Mathematical Analysis: Differentiation and Integration

330 Pages·6.452 MB·English

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

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.GUTER
L.A.LYUSTERNIK • I.L. RAUKHVARGER
M. I. SKANAVI • A. R.YANPOL’SKII
Translated by
H. MOSS
English edition edited by
I.N. SNEDDON
Simson Professor of Mathematics
University of Glasgow
PERGAMON PRESS
OXFORD • LONDON • EDINBURGH • NEW YORK
PARIS • FRANKFURT
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
Pergamon Press Ltd.
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 Mate-
maticheskaya Biblioteka under the editors!) ip of L. A. Ly u-
sternik and A. R. Yanpol’skii, and published by Fizmalgiz,
Moscow, 1961.
2073
FOREWORD
The 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 fix) 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 fiX), 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
Cn — Cn,G
vatives in the region G are defined and continuous
4i/(*o) the increment of the function at the point x0
the partial increment of a function
tffixo), At,fixo) the second and the nth difference at the point x0
, dy dfix) d
the derivative of a function
y’d-x’- * T ’dxf{x)'
Dfix), lift)
fLix 0),/«(*o) the first and the rth left-hand derivative at the
point x0
mx0),n\x0) the first and the rth right-hand derivative at the
point x0
d2y
fix o) = the second derivative of a function at point jc0
dx2 = *o
d"fjx) I
fix o) - the nth derivative of a function at point x0
dx?
, X A
rix0),f\x 0) the second and the nth differential derivatives of a
function at the point x0
f l'\x0) ,f\x 0),f"Kx0) Schwartzian derivatives at the point x0
£-fiX°) the partial derivative at the point A-0
dx,
IX
X NOTATION
P,AD) differential polynomial (polynomial of the opera­
tor D)
£(jt , yi, ■ ■ •. Xi.)
Jacobian
D(Xi, x2, .... X„)
3(xi, y , ■■■,>■„)
2
d(xi ,x2,..., X„)
dy the first differential
d2y. the second and the nth differential
df(x0), df(x0, A) the first differential at the point x0
d2f(x0),d2f(x0, h). the second and the nth differential at the point x0
d'Kxo), d"f<x0,h)
Ax the operator, defined in the set X of elements x
d </"
£> = —, D" = — the operator of differentiation
<4c
df{X°, H) the differential of the operator f(X) at the point X°
A*L the operator of partial increment in x,
hi
the operator of partial differentiation in x,
D‘- k
A, V2 Laplace’s operator
grad/(*°) the gradient of a function f{X) at the point X°
gradr the gradient of the function z
div a divergence of the vector a
rota curl of the vector a
<?(*, y) distance between the points X{x-, ,x2,...,x„) and
Y(yi,yz< ■■■,yn)
iWl the norm of a vector
^U» differential parameters of the first order
E,F,G Gauss’ coefficients
IU9 lVy LU ) IiD , IlMI Lamp’s coefficients
ds an element of length
dq, Aq an element of area for a plane
da, Aa an element of area for a surface
dv an element of volume
(R) \bfix) dx Riemann’s integral
J a
f fix) d(fix) Stieltjes’ integral
y - /w mapping from E„ into Em
NOTATION XI
S(X°,r) a sphere of radius r and centre at point X°
S
the sum sign of several analogous expressions
JWpO 1 functional of X-*)
Euler’s beta function
A*) Euler’s gamma function
IJ(x) pi-function
vW - r 'W
A A
n*) the logarithmic derivative of the pi-function
1 *2
?>(*) = -,= e 2
\J2n
Ei 2: exponential integral
li(*) logarithmic integral
Si (x) sine integral
Ci A) cosine integral
erf (*), tf>(jc) integrals in probability theory
A*, <p) elliptic integral of the first kind
E(k, <p) elliptic integral of the second kind
K-^f)
complete elliptic integral of the first kind
e-e(*'t) complete elliptic integral of the second kind
5W, 5”W Fresnel’s sine-integrals
C{x), C*(x) Fresnel’s cosine-integrals
C Euler’s constant
CHAPTER I
THE DIFFERENTIATION OF FUNCTIONS
OF ONE VARIABLE
The basic operations of mathematical analysis are operations on
functions which are mutually inverse — differentiation and integra­
tion. This chapter is devoted to the operation of the differentiation
of functions of one variable. The concepts of function, limiting
process, the properties of continuous functions and similar topics,
which precede the study of the operation of differentiation in ana­
lysis courses, are dealt with in volume 69 of the series in Pure
and Applied Mathematics called Mathematical Analysis (Functions,
Limits, Series, Continued Fractions) edited by L. A. Lyustemik and
A. R.Yanpol’skii (Pergamon Press, Oxford, 1965).
§ 1. Derivatives and Differentials of the First Order
1. Suppose that the function of one variable, y = fix), is defined
in the set X, which is a line segment or an open interval, or a semi­
open interval. Unless the contrary is stated, the points x0, x0 + h
are assumed to be interior points of the set X.
The derivative of function y = fix), defined in the set X, at
point x = x0 e X is the name given to the limit of the ratio
Ay = fjxo + h) - fjxq)
h h
(of the increment Ay of function y to the increment h of the argu­
ment x) when h -» 0, if this limit exists. The derivative of the func­
tion fix) at the point x = x0 is denoted by fix 0). Thus
fix0 + h) - fixo)
fix 0) = lim = lim = lim
(l.D
a-o h a-o h h
h-*0
2 MATHEMATICAL ANALYSIS
where xQ + he X, Ahf(x0) is the increment of the function /(x)
at the point x = x0.
If the point x0 is the end-point of a segment, the limit (1.1) deter­
mining the derivative is looked upon as one-sided; a right one, when
the point x0 is the left end of the segment; and a left one, when the
point *0 is its right end.
Geometrically, the derivative f'(x0) represents the tangent of the
angle which the tangent at the point x = x0 to the curve y = fix)
makes with the x-axis.
The ratio Ay/h might turn out to be infinitely large in the vicinity of the point
x = x0. If, for h # 0, the ratio Ay/h tends to infinity with a definite sign, it is
said that the function has an infinite derivative fix) = + or f'(x) =
at the given point. Geometrically, this means, that the curve y = /(x) has a
tangent parallel to the axis Oy at the point x = x0. For example, the curve
y = l/x has a vertical tangent at the point x0 = 0, since
2
ix 3 = + OO.
x = 0
1=0
The function /(x) is said to be differentiable at the point x = x0,
if it has a finite derivative at the point x0. If/(x) is differentiable at
every point x of the set X, it is said to be differentiable in the set X;
its derivative /'(x) is a function of the point x of the set X—the derived
function. The operation (or operator) of differentiation relates to the
function fix) its derivative f'{x); the initial function /(x) is called
the antiderivative with respect to its derivative.
Example 1. For the function f(x) = C (C is constant) the derivative equals
zero:
(cy = c.
Example 2. If fix) = x, the derivative equals 1:
to' - 1.
Example 3. The derivative of the function f{x) — sin x equals
sin (x + h) — sin x
(sin x)' = lim cos X.
h
A->o
To denote the derivative of the function y = fix) the following
symbols are also used: y', dyjdx, df(x)jdx, Df{x).
If the argument t denotes time, the derivative of the function u(t)
is also denoted by the symbol u = u(t) (Newton's notation). The
derivative u(t) means the rate of change of the quantity u{t) at the