Sources in the Development of Mathematics
Series and Products from the Fifteenth to the Twentyfirst Century
$90.00 ( ) USD
 Author: Ranjan Roy, Beloit College, Wisconsin
 Date Published: December 2011
 availability: This ISBN is for an eBook version which is distributed on our behalf by a third party.
 format: Adobe eBook Reader
 isbn: 9781139119030
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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 presentday conjectures of Langlands, including that of ShimuraTaniyama, 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.
Read more Presents the evolution of mathematics starting around 1650 that researchers in various areas will read with benefit and enjoyment
 Traces the origins of many ideas in applied areas, which will be of interest to applied mathematicians, scientists and engineers
 Provides detailed proofs for numerous important theorems and formulas
Reviews & endorsements
"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 MagazineSee more reviews"... will provide [Roy] unique recognition for deep scholarship and extraordinary exposition regarding the history of classical mathematical analysis and related algebraic topics. This wellwritten 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 onevolume 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."
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×Product details
 Date Published: December 2011
 format: Adobe eBook Reader
 isbn: 9781139119030
 contains: 44 b/w illus. 379 exercises
 availability: This ISBN is for an eBook version which is distributed on our behalf by a third party.
Table of Contents
1. Power series in fifteenthcentury 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. Lseries
27. The hypergeometric series
28. Orthogonal polynomials
29. qSeries
30. Partitions
31. qSeries and qorthogonal 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.
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