Automorphic Functions

Front Cover
American Mathematical Soc., 1951 - Mathematics - 333 pages
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When published in 1929, Ford's book was the first treatise in Englishon automorphic functions. By this time the field was already fifty yearsold, as marked from the time of Poincare's early Acta papers thatessentially created the subject. The work of Koebe and Poincare onuniformization appeared in 1907. In the seventy years since its firstpublication, Ford's Automorphic Functions has become a classic. Hisapproach to automorphic functions is primarily through the theory ofanalytic functions. He begins with a review of the theory of groups oflinear transformations, especially Fuchsian groups.
  

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Truly a classic. Some of the terminology and methods are a bit outdated, but an excellent reference nonetheless.

Contents

Chapter
1
Symbolic Notation
4
The Fixed Points of the Transformation
6
The Linear Transformation and the Circle
8
Inversion in a Circle
10
The Multiplier K
15
The Hyperbolic Transformation K A
18
The Elliptic Transformation K ei9
19
The Automorphic Functions
144
The Groups with Two Limit Points 62 Determination of the Groups
146
Chapter VII
148
Change of the Primitive Periods
150
The Function Jr
151
Behavior of Jt at the Parabolic Points
153
Further Properties of Jt
155
The Function Xr
157

The Loxodromic Transformation K Aeie
20
The Parabolic Transformation
21
The Isometric Circle
23
The Unit Circle
30
Chapter II
33
Properly Discontinuous Groups
35
Transforming a Group
36
The Fundamental Region
37
The Isometric Circles of a Group
39
The Limit Points of a Group
41
Definition of the Region R
44
The Boundary of R
47
Example A Finite Group
49
Generating Transformations
50
Cyclic Groups
51
The Formation of Groups by the Method of Combination
56
Ordinary Cycles
59
Parabolic Cycles
62
Function Groups
64
Chapter III
67
The Region R and the Region R0
69
Generating Transformations
71
The Cycles
72
Fuchsian Groups of the First and Second Kinds
73
Fixed Points at Infinity Extension of the Method
75
Examples
78
The Modular Group
79
Some Subgroups of the Modular Group
81
Chapter IV
83
Simple Automorphic Functions
86
Behavior at Vertices and Parabolic Points
88
The Poles and Zeros
91
Algebraic Relations
94
Differential Equations
98
Chapter V
102
The Convergence of the Series
104
The Convergence for the Fuchsian Group of the Second Kind
106
Some Properties of the Theta Functions
108
Zeros and Poles of the Theta Functions
112
Series and Products Connected with the Group
115
Chapter VI
117
Stereographic Projection
119
Rotations of the Sphere
120
Groups of the Regular Solids
123
A Study of the Cube
124
The General Regular Solid
127
Determination of All the Finite Groups
129
The Extended Groups
136
The Groups with One Limit Point 59 The Simply and Doubly Periodic Groups
139
Groups Allied to the Periodic Groups
140
The Relation between Xt and Jt
159
Further Properties of Xr
161
Chapter VIII
164
Schwarzs Lemma
165
Area Theorems
167
The Mapping of a Circle on a Plane Finite Region
169
The Deformation Theorem for the Circle
171
A General Deformation Theorem
175
An Application of Poissons Integral
177
The Mapping of a Plane Simply Connected Region on a Circle The Iterative Process
179
The Convergence of the Process
183
The Behavior of the Mapping Function on the Boundary
187
Regions Bounded by Jordan Curves
198
Analytic Arcs and the Continuation of the Mapping Function across the Boundary
201
Circular Arc Boundaries
202
The Mapping of Combined Regions
203
The Mapping of Limit Regions
205
The Mapping of Simply Connected Finitesheeted Regions
213
Conformal Mapping and Groups of Linear Transformations
216
Chapter IX
220
The Connectivity of Regions
221
Algebraic Functions of Genus Zero Uniformization by Means of Rational Functions
229
Algebraic Functions of Genus Greater than Zero Uniformiza tion by Means of Automorphic Functions
233
The Genus of the Fundamental Region of a Group
238
The Cases p 1 and p 1
239
More General Fuchsian Uniformizing Functions
241
The Case p 0
245
Whittakers Groups
247
The Transcendental Functions
249
Chapter X
256
Some Accessory Functions
258
The Mapping of a Multiply Connected Region of Planar Char ante on a Slit Region
262
Application to the Uniformization of Algebraic Functions
266
A Convergence Theorem
267
The Sequence of Mapping Functions
271
The Linearity of Tn
273
An Extension
278
The Mapping of a Multiply Connected Region of Planar Character on a Region Bounded by Complete Circles
279
Chapter XI
284
The Inverse of the Quotient of Two Solutions
287
Regular Singular Points of Differential Equations
293
The Quotient of Two Solutions at a Regular Singular Point
296
Equations with Rational Coefficients
299
The Equation with Two Singular Points
303
The RiemannSchwarz Triangle Functions
305
Equations with Algebraic Coefficients
308
A Bibliography of Automorphic Functions
311
Author Index
325
Subject Index
327
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