Abstract
VOL. 1 of the present treatise is a revised edition of Appell and Goursat's well-known “Theorie des fonctions algebriques et de leurs integrates”. The second volume, contributed by the late M. Pierre Fatou, is an introductory but fairly comprehensive treatise on automorphic functions. One of the most valuable features in the early part of this volume is a new and remarkably simple proof of the theorem, that a group of real linear transformations which has no infinitesimal transformation is properly discontinuous. A determination of the fundamental domains of Fuchsian and Kleinian groups follows, and then the chief known properties of Fuchsian and Kleinian functions, and an account of the theta functions. The only example of a group considered in detail, however, is, as in many previous treatises on automorphic functions, the modular one. No insuperable difficulty exists in the way of extending this list considerably. The whole treatise is written in clear and masterly style; it forms an admirable introduction to the classical work of Klein and Poincare.
Théorie des fonctions algébriques et de leurs intégrales.
Paul
Appell
Prof.
Édouard
Goursat
Par. Deuxième édition, revue et augmentée par Pierre Fatou. Tome 1: Étude des fonctions analytiques sur une surface de Riemann. Pp. xxxv + 526. Tome 2: Théorie des fonctions algébriques d'une variable et des transcendantes qui sy rattachent; fonctions automorphes. Pp. xiv + 521. (Paris: Gauthier-Villars et Cie, 1930.) 200 francs.
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Théorie des fonctions algébriques et de leurs intégrales . Nature 129, 600 (1932). https://doi.org/10.1038/129600a0
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DOI: https://doi.org/10.1038/129600a0