Handbook of Mathematical Functions
The Handbook of Mathematical Functions with
Formulas, Graphs, and Mathematical Tables [1] was the
culmination of a quarter century of NBS work on core
mathematical tools. Evaluating commonly occurring
mathematical functions has been a fundamental need as
long as mathematics has been applied to the solution of
practical problems. In 1938, NBS initiated its Mathe-
matical Tables Project to satisfy the increasing demand
for extensive and accurate tables of functions [2].
Located in New York and administered by the Works
Projects Administration, the project employed not only
mathematicians, but also a large number of additional
staff who carried out hand computations necessary to
produce tables. From 1938 until 1946, 37 volumes of the
NBS Math Tables Series were issued, containing tables
of trigonometric functions, the exponential function,
natural logarithms, probability functions, and related
interpolation formulae. In 1947, the Math Tables Project
was moved to Washington to form the Computation
Laboratory of the new National Applied Mathematics
Laboratories of NBS. Many more tables subsequently
were published in the NBS Applied Mathematics Series;
the first of these, containing tables of Bessel functions
[3], appeared in 1948.
On May 15, 1952, the NBS Applied Mathematics
Division convened a Conference on Tables. Milton
Abramowitz of NBS, who had been a member of the
technical planning staff for the Math Tables Project,
described preliminary plans for a compendium of
mathematical tables and related material. Abramowitz
indicated that the Bureau was in need of both technical
advice and financial support to carry out the project.
With the support of the National Science Foundation
(NSF), a two-day Conference on Tables was held at the
Massachusetts Institute of Technology on September
15-16, 1954, to discuss the prospects for such an under-
taking. Twenty-eight persons attended, including both
table producers and users from the science and engineer-
ing community. The report of the conference concluded
that
“an outstanding need is for a Handbook of
Tables for the Occasional Computer, with
tables of usually encountered functions and a
set of formulas and tables for interpolation
and other techniques useful to the occasional
computer.”
(Note that here the term computer refers to a person
performing a calculation by hand.) The report recom-
mended that NBS manage the production of the Hand
book and that NSF provide financial assistance. The
conference elected the following committee to carry
out its recommendations: P.M. Morse (Chair),
M. Abramowitz, J.H. Curtiss, R.W. Hamming, D. H.
Lehmer, C.B. Tompkins, and J.W. Tukey. The commit-
tee was successful in persuading both NBS and NSF to
of 1956.
The Mathematics Division of the National Research
Council also had an interest in mathematical tables.
Since 1943, they had been publishing a quarterly
journal entitled Mathematical Tables and Other Aids to
Computation ( today known as Mathematics of Compu-
tation). To provide technical assistance to NBS, as well
as independent oversight for NSF, the NRC established
a Committee on Revision of Mathematical Tables. Its
members were P. M. Morse (Chair), A. Erde´lyi, M. C.
Fig. 1. Portrait of Milton Abramowitz.
135
Gray, N. C. Metropolis, J. B. Rosser, H. C. Thacher, Jr.,
John Todd, C. B. Tompkins, and J. W. Tukey. This group
of luminaries in the fields of applied mathematics
and physics provided guidance to NBS throughout the
project to produce the Handbook.
Milton Abramowitz, who was then Chief of the
Computation Laboratory of the NBS Applied Mathe-
matics Division, led the project. Abramowitz was born
in Brooklyn, NY, in 1915. He received a B. A. from
Brooklyn College in 1937 and an M. A. in 1940.
He joined the NBS Math Tables Project in 1938 and in
1948 received a Ph.D. in Mathematics from New York
University. Abramowitz’ dedication, enthusiasm, and
boundless energy led to substantial progress in the
project during its first year. The proposed outline for the
Handbook called for a series of some 20 chapters, each
with a separate author. Authors were drawn from
NBS staff and guest researchers, as well as external
researchers working under contract. Most chapters
would focus on a particular class of functions, providing
formulas, graphs, and tables. Listed formulas would
include differential equations, definite and indefinite
integrals, inequalities, recurrence relations, power
series, asymptotic expansions, and polynomial and
rational approximations. Material would be carefully
selected in order to provide information most important
in applications, especially in physics. Consequently,
the higher mathematical functions, such as Bessel
functions, hypergeometric functions, and elliptic
functions, would form the core of the work. Additional
chapters would provide background on interpolation in
tables and related numerical methods for differentiation
and quadrature.
Philip J. Davis of NBS first prepared Chapter 6, on
the gamma and related functions, to serve as a model for
other authors. This chapter portrayed the telegraphic
style that is a hallmark of the Handbook, i.e., the
material is displayed with a minimum of textual descrip-
tion. In the course of developing his chapter, Davis
became interested in the history of the topic. This led to
a historical profile published in 1959 [4], which won
the prestigious Chauvenet Prize for distinguished
mathematical exposition from the Mathematical
Association of America.
