Theory of Functions of a Complex Variable

Abstract

Volume I, Part 1: Basic Concepts: I.1 Introduction I.2 Complex numbers I.3 Sets and functions. Limits and continuity I.4 Connectedness. Curves and domains I.5. Infinity and stereographic projection I.6 Homeomorphisms Part 2: Differentiation. Elementary Functions: I.7 Differentiation and the Cauchy-Riemann equations I.8 Geometric interpretation of the derivative. Conformal mapping I.9 Elementary entire functions I.10 Elementary meromorphic functions I.11 Elementary multiple-valued functions Part 3: Integration. Power Series: I.12 Rectifiable curves. Complex integrals I.13 Cauchy's integral theorem I.14 Cauchy's integral and related topics I.15 Uniform convergence. Infinite products I.16 Power series: rudiments I.17 Power series: ramifications I.18 Methods for expanding functions in Taylor series Volume II, Part 1: Laurent Series. Calculus of Residues: II.1 Laurent's series. Isolated singular points II.2 The calculus of residues and its applications II.3 Inverse and implicit functions II.4 Univalent functions Part 2: Harmonic and Subharmonic Functions: II.5 Basic properties of harmonic functions II.6 Applications to fluid dynamics II.7 Subharmonic functions II.8 The Poisson-Jensen formula and related topics Part 3: Entire and Meromorphic Functions: II.9 Basic properties of entire functions II.10 Infinite product and partial fraction expansions Volume III, Part 1: Conformal Mapping. Approximation Theory: III.1 Conformal mapping: rudiments III.2 Conformal mapping: ramifications III.3 Approximation by rational functions and polynomials Part 2: Periodic and Elliptic Functions: III.4 Periodic meromorphic functions III.5 Elliptic functions: Weierstrass' theory III.6 Elliptic functions: Jacobi's theory Part 3: Riemann Surfaces. Analytic Continuation: III.7 Riemann surfaces III.8 Analytic continuation III.9 The symmetry principle and its applications Bibliography Index.