On the series expansion method for computing incomplete elliptic integrals of the first and second kinds

Abstract:

In the present paper an attempt is made to improve the series expansion method for computing the incomplete integrals $F(\phi ,k)$ and $E(\phi ,k)$. Therefore the following three pairs of series covering the region $- 1 \leqq k \leqq 1,0 \leqq \phi < \pi /2$ are used: series obtained by a straightforward binomial expansion of the integrands, series valid for ${k’^2}{\tan ^2}\phi < 1$, and new series which converge for $\phi > \pi /4$ and for all values of $k$. Terms of the last two pairs of series can be generated by means of the same recurrence relations, so that the coding of the whole is not longer than that for similar methods using only two pairs of series. Any degree of accuracy can be obtained. In general the method is a little bit slower than Bulirsch’ calculation procedures which are based on the Landen transformation, but it works more quickly in case of large values of ${k^2}$ and/or $\phi$. The new series introduced are also represented in trigonometric form, and the double passage to the limit ${k^2} \to 1$, $\phi \to \pi /2$ is discussed.