Discontinuous Groups and Automorphic Functions
Discontinuous Groups and Automorphic Functions

Author: Joseph Lehner

Publisher: American Mathematical Soc.

Published: 1964-12-31

Total Pages: 440

ISBN-13: 0821815083

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Much has been written on the theory of discontinuous groups and automorphic functions since 1880, when the subject received its first formulation. The purpose of this book is to bring together in one place both the classical and modern aspects of the theory, and to present them clearly and in a modern language and notation. The emphasis in this book is on the fundamental parts of the subject. The book is directed to three classes of readers: graduate students approaching the subject for the first time, mature mathematicians who wish to gain some knowledge and understanding of automorphic function theory, and experts.

Non-Euclidean Geometry in the Theory of Automorphic Functions
Non-Euclidean Geometry in the Theory of Automorphic Functions

Author: Jacques Hadamard

Publisher: American Mathematical Soc.

Published: 1999-01-01

Total Pages: 116

ISBN-13: 9780821890479

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This is the English translation of a volume originally published only in Russian and now out of print. The book was written by Jacques Hadamard on the work of Poincare. Poincare's creation of a theory of automorphic functions in the early 1880s was one of the most significant mathematical achievements of the nineteenth century. It directly inspired the uniformization theorem, led to a class of functions adequate to solve all linear ordinary differential equations, and focused attention on a large new class of discrete groups. It was the first significant application of non-Euclidean geometry. This unique exposition by Hadamard offers a fascinating and intuitive introduction to the subject of automorphic functions and illuminates its connection to differential equations, a connection not often found in other texts.

Automorphic Functions
Automorphic Functions

Author: Lester R. Ford

Publisher: American Mathematical Soc.

Published: 2004

Total Pages: 360

ISBN-13: 9780821837412

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When published in 1929, Ford's book was the first treatise in English on automorphic functions. By this time the field was already fifty years old, as marked from the time of Poincare's early Acta papers that essentially created the subject. The work of Koebe and Poincare on uniformization appeared in 1907. In the seventy years since its first publication, Ford's Automorphic Functions has become a classic. His approach to automorphic functions is primarily through the theory of analytic functions. He begins with a review of the theory of groups of linear transformations, especially Fuchsian groups. He covers the classical elliptic modular functions, as examples of non-elementary automorphic functions and Poincare theta series. Ford includes an extended discussion of conformal mappings from the point of view of functions, which prepares the way for his treatment of uniformization. The final chapter illustrates the connections between automorphic functions and differential equations with regular singular points, such as the hypergeometric equation.

The Scientific Legacy of Poincare
The Scientific Legacy of Poincare

Author: Éric Charpentier

Publisher: American Mathematical Soc.

Published: 2010

Total Pages: 410

ISBN-13: 082184718X

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Henri Poincare (1854-1912) was one of the greatest scientists of his time, perhaps the last one to have mastered and expanded almost all areas in mathematics and theoretical physics. In this book, twenty world experts present one part of Poincare's extraordinary work. Each chapter treats one theme, presenting Poincare's approach, and achievements.

Discontinuous Groups of Isometries in the Hyperbolic Plane
Discontinuous Groups of Isometries in the Hyperbolic Plane

Author: Werner Fenchel

Publisher: Walter de Gruyter

Published: 2011-05-12

Total Pages: 389

ISBN-13: 3110891352

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This is an introductory textbook on isometry groups of the hyperbolic plane. Interest in such groups dates back more than 120 years. Examples appear in number theory (modular groups and triangle groups), the theory of elliptic functions, and the theory of linear differential equations in the complex domain (giving rise to the alternative name Fuchsian groups). The current book is based on what became known as the famous Fenchel-Nielsen manuscript. Jakob Nielsen (1890-1959) started this project well before World War II, and his interest arose through his deep investigations on the topology of Riemann surfaces and from the fact that the fundamental group of a surface of genus greater than one is represented by such a discontinuous group. Werner Fenchel (1905-1988) joined the project later and overtook much of the preparation of the manuscript. The present book is special because of its very complete treatment of groups containing reversions and because it avoids the use of matrices to represent Moebius maps. This text is intended for students and researchers in the many areas of mathematics that involve the use of discontinuous groups.

Non-Euclidean Geometry in the Theory of Automorphic Functions
Non-Euclidean Geometry in the Theory of Automorphic Functions

Author: Jacques Hadamard

Publisher: American Mathematical Soc.

Published: 1999

Total Pages: 109

ISBN-13: 0821820303

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"This unique exposition by Hadamard offers a fascinating and intuitive introduction to the subject of automorphic functions and illuminates its connection to differential equations, a connection not often found in other texts."--Jacket.

A Short Course in Automorphic Functions
A Short Course in Automorphic Functions

Author: Joseph Lehner

Publisher: Courier Corporation

Published: 2015-01-21

Total Pages: 162

ISBN-13: 0486789748

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Concise treatment covers basics of Fuchsian groups, development of Poincaré series and automorphic forms, and the connection between theory of Riemann surfaces with theories of automorphic forms and discontinuous groups. 1966 edition.

Kleinian Groups and Uniformization in Examples and Problems
Kleinian Groups and Uniformization in Examples and Problems

Author: Samuil Le_bovich Krushkal_

Publisher: American Mathematical Soc.

Published: 1986-12-31

Total Pages: 214

ISBN-13: 9780821898123

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Aimed at researchers, graduate students and undergraduates alike, this book presents a unified exposition of all the main areas and methods of the theory of Kleinian groups and the theory of uniformization of manifolds. The past 20 years have seen a rejuvenation of the field, due to the development of powerful new methods in topology, the theory of functions of several complex variables, and the theory of quasiconformal mappings. Thus this new book should provide a valuable resource, listing the basic facts regarding Kleinian groups and serving as a general guide to the primary literature, particularly the Russian literature in the field. In addition, the book includes a large number of examples, problems, and unsolved problems, many of them presented for the first time.