Author(s): Giovanni Sansone, Johan C. H. Gerretsen
Publisher: P. Noordhoff
Year: 1969
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
