Sources in the Development of Mathematics: Series and Products from the Fifteenth to the Twenty-first Century

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. While 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. 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, with many detailed proofs, offering a valuable perspective on modern mathematics. Mathematicians, mathematics students, physicists and engineers will all read this book with benefit and enjoyment.

Table of Contents:
1. Power series in fifteenth-century Kerala; 2. Sums of powers of integers; 3. Infinite product of Wallis; 4. The binomial theorem; 5. The rectification of curves; 6. Inequalities; 7. Geometric calculus; 8. The calculus of Newton and Leibniz; 9. De Analysi per Aequationes Infinitas; 10. Finite differences: interpolation and quadrature; 11. Series transformation by finite differences; 12. The Taylor series; 13. Integration of rational functions; 14. Difference equations; 15. Differential equations; 16. Series and products for elementary functions; 17. Solution of equations by radicals; 18. Symmetric functions; 19. Calculus of several variables; 20. Algebraic analysis: the calculus of operations; 21. Fourier series; 22. Trigonometric series after 1830; 23. The gamma function; 24. The asymptotic series for ln Γ(x); 25. The Euler–Maclaurin summation formula; 26. L-series; 27. The hypergeometric series; 28. Orthogonal polynomials; 29. q-Series; 30. Partitions; 31. q-Series and q-orthogonal polynomials; 32. Primes in arithmetic progressions; 33. Distribution of primes: early results; 34. Invariant theory: Cayley and Sylvester; 35. Summability; 36. Elliptic functions: eighteenth century; 37. Elliptic functions: nineteenth century; 38. Irrational and transcendental numbers; 39. Value distribution theory; 40. Univalent functions; 41. Finite fields.

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.

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