Mathematics of Computation

Author(s): Amparo Gil; Javier Segura; Nico M. Temme.
Journal: Math. Comp. 70 (2001), 1183-1194.
MSC (2000): Primary 33C10, 41A60, 30E10, 65D20
Posted: April 13, 2000
Retrieve article in: PDF
This article is available free of charge

Integral representations are considered of solutions of the inhomogeneous Airy differential equation $w''-z\,w=\pm1/\pi$ . The solutions of these equations are also known as Scorer functions. Certain functional relations for these functions are used to confine the discussion to one function and to a certain sector in the complex plane. By using steepest descent methods from asymptotics, the standard integral representations of the Scorer functions are modified in order to obtain nonoscillating integrals for complex values of

. In this way stable representations for numerical evaluations of the functions are obtained. The methods are illustrated with numerical results.

1.

M. Abramowitz and I.A. Stegun (Eds.), Handbook of Mathematical functions, National Bureau of Standards Applied Mathematics Series No. 55. U.S. Government Printing Office, Washington, DC; many editions. MR 29:4914; MR 94b:00012; etc.

2.

D.E. Amos. ``Algorithm 644: A portable package for Bessel functions of a complex argument and nonnegative order''. ACM Trans. Math. Softw. 12 (1986) 265-273. CMP 19:12

3.

R.M. Corless, D.J. Jeffrey and H. Rasmussen ``Numerical evaluation of Airy functions with complex arguments''. J. Comput. Phys. 99 (1992), 106-114. MR 92k:65028

4.

H. Exton. ``The asymptotic behaviour of the inhomogeneous Airy function $\operatorname{Hi}(z)$ ''. Math. Chronicle 12 (1983), 99-104. MR 84g:33004

5.

B. Fabijonas ``The computation of Scorer functions''. Lecture during the 1998 Annual SIAM Meeting in Toronto, Canada.

6.

GAMS: Guide to available mathematical software. http://gams.nist.gov

7.

S.-Y. Lee ``The inhomogeneous Airy functions, $\operatorname{Gi}(z)$ and $\operatorname{Hi}(z)$ ''. J. Chem. Phys. 72 (1980), 332-336. MR 81b:33015

8.

D.W. Lozier and F.W.J. Olver. ``Numerical evaluation of special functions''. In W. Gautschi (Ed.), AMS Proceedings of Symposia in Applied Mathematics 48 (1998), pp. 79-125. MR 95m:65036

9.

A.J. MacLeod. ``Computation of inhomogeneous Airy functions''. J. Comput. Appl. Math. 53 (1994) 109-116. MR 95k:65023

10.

The National Institute of Standards and Technology has a public web site that includes an extensive treatment of Scorer functions: http://www.nist.gov/DigitalMathLib.

11.

F.W.J. Olver. Asymptotics and Special Functions. Academic Press, New York. Reprinted in 1997 by A.K. Peters. MR 55:8655; MR 97i:41001

12.

R.B. Paris and A.D. Wood. ``Stokes phenomenon demystified'', IMA Bulletin 31 (1995) No.1-2,21-28. CMP 95:10

13.

SLATEC Public Domain Mathematical Library.

gopher://archives.math.utk.edu/11/software/multi-platform/SLATEC

14.

Z. Schulten, D.G.M. Anderson, and R.G. Gordon. ``An algorithm for the evaluation of complex Airy functions''. J. Comput. Phys. 31 (1979) 60-75. MR 80c:65043

15.

R.S. Scorer. ``Numerical evaluation of integrals of the form $I=\int_{x_1}^{x_2}\,f(x)e^{i\phi(x)}\,dx$ and the tabulation of the function $\operatorname{Gi}(z)=(1/\pi) \int_{0}^{\infty}\,\sin\left(uz+\frac{1}{3}u^3\right)\,du$ ''. Quart. J. Mech. Appl. Math. 3 (1950) 107-112. MR 12:287c

16.

N.M. Temme. ``Steepest descent paths for integrals defining the modified Bessel functions of imaginary order''. Methods Appl. Anal. 1 (1994) 14-24. MR 95g:33006

17.

N.M. Temme. Special functions: An introduction to the classical functions of mathematical physics. John Wiley & Sons, New York, 1996. MR 97e:33002

18.

R. Wong Asymptotic approximations of integrals. Academic Press, New York, 1989. MR 90j:41061

Retrieve articles in Mathematics of Computation with MSC (2000): 33C10, 41A60, 30E10, 65D20

Amparo Gil
Affiliation: Instituto de Bioingeniería, Universidad Miguel Hernández, Edificio La Galia. 03202-Elche (Alicante), Spain
Email: amparo@titan.ific.uv.es

Javier Segura
Affiliation: Instituto de Bioingeniería, Universidad Miguel Hernández, Edificio La Galia. 03202-Elche (Alicante), Spain
Email: segura@flamenco.ific.uv.es

Nico M. Temme
Affiliation: CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands
Email: nicot@cwi.nl

DOI: 10.1090/S0025-5718-00-01268-0
PII: S 0025-5718(00)01268-0
Keywords: Inhomogeneous Airy functions, Scorer functions, method of steepest descent, saddle point method, numerical computation of special functions
Received by editor(s): September 11, 1998
Received by editor(s) in revised form: April 27, 1999 and August 25, 1999
Posted: April 13, 2000
Copyright of article: Copyright 2000, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6842(e) ISSN 0025-5718(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS