NIST Handbook of Mathematical Functions

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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. Thes 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.<BR><BR>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. It is destined to replace its predecessor, the classic but long-outdated NBS Handbook of Mathematical Functions, edited by Abramowitz and Stegun.

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Editorial Reviews

From the Publisher
"The NIST Handbook is a handsome product, with large pages and large type. The book is quite heavy; for convenience, one might be inclined to place it on a stand, as with an unabridged dictionary. The book contains numerous graphics, almost all in color. References and cross references to books and articles abound. Applications to both the mathematical and physical sciences are indicated. 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."
Philip J. Davis for SIAM News

"An outstanding group of editors, associate editors and validators updated and extended the classic NBS Handbook of Mathematical Functions, edited by Abramowitz and Stegun. 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, The entire book is contained in a CD-ROM with a searchable PDF. From Leibnitz to Hilbert, from modern science and engineering to other disparate fields of study, functions are ubiquitous , fascinating and beautiful objects of human ingenuity. A prerequisite to their use is to understand their properties, and this handbook provides a direct and concise solution. It contains an extensive bibliography, a list of notations, and an index. The general format for each group of functions includes notation, properties, applications, computation and references. People who work with functions will delight in this handbook."
Barry Masters for Optics & Photonics News

"... an excellent product."
J. H. Davenport, Computing Reviews

"This is like trying to review the bible: it would be eccentric to argue that it is not a “thoroughly good thing”. It’s the modern successor to the wonderful Handbook of Mathematical Functions, edited by Abramowitz and Stegun, and maybe that’s enough said. In summary, this splendid work doesn’t really need the approbation of a mere reviewer. And now I’m off to look up my first unidentified integral to see if it’s a standard form."
Martin Crowder, International Statistical Review

"The editors, associate editors, chapter authors, validators, and NIST staff members deserve our thanks for their very successful and valuable product."
Robert E. O'Malley, SIAM Review

"NHMF and the online version DLMF are a treasure for the mathematical and scientific communities, one that will be used and valued for decades. The organization, presentation, and general appearance are excellent. This beautiful book reflects credit on everyone and every organization involved; NIST; the National Science Foundation for funding; those who organized the project and obtained the funding; the advisors, editors, authors, and validators; and Cambridge University Press. Above all, NHMF and DLMF are a monument to the efforts of the editor-in-chief, author of one chapter of A&S and author or coauthor of five chapters of this successor volume, Frank Olver."
Richard Beals, Notices of the AMS

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Product Details

  • ISBN-13: 9780521140638
  • Publisher: Cambridge University Press
  • Publication date: 5/10/2010
  • Pages: 968
  • Sales rank: 1,422,324
  • Product dimensions: 8.50 (w) x 10.90 (h) x 2.00 (d)

Meet the Author

Frank W. J. Olver is Professor Emeritus in the Institute for Physical Science and Technology and the Department of Mathematics at the University of Maryland. From 1961 to 1986 he was a Mathematician at the National Bureau of Standards in Washington, D.C. Professor Olver has published 76 papers in refereed and leading mathematics journals, and he is the author of Asymptotics and Special Functions (1974). He has served as editor of SIAM Journal on Numerical Analysis, SIAM Journal on Mathematical Analysis, Mathematics of Computation, Methods and Applications of Analysis, and the NBS Journal of Research.

Daniel W. Lozier leads the Mathematical Software Group in the Mathematical and Computational Sciences Division of NIST. In his capacity as General Editor of the Digital Library of Mathematical Functions Project, he has performed most of the administrative functions associated with the project as well as contributing technically. He is an active member of the SIAM Activity Group on Orthogonal Polynomials and Special Functions, having served two terms as chair, one as vice-chair, and currently as secretary. He has been an editor of Mathematics of Computation and the NIST Journal of Research.

