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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
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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.


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  • 1. Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR 0167642
    Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1992. Reprint of the 1972 edition. MR 1225604
  • 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), no. 1, 106–114. MR 1153623, https://doi.org/10.1016/0021-9991(92)90279-8
  • 4. Harold Exton, The asymptotic behaviour of the inhomogeneous Airy function 𝐻𝑖(𝑧), Math. Chronicle 12 (1983), 99–104. MR 706028
  • 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. Soo-Y. Lee, The inhomogeneous Airy functions, 𝐺𝑖(𝑧) and 𝐻𝑖(𝑧), J. Chem. Phys. 72 (1980), no. 1, 332–336. MR 554009, https://doi.org/10.1063/1.438852
  • 8. D. W. Lozier and F. W. J. Olver, Numerical evaluation of special functions, Mathematics of Computation 1943–1993: a half-century of computational mathematics (Vancouver, BC, 1993) Proc. Sympos. Appl. Math., vol. 48, Amer. Math. Soc., Providence, RI, 1994, pp. 79–125. MR 1314844, https://doi.org/10.1090/psapm/048/1314844
  • 9. Allan J. MacLeod, Computation of inhomogeneous Airy functions, J. Comput. Appl. Math. 53 (1994), no. 1, 109–116. MR 1305971, https://doi.org/10.1016/0377-0427(94)90196-1
  • 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 [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Computer Science and Applied Mathematics. MR 0435697
    Frank W. J. Olver, Asymptotics and special functions, AKP Classics, A K Peters, Ltd., Wellesley, MA, 1997. Reprint of the 1974 original [Academic Press, New York; MR0435697 (55 #8655)]. MR 1429619
  • 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 Roy G. Gordon, An algorithm for the evaluation of the complex Airy functions, J. Comput. Phys. 31 (1979), no. 1, 60–75. MR 531124, https://doi.org/10.1016/0021-9991(79)90062-7
  • 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), no. 1, 14–24. MR 1260380, https://doi.org/10.4310/MAA.1994.v1.n1.a2
  • 17. Nico M. Temme, Special functions, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1996. An introduction to the classical functions of mathematical physics. MR 1376370
  • 18. R. Wong, Asymptotic approximations of integrals, Computer Science and Scientific Computing, Academic Press, Inc., Boston, MA, 1989. MR 1016818

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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