Abramowitz and Stegun

Abramowitz and Stegun (AS) is the informal name of a mathematical reference work edited by Milton Abramowitz and Irene Stegun of the United States National Bureau of Standards (NBS), now the National Institute of Standards and Technology (NIST). Its full title is Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. A digital successor to the Handbook was released as the "Digital Library of Mathematical Functions" (DLMF) on May 11, 2010, along with a printed version, the NIST Handbook of Mathematical Functions, published by Cambridge University Press.[1]

Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
Page 97 showing part of a table of common logarithms
AuthorMilton Abramowitz and Irene Stegun
CountryUnited States
LanguageEnglish
GenreMath
PublisherUnited States Department of Commerce, National Bureau of Standards (NBS)
Publication date
1964
ISBN0-486-61272-4
OCLC18003605

Overview

Since it was first published in 1964, the 1046 page Handbook has been one of the most comprehensive sources of information on special functions, containing definitions, identities, approximations, plots, and tables of values of numerous functions used in virtually all fields of applied mathematics.[2][3] The notation used in the Handbook is the de facto standard for much of applied mathematics today.

At the time of its publication, the Handbook was an essential resource for practitioners. Nowadays, computer algebra systems have replaced the function tables, but the Handbook remains an important reference source. The foreword discusses a meeting in 1954 in which it was agreed that "the advent of high-speed computing equipment changed the task of table making but definitely did not remove the need for tables".

More than 1,000 pages long, the Handbook of Mathematical Functions was first published in 1964 and reprinted many times, with yet another reprint in 1999. Its influence on science and engineering is evidenced by its popularity. In fact, when New Scientist magazine recently asked some of the world’s leading scientists what single book they would want if stranded on a desert island, one distinguished British physicist[4] said he would take the Handbook. The Handbook is likely the most widely distributed and most cited NIST technical publication of all time. Government sales exceed 150,000 copies, and an estimated three times as many have been reprinted and sold by commercial publishers since 1965. During the mid-1990s, the book was cited every 1.5 hours of each working day. And its influence will persist as it is currently being updated in digital format by NIST.

— NIST[5]

The chapters are:

  1. Mathematical Constants
  2. Physical Constants and Conversion Factors
  3. Elementary Analytical Methods
  4. Elementary Transcendental Functions
  5. Exponential Integral and Related Functions
  6. Gamma Function and Related Functions
  7. Error Function and Fresnel Integrals
  8. Legendre Functions
  9. Bessel Functions of Integral Order
  10. Bessel Functions of Fractional Order
  11. Integrals of Bessel Functions
  12. Struve Functions and Related Functions
  13. Confluent Hypergeometric Functions
  14. Coulomb Wave Functions
  15. Hypergeometric Functions
  16. Jacobian Elliptic Functions and Theta Functions
  17. Elliptic Integrals
  18. Weierstrass Elliptic and Related Functions
  19. Parabolic Cylinder Functions
  20. Mathieu Functions
  21. Spheroidal Wave Functions
  22. Orthogonal Polynomials
  23. Bernoulli and Euler Polynomials, Riemann Zeta Function
  24. Combinatorial Analysis
  25. Numerical Interpolation, Differentiation, and Integration
  26. Probability Functions
  27. Miscellaneous Functions
  28. Scales of Notation
  29. Laplace Transforms

Editions

Because the Handbook is the work of U.S. federal government employees acting in their official capacity, it is not protected by copyright in the United States. While it could be ordered from the Government Printing Office, it has also been reprinted by commercial publishers, most notably Dover Publications (ISBN 0-486-61272-4), and can be legally viewed on and downloaded from the web.

While there was only one edition of the work, it went through many print runs including a growing number of corrections.

Original NBS edition:

  • 1st printing: June 1964; errata:[6][7][8][9][10][11][12]
  • 2nd printing with corrections: November 1964; errata:[10]
  • 3rd printing with corrections: March 1965; errata:[10][12]
  • 4th printing with corrections: December 1965; errata:[10][12]
  • 5th printing with corrections: August 1966
  • 6th printing with corrections: November 1967
  • 7th printing with corrections: May 1968
  • 8th printing with corrections: 1969
  • 9th printing with corrections: November 1970
  • 10th printing with corrections: December 1972

Reprint edition by Dover Publications:

  • 1st printing: 1965
  • ?
  • 9th printing with additional corrections (based on 10th printing of NBS edition with corrections)

Michael Danos and Johann Rafelski edited the Pocketbook of Mathematical Functions, published by Verlag Harri Deutsch in 1984.[13][14] The book is an abridged version of Abramowitz's and Stegun's Handbook, retaining most of the formulas (except for the first and the two last original chapters, which were dropped), but reducing the numerical tables to a minimum,[13] which, by this time, could be easily calculated with scientific pocket calculators.[14] The references were removed as well.[14] Most known errata were incorporated, the physical constants updated and the now-first chapter saw some slight enlargement compared to the former second chapter.[14] The numbering of formulas was kept for easier cross-reference.[14]

A digital successor to the Handbook, long under development at NIST, was released as the “Digital Library of Mathematical Functions” (DLMF) on May 11, 2010, along with a printed version, the NIST Handbook of Mathematical Functions, published by Cambridge University Press.[1]

