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Special Functions and Orthogonal Polynomials

$89.99 (P)

Part of Cambridge Studies in Advanced Mathematics

  • Date Published: May 2016
  • availability: Available
  • format: Hardback
  • isbn: 9781107106987

$ 89.99 (P)

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About the Authors
  • The subject of special functions is often presented as a collection of disparate results, rarely organized in a coherent way. This book emphasizes general principles that unify and demarcate the subjects of study. The authors' main goals are to provide clear motivation, efficient proofs, and original references for all of the principal results. The book covers standard material, but also much more. It shows how much of the subject can be traced back to two equations - the hypergeometric equation and confluent hypergeometric equation - and it details the ways in which these equations are canonical and special. There is extended coverage of orthogonal polynomials, including connections to approximation theory, continued fractions, and the moment problem, as well as an introduction to new asymptotic methods. There are also chapters on Meijer G-functions and elliptic functions. The final chapter introduces Painlevé transcendents, which have been termed the 'special functions of the twenty-first century'.

    • Covers standard topics from a unified point of view to show how different topics are part of a general scheme
    • Comprehensive but self-contained, covering newer asymptotic methods to give an up-to-date view of an important research area
    • Includes topics such as Painlevé functions and Meijer G-functions, which are not usually treated at this level, to give an understandable and well-motivated introduction to some subjects of great current interest
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    Reviews & endorsements

    '… an excellent graduate textbook, one of the two best available on this subject…' Warren Johnson, MAA Reviews (

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    Product details

    • Date Published: May 2016
    • format: Hardback
    • isbn: 9781107106987
    • length: 488 pages
    • dimensions: 237 x 154 x 32 mm
    • weight: 0.78kg
    • contains: 7 b/w illus. 430 exercises
    • availability: Available
  • Table of Contents

    1. Orientation
    2. Gamma, beta, zeta
    3. Second-order differential equations
    4. Orthogonal polynomials on an interval
    5. The classical orthogonal polynomials
    6. Semiclassical orthogonal polynomials
    7. Asymptotics of orthogonal polynomials: two methods
    8. Confluent hypergeometric functions
    9. Cylinder functions
    10. Hypergeometric functions
    11. Spherical functions
    12. Generalized hypergeometric functions
    13. Asymptotics
    14. Elliptic functions
    15. Painlevé transcendents
    Appendix A. Complex analysis
    Appendix B. Fourier analysis

  • Authors

    Richard Beals, Yale University, Connecticut
    Richard Beals is a former Professor of Mathematics at the University of Chicago and Yale University. He is the author or co-author of books on mathematical analysis, linear operators and inverse scattering theory, and has authored more than 100 research papers in areas including partial differential equations, mathematical economics and mathematical psychology.

    Roderick Wong, City University of Hong Kong
    Roderick Wong is Chair Professor of Mathematics at the City University of Hong Kong. He is the author of books on asymptotic approximations of integrals and applied analysis. He has published over 140 research papers in areas such as asymptotic analysis, singular perturbation theory and special functions.

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