The Handbook project occurred during the period
when general-purpose electronic computing machinery
was first coming into use in government research
laboratories. (Early computer development of SEAC at
NBS is described elsewhere in this volume.) Never-
theless, most of the tables in the Handbook were gener-
ated by hand on desk calculators. However, even at that
time it was clear to the developers of the Handbook
that the need for tables themselves would eventually be
superseded by computer programs which could evaluate
functions for specified arguments on demand.
By the summer of 1958, substantial work had been
completed on the project. Twelve chapters had been
completed, and the remaining ones were well underway.
The project experienced a shocking setback one week-
end in July 1958 when Abramowitz suffered a heart
attack and died. Irene Stegun, who was Assistant Chief
of the Computation Laboratory, took over management
1919, had received an M. A. from Columbia University
in 1941, and joined NBS in 1943. The exacting work of
assembling the many chapters, checking tables and
formulas, and preparing the work for printing took
much longer than anticipated. Nevertheless, the
Handbook of Mathematical Functions, with Formulas,
Graphs, and Mathematical Tables was finally issued as
Applied Mathematics Series Number 55 in June 1964
[1]. The volume, which is still in print at the U.S.
Government Printing Office and stocked by many
bookstores and online booksellers, is 1046 pages in
length. The chapters and authors are as follows.
Fig. 2. Portrait of Irene Stegun.
136
1. Mathematical Constants, D. S. Liepman.
2. Physical Constants and Conversion Factors, A. G.
McNish.
3. Elementary Analytical Methods, M. Abramowitz.
4. Elementary Transcendental Functions, R. Zucker.
5. Exponential Integral and Related Functions, W.
Gautschi (American University) and William F.
Cahill.
6. Gamma Function and Related Functions, P. J .
Davis.
7. Error Function and Fresnel Integrals, W. Gautschi
(American University).
8. Legendre Functions, I. A. Stegun.
9. Bessel Functions of Integer Order, F. W. J. Olver.
10. Bessel Functions of Fractional Order, H. A.
Antosiewicz.
11. Integrals of Bessel Functions, Y. L. Luke.
12. Struve Functions and Related Functions, M.
Abramowitz.
13. Confluent Hypergeometric Functions, L. J. Slater
(Cambridge University).
14. Coulomb Wave Functions, M. Abramowitz.
15. Hypergeometric Functions, F. Oberhettinger.
16. Jacobian Elliptic Functions and Theta Functions, L.
M. Milne-Thomson (University of Arizona).
17. Elliptic Integrals, L. M. Milne-Thomson (Univer-
sity of Arizona).
18. Weierstrass Elliptic and Related Functions, T. H.
Southard.
19. Parabolic Cylinder Functions, J. C. P. Miller (Cam-
bridge University).
20. Mathieu Functions, G. Blanch (Wright-Patterson
Air Force Base).
21. Spheroidal Wave Functions, A. N. Lowan (Yeshiva
University).
22. Orthogonal Polynomials, U. W. Hochstrasser
(American University).
23. Bernoulli and Euler Polynomials—Riemann Zeta
Function, E. V. Haynsworth and K. Goldberg.
24. Combinatorial Analysis, K. Goldberg, M. Newman,
and E. Haynsworth.
25. Numerical Interpolation, Differentiation, and Inte-
gration, P. J. Davis and I. Polonsky.
26. Probability Functions, M. Zelen and N. C. Severo
27. Miscellaneous Functions, I. A. Stegun.
28. Scales of Notation, S. Peavy and A. Schopf
(American University).
29. Laplace Transforms.
The public reaction to the publication of the Hand-
book was overwhelmingly positive. In a preface to the
ninth printing in November 1970, NBS Director Lewis
Branscomb wrote
“The enthusiastic reception accorded the
‘Handbook of Mathematical Functions’ is
little short of unprecedented in the long
history of mathematical tables that began
when John Napier published his tables of
logarithms in 1614. Only four and one-half
years after the first copy came from the press
in 1964, Myron Tribus, the Assistant Secre-
tary for Commerce for Science and Technol-
ogy, presented the 100,000th copy of the
Handbook to Lee A. DuBridge, then Science
Advisor to the President.”
The Handbook has had enormous impact on science
and engineering. Likely the most widely distributed
NBS/NIST technical publication of all time, the govern-
ment edition has never gone out of print, and it has
appeared as a Dover reprint since 1965. It has been
reprinted (in all or part) by other publishers, such as
Moscow Nauka, Verlag Harri Deutsch, and Wiley Inter-
science. Government sales exceed 150,000 copies, with
commercial sales estimated at three to six times this
number. The Handbook’s citation record is also remark-
able. More than 23,000 citations have been logged by
Science Citation Index (SCI) since 1973. Remarkably,
the number of citations to the Handbook continues
to grow, not only in absolute numbers, but also as a
Fig. 3. Photograph of Handbook.
