a logically connected manner, briefly and without proofs, but with many examples worked in detail. Chapter I “The differentiation of functions of one variable” (authors: L. A.Lyusternik and R.S.Guter) and Chapter II “The differentiation of functions of n variables” (author: L. A. Lyusternik) contain a discussion of derivatives and differentials, their properties and their application in investigating the behaviour of functions, Taylor’s formula and series, differential operators and their elemen­ tary properties, stationary points, and also the extrema of functions of one variable (author: I.G.Aramanovich) and of n variables (author: I.L.Raukhvarger). Chapter III “Composite and implicit functions of n variables” (authors: R.S.Guter and I.L.Raukhvarger) contains a discussion of general problems of the theory of functions of n variables in connection with differentiation. Here belong composite and implicit functions, the representation of functions in the form of super­ positions, etc. A separate section (author: V. A. Trenogin) is devoted to Newton’s diagram. In view of the particular importance of functions of two and three variables in their application to problems of analysis, they are vii

viii

FOREWORD

separated out to form a chapter on their own. Chapter IV “ Systems of functions and curvilinear coordinates in a plane and in space” (author: M.I. Skanavi), where a detailed description is given of the properties of mappings of one region into another (in particular, affine mappings) and of different systems of curvilinear coordinates. This chapter (as also Chapter VII) is based on the book by A. F. Bermant [2]. Chapter V “The integration of functions” (authors: R. S. Guter, I. L. Raukhvarger and A. R. Yanpol’skii) contains a discussion of the properties of integrals, methods of integrating elementary functions and the application of integrals to geometrical and mechanical problems. Certain generalizations of the concept of an integral are dealt with in Chapter VI, “Improper integrals; integrals depending on a para­ meter; the integral of Stieltjes” (authors: I.G.Aramanovich, R. S. Guter and I.L. Raukhvarger). Here, a detailed account is given of improper integrals and their properties, the concept of Stieltjes’ integral is given, and also of integrals and derivatives of fractional order. In Chapter VII, “The transformation of differential and integral expressions” (author: M.I.Skanavi) the classification is given of various cases of transformation of the expression named in the heading of the chapter, general rules for the change of variables in the differential and the integral expressions are laid down, and a summary is given of expressions for the basic differential operations (gradient, divergence, curl, Laplacian) in the transformation of rectangular cartesian coordinates to various curvilinear orthogonal coordinates (compiled by V. I. Bityutzkov). Here also the discussion of surface integrals is systematized, and Green’s formulae with various generalizations are given. In the appendixes there are given tables of derivatives of the first and the nth order of elementary functions, the expansion of func­ tions into power series and of integrals (indefinite, definite and multiple). Tables are also to be found of special functions, functions defined by means of integrals of elementary functions (elliptic integrals, integral functions, Fresnel integrals, gamma-functions, etc.).

NOTATION M

operator of multiplication by the argument

C = C[X]

the class of functions fix ) defined and continuous in the set X

c„ - C,Pf]

the class of functions f i x ) defined and continuously differentiable n times in the set X

Cq

the class of functions fiX ) defined and continuous in the region G

O — C1>c

the class of functions f i X ) , all of whose first partial derivatives in the region G are defined and con­ tinuous the class of functions, all of whose nth partial deri­ vatives in the region G are defined and continuous

C n — C n ,G

the increment of the function at the point x 0

4i/(*o)

the partial increment of a function the second and the nth difference at the point x0

t f f i x o), At,fixo) , dy

d fix )

d

y ’d-x’ - * T ’ dxf{x)'

the derivative of a function

D fix ), lift) fL ix 0) ,/« (* o )

the first and the rth left-hand derivative at the point x 0

m x 0) , n \ x 0)

the first and the rth right-hand derivative at the point x 0

f i x o) =

d 2y dx2

f i x o) -

d"fjx) I dx? , X

the second derivative of a function at point jc0

= *o

the nth derivative of a function at point x 0 A

r i x 0) , f \ x 0)

the second and the nth differential derivatives of a function at the point x0

f l' \ x 0) , f \ x 0) ,f " K x 0)

Schwartzian derivatives at the point x 0

£ -fiX ° ) dx,

the partial derivative at the point A-0 IX

NOTATION

X

differential polynomial (polynomial of the opera­ tor D)

P,AD) £ ( j t , y i , ■■•. Xi.) D(Xi, x 2, .... X„)

Jacobian

3 (x i, y 2 , ■■■, >■„) d(xi , x 2, . . . , X„) dy d 2y . d f(x 0), d f(x 0, A) d 2f( x 0) , d 2f ( x 0, h). d 'K x o), d"f