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

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  1. Feb 3, 2013 #1


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    Table of Contents:
    Code (Text):

    [*] 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
    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. Feb 5, 2013 #2
    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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