The NIST Handbook of Mathematical Functions, together with its Web counterpart, the NIST Digital Library of Mathematical Functions (DLMF), is the culmination of a project that was conceived in 1996 at the National Institute of Standards and Technology (NIST). The project had two equally important goals: to develop an authoritative replacement for the highly successful Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, published in 1964 by the National Bureau of Standards (M. Abramowitz and I. A. Stegun, editors); and to disseminate essentially the same information from a public Web site operated by NIST. The new Handbook and DLMF are the work of many hands: editors, associate editors, authors, validators, and numerous technical experts. A summary of the responsibilities of these groups may help in understanding the structure and results of this project.
Executive responsibility was vested in the editors: Frank W. J. Olver (University of Maryland, College Park, and NIST), Daniel W. Lozier (NIST), Ronald F. Boisvert (NIST), and Charles W. Clark (NIST). Olver was responsible for organizing and editing the mathematical content after receiving it from the authors; for communicating with the associate editors, authors, validators, and other technical experts; and for assembling the Notations section and the Index. In addition, Olver was author or co-author of five chapters. Lozier directed the NIST research, technical, and support staff associated with the project, administered grants and contracts, together with Boisvert compiled the Software sections for the Web version of the chapters, conducted editorial and staff meetings, represented the project within NIST and at professional meetings in the United States and abroad, and together with Olver carried out the day-to-day development of the project. Boisvert and Clark were responsible for advising and assisting in matters related to the use of information technology and applications of special functions in the physical sciences (and elsewhere); they also participated in the resolution of major administrative problems when they arose.
The associate editors are eminent domain experts who were recruited to advise the project on strategy, execution, subject content, format, and presentation, and to help identify and recruit suitable candidate authors and validators. The associate editors were:
Richard A. Askey
University of Wisconsin, Madison
Michael V. Berry
University of Bristol
Walter Gautschi (resigned 2002)
Purdue University
Leonard C. Maximon
George Washington University
Morris Newman
University of California, Santa Barbara
Ingram Olkin
Stanford University
Peter Paule
Johannes Kepler University
William P. Reinhardt
University of Washington
Nico M. Temme
Centrum voor Wiskunde en Informatica
Jet Wimp (resigned 2001)
Drexel University
The technical information provided in the Handbook and DLMF was prepared by subject experts from around the world. They are identified on the title pages of the chapters for which they served as authors and in the table of Contents.
The validators played a critical role in the project, one that was absent in its 1964 counterpart: to provide critical, independent reviews during the development of each chapter, with attention to accuracy and appropriateness of subject coverage. These reviews have contributed greatly to the quality of the product. The validators were:
T. M. Apostol
California Institute of Technology
A. R. Barnett
University of Waikato, New Zealand
A. I. Bobenko
Technische Universität, Berlin
B. B. L. Braaksma
University of Groningen
D. M. Bressoud
Macalester College
B. C. Carlson
Iowa State University
B. Deconinck
University of Washington
T. M. Dunster
University of California, San Diego
A. Gil
Universidad de Cantabria
A. R. Its
Indiana University–Purdue University, Indianapolis
B. R. Judd
Johns Hopkins University
R. Koekoek
Delft University of Technology
T. H. Koornwinder
University of Amsterdam
R. J. Muirhead
Pfizer Global R&D
E. Neuman
Southern Illinois University, Carbondale
A. B. Olde Daalhuis
University of Edinburgh
R. B. Paris
University of Abertay Dundee
R. Roy
Beloit College
S. N. M. Ruijsenaars
University of Leeds
J. Segura
Universidad de Cantabria
R. F. Swarttouw
Vrije Universiteit Amsterdam
N. M. Temme
Centrum voor Wiskunde en Informatica
H. Volkmer
University of Wisconsin, Milwaukee
G. Wolf
Universität Duisberg-Essen
R. Wong
City University of Hong Kong
All of the mathematical information contained in the Handbook is also contained in the DLMF, along with additional features such as more graphics, expanded tables, and higher members of some families of formulas; in consequence, in the Handbook there are occasional gaps in the numbering sequences of equations, tables, and figures. The Web address where additional DLMF content can be found is printed in blue at appropriate places in the Handbook. The home page of the DLMF is accessible at DLMF.
The DLMF has been constructed specifically for effective Web usage and contains features unique to Web presentation. The Web pages contain many active links, for example, to the definitions of symbols within the DLMF, and to external sources of reviews, full texts of articles, and items of mathematical software. Advanced capabilities have been developed at NIST for the DLMF, and also as part of a larger research effort intended to promote the use of the Web as a tool for doing mathematics. Among these capabilities are: a facility to allow users to download LaTeX and MathML encodings of every formula into document processors and software packages (eventually, a fully semantic downloading capability may be possible); a search engine that allows users to locate formulas based on queries expressed in mathematical notation; and user-manipulable 3-dimensional color graphics.
