Lectures on the Theory of Functions of a Complex Variable: Volume II: Geometric Theory

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Geometric Theory LECTURES ON THE THEORY OE FUNCTIONS OF A COMPLEX VARIABLE II GEOMETRIC THEORY PREFACE CONTENTS Chapter 9 APPLICATIONS OF GENERAL METRICS TO THE THEORY OF FUNCTIONS 9.1. - Topological considerations 9.2 - Conformal mapping 9.3 - Automorphisms of the extended plane 9.4 - Mobius geometry 9.5 - Hyperbolic geometry 9.6 - Elliptic and absolute geometry 9.7 - Blaschke’s theorems 9.8 - Schwarz’s lemma 9.9 - The theorem of Bloch Chapter 10 CONFORMAL MAPPING OF SIMPLY CONNECTED REGIONS 10.1 - The principle of symmetry 10.2 - Examples of conformal mapping 10.3 - The mapping of a polygon 10.4 - Functions related to the mapping of a square 10.5 - Riemann’s theorem Chapter 11 UNIVALENT FUNCTIONS 11.1 - Preliminary lemmas 11.2 - Distortion theorems 11.3 - Estimates of coefficients 11.4 - Lowner’s theory 11.5 - Applications of Lowner’s theory Chapter 12 ANALYTIC FUNCTIONS - RIEMANN SURFACES 12.1- Analytic continuation 12.2 - Analytic functions 12.3 - Algebraic functions 12.4-Riemann surfaces 12.5 - Classification of algebraic Riemann surfaces 12.6 - Uniformization 12.7 - Deformation of paths Chapter 13 AUTOMORPHIC FUNCTIONS 13.1 - Groups of linear transformations 13.2 - The fundamental domain 13.3 - Fuchsian groups 13.4 - Automorphic functions 13.5 - The Poincare theta series Chapter 14 THE SCHWARZIAN TRIANGLE FUNCTIONS AND THEIR INVERSES 14.1 - The mapping of a curvilinear polygon 14.2 - The Schwarzian triangles and their associated groups 14.3 - Inverses of the Schwarzian triangle functions 14.4 - Picard’s theorem and related theorems 14.5 - The elliptic modular function Chapter 15 LINEAR HOMOGENEOUS DIFFERENTIAL EQUATIONS 15.1 - General theory 15.2 - The theory of Fuchs 15.3 - Bessel functions 15.4 - Legendre’s functions 15.5 - Fuchsian equations 15.6 - Riemann’s equation Chapter 16 THE HYPERGEOMETRIC DIFFERENTIAL EQUATION 16.1 - The hypergeometric series 16.2 - Hypergeometric polynomials 16.3 - The hypergeometric series as functions of the parameters 16.4 - The fundamental system in the case that the third parameter is an integer 16.5 - Barnes’s contour integrals 16.6 - Conformal mapping 16.7 - Confluent hypergeometric functions 16.8 - Confluent hypergeometric polynomials INDEX