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

TL;DR: In this paper, the authors used Jacobi's imaginary transformation to obtain alternative expressions which converge most rapidly in the limit as m → 1, where no more than three terms of any series need be used to ensure eight-figure accuracy.

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 6 1/2, and those obtained here for m > 1/2, 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. The Jacobian elliptic functions sn(u|m), cn(u|m),dn(u|m), etc., where u is the argument and m the parameter, and the complete elliptic integrals K(m) and E(m) can be calculated in a number of ways. Methods include the use of power series, Fourier series, Landen transformations, and theta functions, for which various methods exist including the use of infinite series and products. Most of these are presented in the chapters by L. M. Milne-Thomson in [1]. These methods are not useful for all values of argument and parameter. For example, the power series are useful only for small arguments, and the Fourier series are not convergent if the parameter approaches unity. The Landen transformations are rapidly convergent, but are non-trivial to apply. If theta functions are used, the series and products for these are the most convenient of all for small values of the parameter. However, they too do not converge if the parameter approaches unity. It is known that Jacobi’s imaginary transformation may be simply applied to recast the expressions for theta functions so that they are most rapidly convergent in the limit as the parameter tends to unity. However, explicit presentations of these recast series seem not to have been given, except by Eagle [2, Section 3.53] who considered non-standard functions. The existence of these alternative expressions seems to be almost unknown. In this paper, alternative series and products for theta functions are obtained using the imaginary transformation. These results are then used to give alternative expressions for the elliptic functions which also converge most rapidly in the limit where previously-presented expressions do not converge. Finally, alternative methods for the calculation of complete elliptic integrals are developed. These are shown to be the simple complement of well-known methods but, remarkably, seem to be unknown.