A course of modern analysis : an introduction to the general theory of infinite processes and of analytic functions with an account of the principal transcendental functions / E.T. Whittaker and G.N. Watson ; fifth edition edited and prepared for publication by Victor H. Moll.

Format:

Book

Author/Creator:

Whittaker, E. T. (Edmund Taylor), 1873-1956, author.

Watson, G. N. (George Neville), 1886-1965, author.

Contributor:

Moll, Victor H., 1956- editor.

Language:

English

Subjects (All):

Series, Infinite.

Harmonic analysis.

Functions.

Physical Description:

1 online resource (liii, 668 pages) : digital, PDF file(s).

Edition:

Fifth edition.

Place of Publication:

Cambridge : Cambridge University Press, 2021.

Summary:

This classic work has been a unique resource for thousands of mathematicians, scientists and engineers since its first appearance in 1902. Never out of print, its continuing value lies in its thorough and exhaustive treatment of special functions of mathematical physics and the analysis of differential equations from which they emerge. The book also is of historical value as it was the first book in English to introduce the then modern methods of complex analysis. This fifth edition preserves the style and content of the original, but it has been supplemented with more recent results and references where appropriate. All the formulas have been checked and many corrections made. A complete bibliographical search has been conducted to present the references in modern form for ease of use. A new foreword by Professor S.J. Patterson sketches the circumstances of the book's genesis and explains the reasons for its longevity. A welcome addition to any mathematician's bookshelf, this will allow a whole new generation to experience the beauty contained in this text.

Contents:

Cover

Half-title

Frontispiece

Title page

Copyright information

Contents

Foreword

Preface to the Fifth Edition

Preface to the Fourth Edition

Preface to the Third Edition

Preface to the Second Edition

Preface to the First Edition

Introduction

Part I The Process of Analysis

1 Complex Numbers

1.1 Rational numbers

1.2 Dedekind's theory of irrational numbers

1.3 Complex numbers

1.4 The modulus of a complex number

1.5 The Argand diagram

1.6 Miscellaneous examples

2 The Theory of Convergence

2.1 The definition of the limit of a sequence

2.11 Definition of the phrase 'of the order of'

2.2 The limit of an increasing sequence

2.21 Limit-points and the Bolzano-Weierstrass theorem

2.22 Cauchy's theorem on the necessary and sufficient condition for the existence of a limit [120, p. 125]

2.3 Convergence of an infinite series

2.31 Dirichlet's test for convergence

2.32 Absolute and conditional convergence

2.33 The geometric series, and the series [sum[sup(infty)sub(n=1)]frac[sup(1)sub(n[sup(x)])]]

2.34 The comparison theorem

2.35 Cauchy's test for absolute convergence

2.36 D'Alembert's ratio test for absolute convergence

2.37 A general theorem on series for which [lim[sub(n[rightarrow]infty)]|frac[u[sub(n)+1]][u[sub(n)]|=1]

2.38 Convergence of the hypergeometric series

2.4 Effect of changing the order of the terms in a series

2.41 The fundamental property of absolutely convergent series

2.5 Double series

2.51 Methods of summing a double series

2.52 Absolutely convergent double series

2.53 Cauchy's theorem on the multiplication of absolutely convergent series

2.6 Power series

2.61 Convergence of series derived from a power series

2.7 Infinite products

2.71 Some examples of infinite products

2.8 Infinite determinants.

2.81 Convergence of an infinite determinant

2.82 The rearrangement theorem for convergent infinite determinants

2.9 Miscellaneous examples

3 Continuous Functions and Uniform Convergence

3.1 The dependence of one complex number on another

3.2 Continuity of functions of real variables

3.21 Simple curves. Continua

3.22 Continuous functions of complex variables

3.3 Series of variable terms. Uniformity of convergence

3.31 On the condition for uniformity of convergence

3.32 Connexion of discontinuity with non-uniform convergence

3.33 The distinction between absolute and uniform convergence

3.34 A condition, due to Weierstrass, for uniform convergence

3.35 Hardy's tests for uniform convergence

3.4 Discussion of a particular double series

3.5 The concept of uniformity

3.6 The modified Heine-Borel theorem

3.61 Uniformity of continuity

3.62 A real function, of a real variable, continuous in a closed interval, attains its upper bound

3.63 A real function, of a real variable, continuous in a closed interval, attains all values between its upper and lower bounds

