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Product Code:  EULER 
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Electronic ISBN:  9781470424886 
Product Code:  EULER.E 
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Book Details2006; 302 ppMSC: Primary 01; Secondary 11; 40;
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 HarishChandra: A Celebration of Representation Theory and Harmonic Analysis, The Selected Works of V.S. Varadarajan, and Algebra in Ancient and Modern Times.ReadershipUndergraduates, graduate students, and research mathematicians interested in the history of mathematics and Euler's influence on modern mathematics.

Table of Contents

Cover

Title

Copyright

Contents

Preface

Chapter 1. Leonhard Euler (17071783)

1.1 Introduction

1.2. Early life

1.3. The first stay in St. Petersburg: 17271741

1.4. The Berlin years: 17411766

1.5. The second St. Petersburg stay and the last years: 17661783

1.6. Opera Omnia

1.7. The personality of Euler

Notes and references

Chapter 2. The Universal Mathematician

2.1. Introduction

2.2. Calculus

2.3. Elliptic integrals

2.4. Calculus of variations

2.5. Number theory

Notes and references

Chapter 3. Zeta Values

3.1. Summary

3.2. Some remarks on infinite series and products and their values

3.3. Evaluation of ζ(2) and ζ(4)

3.4. Infinite products for circular and hyperbolic functions

3.5. The infinite partial fractions for (sinx)[sup(…1)] and cot x. Evaluation of ζ(2k) and L(2k + 1)

3.6. Partial fraction expansions as integrals

3.7. Multizeta values

Notes and references

Chapter 4. EulerMaclaurin Sum Formula

4.1. Formal derivation

4.2. The case when the function is a polynomial

4.3. Summation formula with remainder terms

4.4. Applications

Notes and references

Chapter 5. Divergent Series and Integrals

5.1. Divergent series and Euler's ideas about summing them

5.2. Euler's derivation of the functional equation of the zeta function

5.3. Euler's summation of the factorial series

5.4. The general theory of summation of divergent series

5.5. Borel summation

5.6. Tauberian theorems

5.7. Some applications

5.8. Fourier integral, Wiener Tauberian theorem, and Gel'fand transform on commutative Banach algebras

5.9. Generalized functions and smeared summation

5.10. Gaussian integrals, Wiener measure and the path integral formulae of Feynman and Kac

Notes and references

Chapter 6. Euler Products

6.1. Euler's product formula for the zeta function and others

6.2. Euler products from Dirichlet to Hecke

6.3. Euler products from Ramanujan and Hecke to Langlands

6.4. Abelian extensions and class field theory

6.5. Artin nonabelian Lfunctions

6.6. The Langlands program

Notes and references

Gallery

Sample Pages from Opera Omnia

Index

Back Cover


Additional Material

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 nonspecialist 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


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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 HarishChandra: A Celebration of Representation Theory and Harmonic Analysis, The Selected Works of V.S. Varadarajan, and Algebra in Ancient and Modern Times.
Undergraduates, graduate students, and research mathematicians interested in the history of mathematics and Euler's influence on modern mathematics.

Cover

Title

Copyright

Contents

Preface

Chapter 1. Leonhard Euler (17071783)

1.1 Introduction

1.2. Early life

1.3. The first stay in St. Petersburg: 17271741

1.4. The Berlin years: 17411766

1.5. The second St. Petersburg stay and the last years: 17661783

1.6. Opera Omnia

1.7. The personality of Euler

Notes and references

Chapter 2. The Universal Mathematician

2.1. Introduction

2.2. Calculus

2.3. Elliptic integrals

2.4. Calculus of variations

2.5. Number theory

Notes and references

Chapter 3. Zeta Values

3.1. Summary

3.2. Some remarks on infinite series and products and their values

3.3. Evaluation of ζ(2) and ζ(4)

3.4. Infinite products for circular and hyperbolic functions

3.5. The infinite partial fractions for (sinx)[sup(…1)] and cot x. Evaluation of ζ(2k) and L(2k + 1)

3.6. Partial fraction expansions as integrals

3.7. Multizeta values

Notes and references

Chapter 4. EulerMaclaurin Sum Formula

4.1. Formal derivation

4.2. The case when the function is a polynomial

4.3. Summation formula with remainder terms

4.4. Applications

Notes and references

Chapter 5. Divergent Series and Integrals

5.1. Divergent series and Euler's ideas about summing them

5.2. Euler's derivation of the functional equation of the zeta function

5.3. Euler's summation of the factorial series

5.4. The general theory of summation of divergent series

5.5. Borel summation

5.6. Tauberian theorems

5.7. Some applications

5.8. Fourier integral, Wiener Tauberian theorem, and Gel'fand transform on commutative Banach algebras

5.9. Generalized functions and smeared summation

5.10. Gaussian integrals, Wiener measure and the path integral formulae of Feynman and Kac

Notes and references

Chapter 6. Euler Products

6.1. Euler's product formula for the zeta function and others

6.2. Euler products from Dirichlet to Hecke

6.3. Euler products from Ramanujan and Hecke to Langlands

6.4. Abelian extensions and class field theory

6.5. Artin nonabelian Lfunctions

6.6. The Langlands program

Notes and references

Gallery

Sample Pages from Opera Omnia

Index

Back Cover

...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 nonspecialist 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