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. The isometric circle, a concept introduced at an early stage, proves to be a valuable tool.
The third chapter is devoted to "Fuchsian Groups," and the fourth to some of the fundamental properties of automorphic functions.
In the fifth chapter existence theorems are established by means of the Poincaré theta series, and some properties of the theta functions are proved.
The sixth chapter, "The Elementary Groups," contains also a treatment of inversion in a sphere and of stereographic projection, as well as an interesting discussion of the regular solids.
Chapter VII, "The Elliptic Modular Functions," contains the definitions and some of the properties of the functions J(τ) and λ(τ) and a discussion of their relation to each other.
Chapter VIII gives a thorough treatment of conformal mapping, including the mapping of a plane simply connected region on a circle (with particular attention to the behaviour of the mapping function on the boundary), the mapping of limit regions, and of simply connected finite-sheeted regions. These results are applied in Chapters IX and X to problems in uniformization.
The text closes with a brief discussion of the relation between the theory of automorphic functions and certain parts of the theory of linear homogeneous differential equations of the second order.
Not the least valuable part of the book is an excellent bibliography containing more than three hundred titles. The arrangement is chronological, but it is easy to locate all the references to a particular author, if desired, since the Author Index refers to the bibliography as well as to the text itself.
Fred W Perkins