Production of the Handbook and DLMF was a mammoth undertaking, made possible by the dedicated leadership of Bruce R. Miller (NIST), Bonita V. Saunders (NIST), and Abdou S. Youssef (George Washington University and NIST). Miller was responsible for information architecture, specializing LaTeX for the needs of the project, translation from LaTeX to MathML, and the search interface. Saunders was responsible for mesh generation for curves and surfaces, data computation and validation, graphics production, and interactive Web visualization. Youssef was responsible for mathematics search indexing and query processing. They were assisted by the following NIST staff: Marjorie A. McClain (LaTeX, bibliography), Joyce E. Conlon (bibliography), Gloria Wiersma (LaTeX), Qiming Wang (graphics generation, graphics viewers), and Brian Antonishek (graphics viewers).
The editors acknowledge the many other individuals who contributed to the project in a variety of ways. Among the research, technical, and support staff at NIST these are B. K. Alpert, T. M. G. Arrington, R. Bickel, B. Blaser, P. T. Boggs, S. Burley, G. Chu, A. Dienstfrey, M. J. Donahue, K. R. Eberhardt, B. R. Fabijonas, M. Fancher, S. Fletcher, J. Fowler, S. P. Frechette, C. M. Furlani, K. B. Gebbie, C. R. Hagwood, A. N. Heckert, M. Huber, P. K. Janert, R. N. Kacker, R. F. Kayser, P. M. Ketcham, E. Kim, M. J. Lieberman, R. R. Lipman, M. S. Madsen, E. A. P. Mai, W. Mehuron, P. J. Mohr, S. Olver, D. R. Penn, S. Phoha, A. Possolo, S. P. Ressler, M. Rubin, J. Rumble, C. A. Schanzle, B. I. Schneider, N. Sedransk, E. L. Shirley, G. W. Stewart, C. P. Sturrock, G. Thakur, S. Wakid, and S. F. Zevin. Individuals from outside NIST are S. S. Antman, A. M. Ashton, C. M. Bender, J. J. Benedetto, R. L. Bishop, J. M. Borwein, H. W. Braden, C. Brezinski, F. Chyzak, J. N. L. Connor, R. Cools, A. Cuyt, I. Daubechies, P. J. Davis, C. F. Dunkl, J. P. Goedbloed, B. Gordon, J. W. Jenkins, L. H. Kellogg, C. D. Kemp, K. S. Kölbig, S. G. Krantz, M. D. Kruskal, W. Lay, D. A. Lutz, E. L. Mansfield, G. Marsaglia, B. M. McCoy, W. Miller, Jr., M. E. Muldoon, S. P. Novikov, P. J. Olver, W. C. Parke, M. Petkovsek, W. H. Reid, B. Salvy, C. Schneider, M. J. Seaton, N. C. Severo, I. A. Stegun, F. Stenger, M. Steuerwalt, W. G. Strang, P. R. Turner, J. Van Deun, M. Vuorinen, E. J. Weniger, H. Wiersma, R. C. Winther, D. B. Zagier, and M. Zelen. Undoubtedly, the editors have overlooked some individuals who contributed, as is inevitable in a large long-lasting project. Any oversight is unintentional, and the editors apologize in advance.
The project was funded in part by NSF Award 9980036, administered by the NSF’s Knowledge and Distributed Intelligence Program. Within NIST financial resources and staff were committed by the Information Technology Laboratory, Physics Laboratory, Systems Integration for Manufacturing Applications Program of the Manufacturing Engineering Laboratory, Standard Reference Data Program, and Advanced Technology Program.
Notwithstanding the great care that has been exercised by the editors, authors, validators, and the NIST staff, it is almost inevitable that in a work of the magnitude and scope of the NIST Handbook and DLMF errors will still be present. Users need to be aware that none of these individuals nor the National Institute of Standards and Technology can assume responsibility for any possible consequences of such errors.
Lastly, the editors appreciate the skill, and long experience, that was brought to bear by the publisher, Cambridge University Press, on the production and publication of the new Handbook.
Adri B. Olde Daalhuis
Mathematics Editor
Daniel W. Lozier, Barry I. Schneider
General Editors
Ronald F. Boisvert
Editor at Large
Charles W. Clark
Physical Sciences Editor
Bruce R. Miller
Information Technology Editor
Bonita V. Saunders
Visualization Editor
Dr. Adri B. Olde Daalhuis of the University of Edinburgh was named a Mathematics Editor on August 24, 2012. Frank W. J. Olver served as Editor-in-Chief and Mathematics Editor for the DLMF project from its beginning until his death on April 23, 2013. The honorary title Editor-in Chief has been been dropped.