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See also

References
  1. Olver, Frank William John; Lozier, Daniel W.; Boisvert, Ronald F.; Clark, Charles W., eds. (2010). NIST Handbook of Mathematical Functions. National Institute of Standards and Technology (NIST), U.S. Department of Commerce, Cambridge University Press. ISBN 978-0-521-19225-5. MR 2723248.
  2. Wrench, Jr., John William (April 1965). "Reviews and Descriptions of Tables and Books 1: Milton Abramowitz & Irene A. Stegun (editors). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables 1964" (PDF). Mathematics of Computation. 19 (89): 147–149. doi:10.2307/2004114. eISSN 1088-6842. ISSN 0025-5718. JSTOR 2004114. MSC:A-F,H-M,R,SX. Retrieved 2016-03-13.
  3. Ronald F., Boisvert; Clark, Charles W.; Lozier, Daniel W.; Olver, Frank William John (2011). "A Special Functions Handbook for the Digital Age" (PDF). Notices of the American Mathematical Society (NAMS). 58 (7): 905–911. Retrieved 2016-03-13.
  4. Berry, Michael (1997-11-22). "Christmas books: ... Or maybe two or three ..." New Scientist. Retrieved 2016-03-13.
  5. "NIST at 100: Foundations for Progress, 1964: Mathematics Handbook Becomes Best Seller". 2001. Archived from the original on 2016-03-13. Retrieved 2016-03-13.
  6. Cloud, J. D.; Denman, H. H.; Wrench, Jr., John William (April 1965). "Table Errata 362: Milton Abramowitz & Irene A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D. C, 1964" (PDF). Mathematics of Computation. 19 (89): 174–175. eISSN 1088-6842. ISSN 0025-5718. JSTOR 2004148. MTE:362. Retrieved 2016-03-13.
  7. Stegun, Irene Ann; Aggarwal, H. R.; Ablow, C. M. (April 1965). "Table Errata 365: Milton Abramowitz & Irene A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D. C, 1964" (PDF). Mathematics of Computation. 19 (90): 360–363. eISSN 1088-6842. ISSN 0025-5718. JSTOR 2003399. MTE:365. Retrieved 2016-03-13.
  8. Thacher, Jr., Henry C. (July 1965). "Errata 373: Milton Abramowitz & Irene A. Stegun, Editors, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1964" (PDF). Mathematics of Computation. 19 (91): 527–528. doi:10.2307/2003725. eISSN 1088-6842. ISSN 0025-5718. JSTOR 2003725. MTE:373. Retrieved 2016-03-13.
  9. Manning, Irwin; Templeton, J. G. C.; Thacher, Jr., Henry C. (October 1965). "Errata 376: Milton Abramowitz & Irene A. Stegun, Editors, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1964" (PDF). Mathematics of Computation. 19 (92): 705–706. doi:10.2307/2003991. eISSN 1088-6842. ISSN 0025-5718. JSTOR 2003991. MTE:376. Retrieved 2016-03-13.
  10. Shanks, Daniel; Greenebaum, Michael (January 1966). "Errata 379: Milton Abramowitz & Irene A. Stegun, Editors, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1964" (PDF). Mathematics of Computation. 20 (93): 202–206. eISSN 1088-6842. ISSN 0025-5718. JSTOR 2004328. MTE:379. Retrieved 2016-03-13.
  11. Wood, Van E.; Gamper, René (April 1966). "Errata 388: Milton Abramowitz & Irene A. Stegun, Editors, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1964" (PDF). Mathematics of Computation. 20 (94): 344. eISSN 1088-6842. ISSN 0025-5718. JSTOR 2003533. MTE:388. Retrieved 2016-03-13.
  12. Salzer, Herbert E.; Johnson, Robert S.; Fettis, Henry E.; Caslin, James C.; Concus, Paul (July 1966). "Errata 393: Milton Abramowitz & Irene A. Stegun, Editors, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1964" (PDF). Mathematics of Computation. 20 (95): 469–471. doi:10.2307/2003630. eISSN 1088-6842. ISSN 0025-5718. JSTOR 2003630. MTE:393. Retrieved 2016-03-13.
  13. Danos, Michael; Rafelski, Johann; Abramowitz, Milton; Stegun, Irene Ann, eds. (1984). Pocketbook of Mathematical Functions. Thun, Frankfurt/Main: Verlag Harri Deutsch. ISBN 978-3-87144-818-8. Retrieved 2016-03-13. (468 pages)
  14. Wrench, Jr., John William (April 1988). "Reviews and Descriptions of Tables and Books 18: Milton Abramowitz and Irene A. Stegun. Pocketbook of Mathematical Functions - Abridged Edition of Handbook of Mathematical Functions. Material selected by Michael Danos and Johann Rafelski" (PDF). Mathematics of Computation. 50 (182): 646–647. doi:10.2307/2008636. eISSN 1088-6842. ISSN 0025-5718. JSTOR 2008636. MSC:33-00,65A05. Retrieved 2016-03-13.
  15. "Boole's Rule - from Wolfram MathWorld". Mathworld.wolfram.com. 2009-10-27. Retrieved 2009-11-13.
  16. Zucker, Ruth (1983) [June 1964]. "Chapter 25.4.14: Numerical Interpolation, Differentiation, and Integration - Integration - Numerical Analysis". In Abramowitz, Milton; Stegun, Irene Ann (eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 886. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.

Further reading
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