Some years before his death, which occurred in 1929, the brilliant analyst P Fatou undertook the preparation of a revised edition of Appell and Goursat's Fonctions Algébriques. Fatou was invited by Professor Goursat to add to the original edition a treatment of automorphic functions. He responded by writing a comprehensive exposition of that subject, which it was found desirable to publish in a separate volume. Thus, Chapters 13 to 16 of the Fonctions Algébriques, which make up the second volume of that work, really form an independent treatise, due entirely to Fatou.
Chapter 13 begins with a study of the elementary properties of linear transformations of a complex variable z. With a free use of projective geometry, a knowledge of which is assumed, groups of linear transformations of z are shown to be isomorphic with groups of motions in non-euclidean spaces. Fatou uses Montel's theory of normal families of functions to obtain a new and very simple proof of the theorem that a group of real linear transformations which has no infinitesimal transformation is properly discontinuous. It is also proved that every group which has no infinitesimal transformation induces a properly discontinuous group of transformations of three-dimensional space.
Chapter 14 deals with the determination of the fundamental domains of Fuchsian and of Kleinian groups, and with the construction of such groups through the stipulation, in advance, of their fundamental domains. Chapter 15 presents the chief known properties of Fuchsian and of Kleinian functions. The theta series are studied in detail. Chapter 16 gives an interesting account of conformal mapping and uniformization.
Fatou's style is rapid, but clear. One sees that he insisted everywhere on complete rigor, and that he refrained from presenting conjectures as facts. His masterly treatise will sustain, and encourage the advancement of, the elegant theory of Schwarz, Schottky, Klein, and Poincaré.