I'm interested in doing something similar to this question, however I'm interested in the complete elliptic integral of the third kind $$\Pi(n,k)=\int_0^{\pi/2} \frac{d\theta}{(1-n\sin^2\theta)\sqrt{1-k^2\sin^2\theta}}$$ for $k>1$. Playing around with the definition a bit a bit, $$\Pi(n,k)=\int_0^{\arcsin(1/k)} \frac{d\theta}{(1-n\sin^2\theta)\sqrt{1-k^2\sin^2\theta}}-i\int_{\arcsin(1/k)}^{\pi/2} \frac{d\theta}{(1-n\sin^2\theta)\sqrt{k^2\sin^2\theta-1}},$$ I believe that it should be expressible in terms of incomplete elliptic integrals of the third kind as $$\Pi(n,k)=\Pi\left(n,k\mid\arcsin\left(\tfrac{1}{k}\right)\right)+\frac{i}{\sqrt{k^2-1}(n-1)}\Pi\left(\tfrac{n}{n-1},\tfrac{\pi}{2}-\arcsin\left(\tfrac{1}{k}\right)\mid\tfrac{k^2}{k^2-1}\right).$$ (I'm not 100% certain about the second term, however; I obtained it via Mathematica.) However, I don't find this very satisfying since it consists of a bunch of incomplete elliptic integrals. So, my question is, is it possible to express $\Pi(n,k)$ for $k>1$ in terms of complete elliptic integrals, such that it is separated into real and imaginary parts (as in the linked question)?
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