This comprehensive collection contains over 1,500 problems on the theory of functions of the complex variable and covers nearly every branch of classical function theory. It will be of special interest to practicing engineers and researchers in the physical sciences, for considerable attention is given to those problems which illustrate the application of the theory of functions of a complex variable to problems dealing with the mechanics of continuous media and electrical engineering.
The problems themselves range in difficulty from elementary to those encountered only in more advanced treatments.
The first four chapters involve complex numbers and functions of a complex variable; conformal mappings connected with elementary functions; supplementary geometrical questions and generalized analytic functions; and integrals and power series.
Chapters V through VIII cover the Laurent series, singularities of single-valued functions, and integral functions; various series of functions, parametric integrals, and infinite products; residues and their applications; integrals of the Cauchy type;and integral formulae of Poisson and Schwarz.
The final three chapters discuss analytic continuation, singularities of many-valued character, and Riemann Surfaces; conformal mappings; and applications to mechanics and physics. Answers and solutions are grouped at the end of the text.
Unabridged Dover (1991) republication of the edition published by Pergamon Press, Oxford, England, 1965
Author(s): L. I. Volkovyskii, G. L. Lunts, I. G. Aramanovich
Series: Dover Books on Mathematics
Edition: Revised ed.
Publisher: Dover Publications
Year: 1991
Language: English
Pages: C, X, 426
FOREWORD
CHAPTER I COMPLEX NUMBERS AND FUNCTIONS OF A COMPLEX VARIABLE
§1. Complex numbers
§2. Elementary transcendental functions
§3. Functions of a complex variable
§4. Analytic and harmonic functions
CHAPTER II CONFORMAL MAPPINGS CONNECTED WITH ELEMENTARY FUNCTIONS
§ 1. Linear functions
§ 2. Supplementary questions of the theory of linear transformations
§ 3. Rational and algebraic functions
§ 4. Elementary transcendental functions
§ 5. Boundaries of univalency, convexity and starlikeness
CHAPTER Ill SUPPLEMENTARY GEOMETRICAL QUESTIONS. GENERALISED ANALYTIC FUNCTIONS
§ 1. Some properties of domains and their boundaries. Mappings of domains
§ 2. Quasi-conformal mappings. Generalised analytic functions
CHAPTER IV INTEGRALS AND POWER SERIES
§ 1. The integration of functions of a complex variable
§ 2. Cauchy's integral theorem
§ 3. Cauchy's integral formula
§ 4. Numerical series
§ 5. Power series
§ 6. The Taylor series
§ 7. Some applications of Cauchy's integral formula and power series
CHAPTER V LAURENT SERIES, SINGULARITIES OF SINGLE-VALUED FUNCTIONS. INTEGRAL FUNCTIONS
§ 1. Laurent series
§ 2. Singular points of single-valued analytic functions
§ 3. Integral functions
CHAPTER VI VARIOUS SERIES OF FUNCTIONS. PARAMETRIC INTEGRALS. INFINITE PRODUCTS
§ 1. Series of functions
§ 2. Dirichlet seriest
§ 3. Parametric integrals
§ 4. Infinite products
CHAPTER Vll RESIDUES AND THEIR APPLICATIONS
§ 1. The calculus of residues
§ 2. The evaluation of integrals
§ 3. The distribution of zeros. The inversion of series
§ 4. Partial fraction and infinite product expansions. The summation of series
CHAPTER VIII INTEGRALS OF CAUCHY TYPE. THE INTEGRAL FORMULAE OF POISSON AND SCHWARZ. SINGULAR INTEGRALS
§ 1. Integrals of Cauchy type
§ 2. Some integral relations and double integrals
§ 3. Dirichlet´s integral, harmonic functions, the logarithmic potential and Green´s function
§ 4. Poisson's integral, Schwarz's formula, harmonic measure
§ 5. Some singular integrals
CHAPTER IX ANALYTIC CONTINUATION. SINGULARITIES OF MANY-VALUED CHARACTER. RIEMANN SURFACES
§ 1. Analytic continuation
§ 2. Singularities of many-valued character. Riemann surfaces
§ 3. Some classes of analytic functions with non-isolated singularities
CHAPTER X CONFORMAL MAPPINGS (CONTINUATION)
§ 1. The Schwarz-Christoffel formula
§ 2. Conformal mappings involving the use of elliptic functions
CHAPTER XI APPLICATIONS TO MECHANICS AND PHYSICS
§ 1. Applications to hydrodynamics
§ 2. Applications to electrostatics
§ 3. Applications to the plane problem of heat conduction
ANSWERS AND SOLUTIONS
CHAPTER I
CHAPTER II
CHAPTER III
CHAPTER IV
CHAPTER V
CHAPTER VI
CHAPTER VII
CHAPTER VIII
CHAPTER IX
CHAPTER X
CHAPTER XI
