Euler Through Time: A New Look at Old Themes 1st Edition by V. S. Varadarajan (PDF)

Description

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.

User’s Reviews

Editorial Reviews: Review “…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

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Euler is the one of the 4 greastest mathematician of all times. ( The other three are Archaemides, Newton and Gauss. But he is also the most popular mathamticians. He actually had no foes, no controversy, no priorities disputes. He just enjoyed mathematics. In this book, we have a glimpse of part of Euler’s work which is presented in a even modern way.( though Euler’s writting already quite modern, unlikely Newton etc )I really enjoy reading that book. Strongly recommended.

⭐This book talks more about modern research inspired by Euler’s work than about Euler’s work itself. The book presents Euler’s results in modern language and also goes far beyond what Euler did to talk about how these themes were developed. It has no substantial material on Euler’s work in mechanics or differential equations (except as this relates to elliptic integrals), but this is the perfect book to get a better understanding of Euler’s analytic number theory.A lot of the book is about infinite series: the zeta function, how to sum series, and how to make sense of divergent series. Varadarajan does a good job explaining Euler’s two biggest discoveries in analytic number theory: the Euler product for the zeta function and the functional equation for the zeta function. To talk about the latter we have to talk about divergent series too, and this book has a chapter on divergent series that includes a readable exposition of Tauberian theory.The author has nothing close to the historical knowledge or sensitivity of André Weil, and quotes Weil many times. For historical details I recommend going straight to Weil’s papers and book on the history of number theory. In my mind the ideal historical work in mathematics is Weil’s “Number theory: An approach through history”, but this complete familiarity with the original sources and the general history of mathematics (to know what problems people were working on and what had already been done) on the one hand, and the modern significance of the work on the other hand is an heroic accomplishment, and to measure everything by this is to have too high a standard.

⭐one great mathematician writes about an all time great mathematician and what do you have – this treasure!! get it.

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