Lectures on the Theory of Functions of a Complex Variable, Vol. 2: Geometric Theory

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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