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


by micromass
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micromass
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Feb3-13, 05:00 PM
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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
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fourier jr
fourier jr is offline
#2
Feb5-13, 03:48 PM
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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