Description
Book SynopsisIncludes over 150 illustrations and 700 exercises.
Table of ContentsVolume 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.
