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

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 Mentor P: 16,692 Author: L. I. Volkovyskii, G. L. Lunts, I. G. Aramanovich Title: A Collection of Problems on Complex Analysis Amazon link: http://www.amazon.com/Collection-Pro...dp/0486669130/ Table of Contents:  Foreword Complex numbers and functions of a complex variable 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) Conformal mappings connected with elementary functions 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 Supplementary geometrical questions. Generalised analytic functions Some properties of domains and their boundaries. Mappings of domains Quasi-conformal mappings. Generalised analytic functions Integrals and power series 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) Laurent series, singularities of single-valued functions. Integral functions 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) Various series of functions. Parametric integrals. Infinite products Series of functions Dirichlet series Parametric integrals (convergence of integrals; Laplace's integral) Infinite products Residues and their applications 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 Integrals of Cauchy type. The integral formulae of Poisson and Schwarz. Singular integrals 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 Analytic continuation. Singularities of many-valued character. Riemann surfaces Analytic continuation Singularities of many-valued character. Riemann surfaces Some classes of analytic functions with non-isolated singularities Conformal mappings (continuation) The Schwarz-Christoffel formula Conformal mappings involving the use of elliptic functions Applications to mechanics and physics Applications to hydrodynamics Applications to electrostatics Applications to the plane problem of heat conduction Answers and Solutions 
 P: 943 1510 probs on complex analysis. Answers to most of the computational problems, & also the occasional complete solution. Markushevich & Brown/Churchill are good references.

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