# Are there any identities for incomplete elliptic integrals of the third kind with complex arguments?

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.