Press here to get the full document in PostScript format.

Press here to get this subdocument in PostScript format.

Numerical Evaluation of Special Functions

D. W. Lozier and F. W. J. Olver

2. Mathematical Developments

Comprehensive compendia of mathematical properties of the special functions are provided by the National Bureau of Standards' Handbook of Mathematical Functions [ AS64] , published originally in 1964, and the 3-volume set that resulted from the Bateman Manuscript Project [ EMOT53a,EMOT53b,EMOT55] , published in 1953 and 1955. These references employ the same notation for the special functions, and we shall follow them. The NBS Handbook has been reprinted many times by the U. S. Government Printing Office and has also been issued in whole, or in part, by other publishers including Dover Publications, Moscow Nauka, Verlag Harri Deutsch and Wiley-Interscience.

The forerunner of [ AS64] is the book of Jahnke and Emde [ JE45] , published originally in 1909, and still in printgif. It continues to be useful, especially for its collection of graphs. Other useful compendia include those of Magnus, Oberhettinger and Soni [ MOS66] , and (from the standpoint of hypergeometric functions) Luke [ Luk69a] . For an introductory compendium, see the recent ``atlas'' of Spanier and Oldham [ SO87] .

Books and articles that include descriptions or surveys of general methods for computing special functions include [ Bre78b,DKK81,Gau75,HCL+68,Luk69b,Luk77b,PT84,PTVF92,Tem78,vdLT84] .

Other books and articles that provide indepth coverage of pertinent topics include:

[ Ask89, survey of compendia] , [ BG81a,BG81b, Padé approximations] , [ BH75, asymptotic approximations] , [ Bre91, continued fractions, Padé approximations] , [ BvI93, Padé approximations] , [ Cod70, polynomial and rational approximations] , [ Fik68, polynomial and rational approximations] , [ FP68, Chebyshev polynomials] , [ JT80, continued fractions] , [ Kar91, power series] , [ KG80, statistical computations] , [ Luk75, supplement to AS64---especially for functions of hypergeometric type] , [ Mor80, power series] , [ Olv74, asymptotic approximations] , [ Riv90, Chebyshev polynomials] , [ Tem77, integral representations] , [ Tem85, asymptotic approximations] , [ Wim84, recurrence relations] , [ Won89, asymptotic approximations] .


M. Abramowitz and I. A. Stegun (eds.), Handbook of mathematical functions with formulas, graphs and mathematical tables, National Bureau of Standards Applied Mathematics Series, vol. 55, U. S. Government Printing Office, Washington, D. C., 1964.

R. A. Askey, Handbooks of special functions, A Century of Mathematics in America, Part III, Hist. Math., vol. 3, American Mathematical Society, Providence, Rhode Island, 1989, pp. 369--391.

G. A. Baker, Jr. and P. Graves-Morris, Padé approximants, part I, Encyclopedia of Mathematics and its Applications, vol. 13, Addison-Wesley Publishing Company, Reading, Massachusetts, 1981.

G. A. Baker, Jr. and P. Graves-Morris, Padé approximants, part II, Encyclopedia of Mathematics and its Applications, vol. 14, Addison-Wesley Publishing Company, Reading, Massachusetts, 1981.

N. Bleistein and R. A. Handelsman, Asymptotic expansions of integrals, Holt, Rinehart and Winston, New York, 1975.

R. P. Brent, A Fortran multiple-precision arithmetic package, ACM Trans. Math. Software 4 (1978), 57--70.

C. Brezinski, History of continued fractions and Padé approximants, Springer Series in Computational Mathematics, vol. 12, Springer-Verlag, Berlin, 1991.

C. Brezinski and J. van Iseghem, Padé approximations, Handbook of Numerical Analysis (P. G. Ciarlet and J. L. Lions, eds.), vol. 3, North-Holland, Amsterdam, 1993, in press.

W. J. Cody, A survey of practical rational and polynomial approximation of functions, SIAM Rev. 12 (1970), 400--423.

V. A. Ditkin, K. A. Karpov, and M. K. Kerimov, The computation of special functions, U.S.S.R. Comput. Math. and Math. Phys. 20 (1981), no. 5, 3--12.

A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher transcendental functions, vol. 1, McGraw-Hill, New York, 1953, reprinted and published in 1981 by Krieger Publishing Company, Melbourne, Florida.

A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher transcendental functions, vol. 2, McGraw-Hill, New York, 1953, reprinted and published in 1981 by Krieger Publishing Company, Melbourne, Florida.

A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher transcendental functions, vol. 3, McGraw-Hill, New York, 1955, reprinted and published in 1981 by Krieger Publishing Company, Melbourne, Florida.

