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Numerical Evaluation of Special Functions

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

5.4. Fermi-Dirac, Bose-Einstein, and Debye Integrals .

This section includes the Lerch transcendent.

5.4.1. Real Parameter and Argument.


[ CT67] , [ NDT69] .

Software Packages:

[ BDM81, Fortran] , [ FR86, Fortran] .

Intermediate Libraries:

[ Bak92] .

5.4.2. Complex Argument and/or Parameters.

Interactive Systems:


5.4.3. Articles.

[ Bui91] , [ Gau93a] , [ Gau93b] , [ LS91] , [ NM93] , [ Pas88] , [ Pas91] , [ Pic89] , [ Sag91a] , [ Sag91b] .


L. Baker, C mathematical function handbook, McGraw-Hill, Inc., New York, 1992, includes diskette.

A. Bañuelos, R. A. Depine, and R. C. Mancini, A program for computing the Fermi-Dirac functions, Comput. Phys. Comm. 21 (1981), 315--322.

Bui Doan Khanh, A computation of the Fermi-Dirac integrals by asymptotics and the Hermite corrector formula, Appl. Math. Comput. 41 (1991), 61--68.

W. J. Cody and H. C. Thacher, Jr., Rational Chebyshev approximations for Fermi-Dirac integrals of orders , and , Math. Comp. 21 (1967), 30--40.

L. W. Fullerton and G. A. Rinker, Generalized Fermi-Dirac integrals---FD, FDG, FDH, Comput. Phys. Comm. 39 (1986), 181--185.

W. Gautschi, Gauss-type quadrature rules for rational functions, International Series of Numerical Mathematics, vol. 112, Birkhäuser Verlag, Basel, 1993, pp. 111--130.

W. Gautschi, On the computation of generalized Fermi-Dirac and Bose-Einstein integrals, Comput. Phys. Comm. 74 (1993), 233--238.

A. I. Litvin and S. D. Simonzhenkov, Computation of Fermi-Dirac functions, Comput. Math. and Math. Phys. 31 (1991), no. 8, 100--103.

E. W. Ng, C. J. Devine, and R. F. Tooper, Chebyshev polynomial expansion of Bose-Einstein functions of orders 1 to 10, Math. Comp. 23 (1969), 639--643.

A. Natarajan and N. Mohan Kumar, On the numerical evaluation of the generalised Fermi-Dirac integrals, Comput. Phys. Comm. 76 (1993), 48--50.

S. Paszkowski, Evaluation of Fermi-Dirac integral, Nonlinear Numerical Methods and Rational Approximation (A. Cuyt, ed.), D. Reidel Publishing Company, Dordrecht, 1988, pp. 435--444.

S. Paszkowski, Evaluation of the Fermi-Dirac integral of half-integer order, Zastos. Mat. 21 (1991), 289--301.

B. Pichon, Numerical calculation of the generalized Fermi-Dirac integrals, Comput. Phys. Comm. 55 (1989), 127--136.

R. P. Sagar, A Gaussian quadrature for the calculation of generalized Fermi-Dirac integrals, Comput. Phys. Comm. 66 (1991), 271--275.

R. P. Sagar, On the evaluation of the Fermi-Dirac integrals, Astrophys. J. 376 (1991), 364--366.


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.

Daniel W Lozier
Fri Apr 7 14:11:23 EDT 1995