Table Of ContentA COURSE OF
HIGHER
MATHEMATICS
V. I. Smirnov
Volume III Part 2
COMPLEX
VARIABLES
- SPECIAL
FUNCTIONS
A D I W E S I N T E R N A T IO N A L SE RI ES
IN MATHEMATICS
A. J. Lohwater
Consulting Editor
A COURSE OF
Higher Mathematics
VOLUME III
PART TWO
V. I. SMIRNOV
Translated by
D. E. BROWN
Translation edited by
I. N. SNEDDON
Simson Professor in Mathematics
University of Glasgow
PERGAMON PRESS
OXFORD • LONDON • EDINBURGH • NEW YORK
PARIS • FRANKFURT
ADDISON-WESLEY PUBLISHING COMPANY, INC.
READING, MASSACHUSETTS • PALO ALTO ■ LONDON
1964
Copyright (c) 1964
PERGAMON PRESS LTD.
U. S. A. Edition distributed by
ADDISON-WESLEY PUBLISHING COMPANY, INC.
Reading, Massachusetts • Palo A ho ■ London
PERGAMON PRESS
International Series of Monographs in
PURE AND APPLIED MATHEMATICS
Volume 60
Library of Congress Catalog Card No. 63-10134
This translation has been made from
the Russian Edition of
V. I. Smirnov’s book
Kypc ebicuieu MameMamuKU (Kurs vysshei maternal iki),
published in 1957 by Fizmalgiz, Moscow
HADE IN GREAT BRITAIN
CONTENTS
Introduction ix
Foreword to the Fourth Edition x
CHAPTER I
THE BASIS OF THE THEORY OF FUNCTIONS
OF A COMPLEX VARIABLE 1
1. Functions of a complex variable 2. The derivative. 3. Conformal
transformation. 4. The integral. 5. Cauchy’s theorem. 6.The fundamen
tal formula of the integral calculus. 7. Cauchy’s formula. 8. Integrals of
Cauchy’s type. 9. Corollaries of Cauchy’s formula. 10. Isolated sin
gularities. 11. Infinite series with complex terms. 12. The Weierstrass
theorem. 13. Power series. 14. Taylor’s series. 15. Laurent’s series.
16. Examples. 17. Isolated singularities. Point at infinity. 18. Analytic
continuation. 19. Examples of many-valued functions. 20. Singularities
of analytical functions and Riemann surfaces. 21. The theorem of
residues. 22. Theorem on the number of zeros. 23. The inversion of a
power series. 24. The principle of symmetry. 25. Taylor’s series on the
circumference of the circle of convergence. 26. The principal value of
an integral. 27. The principal value of an integral (continuation).
28. Integrals of Cauchy’s type.
CHAPTER II
CONFORMAL TRANSFORMATION
AND THE TWO-DIMENSIONAL FIELD 120
29. Conformal transformation. 30. Linear transformation. 31. Bilinear
transformation. 32. The function w—zz.33. The function w=lc{z-\-\jz)l2
34. The biangular figure and the strip. 35. The fundamental theorem.
36. Christoffel’s formula. 37. Individual cases. 38. The exterior of the
polygon. 39. The minimum property of the transformation into a circle.
40. The method of conjugate trigonometric series. 41. The two-dimen
sional established flow of liquids. 42. Examples. 43. The problem of
flow round a contour. 44. N. E. Zhukovskij’s formula. 45. The two-
dimensional problem of electrostatics. 46. Examples. 47. The two-
dimensional magnetic field. 48. Schwarz’s formula. 49. The core cot
(«—<)/2. 50. Limiting problems. 51. The biharmonic equation. 52.The
wave equation and analytio functions. 53. The fundamental theorem.
54. The diffraction of a two-dimensional wave. 55. The reflection of
elastic waves from rectilinear objects.
vi
CONTENTS
CHAPTER III
THE APPLICATION OF THE THEORY OF RESIDUES,
INTEGRAL AND FRACTIONAL FUNCTIONS 223
56. Fresnel’s integral. 57. Integration of expressions containing trigono
metric functions. 58. Integration of a rational fraction. 59. Certain
new types of integrals containing trigonometric functions. 60. Jordan’s
lemma. 61. Contour integrals of certain functions. 62. Examples of
integrals of many-valued functions. 63. Integration of a system of
linear equations with constant coefficients. 64. The expansion of a
fractional function into partial fractions. 65. The function cot z. 66. The
construction of meromorphic functions. 67. Integral functions. 68. Infi
nite products. 69. The construction of an integral function from its
given zeros. 70. Integrals which depend on parameters. 71. Euler’s
integral of the second class. 72. Euler’s integral of the first class.
73. The infinite product of the function [F(z)]“ *. 74. The representation
of r(z) by a contour integral. 75. Stirling’s formula. 76. Euler’s summa
tion formula. 77. Bernoulli numbers. 78. Method of the steepest
descent. 79. Isolation of the principal part of an integral. 80. Examples.
CHAPTER IV
FUNCTIONS OF SEVERAL VARIABLES AND MATRIX
FUNCTIONS 313
81. Regular functions of several variables. 82. The double integral and
Cauchy’s formula. 83. Power series. 84. Analytic continuation. 85. Mat
rix functions. Preliminary propositions. 86. Power series of one matrix.
87. Multiplication of power series. Conversion of power series. 88. Fur
ther investigations of convergence. 89. Interpolation polynomials.
90.Cayley’s identity and Sylvester’s formula. 91. Analytic continuation.
92. Examples of many-valued functions. 93. Systems of linear equ
ations with constant coefficients. 94. Functions of several matrices.
CHAPTER V
LINEAR DIFFERENTIAL EQUATIONS 357
95. The expansion of a solution into a power series. 96. The analytic
continuation of the solution. 97. The neighbourhood of a singularity.
98. Regular singularity. 99. Equations of Fuchs’s class. 100. The Gauss
equation 101. The hypergeometric series. 102. The Legendre poly
nomials. 103. Jacobian polynomials. 104. Conformal transformation
and the formula of Gauss. 105. Irregular singularities. 106. Asymptotic
expansion. 107. The Laplace transformation. 108. The choice of solu
tions. 109. The asymptotic representation of solutions. 110. Compari
son of results. 111. The Bessel equation. 112. The Hankel function.
113. The Bessel functions. 114. The Laplace transformation in more
general cases. 115. The generalized Laguerre polynomials. 116. Positive
Vll
CONTENTS
values of the parameter. 117. The degeneration of the equation of Gauss.
118. Equations with periodic coefficients. 119. The case of analytic
coefficients. 120. Systems of linear differential equations. 121. Regular
singularities. 122. Regular systems. 123. The form of the solution in the
neighbourhood of a singularity. 124. Canonical solutions. 125. The
connection with regular solutions of Fuchs’s type. 126. The case of the
arbitrary Us. 127. Expansion in the neighbourhood of an irregular sin
gularity. 128. Expansions into uniformly convergent series.
CIIAPTER VJ
SPECIAL FUNCTIONS
§ 1. Spherical functions 493
129. The determination of spherical functions. 130. The definite expres
sion for spherical functions. 131. The orthogonal properties. 132. The
Legendre polynomials. 133. The expansion in terms of spherical func
tions. 134. Proof of convergence. 135. The connection between spherical
functions and limit problems. 136. The Dirichlet and Neumann problems.
137. The potential of voluminous masses. 138. The potential of a
spherical shell. 139. The electron in a central field. 140. Spherical
functions and the linear representation of rotating groups. 141. The
Legendre function. 142. The Legendre functions of the second kind.
§ 2. Bessel functions 537
143. The determination of Bessel functions. 144. Relationships between
Bessel functions. 145. The orthogonality of Bessel functions and their
zeros. 146. Converting function and integral representation. 147. The
Fourier—Bessel formula. 148. The Hankel and Neumann functions. 149.
The expansion of the Neumann function with an integer subscript. 150.