C. T. Fike, Computer evaluation of mathematical functions, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1968.

L. Fox and I. B. Parker, Chebyshev polynomials in numerical analysis, Oxford University Press, London, 1968.

W. Gautschi, Computational methods in special functions---a survey, Theory and application of special functions, Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., Academic Press, New York, 1975, pp. 1--98.

J. F. Hart, E. W. Cheney, C. L. Lawson, H. J. Maehly, C. K. Mesztenyi, J. R. Rice, H. C. Thacher, Jr., and C. Witzgall, Computer approximations, John Wiley and Sons, Inc., New York, 1968.

E. Jahnke and F. Emde, Tables of functions with formulae and curves, fourth ed., Dover Publications, Inc., New York, 1945.

E. Jahnke, F. Emde, and F. Lösch, Tables of higher functions, sixth ed., McGraw-Hill, New York, 1960.

W. B. Jones and W. J. Thron, Continued fractions : analytic theory and applications, Encyclopedia of Mathematics and its Applications, vol. 11, Addison-Wesley Publishing Company, Reading, Massachusetts, 1980.

E. A. Karatsuba, Fast evaluation of transcendental functions, Problems Inform. Transmission 27 (1991), 339--360.

W. J. Kennedy, Jr. and J. E. Gentle, Statistical computing, Marcel Dekker, New York, 1980.

Y. L. Luke, The special functions and their approximations, vol. 1, Academic Press, New York, 1969.

Y. L. Luke, The special functions and their approximations, vol. 2, Academic Press, New York, 1969.

Y. L. Luke, Mathematical functions and their approximations, Academic Press, New York, 1975.

Y. L. Luke, Algorithms for the computation of mathematical functions, Academic Press, New York, 1977.

M. Mori, Analytic representations suitable for numerical computation of some special functions, Numer. Math. 35 (1980), 163--174.

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and theorems for the special functions of mathematical physics, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, vol. 52, Springer-Verlag, New York, 1966.

F. W. J. Olver, Asymptotics and special functions, Academic Press, New York, 1974.

B. A. Popov and G. S. Tesler, Computation of functions on electronic computers---handbook, Naukova Dumka, Kiev, 1984 (Russian), see review by K. S. Kölbig in Math. Comp. v. 55 (1990), pp. 395--397.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipes. The art of scientific computing, second ed., Cambridge University Press, 1992, diskettes and example books available. Editions exist in Basic (1991), C (1992), Fortran (1992), Macintosh Fortran (1988) and Pascal (1989).

T. J. Rivlin, Chebyshev polynomials. From approximation theory to algebra and number theory, second ed., John Wiley and Sons, Inc., New York, 1990.

J. Spanier and K. B. Oldham, An atlas of functions, Hemisphere Publishing Corporation, Washington, D. C., 1987.

N. M. Temme, The numerical computation of special functions by use of quadrature rules for saddle point integrals. I. Trapezoidal integration rules, Tech. Report TW 164/77, Mathematisch Centrum, Amsterdam, 1977.

N. M. Temme, Some aspects of applied analysis : asymptotics, special functions and their numerical computation, Mathematisch Centrum, Amsterdam, 1978.

N. M. Temme, Special functions as approximants in uniform asymptotic expansions of integrals ; A survey, Rend. Sem. Mat. Univ. Politec. Torino Fascicolo Speciale. Special Functions: Theory and Computation (1985), 289--317.

C. G. van der Laan and N. M. Temme, Calculation of special functions : The gamma function, the exponential integrals and error-like functions, CWI Tract, vol. 10, Centrum voor Wiskunde en Informatica, Amsterdam, 1984.

J. Wimp, Computation with recurrence relations, Pitman, London, 1984.

R. Wong, Asymptotic approximations of integrals, Academic Press, New York, 1989.


This document is an excerpt from the current hypertext version of an article that appeared in Walter Gautschi (ed.), Mathematics of Computation 1943--1993: A Half-Century of Computational Mathematics, Proceedings of Symposia in Applied Mathematics 48, American Mathematical Society, Providence, RI 02940, 1994. The symposium was held at the University of British Columbia August 9--13, 1993, in honor of the fiftieth anniversary of the journal Mathematics of Computation.

The original abstract follows.

Higher transcendental functions continue to play varied and important roles in investigations by engineers, mathematicians, scientists and statisticians. The purpose of this paper is to assist in locating useful approximations and software for the numerical generation of these functions, and to offer some suggestions for future developments in this field.

Applied and Computational Mathematics Division, National Institute of Standards and Technology, Gaithersburg, Md 20899

E-mail address:

Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742

E-mail address:

The research of the second author has been supported by NSF Grant CCR 89-14933.

1991 Mathematics Subject Classification. Primary 65D20; Secondary 33-00.

A more recent edition, with F. Lösch added as author [ JEL60] , is no longer in print.

Daniel W Lozier
Fri Apr 7 13:32:38 EDT 1995