Ronald F. Boisvert leads the Mathematical and Computational Sciences Division of the Information Technology Laboratory at NIST. He received his Ph.D. in computer science from Purdue University in 1979 and has been at NIST since then. He has served as editor-in-chief of the ACM Transactions on Mathematical Software. He is currently co-chair of the Publications Board of the Association for Computing Machinery (ACM) and chair of the International Federation for Information Processing (IFIP) Working Group 2.5 (Numerical Software).

Charles W. Clark received his Ph.D. in physics from the University of Chicago in 1979. He is a member of the U.S. Senior Executive Service and is Chief of the Electron and Optical Physics Division and acting Group Leader of the NIST Synchrotron Ultraviolet Radiation Facility (SURF III). Clark serves as Program Manager for Atomic and Molecular Physics at the U.S. Office of Naval Research and is a Fellow of the Joint Quantum Institute of NIST and the University of Maryland at College Park and a Visiting Professor at the National University of Singapore.

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Table of Contents



Mathematical Introduction

1 Algebraic and Analytic Methods R. Wong Wong, R. 1

2 Asymptotic Approximations R. Wong Wong, R. 41

3 Numerical Methods N. M. Temme Temme, N. M. 71

4 Elementary Functions F. W. J. Olver Olver, F. W. J. 103

5 Gamma Function R. Roy Roy, R. 135

6 Exponential, Logarithmic, Sine, and Cosine Integrals N. M. Temme Temme, N. M. 149

7 Error Functions, Dawson's and Fresnel Integrals N. M. Temme Temme, N. M. 159

8 Incomplete Gamma and Related Functions R. B. Paris Paris, R. B. 173

9 Airy and Related Functions F. W. J. Olver Olver, F. W. J. 193

10 Bessel Functions L. C. Maximon Maximon, L. C. 215

11 Struve and Related Functions R. B. Paris Paris, R. B. 287

12 Parabolic Cylinder Functions N. M. Temme Temme, N. M. 303

13 Confluent Hypergeometric Functions A. B. Olde Daalhuis Daalhuis, A. B. Olde 321

14 Legendre and Related Functions T. M. Dunster Dunster, T. M. 351

15 Hypergeometric Function A. B. Olde Daalhuis Daalhuis, A. B. Olde 383

16 Generalized Hypergeometric Functions and Meijer G-Function A. B. Olde Daalhuis Daalhuis, A. B. Olde 403

17 q-Hypergeometric and Related Functions G. E. Andrews Andrews, G. E. 419

18 Orthogonal Polynomials R. F. Swarttouw Swarttouw, R. F. 435

19 Elliptic Integrals B. C. Carlson Carlson, B. C. 485

20 Theta Functions P. L. Walker Walker, P. L. 523

21 Multidimensional Theta Functions B. Deconinck Deconinck, B. 537

22 Jacobian Elliptic Functions P. L. Walker Walker, P. L. 549

23 Weierstrass Elliptic and Modular Functions P. L. Walker Walker, P. L. 569

24 Bernoulli and Euler Polynomials K. Dilcher Dilcher, K. 587

25 Zeta and Related Functions T. M. Apostol Apostol, T. M. 601

26 Combinatorial Analysis D. M. Bressoud Bressoud, D. M. 617

27 Functions of Number Theory T. M. Apostol Apostol, T. M. 637

28 Mathieu Functions and Hill's Equation G. Wolf Wolf, G. 651

29 Lame Functions H. Volkmer Volkmer, H. 683

30 Spheroidal Wave Functions H. Volkmer Volkmer, H. 697

31 Heun Functions V. B. Kuznetsov Kuznetsov, V. B. 709

32 Painleve Transcendents P. A. Clarkson Clarkson, P. A. 723

33 Coulomb Functions I. J. Thompson Thompson, I. J. 741

34 3j, 6j, 9j Symbols L. C. Maximon Maximon, L. C. 757

35 Functions of Matrix Argument D. St. P. Richards Richards, D. St. P. 767

36 Integrals with Coalescing Saddles C. J. Howls Howls, C. J. 775

Bibliography 795

Notations 873

Index 887

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