# Reduction of integral for geodesic area to elliptic integrals

In my paper on geodesics on an ellipsoid, I express the area between a geodesic segment and the equator in terms of an indefinite integral $$\int \frac{t(e'^2) - t(k^2\sin^2\sigma)}{e'^2-k^2\sin^2\sigma} \frac{\sin\sigma}2 \,d\sigma,$$ where $$t(x) = x + \sqrt{x^{-1} + 1}\,\sinh^{-1}\!\sqrt x,$$ $e'$ is the second eccentricity, $k = e'\cos\alpha_0$, and $\alpha_0$ is the azimuth of the geodesic when crossing the equator. For oblate ellipsoids, we have $0 < k \le e'$.

In the paper, I evaluate this integral by Taylor expanding the integrand in the limit that $e' \rightarrow 0$. I would like to relax this assumption. I have made some unsystematic (and unsuccessful) stabs at expressing the integral in terms of elliptic integrals.

I would appreciate help with

• expressing the integral in terms of elliptic integrals,
• pointing me to a systematic procedure for doing this,
• proving that the integral can't be expressed in terms of elliptic integrals, or
• expressing the integral in terms of other special functions (especially those which can be numerical evaluated easily).

I have seen your paper. You have worked out a series with $e'\rightarrow 0$. You can do the same on the other side $e'\rightarrow\infty$. Also for this series the integrals can be evaluated. In this way, you will be able to get a satisfactory numerical evaluation of the integral for a wide range of values of $e'$. That this can be done can be easily seen by noting the asymptotic series $$t(x)=x+\ln 2+\frac{1}{2}\ln x+\left(\frac{1}{2}\ln 2+\ln x+\frac{1}{4}\right)\frac{1}{x}+\ldots$$ that entails both logarithm and power terms. For example, for your integral you will get $$\int\frac{t(e'^2)-t(k^2\sin^2\sigma)}{e'^2-k^2\sin^2\sigma}\frac{\sin\sigma}{2}d\sigma= -\frac{1}{2}\cos\sigma+O\left(\frac{1}{e'^2}\right)$$ and you can evaluate the higher order terms by any computer based program you prefer.