Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-22T07:41:06.108Z Has data issue: false hasContentIssue false
  • English
  • Français

Rapidly-convergent methods for evaluating elliptic integrals and theta and elliptic functions

Published online by Cambridge University Press:  17 February 2009

J. D. Fenton  and
R. S. Gardiner-Garden
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The expressions for elliptic integrals, elliptic functions and theta functions given in standard reference books are slowly convergent as the parameter m approaches unity, and in the limit do not converge. In this paper we use Jacobi's imaginary transformation to obtain alternative expressions which converge most rapidly in the limit as m → 1. With the freedom to use the traditional formulae for m ≤ ½ and those obtained here for m ≥ ½, extraordinarily rapidly-convergent methods may be used for all values of m; no more than three terms of any series need be used to ensure eight-figure accuracy.

Type
Research Article
Information
The ANZIAM Journal , Volume 24 , Issue 1 , July 1982 , pp. 47 - 58
Copyright
Copyright © Australian Mathematical Society 1982

References

[1]Abramowitz, M. and Stegun, I. A., (eds.), Handbook of mathematical functions (Dover, New York, 1968).Google Scholar
[2]Eagle, A., The elliptic functions as they should be (Galloway and Porter, Cambridge, 1958).Google Scholar
[3]Neville, E. H., Jacobian elliptic functions (Oxford University Press, 1951).Google Scholar
[4]Whittaker, E. T. and Watson, G. N., A course of modern analysis (Cambridge University Press, 1952).Google Scholar