Hostname: page-component-745bb68f8f-grxwn Total loading time: 0 Render date: 2025-02-10T20:56:24.716Z Has data issue: false hasContentIssue false
  • English
  • Français

Integrals with a large parameter: Legendre functions of large degree and fixed order

Published online by Cambridge University Press:  24 October 2008

F. Ursell
F. Ursell
Affiliation:
Department of Mathematics, University of Manchester, Manchester M13 9PL
Get access Rights & Permissions [Opens in a new window]

Abstract

Suppose that a function f(z, n) depends on a large parameter n. A proposed expression g(z, n) is an asymptotic approximation for f(z, n) if it can be shown that the error (i.e. the difference between f(z, n) and g(z, n)) is small of an appropriate order when n → ∞. Effective error bounds are particularly useful in numerical work with asymptotic expansions. Most of the existing derivations of error bounds involve complicated calculations.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 95 , Issue 2 , March 1984 , pp. 367 - 380
Copyright
Copyright © Cambridge Philosophical Society 1984

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.)

References

REFERENCES

[1]A., Erdélyi (ed.) Higher Transcendental Functions: Bateman Manuscript Project. (McGraw-Hill, 1953.)Google Scholar
[2]Martin, J.. Integrals with a large parameter and several nearly coincident saddle points; the continuation of uniformly asymptotic expansions. Proc. Cambridge Philos. Soc. 76 (1974), 211–231.CrossRefGoogle Scholar
[3]Olver, F. W. J.. Asymptotics and Special Functions. (Academic Press, 1974.)Google Scholar
[4]Szegö, G.. Über einige asymptotische Entwicklungen der Legendreschen Funktionen. Proc. London Math. Soc. 36 (1933), 427–450.Google Scholar
[5]Titchmarsh, E. C.. Theory of Functions, 2nd ed. (Oxford University Press, 1939.)Google Scholar
[6]Ursell, F.. Integrals with a large parameter. The continuation of uniformly asymptotic expansions. Proc. Cambridge Philos. Soc. 61 (1965), 113–128.CrossRefGoogle Scholar
[7]Ursell, F.. Integrals with a large parameter. Several nearly coincident saddle points. Proc. Cambridge Philos. Soc. 72 (1972), 49.65.CrossRefGoogle Scholar
[8]Watson, G. N.. Bessel Functions, 2nd ed. (Cambridge University Press, 1944.)Google Scholar