The case of the purely imaginary argument. 151. Integral representation
152. The asymptotic representation of Hankel functions. 153. Bessel
functions and the Laplace equation. 154. The wave equation in cylindri
cal coordinates. 155. The wave equation in spherical coordinates.
§ 3. The Hermitian and Laguerre polynomials 584
156. The linear oscillator and the Hermitian polynomial. 157. Ortho
gonality. 158. The conversion function. 159. Parabolic coordinates and
Hermitian functions. 160.The Laguerre polynomials. 161. The connection
between Hermitian and Laguerre polynomials. 162. The asymptotic
expression for Hermitian polynomials. 163. The asymptotic expression
for Legendre polynomials.
§ 4. Elliptic integrals and elliptic functions 604
164. The transformation of elliptic integrals into normal form. 165.
The conversionof the integrals into a trigonometric form. 166. Examples.
167. The conversion of elliptic integrals. 168. General properties of
elliptic functions. 169. Fundamental lemma. 170. The Weierstrass
CONTENTS
function. 171. The differential equation for J)(m) (608). 172. The func
tions ak(u). 173. The expansion of a periodic integral function. 174. The
new notation. 175. The function #,(«). 176. The function &k{v). 177. The
properties of theta-functions. 178. An expression for the numbers ek in
terms of &s. 179. The elliptic Jacobian functions. 180. The fundamental
properties of Jacobian functions. 181. The differential equation for
Jacobian functions. 182. Addition formulae. 183. The connection be
tween the functions ty(u) and sn(tt). 184. Elliptic coordinates. 185. The
introduction of elliptic functions. 186. The Lam6 equation. 187. The
simple pendulum. 188. An example of conformal transformation.
SUPPLEMENT
THE CONVERSION OF MATRICES INTO THE
CANONICAL FORM 670
189. Auxiliary hypothesis. 190. The case of simple zeros. 191. The first
stage of the transformation in the case of repeated zeros. 192. Convers
ion into the canonical form. 193. The determination of the structure of
the canonical form. 194. Examples.
Index 697
INTRODUCTION
Some observations on the aims and history of Prof. Smirnov’s five-
volume course on higher mathematics have been made in the Intro
duction to the first volume of the present English edition.
In the present volume which forms the second part of Vol. Ill the
author comes to the discussion of one of the central subjects of modern
pure mathematics, an understanding of which is essential also to
engineers and physicists — the theory of functions of a complex
variable. Prof. Smirnov’s approach is classical and for that reason is an
excellent introduction for those students who will go on to study the
more modern developments of the theory of functions while at the
same time presenting a complete picture of those aspects of the
theory (conformal mappings, differential equations in the complex
plane, calculus of residues) which are of most direct interest to applied
mathematicians. An interesting feature of the book is that it was the first
textbook at this level to include a chapter on the theory of functions
of several complex variables, a subject which forms an essential tool
to workers in quantum field theory.
Special functions continue to play an important part in the edu
cation of both pure and applied mathematicians. This is recognized
by Prof. Smirnov. He follows his lucid account of the general theory of
functions with a full treatment of those properties of special functions
which are necessary for the proper understanding of mathematical
physics and engineering.
Both sides of the coin are engraved with the author’s characteristic
style and the result is a work of great interest which has already been
acknowledged as a classic in the countries of Eastern Europe and is
now whole heartedly commended to students in the English-speaking
world.
I. N. Sneddon
FOREWORD TO THE FOURTH EDITION
In this edition Volume III was divided into two parts. The second
part contains material from the former Volume III, beginning with
the chapter dealing with the principles of the theory of the functions
of the complex variable. It was slightly rearranged and new material
was added. This new material is related mainly to the treatise of the
integrals of Cauchy’s type and to the approximate calculation of in
tegrals by the method of the steepest descent.
I was greatly assisted in my treatment of the latter by Professor
G. J. Petrashen’, to whom I am deeply grateful.
References to the first part of Volume III are indicated, for
example: [IIIj, 44].
V. Smirnov
