Sources in the Development of Mathematics: Series and Products from the Fifteenth to the Twenty-first Century 1st Edition by Ranjan Roy (PDF)

Description

The discovery of infinite products by Wallis and infinite series by Newton marked the beginning of the modern mathematical era. It allowed Newton to solve the problem of finding areas under curves defined by algebraic equations, an achievement beyond the scope of the earlier methods of Torricelli, Fermat, and Pascal. Newton and his contemporaries, including Leibniz and the Bernoullis, concentrated on mathematical analysis and physics. Euler’s prodigious accomplishments demonstrated that series and products could also address problems in algebra, combinatorics, and number theory. Series and products have continued to be pivotal mathematical tools in the work of Gauss, Abel, and Jacobi in elliptic functions; in Boole’s and Lagrange’s infinite series and products of operators; in work by Cayley, Sylvester, and Hilbert in invariant theory; and in the present-day conjectures of Langlands, including that of Shimura-Taniyama, leading to Wiles’s proof of Fermat’s last theorem. In this book, Ranjan Roy describes many facets of the discovery and use of infinite series and products as worked out by their originators, including mathematicians from Asia, Europe, and America. The text provides context and motivation for these discoveries; the original notation and diagrams are presented when practical. Multiple derivations are given for many results, and detailed proofs are offered for important theorems and formulas. Each chapter includes interesting exercises and bibliographic notes, supplementing the results of the chapter. These original mathematical insights offer a valuable perspective on modern mathematics. Mathematicians, mathematics students, physicists, and engineers will all read this book with benefit and enjoyment.

User’s Reviews

Editorial Reviews: Review “This work is unbelievably thorough. Roy includes not just results but also many proofs, historic contexts, references, and exercises. Is it the sort of encyclopedic effort that one typically associates with a group of authors rather than an individual. Roy has made an important contribution with this book.” C. Bauer, Choice Magazine”… will provide [Roy] unique recognition for deep scholarship and extraordinary exposition regarding the history of classical mathematical analysis and related algebraic topics. This well-written book will be a valuable source of fresh information on the wide range of topics covered. It can be expected to have great positive impact on pedagogy and understanding. It certainly seems to be the best one-volume history of mathematics I know…” Robert E. O’Malley, SIAM Review”I recommend this book to a wide audience. Undergraduates can learn of the truly vast amount of material that lies alongside some of their more standard endeavors, many of which involve only elementary matters: sums, products, limits, calculus. Graduate students and nonspecialist faculty can wonder at the ingenuity of their predecessors and the connections between now disparate areas that are afforded by this very classical view. They’ll also get lots of good ideas for teaching (and they may waste a good deal of time on the problems, as well). Historians, philosophers, and others should read this book, if only for the view of mathematics it propounds. And specialized researchers in the area of special functions and related fields should simply have a good time. All of these readers can benefit from the remarkable expository talents of the author and his careful choice of material. Among personal views of mathematics that use history as a key to understanding, Roy’s book stands out as a model.” Tom Archibald, Notices of the AMS Book Description A look at the discovery and use of infinite series and products from Wallis and Newton through Euler and Gauss to the present day. About the Author Ranjan Roy is the Ralph C. Huffer Professor of Mathematics and Astronomy at Beloit College. Roy has published papers and reviews in differential equations, fluid mechanics, Kleinian groups, and the development of mathematics. He co-authored Special Functions (2001) with George Andrews and Richard Askey, and authored chapters in the NIST Handbook of Mathematical Functions (2010). He has received the Allendoerfer prize, the Wisconsin MAA teaching award, and the MAA Haimo award for distinguished mathematics teaching. Read more

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

⭐There are books on history of math, but they lack in detail! and there are books that treat math in detail, but they lack historical perspective and concepts spring up out of thin air, as if without roots or parentage!Roy’s book fills this void; it provides us the details as well as their historical development!A word about the subtitle of the book concerning series and products: As many of us know, the finite and infinite series and products have played an indispensable role in the understanding and development of mathematical ideas; without them, it is impossible to understand even the simplest of mathematical numbers or functions, like pi, or sin(x).Why and how did these mathematical series arose and were manipulated by the original mathematicians of the bygone era? How their mind worked? and how they surmounted problems without adequate tools at their disposal? how their insight led to great mathematical ideas for successive mathematicians to develop the subject further? and finally how and why we have reached where we are today?If you are interested in or curious to know all these in sufficient (and not just superficial or anecdotal) detail, I strongly recommend that you must read and digest this book; you would enjoy it thoroughly, it would brighten you, it would make the famous mathematicians of yore more near and real to you – almost like they are your friends -; and you may not need to go anywhere else!Roy’s book spares no details! We all know about the most famous mathematicians – Archimedes, Newton, Cauchy, Euler ….et. al. , but how many of us know that it was an Italian mathematician named MAUROLICO, who in 1575 first gave us the Principle of mathematical Induction!Roy’s ” Sources in the development of mathematics” is truly a detailed study and development of modern mathematical ideas through the march of history in their full glory. Roy deals with each subject in a manner, so that a modern student or teacher today can experience the pleasure of working out the details of the original formalisms without any difficulty like having to read the original papers in Latin or Greek or Arabic or Sanskrit and understanding them in a modern context). And there are plenty of exercises to play around with in each chapter.Could there be a better way to truly understand the mathematical concepts and, kind of, “own” them as your own?Undoubtedly the book will serve as a great reference book for a long long time – probably the ultimate one in our era.

⭐The book covered a huge amount of contents and had great exercises. It is very helpful for students to go into more indepth topics. Great book.

⭐This carefully written, well-organized work is now one of my favorite mathematics books of all time. I deeply appreciate the effort that this undertaking must have entailed. The author does a marvelous job of balancing historical information with mathematical results, leaving just the right amount of detail to the reader — assuming that the reader is well-versed in the subject. In a nutshell, I’m in awe.

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