Automorphic Functions

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 Poincaré's early Acta papers that essentially created the subject. The work of Koebe and Poincaré 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 Poincaré 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.

This book is the first containing an extensive systematic treatment of the theory of automorphic functions in English. The author has succeeded in presenting this difficult subject in a manner which makes it accessible to those who are familiar with the fundamentals of the theory of functions of a complex variable. ... The first chapter, entitled 'Linear Transformations,' and the second, 'Groups of Linear Transformations,' give an especially clear treatment of topics which are of great importance in later work. --Fred W. Perkins, American Mathematical Monthly

The exposition is remarkably clear and explicit ... a very simple and elegant treatment of groups of linear transformations, their fundamental regions and the functions invariant under the groups ... Professor Ford's work, in the first part of his book, is not mere exposition. The methods which he creates are original, and of permanent scientific value ... The second half of the book gives a detailed account of conformal mapping and uniformization, and considers some of the relations of automorphic functions to differential equations. --Bulletin of the AMS