Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

On nonoscillating integrals for computing inhomogeneous Airy functions


Authors: Amparo Gil, Javier Segura and Nico M. Temme
Journal: Math. Comp. 70 (2001), 1183-1194
MSC (2000): Primary 33C10, 41A60, 30E10, 65D20
DOI: https://doi.org/10.1090/S0025-5718-00-01268-0
Published electronically: April 13, 2000
MathSciNet review: 1826580
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract:

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


References [Enhancements On Off] (What's this?)

  • 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

Similar Articles

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

Retrieve articles in all journals with MSC (2000): 33C10, 41A60, 30E10, 65D20


Additional Information

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: https://doi.org/10.1090/S0025-5718-00-01268-0
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
Published electronically: April 13, 2000
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society