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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)?
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  • $\begingroup$ Any restrictions on $m\sinh^2 t$? $\endgroup$ Commented Feb 10 at 16:15
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    $\begingroup$ 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 ..." $\endgroup$
    – njuffa
    Commented Feb 10 at 23:03
  • $\begingroup$ Isn't $E$ implemented for complex numbers in mpmath? $\endgroup$ Commented Feb 14 at 10:37

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