The InverseJacobiAM and the twelve InverseJacobiPQ functions, where P and Q are any two of {C,D,N,S}, are all elliptic integrals related to the incomplete elliptic integral of the first kind EllipticF. The exact definition for each InverseJacobiPQ can be seen, for instance, via:

FunctionAdvisor( definition, InverseJacobiAM);

$\left[\mathrm{InverseJacobiAM}\left(\mathrm{\phi }\,k\right)={\int }_0^{\mathrm{\phi }}\frac{1}{\sqrt{1-k^2{\sin \left(\mathrm{\_\θ1}\right)}^2}}\ⅆ\mathrm{\_\θ1}\,\mathrm{with\; no\; restrictions\; on}\left(\mathrm{\phi }\,k\right)\right]$

FunctionAdvisor( definition, InverseJacobiSN);

$\left[\mathrm{InverseJacobiSN}\left(z\,k\right)={\int }_0^z\frac{1}{\sqrt{-{\mathrm{\_\α1}}^2+1}\sqrt{-k^2{\mathrm{\_\α1}}^2+1}}\ⅆ\mathrm{\_\α1}\,\mathrm{with\; no\; restrictions\; on}\left(z\,k\right)\right]$

From these definitions, InverseJacobiAM represents the trigonometric form of the incomplete Elliptic integral of the first kind (A&S 16.1.3, 17.2.6; G&R, 8.111, 8.141 - see References section) and InverseJacobiSN equals EllipticF,

FunctionAdvisor( definition, EllipticF);

$\left[\mathrm{EllipticF}\left(z\,k\right)={\int }_0^z\frac{1}{\sqrt{-{\mathrm{\_\α1}}^2+1}\sqrt{-k^2{\mathrm{\_\α1}}^2+1}}\ⅆ\mathrm{\_\α1}\,\mathrm{with\; no\; restrictions\; on}\left(z\,k\right)\right]$

EllipticF in turn is the Legendre normal form of the incomplete Elliptic integral of the first kind, (A&S 17.2.7; G&R 8.111). Note however that, unlike in those formulas, in the definition above the square root in the denominator of the integrand is split. The general relation between the Legendre and the trigonometric form of the incomplete Elliptic integral of the first kind, that is, between $\mathrm{InverseJacobiAM}\left(\mathrm{\phi }\,k\right)$ and $\mathrm{EllipticF}\left(z\,k\right)$, as well as the restrictions on the parameters such that the relations are valid are given by:

FunctionAdvisor( specialize, EllipticF, InverseJacobiAM);

$\left[\mathrm{EllipticF}\left(z\,k\right)=\mathrm{InverseJacobiAM}\left(\arcsin \left(z\right)\,k\right)\,\mathrm{with\; no\; restrictions\; on}\left(z\,k\right)\right]$

FunctionAdvisor( specialize, InverseJacobiAM(phi,k), EllipticF);

$\left[\mathrm{InverseJacobiAM}\left(\mathrm{\phi }\&comma;k\right)=\mathrm{EllipticF}\left(\sin \left(\mathrm{\pi }⌊\frac{1}{2}-\frac{\mathrm{\Re }\left(\mathrm{\phi }\right)}{\mathrm{\pi }}⌋+\mathrm{\phi }\right)\&comma;k\right)-2⌊\frac{1}{2}-\frac{\mathrm{\Re }\left(\mathrm{\phi }\right)}{\mathrm{\pi }}⌋\mathrm{EllipticK}\left(k\right)\&comma;\mathrm{with\; no\; restrictions\; on}\left(\mathrm{\phi }\&comma;k\right)\right],\left[\mathrm{InverseJacobiAM}\left(\mathrm{\phi }\&comma;k\right)=\mathrm{EllipticF}\left(\sin \left(\mathrm{\phi }\right)\&comma;k\right)\&comma;\left(-\frac{\mathrm{\pi }}{2}<\mathrm{\Re }\left(\mathrm{\phi }\right)\wedge \mathrm{\Re }\left(\mathrm{\phi }\right)<\frac{\mathrm{\pi }}{2}\right)\vee \left(-\frac{\mathrm{\pi }}{2}=\mathrm{\Re }\left(\mathrm{\phi }\right)\wedge 0\le \mathrm{\Im }\left(\mathrm{\phi }\right)\right)\vee \left(\frac{\mathrm{\pi }}{2}=\mathrm{\Re }\left(\mathrm{\phi }\right)\wedge \mathrm{\Im }\left(\mathrm{\phi }\right)\le 0\right)\right]$

