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Elliptic Functions According to Eisenstein and Kronecker (Classics in Mathematics)by Andre Weil
Synopses & ReviewsPublisher Comments:"As a contribution to the history of mathematics, this is a model of its kind. While adhering to the basic outlook of Eisenstein and Kronecker, it provides new insight into their work in the light of subsequent developments, right up to the present day. As one would expect from this author, it also contains some pertinent comments looking into the future. It is not however just a chapter in the history of our subject, but a wideranging survey of one of the most active branches of mathematics at the present time. The book has its own very individual flavour, reflecting a sort of combined EisensteinKroneckerWeil personality. Based essentially on Eisenstein's approach to elliptic functions via infinite series over lattices in the complex plane, it stretches back to the very beginnings on the one hand and reaches forward to some of the most recent research work on the other. (...) The persistent reader will be richly rewarded."A. Fröhlich, Bulletin of the London Mathematical Society, 1978
Synopsis: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 selfcontained. 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 thetafunction) this presupposes the calculation of some classical integrals. (...) As to the final chapter, it concerns applications to number theory (...).
About the AuthorBiography of André Weil André Weil was born on May 6, 1906 in Paris. After studying mathematics at the École Normale Supérieure 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. André 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 multivolume treatise Eléments de mathématique.
Table of ContentsPart 1 Eisenstein: Introduction. Trigonometric functions. The basic elliptic functions. Basic relations and infinite products. Variation I. Variation II. Part 2 Kronecker: Prelude to Kronecker. Kronecker's double series. Finale: Allegro con brio (Pell's equation and the ChowlaSelberg formula). Index of Notations.
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