137
fraction of the total number of citations made in the
sciences and engineering. During the mid-1990s, for
example, about once every 1.5 hours of each working
day some author, somewhere, made sufficient use of the
Handbook to list it as a reference. The success of the
Handbook was due to several factors. It collected in one
place, and in a well-organized way, the most important
information needed to make use of mathematical func-
tions in practical applications. It served to standardize
notations and normalizations for the special functions of
applied mathematics, thus easing the communication of
scientific results. In 1965, Irene Stegun was awarded a
Gold Medal from the Department of Commerce for her
efforts in completing the project.
A number of difficult mathematical problems that
emerged in the course of developing the Handbook
engaged researchers in the NBS Applied Mathematics
Division for a number of years after its publication.
to do with stability of computations and the second with
precision.
Mathematical functions often satisfy recurrence
relations (difference equations) that have great potential
for use in computations. However, if used improperly,
recurrence relations can quickly lead to ruinous errors.
This phenomenon, known as instability, has tripped up
many a computation that appeared, superficially, to be
straightforward. The errors are the result of subtle
interactions in the set of all possible solutions of the
difference equation. Frank Olver, who wrote the
Handbook’s chapter on Bessel functions of integer
order, studied this problem in great detail. In a paper
published in 1967 [5], Olver provided the first (and only)
stable algorithm for computing all types of solutions of
a difference equation with three different kinds of
behavior: strongly growing, strongly decaying, and
showing moderate growth or decay. Part of the impact of
this work is reflected today in the existence of robust
software for higher mathematical functions. Olver
worked on such topics in the Mathematical Analysis
Division of NBS, and this work provided the foundation
for his very influential later book on asymptotic analysis
Fig. 4. Screen shot of the NIST Digital Library of Mathematical Functions.
138
and special functions [6]. This book has been cited more
than 800 times, according to SCI.
Another important problem in mathematical compu-
tation is the catastrophic loss of significance caused by
the fixed length requirement for numbers stored in
computer memory. Morris Newman, who co-authored
the Handbook’s chapter on combinatorial analysis,
sought to remedy this situation. He proposed storing
numbers in a computer as integers and performing oper-
ations on them exactly. This contrasts with the standard
approach in which rounding errors accumulate with
each arithmetic operation. Newman’s approach had its
roots in classical number theory: First perform the
computations modulo a selected set of small prime
numbers, where the number of primes required is deter-
mined by the problem. These computations furnish
a number of local solutions, done using computer
numbers represented in the normal way. At the end, only
one multilength computation is required to construct the
global solution (the exact answer) by means of the
Chinese Remainder Theorem. This technique was first
described in a paper by Newman in 1967 [7]; it was
employed with great success in computing and checking
the tables in Chapter 24 of the Handbook. Today, this
technique remains a standard method by which exact
computations are performed. Newman’s research on this
and other topics, performed at NBS, formed the basis
for his 1972 book [8], which quickly became a standard
reference in the applications of number theory to
computation.
Research into the functions of applied mathematics
has continued actively in the 36 years since the Hand-
book appeared. New functions have emerged in impor-
tance, and new properties of well-known functions have
been discovered. In spite of the fact that sophisticated
numerical methods have been embodied in well-
designed commercial software for many functions, there
continues to be a need for a compendium of information
this need, NIST is currently developing a successor to
the Handbook to be known as the Digital Library of
Mathematical Functions (DLMF) [9]. Based upon a
completely new survey of the literature, the DLMF will
provide reference data in the style of the Handbook
in a freely available online format, with sophisticated
mathematical search facilities and interactive three-
dimensional graphics.
Prepared by Ronald F. Boisvert and Daniel W. Lozier.
Bibliography
[1] Milton Abramowitz and Irene A. Stegun, eds., Handbook of Math-
ematical Functions With Formulas, Graphs, and Mathematical
Tables, NBS Applied Mathematics Series 55, National Bureau of
Standards, Washington, DC (1964).
[2] Arnold N. Lowan, The Computation Laboratory of the National
Bureau of Standards, Scripta Math. 15, 33-63 (1949).
[3] Tables of the Bessel Functions Y
0
(x), Y
1
(x), K
0
(x), K
1
(x) 0
<
=
x
<
=
1,
NBS Applied Mathematics Series 1, National Bureau of Stan-
dards, Washington, DC (1948).
[4] Philip J. Davis, Leonhard Euler’s Integral: A Historical Profile of
the Gamma Function, Am. Math. Monthly 66, 849-869 (1959).
[5] F. W. J. Olver, Numerical Solution of Second-Order Linear Differ-
ence Equations, J. Res. Natl. Bur. Stand. 71B, 111-129 (1967).
[6] F. W. J. Olver, Asymptotics and Special Functions, Academic
Press, New York (1974). Reprinted by A. K. Peters, Wellesley, MA
(1997).
[7] Morris Newman, Solving Equations Exactly, J. Res. Natl. Bur.
Stand. 71B, 171-179 (1967).
[8] Morris Newman, Integral Matrices, Academic Press, New York
(1972).
[9] D. Lozier, F. W. J. Olver, C. Clark, and R. Boisvert (eds.),
Digital Library of Mathematical Functions, (http://dlmf.nist.gov)
National Institute of Standards and Technology, .
139
