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

 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