It really is a gem, both in terms of its table of contents and the level of discussion. The exercises also look very good. --Clifford Earle, Cornell University This book has a soul and has passion. --William Abikoff, University of Connecticut This classic book gives an excellent presentation of topics usually treated in a complex analysis course, starting with basic notions (rational functions, linear transformations, analytic function), and culminating in the discussion of conformal mappings, including the Riemann mapping theorem and the Picard theorem. The two quotes above confirm that the book can be successfully used as a text for a class or for self-study.
Rolf Nevanlinna and Veikko Paatero
Chapter 1 The Concept of an Analytic Function §1 The complex numbers §2 Point sets in the complex plane §3 Functions of a complex variable Chapter 2 General Properties of Rational Functions §1 The n-th power §2 Polynomials §3 Rational functions Chapter 3 Linear Transformations §1 Basic properties of linear transformations §2 Mapping problems §3 Stereographic projection Chapter 4 Mapping by Rational Functions of Second Order Chapter 5 The Exponential Function and its Inverse. The General Power §1 Definition and basic properties of the exponential function §2 Mapping by means of the exponential function. The logarithm §3 The general power Chapter 6 The Trigonometric Functions §1 The sine and cosine §2 The tangent and the cotangent §3 The mappings given by the functions tan z and cot z. Their inverse functions §4 The mappings given by the functions sin z and cos z. The functions arc sin z and arc cos z §5 Survey of the Riemann surfaces of the elementary functions Chapter 7 Infinite Series with Complex Terms §1 General theorems §2 Power series Chapter 8 Integration in the Complex Domain. Cauchy’s Theorem §1 Complex line integrals §2 The primitive function §3 Cauchy’s theorem §4 The general formulation of Cauchy’s theorem Chapter 9 Cauchy’s Integral Formula and its Applications §1 Cauchy’s formula §2 The Taylor expansion of an analytic function §3 Consequences of Cauchy’s integral formula §4 The Laurent expansion §5 Isolated singularities of an analytic function §6 The inverse of an analytic function §7 Mapping by a rational function §8 Normal Families Chapter 10 The Residue Theorem and its Applications §1 The residue theorem §2 Application of the residue theorem to the evaluation of definite integrals §3 The partial-fraction expansion of cot ttz §4 The argument principle §5 Applications of the argument principle Chapter 11 Harmonic Functions §1 Preliminary considerations §2 Gauss’s mean-value theorem. The maximum and minimum principles §3 Poisson’s formula §4 The harmonic measure §5 The Dirichlet problem §6 Harnack’s principle Chapter 12 Analytic Continuation §1 The principle of analytic continuation §2 The monodromy theorem §3 The inverse of a rational function §4 Harmonic continuation. The reflection principle Chapter 13 Entire Functions §1 Infinite products §2 Product representation of the function w sin nz §3 The Weierstrass factorization theorem §4 Jensen’s formula. The growth of entire functions Chapter 14 Periodic Functions §1 Definitions of simply and doubly periodic functions §2 Reduction of simply periodic functions to the exponential function §3 The basic properties of doubly periodic functions §4 The Weierstrass ℘-function §5 The Weierstrass ζ- and cr-functions §6 Representation of doubly periodic functions by means of the σ-function §7 The differential equation of the function ℘(z) §8 Representation of doubly periodic functions as rational functions of ℘ and ℘’ §9 Addition theorem for doubly periodic functions §10 Determination of a doubly periodic function with prescribed principal parts §11 Mapping by a doubly periodic function of order 2 §12 Elliptic integrals
Chapter 15 The Euler Γ- Function §1 Definition of the Γ-function §2 Stirling’s formula §3 The product representation of the r-function Chapter 16 The Riemann ζ-Function §1 Definition and the Euler product formula §2 Integral representation of the ζ-function §3 Analytic continuation of the ζ-function §4 Riemann’s functional equation §5 The zeros of the ζ-function and the distribution of prime numbers Chapter 17 The Theory of Conformal Mapping §1 The Riemann mapping theorem §2 Construction of the solution §3 Boundary correspondence under conformal mapping §4 The connection between conformal mapping and the Dirichlet problem §5 The conformal mapping of polygons §6 Triangle functions §7 The Picard theorem Index