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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 $\Pi(n>1,\phi|m=1)$:

real EllipticPi for "m" equals one

Which is symmetric around $\phi=\pi/2$ but anti-symmetric around $\phi=0$. But here's the strange one, the imaginary part:

alt text

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 $0<\phi<\pi/2$?

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One definition given by Wikipedia for the incomplete elliptic integral of the third kind is: $$ \Pi(n; \phi \vert m) = \int_0^{\sin\phi} \frac{1}{1-nt^2}\frac{dt}{\sqrt{(1-mt^2)(1-t^2)}}. $$ This indicates that the function is periodic with period $2\pi$, is symmetric about $\phi=\pi/2$, and is antisymmetric about $\phi=0$. So the values for $0\le\phi < \pi/2$ define the function for all $\phi$.

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Ah, I see now—I'd missed the symmetry around $\pi/2$. Thanks! – Will Robertson Jan 23 '11 at 6:00

(Not a full answer, but only some preliminary comments, since I've no reputation to do so.)

The identity in the Wolfram Functions site is, as mentioned, valid for characteristic $n$ in the interval $[-1,1)$ ; for $n=1$, your elliptic integral of the third kind degenerates into a particular combination of the elliptic integrals of the other two kinds and trigonometric functions:


such that you can use the properties of the other elliptic integrals you already know.

That leaves the circular case $n<-1$ and the hyperbolic case $n>1$; in the sequel I assume that the parameter $m$ is in the usual interval $[0,1]$ (if you need to deal with parameters outside the usual range, I'll have to dig further into my references).

For the circular case, the identity


can be used to express the circular case of an incomplete elliptic integral of the third kind back into the case where the characteristic is in the interval $(m,1)$

In the hyperbolic case, the identity you will need is


such that only characteristics in the interval $(0,m)$ are evaluated; the function is $2\pi$-periodic in $\phi$ for the hyperbolic case, so that if your amplitude $\phi$ is outside the usual interval $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$, either add/subtract appropriate multiples of $2\pi$ from the amplitude, or add/subtract appropriate multiples of $\Pi(n|m)=K(m)-\Pi\left(\frac{m}{n}|m\right)$ from the right hand side of that identity.

Alternatively, if you can help it, you might do better to just use the Carlson integral of the third kind instead; there is less worry here of having to split into "circular" and "hyperbolic" cases. The relations to interconvert Carlson and Legendre-Jacobi integrals are in the DLMF.

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