We are trying to implement transformations to evaluate the incomplete integral of the third kind $\Pi(n;\phi|m)$ for arbitrary inputs, and I can't find any references for how to calculate this function with phase $\phi$ less than zero or greater than $\pi/2$.
Are there periodic identities for this function for various ranges of $m$ and $n$? Plotting it in Mathematica tends to suggest there is, but the DLMF only lists periodicity for elliptic integrals of the first two kinds.
Update: Wolfram has a periodicity equation for $-1\le n\le 1$ although in practise it seems to work for $n<-1$ as well. But I'm still looking for a similar equation for $n>1$.
2nd update: mjqxxxx below gives periodicity for $n>1$, but for $m=1$ things are slightly different. (I'm trying to be complete.) Here's the real part for $\Pi(n>1,\phi|m=1)$:
Which is symmetric around $\phi=\pi/2$ but anti-symmetric around $\phi=0$. But here's the strange one, the imaginary part:
Symmetric around $\phi=0$ but around $\phi=\pi/2$ the symmetry breaks down. Can the function be mapped from the region $\pi/2<\phi<\pi$ back to $0<\phi<\pi/2$?