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## closed-form of an integral relating to the elliptic integral

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### closed-form of an integral relating to the elliptic integral

Fri Dec 09, 2016 7:08 am

Posts: 45
Location: China
I want to get the closed-form of $F=F_++F_-$, where $F_{\nu}$ is the integral of the form
$$F_{\nu} = \frac{1}{\pi}\int_0^{\pi}\frac{1}{\sqrt{1+\nu\sqrt{q^2+d^2\cos^2k}}}\;dk$$
where $q\in(-1,1)$, $d\in[0,1)$, and $\nu=\pm1$. It is alse assumed that $q^2+d^2<1$, so do not warry about the complex number.

If $q=0$, the integral above relates to the elliptic integral(https://en.wikipedia.org/wiki/Elliptic_integral).