Theory of Functions of a Complex Variable, Volume Two

Title......Page 1
Copyright......Page 2
TRANSLATOR\'S PREFACE......Page 3
EDITOR\'S PREFACE......Page 5
CONTENTS......Page 7
PART SIX: FOUNDATIONS OF GEOMETRIC FUNCTION THEORY......Page 10
Functions of Bound One (§ 281 )......Page 11
Unit Functions (§§ 282.285)......Page 12
G. Pick\'s Theorem (§§ 286-289)......Page 14
The Derivative of a Bounded Function (§§ 290-291 )......Page 18
A Distortion Theorem (§ 292)......Page 20
Jensen\'s Theorem (§ 293)......Page 22
Julia\'s Theorem (§§ 295.297)......Page 23
The Angular Derivative (§§ 298-299)......Page 28
Further Properties of the Angular Derivative (§ 300)......Page 32
The Angular Derivative and the Zero8 of a Function (§§ 301.304)......Page 34
An Application of the Poisson Integral (§§ 305·306 )......Page 38
Continuity of Bounded Functions under Approach Within a Sector (§§ 307.308)......Page 39
Two Theorems of E. Lindellif (§ 309)......Page 42
Fatou\'s Theorem (§§ 310-311)......Page 43
Generalization of Poisson\'s Int~gral (§ 312)......Page 49
The Theorem of F. and M. Riesz (§ 313)......Page 50
Elementary Mappings (§ 314)......Page 52
The Riemann Mapping Theorem for Bounded Regions (§§ 315·323 )......Page 54
The Koebe-Faber Distortion Theorem (§§ 324-329)......Page 62
The Mapping of Doubly-Connected Regions (§l§ 330-331 )......Page 69
The Group of Congruence Transformations (§§ 332-335)......Page 73
The Universal Covering Surface (§§ 336-338)......Page 80
Simultaneous Conformal Mapping of Nested Annular Regions (§§ 339·340)......Page 84
The Schwarz Reflection Principle (§\'§ 341·342)......Page 88
Conformal Mapping of the Region In8ide a Jordan Curve (§.§ 343·346)......Page 92
The Concept of Free Boundary Arc (§§ 347-348 )......Page 99
Reflection in Analytic Curves (§ 349)......Page 101
The Mapping of the Frontier in the Neighborhood of a Free Analytic Boundary Arc (§ 350)......Page 102
The Mapping at a Corner (§§ 351-353 )......Page 104
Milloux\'s Theorem (§ 354)......Page 107
PART SEVEN: THE TRIANGLE FUNCTIONS AND PICARD\'S THEOREM......Page 112
Definition of an Analytic Function of Two Variables (§§ 355-356)......Page 113
Proof of Hartogs\' 11teorem for Bounded Functions (§§ 357.358)......Page 114
Proof of Hartogs\' Theorem in the General Case (§§ 359.361)......Page 116
Properties of the Function F(x, y) (§§ 362-363)......Page 119
Analytic Functions of More Than Two Variables (§§ 364-365)......Page 120
Some Theorems on Dift\'erential Equations (§§ 366-368)......Page 123
Systems of Difterential Equations (§ 369)......Page 127
The Schwarzian Derivative (§§ 370-374)......Page 129
Reduction to the Hypergeometric Differential Equation (§§ 375.377)......Page 135
The Hypergeometric Series (§§ 378-380)......Page 138
Calculation of F(\\alpha, \\beta, \\ghama; 1) (§ 381)......Page 141
Kummer\'s Differential Equation (§§ 382-383)......Page 143
The Twenty-Four Integrals of Kummer (§ 384)......Page 148
The Fundamental Solutions of the Hypergeometric Differential Equation (§ 385)......Page 149
The Exceptional Cases and the Function F*(a, P, Y; z) (§§ 386-388)......Page 151
Connecting Formulas (§§ 389·391 )......Page 155
Explicit Calculation of the Mapping Function of a Circular-Arc Triangle (§ 392)......Page 162
The Derivative of the Mapping Function (§ 393 )......Page 164
The Case \\lambda = 0 (§ 394)......Page 165
Calculation of the Derivative in the Case \\lambda = 0 (§ 395)......Page 167
Tables for the Hypergeometric Differential Equation......Page 168
Description of the Schwarz Triangles (§§ 396400)......Page 173
Schwarz Triangles with a Right Angle (§ 401)......Page 181
Schwarz Triangles with Cusps (§ 402)......Page 183
The Modular Configuration (§ 403 )......Page 184
Conformal Mapping of the Schwarz Triangle Nets (§ 404)......Page 185
The Inverse Function \\Omega (w) as an Automorphic Function (§ 405)......Page 186
The Modular Function (§§ 4(6407)......Page 187
Landau\'s Theorem and Picard\'s First Theorem (§§ 411.413)......Page 195
Schottky\'s Theorem (§§ 414-415 )......Page 199
Montel\'s Theorem (§ 416 )......Page 201
Picard\'s Seeond Theorem (§ 417)......Page 202
A Corollary of Montel\'s Theorem (§ 418)......Page 203
A Generalization (§ 419)......Page 204
The Essential Singularities of Meromorphic Functions (§§ 420.424)......Page 205
An Application to the Reflection Principle (§ 425 )......Page 211
Examples (§ 426)......Page 212
An Application to Measurable Real Functions (§§ 427-428)......Page 214
INDEX......Page 218
INDEX......Page 219