How to evaluate incomplete elliptic integrals?

Question

From what I learned here, for evaluating complete elliptic integrals, using the binomial theorem is quite effective as it puts the elliptic integral in the form of. $$\sum_{n=1}^{\infty} some-constants\int_0^{\dfrac{\pi}{2}} \sin^{2n} \phi d\phi $$ which turns out to be easy to evaluate. Yet when it turns in to the form $$\sum_{n=1}^{\infty} some-constants\int_0^{\theta} \sin^{2n} \phi d\phi $$ which is quite common among elliptic integrals, I start to not know what to do. I resolved to go on with conducting a Taylor expansion where the derivatives(as I do not know how to evaluate the derivatives) I computed using Matlab. Yet what is the best way or the standard way(if there is such a way) to conduct/approximate these kinds of incomplete elliptic integrals? As a reference, I want am trying to compute $$ F(\dfrac{\theta + \gamma}{2}|\dfrac{2a}{a-1})$$

Answer 1

If you want to compute $$y=F(x|k)$$ for small values of $k$ and $x$, you could use series expansion $$y=\sum_{n=1}^p a_n\, x^{2n-1}+O(x^{2p+1})$$ where the first coefficients would be $$a_1=1 \qquad a_2=\frac{k}{6}\qquad a_3=\frac{ k (9 k-4)}{120}\qquad a_4=\frac{k \left(225 k^2-180 k+16\right)}{5040}$$

Answer 2

The modern way to compute incomplete elliptic integrals is via the Carlson symmetric form: $$F(\varphi,m)=\sin\varphi R_F(\cos^2\varphi,1-m\sin^2\varphi,1)$$ $$R_F(x,y,z)=\frac12\int_0^\infty\frac1{\sqrt{(t+x)(t+y)(t+z)}}\,dt$$ $R_F$ itself can be computed efficiently by a so-called duplication theorem, as detailed in the "Numerical evaluation" section of the linked Wikipedia article. The process is reminiscent of the AGM formula for computing complete elliptic integrals.