Rapidly-convergent methods for evaluating elliptic integrals and theta and elliptic functions
- 1 July 1982
- journal article
- research article
- Published by Cambridge University Press (CUP) in The Journal of the Australian Mathematical Society. Series B. Applied Mathematics
- Vol. 24 (1) , 47-58
- https://doi.org/10.1017/s0334270000003301
Abstract
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.Keywords
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