Elliptic Functions according to Eisenstein and Kronecker: (88 Grundlehren der Mathematischen Wissenschaften)

Drawn from the Foreword: (...) On the other hand, since much of the material in this volume seems suitable for inclusion in elementary courses, it may not be superfluous to point out that it is almost entirely self-contained. Even the basic facts about trigonometric functions are treated ab initio in Ch. II, according to Eisenstein's method. It would have been both logical and convenient to treat the gamma -function similarly in Ch. VII; for the sake of brevity, this has not been done, and a knowledge of some elementary properties of T(s) has been assumed. One further prerequisite in Part II is Dirichlet's theorem on Fourier series, together with the method of Poisson summation which is only a special case of that theorem; in the case under consideration (essentially no more than the transformation formula for the theta-function) this presupposes the calculation of some classical integrals. (...) As to the final chapter, it concerns applications to number theory (...).

Table of Contents:
I EISENSTEIN.- I Introduction.- II Trigonometric functions.- III The basic elliptic functions.- IV Basic relations and infinite products.- V Variation I.- VI Variation II.- II KRONECKER.- VII Prelude to Kronecker.- VIII Kronecker's double series.- IX Finale: Allegro con brio (Pell's equation and the Chowla-Selberg formula).- Index of Notations.

About the Author :
Biography of Andre Weil Andre Weil was born on May 6, 1906 in Paris. After studying mathematics at the Ecole Normale Superieure and receiving a doctoral degree from the University of Paris in 1928, he held professorial positions in India, France, the United States and Brazil before being appointed to the Institute for Advanced Study, Princeton in 1958, where he remained until he died on August 6, 1998. Andre Weil's work laid the foundation for abstract algebraic geometry and the modern theory of abelian varieties. A great deal of his work was directed towards establishing the links between number theory and algebraic geometry and devising modern methods in analytic number theory. Weil was one of the founders, around 1934, of the group that published, under the collective name of N. Bourbaki, the highly influential multi-volume treatise Elements de mathematique.

Review :
E. Hlwka, Monatshefte fur Mathematik, Bd. 83, 1977, Heft 3: "This book does not fit into any normal framework and is not, as one might think, a historical work. It is one that contains too personal thoughts of A. Weil and that will most certainly have a fundamental role to play in the history of mathematics. It is warmly recommended reading for anyone interested to learn what is happening in mathematics now."