2006;
302 pp;
Hardcover

MSC: Primary 01;
Secondary 11; 40

Print ISBN: 978-0-8218-3580-7

Product Code: EULER

List Price: $65.00

Individual Member Price: $52.00

**Electronic ISBN: 978-1-4704-2488-6
Product Code: EULER.E**

List Price: $65.00

Individual Member Price: $52.00

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#### Supplemental Materials

# Euler Through Time: A New Look at Old Themes

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*V. S. Varadarajan*

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*.

#### Table of Contents

# Table of Contents

## Euler Through Time: A New Look at Old Themes

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- Preface vii8 free
- Chapter 1. Leonhard Euler (1707-1783) 110 free
- Chapter 2. The Universal Mathematician 2130
- Chapter 3. Zeta Values 5968
- 3.1. Summary 5968
- 3.2. Some remarks on infinite series and products and their values 6473
- 3.3. Evaluation of ζ(2) and ζ(4) 6877
- 3.4. Infinite products for circular and hyperbolic functions 7786
- 3.5. The infinite partial fractions for (sinx)[sup(…1)] and cot x. Evaluation of ζ(2k) and L(2k + 1) 8796
- 3.6. Partial fraction expansions as integrals 94103
- 3.7. Multizeta values 105114
- Notes and references 110119

- Chapter 4. Euler-Maclaurin Sum Formula 113122
- Chapter 5. Divergent Series and Integrals 125134
- 5.1. Divergent series and Euler's ideas about summing them 125134
- 5.2. Euler's derivation of the functional equation of the zeta function 131140
- 5.3. Euler's summation of the factorial series 138147
- 5.4. The general theory of summation of divergent series 145154
- 5.5. Borel summation 152161
- 5.6. Tauberian theorems 158167
- 5.7. Some applications 163172
- 5.8. Fourier integral, Wiener Tauberian theorem, and Gel'fand transform on commutative Banach algebras 171180
- 5.9. Generalized functions and smeared summation 185194
- 5.10. Gaussian integrals, Wiener measure and the path integral formulae of Feynman and Kac 191200
- Notes and references 206215

- Chapter 6. Euler Products 211220
- 6.1. Euler's product formula for the zeta function and others 211220
- 6.2. Euler products from Dirichlet to Hecke 217226
- 6.3. Euler products from Ramanujan and Hecke to Langlands 238247
- 6.4. Abelian extensions and class field theory 251260
- 6.5. Artin nonabelian L-functions 262271
- 6.6. The Langlands program 264273
- Notes and references 265274

- Gallery 269278
- Sample Pages from Opera Omnia 295304
- Index 301310 free
- Back Cover Back Cover1312

#### Readership

Undergraduates, graduate students, and research mathematicians interested in the history of mathematics and Euler's influence on modern mathematics.

#### Reviews

...something truly special...Varadarajan has provided us with a useful guide to certain portions of Euler's work and with interesting surveys of the mathematics to which that work led over the centuries.

-- MAA Reviews

...the author has admirablly managed to organize the text in such a manner that an interested non-specialist will find the whole story comprehensible, absorbing, and enjoyable. This book has been written with the greatest insight, expertise, experience, and passion on the part of the author's, and it should be seen as what it really is: a cultural jewel in the mathematical literature as a whole!

-- Zentralblatt MATH

By taking some of Euler's most important insights, developing them, and showing their connection to contemporary research, this book offers a profound understanding of Euler's achievements and their role in the development of mathematics as we now know it.

-- Mathematical Review