Abramowitz and Stegun provide identites for dealing with incomplete elliptic integrals of the first and second kinds with complex arguments:

For $\tan\theta = \sinh\phi$

17.4.8 $F(i\phi | \alpha) = iF(\theta | \frac{\pi}{2} - \alpha)$

17.4.9 $E(i\phi | \alpha) = -i E(\theta | \frac{\pi}{2} - \alpha) + i F(\theta \frac{\pi}{2} - \alpha) + i\tan\theta(1-\cos^2\alpha \sin^2\theta)^{1/2}$

Are there any useful identities for incomplete elliptic integrals of the third kind, $\Pi(n;\phi | \alpha)$?

I would like to evaluate such integrals numerically, but many libraries such as GSL only handle real arguments.


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