# Hyperbolic integral of the second kind

The elliptic integral of the second kind is given by $$E(t,m) = \int_{0}^t \sqrt{1-m \sin(s)^2} \operatorname{ds}$$ and there is for instance a scipy function ellipeinc that computes it.

The following formula comes up in the parametrization of constant negatively curved surfaces of rotation. $$H(t,m) := \int_{0}^t \sqrt{1-m \sinh(s)^2} \operatorname{ds}$$

Note that if we allow complex numbers, then $$H(t,m) = E(it,im)$$. But ellipeinc does not allow complex inputs.

• Is there a commonly accepted name for this "hyperbolic integral of the second kind" $$H$$?
• Is there a way to compute $$H(t,1)$$ using scipy functions, or other methods? How would one compute it efficiently (in Python)?
• Any restrictions on $m\sinh^2 t$? Commented Feb 10 at 16:15
• The Arb library offers acb_elliptic_e_inc(), which evaluates the Legendre incomplete elliptic integral of the second kind. The function's arguments are of type acb_t, i.e. complex numbers. The library's author notes in a relevant publication: "The definitions for complex variables do not appear to be standardized in the literature, but following the conventions used in Mathematica [31], we may fix an interpretation ..." Commented Feb 10 at 23:03
• Isn't $E$ implemented for complex numbers in mpmath? Commented Feb 14 at 10:37