Euler through time: a new look at old themes (AMS 2006)

V. S. Varadarajan0821835807, 9780821835807

Euler is one of the greatest and most prolific mathematicians of all time. He wrote the first accessible books on calculus, created the theory of circular functions, and discovered new areas of research such as elliptic integrals, the calculus of variations, graph theory, divergent series, and so on. It took hundreds of years for his successors to develop in full the theories he began, and some of his themes are still at the center of today’s mathematics. It is of great interest therefore to examine his work and its relation to current mathematics. This book attempts to do that. In number theory the discoveries he made empirically would require for their eventual understanding such sophisticated developments as the reciprocity laws and class field theory. His pioneering work on elliptic integrals is the precursor of the modern theory of abelian functions and abelian integrals. His evaluation of zeta and multizeta values is not only a fantastic and exciting story but very relevant to us, because they are at the confluence of much research in algebraic geometry and number theory today (Chapters 2 and 3 of the book). Anticipating his successors by more than a century, Euler created a theory of summation of series that do not converge in the traditional manner. Chapter 5 of the book treats the progression of ideas regarding divergent series from Euler to many parts of modern analysis and quantum physics. The last chapter contains a brief treatment of Euler products. Euler discovered the product formula over the primes for the zeta function as well as for a small number of what are now called Dirichlet $L$-functions. Here the book goes into the development of the theory of such Euler products and the role they play in number theory, thus offering the reader a glimpse of current developments (the Langlands program). For other wonderful titles written by this author see: Supersymmetry for Mathematicians: An Introduction, The Mathematical Legacy of Harish-Chandra: A Celebration of Representation Theory and Harmonic Analysis, The Selected Works of V.S. Varadarajan, and Algebra in Ancient and Modern Times.

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Table of contents :
Contents……Page 3
Preface……Page 5
1.1. Introduction……Page 7
1.2. Early life……Page 11
1.3. The first stay in St. Petersburg: 1727-1741……Page 14
1.4. The Berlin years: 1741-1766……Page 17
1.5. The second St. Petersburg stay and the last years: 1766-1783……Page 18
1.6. Opera Omnia……Page 19
1.7. The personality of Euler……Page 20
Notes and references……Page 21
2.2. Calculus……Page 27
2.3. Elliptic integrals……Page 29
2.4. Calculus of variations……Page 39
2.5. Number theory……Page 43
Notes and references……Page 63
3.1. Summary……Page 65
3.2. Some remarks on infinite series and products and their values……Page 70
3.3. Evaluation of zeta(2) and zeta(4)……Page 74
3.4. Infinite products for circular and hyperbolic functions……Page 83
3.5. The infinite partial fractions for (sin x)^-1 and cot x. Evaluation of zeta(2k) and L(2k+1)……Page 93
3.6. Partial fraction expansions as integrals……Page 100
3.7. Multizeta values……Page 111
Notes and references……Page 116
4.1. Formal derivation……Page 119
4.2. The case when the function is a polynomial……Page 122
4.3. Summation formula with remainder terms……Page 123
4.4. Applications……Page 127
Notes and references……Page 130
5.1. Divergent series and Euler’s ideas about summing them……Page 131
5.2. Euler’s derivation of the functional equation of the zeta function……Page 137
5.3. Euler’s summation of the factorial series……Page 144
5.4. The general theory of summation of divergent series……Page 151
5.5. Borel summation……Page 158
5.6. Tauberian theorems……Page 164
5.7. Some applications……Page 169
5.8. Fourier integral, Wiener Tauberian theorem, and Gel’fand transform on commutative Banach algebras……Page 177
5.9. Generalized functions and smeared summation……Page 191
5.10. Gaussian integrals, Wiener measure and the path integral formulae of Feynman and Kac……Page 197
Notes and references……Page 212
6.1. Euler’s product formula for the zeta function and others……Page 217
6.2. Euler products from Dirichlet to Hecke……Page 223
6.3. Euler products from Ramanujan and Hecke to Langlands……Page 244
6.4. Abelian extensions and class field theory……Page 257
6.5. Artin nonabelian L-functions……Page 268
6.6. The Langlands program……Page 270
Notes and references……Page 271
Gallery……Page 275
Sample Pages from Opera Omnia……Page 301
Index……Page 307