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

$90.00 ( ) USD

  • 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

$ 90.00 USD ( )
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About the Authors
  • 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.

    • 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
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    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 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

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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 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.

  • Author

    Ranjan Roy, Beloit College, Wisconsin
    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.

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