3.64 The fluctuation of a function of a real variable

3.7 Uniformity of convergence of power series

3.71 Abel's theorem

3.72 Abel's theorem on multiplication of convergent series

3.73 Power series which vanish identically

3.8 Miscellaneous examples

4 The Theory of Riemann Integration

4.1 The concept of integration

4.11 Upper and lower integrals

4.12 Riemann's condition of integrability

4.13 A general theorem on integration

4.14 Mean-value theorems

4.2 Differentiation of integrals containing a parameter

4.3 Double integrals and repeated integrals

4.4 Infinite integrals

4.41 Infinite integrals of continuous functions. Conditions for convergence

4.42 Uniformity of convergence of an infinite integral.

4.43 Tests for the convergence of an infinite integral

4.44 Theorems concerning uniformly convergent infinite integrals

4.5 Improper integrals. Principal values

4.51 The inversion of the order of integration of a certain repeated integral

4.6 Complex integration

4.61 The fundamental theorem of complex integration

4.62 An upper limit to the value of a complex integral

4.7 Integration of infinite series

4.8 Miscellaneous examples

5 The Fundamental Properties of Analytic Functions

Taylor's, Laurent's and Liouville's Theorems

5.1 Property of the elementary functions

5.11 Occasional failure of the property

5.12 Cauchy's definition of an analytic function of a complex variable

5.13 An application of the modified Heine-Borel theorem

5.2 Cauchy's theorem on the integral of a function round a contour

5.21 The value of an analytic function at a point, expressed as an integral taken round a contour enclosing the point

5.22 The derivatives of an analytic function f(z)

5.23 Cauchy's inequality for f[sup((n))](a)

5.3 Analytic functions represented by uniformly convergent series

5.31 Analytic functions represented by integrals

5.32 Analytic functions represented by infinite integrals

5.4 Taylor's theorem

5.41 Forms of the remainder in Taylor's series

5.5 The process of continuation

5.51 The identity of two functions

5.6 Laurent's theorem

5.61 The nature of the singularities of one-valued functions

5.62 The 'point at infinity'

5.63 Liouvillle's theorem

5.64 Functions with no essential singularities

5.7 Many-valued functions

5.8 Miscellaneous examples

6 The Theory of Residues

Application to the Evaluation of Definite Integrals

6.1 Residues

6.2 The evaluation of definite integrals.

6.21 The evaluation of the integrals of certain periodic functions taken between the limits 0 and 2

6.22 The evaluation of certain types of integrals taken between the limits -[infty] and +[infty]

6.23 Principal values of integrals

6.24 Evaluation of integrals of the form [int[sup(infty)sub(0)]x[sup(a-1)]Q(x)dx]

6.3 Cauchy's integral

6.31 The number of roots of an equation contained within a contour

6.4 Connexion between the zeros of a function and the zeros of its derivative

6.5 Miscellaneous examples

7 The Expansion of Functions in Infinite Series

7.1 A formula due to Darboux

7.2 The Bernoullian numbers and the Bernoullian polynomials

7.21 The Euler-Maclaurin expansion

7.3 Bürmann's theorem

7.31 Teixeira's extended form of Bürmann's theorem

7.32 Lagrange's theorem

7.4 The expansion of a class of functions in rational fractions

7.5 The expansion of a class of functions as infinite products

7.6 The factor theorem of Weierstrass

7.7 The expansion of a class of periodic functions in a series of cotangents

7.8 Borel's theorem

7.81 Borel's integral and analytic continuation

7.82 Expansions in series of inverse factorials

7.9 Miscellaneous examples

8 Asymptotic Expansions and Summable Series

8.1 Simple example of an asymptotic expansion

8.2 Definition of an asymptotic expansion

8.21 Another example of an asymptotic expansion

8.3 Multiplication of asymptotic expansions

8.31 Integration of asymptotic expansions

8.32 Uniqueness of an asymptotic expansion

8.4 Methods of summing series

8.41 Borel's method of summation [85, p. 97-115]

8.42 Euler's method of summation [85, 201]

8.43 Cesàro's method of summation [141]

8.44 The method of summation of Riesz [559]

8.5 Hardy's convergence theorem

8.6 Miscellaneous examples.

9 Fourier Series and Trigonometric Series

9.1 Definition of Fourier series

9.11 Nature of the region within which a trigonometrical series converges

9.12 Values of the coefficients in terms of the sum of a trigonometrical series

9.2 On Dirichlet's conditions and Fourier's theorem

9.21 The representation of a function by Fourier series for ranges other than (-[pi],[pi])

9.22 The cosine series and the sine series

9.3 The nature of the coefficients in a Fourier series

9.31 Differentiation of Fourier series

9.32 Determination of points of discontinuity

9.4 Fejér's theorem

9.41 The Riemann-Lebesgue lemmas

9.42 The proof of Fourier's theorem

9.43 The Dirichlet-Bonnet proof of Fourier's theorem

9.44 The uniformity of the convergence of Fourier series

9.5 The Hurwitz-Liapounoff theorem concerning Fourier constants

9.6 Riemann's theory of trigonometrical series

9.61 Riemann's associated function

9.62 Properties of Riemann's associated function

Riemann's first lemma

9.63 Riemann's theorem on trigonometrical series

9.7 Fourier's representation of a function by an integral

9.8 Miscellaneous examples

10 Linear Differential Equations

10.1 Linear differential equations

10.2 Solution of a differential equation valid in the vicinity of an ordinary point

10.21 Uniqueness of the solution

10.3 Points which are regular for a differential equation

10.31 Convergence of the expansion of 10.3

10.32 Derivation of a second solution in the case when the difference of the exponents is an integer or zero

10.4 Solutions valid for large values of |z|

10.5 Irregular singularities and confluence

10.6 The differential equations of mathematical physics

10.7 Linear differential equations with three singularities

10.71 Transformations of Riemann's P-equation.

10.72 The connexion of Riemann's P-equation with the hypergeometric equation.

Notes:

Title from publisher's bibliographic system (viewed on 06 Aug 2021).

Other Format:

Print version:

ISBN:

1-009-00859-5

1-009-00409-3

OCLC:

1281960119