With some restrictions on the values of the function parameters, InverseJacobiAM is the inverse of the amplitude JacobiAM function:

FunctionAdvisor( definition, JacobiAM);

$\left[z=\mathrm{JacobiAM}\left({\int }_0^z\frac{1}{\sqrt{1-k^2{\sin \left(\mathrm{\theta }\right)}^2}}\&DifferentialD;\mathrm{\theta }\&comma;k\right)\&comma;z::\left[-\frac{3}{2}\&comma;\frac{3}{2}\right]\right]$

where in above the integral is equal to InverseJacobiAM(z,k). For all the other InverseJacobiPQ, where P and Q are any two of {C, D, N, S}, the InverseJacobiPQ function is the exact inverse of the corresponding JacobiPQ function for all values of the function parameters, so for instance,

JacobiSN( InverseJacobiSN(z,k), k);

$z$

JacobiCN( InverseJacobiCN(z,k), k);

$z$

JacobiDN( InverseJacobiDN(z,k), k);

$z$

All InverseJacobiPQ (excluding InverseJacobiAM) satisfy $\mathrm{InverseJacobiPQ}\left(z\&comma;k\right)=\mathrm{InverseJacobiQP}\left(\frac{1}{z}\&comma;k\right)$ For k = infinity, all the InverseJacobiPQ functions have the value zero, independent of the value of the first parameter z. For z = infinity, however, most of these functions have finite values (see below).

$\mathrm{FunctionAdvisor}\left(\mathrm{special\_values}\&comma;\mathrm{InverseJacobiAM}\right)$

$\left[\mathrm{InverseJacobiAM}\left(-\mathrm{\phi }\&comma;k\right)=-\mathrm{InverseJacobiAM}\left(\mathrm{\phi }\&comma;k\right)\&comma;\mathrm{InverseJacobiAM}\left(\mathrm{\phi }\&comma;-k\right)=\mathrm{InverseJacobiAM}\left(\mathrm{\phi }\&comma;k\right)\&comma;\mathrm{InverseJacobiAM}\left(0\&comma;k\right)=0\&comma;\mathrm{InverseJacobiAM}\left(\mathrm{\phi }\&comma;0\right)=\mathrm{\phi }\&comma;\left[\mathrm{InverseJacobiAM}\left(\frac{n\mathrm{\pi }}{2}\&comma;k\right)=n\mathrm{EllipticK}\left(k\right)\&comma;n::\mathbb{Z}\right]\&comma;\left[\mathrm{InverseJacobiAM}\left(\mathrm{\phi }\&comma;k\right)=\ln \left(\sec \left(\mathrm{\phi }\right)+\tan \left(\mathrm{\phi }\right)\right)\&comma;\left|\mathrm{\Re }\left(\mathrm{\phi }\right)\right|<\frac{\mathrm{\pi }}{2}\wedge k\in \left\{\mathrm{-1}\&comma;1\right\}\right]\&comma;\left[\mathrm{InverseJacobiAM}\left(\mathrm{\phi }\&comma;k\right)=0\&comma;k\in \left\{\mathrm{\infty }\&comma;-\mathrm{\infty }\right\}\right]\&comma;\left[\mathrm{InverseJacobiAM}\left(\mathrm{\infty }\mathrm{I}\&comma;k\right)=\mathrm{EllipticK}\left(k\right)-\frac{\mathrm{EllipticK}\left(\frac{1}{k}\right)}{\sqrt{k}}\&comma;k^2\in \left(0\&comma;1\right)\right]\right]$

