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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
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Abstract

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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

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