Modern developments in theoretical and applied science depend on knowledge of the properties of mathematical functions, from elementary trigonometric functions to the multitude of special functions. These functions appear whenever natural phenomena are studied, engineering problems are formulated, and numerical simulations are performed. They also crop up in statistics, financial models, and economic analysis. Using them effectively requires practitioners to have ready access to a reliable collection of their properties. This handbook results from a 10-year project conducted by the National Institute of Standards and Technology with an international group of expert authors and validators. Printed in full colour, it is destined to replace its predecessor, the classic but long-outdated Handbook of Mathematical Functions, edited by Abramowitz and Stegun. Includes a DVD with a searchable PDF of each chapter.

• Compendium of properties of mathematical special functions • Developed by expert authors, editors, and validators • Carefully edited for uniform treatment of technical content

### Contents

1. Algebraic and analytic methods Ranjan Roy, Frank W. J. Olver, Richard A. Askey and Roderick S. C. Wong; 2. Asymptotic approximations Frank W. J. Olver and Roderick S. C. Wong; 3. Numerical methods Nico M. Temme; 4. Elementary functions Ranjan Roy and Frank W. J. Olver; 5. Gamma function Richard A. Askey and Ranjan Roy; 6. Exponential, logarithmic, sine and cosine integrals Nico M. Temme; 7. Error functions, Dawson's and Fresnel integrals Nico M. Temme; 8. Incomplete gamma and related functions Richard B. Paris; 9. Airy and related functions Frank W. J. Olver; 10. Bessel functions Frank W. J. Olver and Leonard C. Maximon; 11. Struve and related functions Richard B. Paris; 12. Parabolic cylinder functions Nico M. Temme; 13. Confluent hypergeometric functions Adri B. Olde Daalhuis; 14. Legendre and related functions T. Mark Dunster; 15. Hypergeometric function Adri B. Olde Daalhuis; 16. Generalized hypergeometric functions and Meijer G-function Richard A. Askey and Adri B. Olde Daalhuis; 17. q-Hypergeometric and related functions George E. Andrews; 18. Orthogonal polynomials Tom H. Koornwinder, Roderick S. C. Wong, Roelof Koekoek and Rene F. Swarttouw; 19. Elliptic integrals Bille C. Carlson; 20. Theta functions William P. Reinhardt and Peter L. Walker; 21. Multidimensional theta functions Bernard Deconinck; 22. Jacobian elliptic functions William P. Reinhardt and Peter L. Walker; 23. Weierstrass elliptic and modular functions William P. Reinhardt and Peter L. Walker; 24. Bernoulli and Euler polynomials Karl Dilcher; 25. Zeta and related functions Tom M. Apostol; 26. Combinatorial analysis David M. Bressoud; 27. Functions of number theory Tom M. Apostol; 28. Mathieu functions and Hill's equation Gerhard Wolf; 29. Lamé functions Hans Volkmer; 30. Spheroidal wave functions Hans Volkmer; 31. Heun functions Brian D. Sleeman and Vadim Kuznetsov; 32. Painlevé transcendents Peter A. Clarkson; 33. Coulomb functions Ian J. Thompson; 34. 3j,6j,9j symbols Leonard C. Maximon; 35. Functions of matrix argument Donald St. P. Richards; 36. Integrals with coalescing saddles Michael V. Berry and Chris Howls.

### Reviews

'The NIST Handbook is indeed a monumental achievement, and the many, many individuals who participated in its creation and dissemination are to be congratulated and thanked.' SIAM News

'The National Institute of Standards and Technology (NIST) and Cambridge University Press are to be congratulated for publishing a treasury. It is eminently readable with clear, sharp, high-contrast text, mathematical notation and colored graphs and figures. … People who work with functions will delight in this handbook.' Optics and Photonics News

'… distinguished collection of chapter authors … To find and effectively utilize such a collection of experts seems deserving of an Olympic medal!' Robert E. O'Malley, Jr, SIAM Review

'This book is essentially an expanded and updated version of [Abramowitz and Stegun's Handbook of Mathematical Functions], but it also comes with a CD, and with weblinks, which enable one readily to access far more material, including some of the original references. As such, it is a welcome addition to one's reference collection. It contains far more material than [Abramowitz and Stegun], especially welcome being an up-to-date chapter on numerical methods and approximations.' The Observatory

'The NIST Handbook provides comprehensive information on hundreds of mathematical functions … Their qualitative features are illustrated by numerous color figures in two or three dimensions. This is a timely and authoritative modern replacement of the classic [A and S] …The associated DLMF may well serve as a model for the effective presentation of highly mathematical reference material on the Web. The exposition is eminently readable and delightful, and everyone who works with or applies special mathematical functions will profit definitely from this impressive handbook.' Journal of Geometry and Symmetry in Physics

'… a concise and well-structured format … there is no doubting the quality of this book … its content will be useful to anyone working with special functions.' Contemporary Physics