Analysis A Collection of Problems on Complex Analysis by Volkovyskii, Lunts, Aramanovich

[micromass]

• Author: L. I. Volkovyskii, G. L. Lunts, I. G. Aramanovich
• Title: A Collection of Problems on Complex Analysis
• Amazon link: https://www.amazon.com/dp/0486669130/?tag=pfamazon01-20

Table of Contents:

[LIST]
[*] Foreword
[*] Complex numbers and functions of a complex variable
[LIST]
[*] Complex numbers (complex numbers; geometrical interpretation; stereographic projection; quaternions)
[*] Elementary transcendental functions
[*] Functions of a complex variable (complex functions of a real variable; functions of a complex variable; limits and continuity)
[*] Analytic and harmonic functions (the Cauchy-Riemann equations; harmonic functions; the geometrical meaning of the modulus and argument of a derivative)
[/LIST]
[*] Conformal mappings connected with elementary functions
[LIST]
[*] Linear functions (linear functions; bilinear functions)
[*] Supplementary questions of the theory of linear transformations (canonical forms of linear transformations; some approximate formulae for linear transformations; mappings of simply connected domains; group properties of bilinear transformations ; linear transformations and non-Euclidean geometry)
[*] Rational and algebraic functions (some rational functions; mappings of circular lunes and domains with cuts; the function 1/2 (z + 1/z ); application of the principle of symmetry; the simplest non-schlicht mappings)
[*] Elementary transcendental functions (the fundamental transcendental functions; mappings leading to mappings of the strip and half-strip; the application of the symmetry principle; the simplest many-sheeted mappings)
[*] Boundaries of univalency, convexity and starlikeness
[/LIST]
[*] Supplementary geometrical questions. Generalised analytic functions
[LIST]
[*] Some properties of domains and their boundaries. Mappings of domains
[*] Quasi-conformal mappings. Generalised analytic functions
[/LIST]
[*] Integrals and power series
[LIST]
[*] The integration of functions of a complex variable 
[*] Cauchy's integral theorem
[*] Cauchy's integral formula
[*] Numerical series
[*] Power series (determination of the radius of convergence; behaviour on the boundary; Abel's theorem)
[*] The Taylor series (the expansion of functions in Taylor series; generating functions of systems of polynomials; the solution of differential equations)
[*] Some applications of Cauchy's integral formula and power series (Cauchy's inequalities; area theorems for univalent functions; the maximum principle; zeros of analytic functions; the uniqueness theorem; the expression of an analytic function in terms of its real or imaginary part)
[/LIST]
[*] Laurent series, singularities of single-valued functions. Integral functions
[LIST]
[*] Laurent series (the expansion of functions in Laurent series; some properties of univalent functions)
[*] Singular points of single-valued analytic functions (singular points; Picard's theorem; power series with singularities on the boundary of the circle of convergence) 
[*] Integral functions (order; type; indicator function)
[/LIST]
[*] Various series of functions. Parametric integrals. Infinite products
[LIST]
[*] Series of functions
[*] Dirichlet series
[*] Parametric integrals (convergence of integrals; Laplace's integral)
[*] Infinite products
[/LIST]
[*] Residues and their applications
[LIST]
[*] The calculus of residues
[*] The evaluation of integrals (the direct application of the theorem of residues; definite integrals; integrals connected with the inversion of Laplace's integral; the asymptotic behaviour of integrals)
[*] The distribution of zeros. The inversion of series (Rouche's theorem; the argument principle; the inversion of series)
[*] Partial fraction and infinite product expansions. The summation of series
[/LIST]
[*] Integrals of Cauchy type. The integral formulae of Poisson and Schwarz. Singular integrals
[LIST]
[*] Integrals of Cauchy type
[*] Some integral relations and double integrals
[*] Dirichlet's integral, harmonic functions, the logarithmic potential and Green's function
[*] Poisson's integral, Schwarz's formula, harmonic measure
[*] Some singular integrals
[/LIST]
[*] Analytic continuation. Singularities of many-valued character. Riemann surfaces
[LIST]
[*] Analytic continuation
[*] Singularities of many-valued character. Riemann surfaces
[*] Some classes of analytic functions with non-isolated singularities
[/LIST]
[*] Conformal mappings (continuation)
[LIST]
[*] The Schwarz-Christoffel formula
[*] Conformal mappings involving the use of elliptic functions
[/LIST]
[*] Applications to mechanics and physics
[LIST]
[*] Applications to hydrodynamics
[*] Applications to electrostatics
[*] Applications to the plane problem of heat conduction
[/LIST]
[*] Answers and Solutions
[/LIST]

 

[fourier jr]

1510 probs on complex analysis. Answers to most of the computational problems, & also the occasional complete solution. Markushevich & Brown/Churchill are good references.