## Automorphic FunctionsWhen 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 Some of the terminology and methods are a bit outdated, but an excellent reference nonetheless.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 |

327 | |

### Common terms and phrases

algebraic function angle automorphic with respect belonging boundary point bounded branch points circle Q0 cluster point congruent points congruent sides consider contains converges cross-cut curve cycle cyclic group domain of existence elliptic transformation exterior finite number finite plane fixed circle fixed points fn(z following theorem Fuchsian group fundamental region Hence inequality inner point integer interior point inverse isometric circles joining limit points linear transformation loop-cut loxodromic lying mapped conformally mapping function neighborhood one-to-one ordinary point origin pair parabolic points plane region point at infinity points congruent principal circle properly discontinuous radius rational function real axis Riemann surface rotation sequence sheets simple automorphic function simply connected simply connected region single single-valued solutions sphere square elements takes tion trans transformation carrying triangle unit circle upper half plane vertex vertices whence z-plane