$\mathrm{FunctionAdvisor}\left(\mathrm{special\_values}\&comma;\mathrm{InverseJacobiSN}\right)$

$\left[\mathrm{InverseJacobiSN}\left(0\&comma;k\right)=0\&comma;\mathrm{InverseJacobiSN}\left(1\&comma;k\right)=\mathrm{EllipticK}\left(k\right)\&comma;\mathrm{InverseJacobiSN}\left(z\&comma;0\right)=\arcsin \left(z\right)\&comma;\mathrm{InverseJacobiSN}\left(z\&comma;1\right)=\mathrm{arctanh}\left(z\right)\&comma;\mathrm{InverseJacobiSN}\left(z\&comma;\mathrm{\infty }\right)=0\&comma;\left[\mathrm{InverseJacobiSN}\left(\mathrm{\infty }\&comma;k\right)=\mathrm{EllipticK}\left(k\right)-\frac{\mathrm{EllipticK}\left(\frac{1}{k}\right)}{k}\&comma;1<k\right]\right]$

$\mathrm{FunctionAdvisor}\left(\mathrm{branch\_points}\&comma;\mathrm{InverseJacobiAM}\right)$

$\left[\mathrm{InverseJacobiAM}\left(\mathrm{\phi }\&comma;k\right)\&comma;\left(\mathrm{\phi }=\arcsin \left(\frac{1}{k}\right)+n\mathrm{\pi }\wedge n::\mathbb{Z}\right)\vee \left(\mathrm{\phi }=-\arcsin \left(\frac{1}{k}\right)+n\mathrm{\pi }\wedge n::\mathbb{Z}\right)\vee \mathrm{\phi }=\mathrm{\infty }+\mathrm{\infty }\mathrm{I}\vee k=\csc \left(\mathrm{\phi }\right)\vee k=-\csc \left(\mathrm{\phi }\right)\vee k=\mathrm{\infty }+\mathrm{\infty }\mathrm{I}\right]$

$\mathrm{JacobiSN}\left(\mathrm{InverseJacobiAM}\left(z\&comma;k\right)\&comma;k\right)$

$\sin \left(z\right)$

$\mathrm{JacobiCN}\left(\mathrm{InverseJacobiAM}\left(z\&comma;k\right)\&comma;k\right)$

$\cos \left(z\right)$

$\mathrm{JacobiCN}\left(\mathrm{InverseJacobiSN}\left(z\&comma;k\right)\&comma;k\right)$

$\sqrt{-z^2+1}$

$\mathrm{JacobiSN}\left(\mathrm{InverseJacobiCN}\left(z\&comma;k\right)\&comma;k\right)$

$\sqrt{-z^2+1}$

$\mathrm{JacobiCN}\left(\mathrm{InverseJacobiSN}\left(z\&comma;k\right)\&comma;k\right)$

$\sqrt{-z^2+1}$

$\mathrm{InverseJacobiDN}\left(z\&comma;k\right)=\mathrm{convert}\left(\mathrm{InverseJacobiDN}\left(z\&comma;k\right)\&comma;\mathrm{InverseJacobiAM}\right)$

$\mathrm{InverseJacobiDN}\left(z\&comma;k\right)=\frac{\mathrm{InverseJacobiAM}\left(\arccos \left(z\right)\&comma;\frac{1}{k}\right)}{k}$

$\mathrm{InverseJacobiCD}\left(z\&comma;k\right)=\mathrm{convert}\left(\mathrm{InverseJacobiCD}\left(z\&comma;k\right)\&comma;\mathrm{InverseJacobiSN}\right)$

$\mathrm{InverseJacobiCD}\left(z\&comma;k\right)=\mathrm{EllipticK}\left(k\right)-\mathrm{InverseJacobiSN}\left(z\&comma;k\right)$

[A&S] Abramowitz, M., and Stegun, I., eds. Handbook of Mathematical Functions. New York: Dover publications.

[G&R] Gradshteyn, and Ryzhik. Table of Integrals Series and Products. 5th ed. Academic Press.