Hostname: page-component-5b777bbd6c-6lqsf Total loading time: 0 Render date: 2025-06-18T21:35:37.489Z Has data issue: false hasContentIssue false
  • English
  • Français

The asymptotic solution of differential equations with a turning point and singularities†

Published online by Cambridge University Press:  24 October 2008

R. C. Thorne
R. C. Thorne
Affiliation:
California Institute of Technology Pasadena 4, California
Get access Rights & Permissions [Opens in a new window]

Abstract

Asymptotic solutions of the differential equation

for large positive values of u, are examined; z is a complex variable in a domain Dz in which P1(z) and z2q(z) are regular and p1(z) does not vanish. In this paper it is shown that there exist Airy-type expansions of the solutions of this equation which are valid uniformly with respect to z in a domain in which z = 0 and z = z0 are interior points. If Dz is unbounded and the equation has a regular singularity at infinity, Airy-type expansions exist which are valid at z = 0, z = z0 and z = δ. If p(z) = constant + O (│z│-1) as │ z │ → ∞ in Dz, similar expansions also exist. The results given here are new.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 53 , Issue 2 , April 1957 , pp. 382 - 398
Copyright
Copyright © Cambridge Philosophical Society 1957

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

†

Prepared under contract Nonr-220 (11) between the Office of Naval Research and the California Institute of Technology. Reference no. NR 043-121. Reproduction in whole or in part is permitted for any purpose of the United States Government.

References

REFERENCES

(1) Asymptotic solutions of differential equations with turning points. Tech. Rep. no. 1, under contract Nonr-220 (11) (California Institute of Technology, 1953).Google Scholar
(2) Cherry, T. M. Trans. Amer. Math. Soc. 68 (1950), 224–57.Google Scholar
(3) Erdélyi, A. et al. Higher transcendental functions, vol. 1 (New York, 1953).Google Scholar
(4) Olver, F. W. J. Phil. Trans. A, 247 (1954), 307–27.Google Scholar
(5) Olver, F. W. J. Phil. Trans. A, 247 (1954), 328–68.Google Scholar
(6) Olver, F. W. J. Phil. Trans. A, 249 (1956), 65–97.Google Scholar
(7) Watson, G. N. A treatise on the theory of Bessel functions, 2nd ed. (Cambridge, 1944).Google Scholar