I'm re-implementing numeric evaluation of the inverse elliptic Jacobi functions in Sage, since the current implementation uses Maxima with limited precision.
They can be conveniently computed with the incomplete elliptic integral of the first kind with the following table from the AMath and DAMath special functions reference manual by Wolfgang Ehrhardt, licensed under Creative Commons BY-NC-ND 3.0, and based on the trusty Abramowitz & Stegun:
When I coded this to evaluate with mpmath, I was surprised to find that it didn't work; the reason is that the identities above are for $F(\phi, k) = \int_0^\phi \frac{dt}{\sqrt{1 - k^2\sin^2{t}}}$, while mpmath (and indeed also Maxima and Mathematica) define $F$ as $F(\phi|m) = \int_0^\phi \frac{dt}{\sqrt{1 - m\sin^2{t}}}$ (see the Wikipedia article for all the crazy notation).
They can be conveniently computed with the incomplete elliptic integral of the first kind with the following table from the AMath and DAMath special functions reference manual by Wolfgang Ehrhardt, licensed under Creative Commons BY-NC-ND 3.0, and based on the trusty Abramowitz & Stegun:
| __________ | ____________________ |
| $\operatorname{arcsn}(x, k)$ | $F\left(\arcsin(x), k\right)$ |
| $\operatorname{arccn}(x, k)$ | $F\left(\arccos(x), k\right)$ |
| $\operatorname{arcdn}(x, k)$ | $F\left(\arcsin\left(\sqrt{\frac{1 - x^2}{k^2}}\right), k\right)$ |
| $\operatorname{arccd}(x, k)$ | $F\left(\arcsin\left(\sqrt{\frac{1 - x^2}{1 - k^2x^2}}\right), k\right)$ |
| $\operatorname{arcsd}(x, k)$ | $F\left(\arcsin\left(\frac{x}{\sqrt{1 + k^2x^2}}\right), k\right)$ |
| $\operatorname{arcnd}(x, k)$ | $F\left(\arcsin\left(\sqrt{\frac{x^2 - 1}{k^2x^2}}\right), k\right)$ |
| $\operatorname{arcdc}(x, k)$ | $F\left(\arcsin\left(\sqrt{\frac{1 - x^2}{k^2 - x^2}}\right), k\right)$ |
| $\operatorname{arcnc}(x, k)$ | $F\left(\operatorname{arcsec}(x), k\right)$ |
| $\operatorname{arcsc}(x, k)$ | $F\left(\arctan(x), k\right)$ |
| $\operatorname{arcns}(x, k)$ | $F\left(\operatorname{arccsc}(x), k\right)$ |
| $\operatorname{arcds}(x, k)$ | $F\left(\arcsin\left(\frac{1}{\sqrt{k^2 + x^2}}\right), k\right)$ |
| $\operatorname{arccs}(x, k)$ | $F\left(\operatorname{arccot}(x), k\right)$ |
| __________ | ____________________ |
When I coded this to evaluate with mpmath, I was surprised to find that it didn't work; the reason is that the identities above are for $F(\phi, k) = \int_0^\phi \frac{dt}{\sqrt{1 - k^2\sin^2{t}}}$, while mpmath (and indeed also Maxima and Mathematica) define $F$ as $F(\phi|m) = \int_0^\phi \frac{dt}{\sqrt{1 - m\sin^2{t}}}$ (see the Wikipedia article for all the crazy notation).