The first English edition of this magnificent textbook, translated from Russian, was published in three substantial volumes of 459, 347, and 374 pages, respectively. In this second English edition all three volumes have been put together with a new, combined index and bibliography. Some corrections and revisions have been made in the text, primarily in Volume II. Volumes II and III contain numerous references to the earlier volumes, so that the reader is reminded of the exact statements (and proofs) of the more elementary results made use of. The three-volume-in-one format makes it easy to flip back the pages, refresh one's memory, and proceed. The proofs chosen are those that give the student the best 'feel' for the subject. The watchword is clarity and straightforwardness. The author was a leading Soviet function-theorist: It is seldom that an expert of his stature puts himself so wholly at the service of the student. This book includes over 150 illustrations and 700 exercises.
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.