A simple rational approximation to the generalized elliptic integral of the first kind

Abstract

In this paper, we obtained a simple rational approximation for ${\mathcal {K}}_{a}(r)$:

$$\begin{aligned} \frac{2}{\pi }{\mathcal {K}}_{a}(r) \, \mathbf {>}\frac{\left( 1-3a+3a^{2}\right) r\mathbf {^{\prime }}+\left( 1+3a-3a^{2}\right) }{\left( 1+a-a^{2}\right) r\mathbf {^{\prime }}+\left( 1-a+a^{2}\right) }\; \end{aligned}$$

holds for all $r\in (0,1),$ where ${\mathcal {K}}_{a}(r)=$ $\frac{\pi }{2}F\left( a,1-a;1;r^{2}\right) =\frac{\pi }{2}\sum _{n=0}^{\infty }\frac{ \left( a\right) _{n}\left( 1-a\right) _{n}}{(n!)^{2}}r^{2n}$ is the generalized elliptic integral of the first kind, and $r\mathbf {^{\prime }=} \sqrt{1-r^{2}}$. In particular, when ${\small a}$ is taken as 1/2, 1/3, 1/4 and 1/6 respectively, we can obtain the specific lower bound of the corresponding function.