EllipticF is the Incomplete Elliptic integral of the first kind and is defined by

FunctionAdvisor(definition, EllipticF);

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

EllipticK and EllipticCK are respectively the Complete and the Complementary Elliptic integrals of the first kind and are defined by

FunctionAdvisor( definition, EllipticK);

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

FunctionAdvisor( definition, EllipticCK);

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

EllipticK, EllipticCK and EllipticF are related by

FunctionAdvisor( relate, EllipticK,EllipticF);

$\mathrm{EllipticK}\left(k\right)=\mathrm{EllipticF}\left(1\,k\right)$

FunctionAdvisor( relate, EllipticK,EllipticCK);

$\mathrm{EllipticK}\left(k\right)=\mathrm{EllipticCK}\left(\sqrt{-k^2+1}\right)$

EllipticF is also identical to the InverseJacobiSN function

FunctionAdvisor(relate, EllipticF, InverseJacobiSN);

$\mathrm{EllipticF}\left(z\,k\right)=\mathrm{InverseJacobiSN}\left(z\,k\right)$

and therefore can be used to represent all the InverseJacobiPQ functions provided some restrictions on the function parameters hold.

Elliptic integrals and the related functions are well described in the Table of Integrals Series and Products, Gradshteyn and Ryzhik (G&R) and in the popular Handbook of Mathematical Functions edited by Abramowitz and Stegun (A&S). In A&S, these functions are expressed in terms of a parameter m, representing the square of the modulus k entering the definition of the Elliptic, JacobiPQ and InverseJacobiPQ functions in Maple and G&R. For example, the $K\left(m\right)$ function shown in A&S is numerically equal to the Maple $\mathrm{EllipticK}\left(\sqrt{m}\right)$ command.

It is worth noting the difference between the Legendre normal form of the Incomplete Elliptic integral of the first kind (see A&S 17.2.7), in Maple represented by EllipticF(z,k) but for the splitting of the square root in the denominator of the integrand (see definition lines above), and the normal trigonometric form of this elliptic integral (see A&S 17.2.6), in Maple represented by the InverseJacobiAM function

InverseJacobiAM(phi,k);

$\mathrm{InverseJacobiAM}\left(\mathrm{\phi }\,k\right)$

(7) = convert((7), Int);

$\mathrm{InverseJacobiAM}\left(\mathrm{\phi }\,k\right)={\int }_0^{\mathrm{\phi }}\frac{1}{\sqrt{1-k^2{\sin \left(\mathrm{\_\θ1}\right)}^2}}\ⅆ\mathrm{\_\θ1}$

For instance, for -Pi/2 <= phi <= Pi/2 these two forms can be related with ease by changing variables:

EllipticF(z,k);

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

(9) = convert((9), Int);

$\mathrm{EllipticF}\left(z\&comma;k\right)={\int }_0^z\frac{1}{\sqrt{-{\mathrm{\_\&alpha;1}}^2+1}\sqrt{-k^2{\mathrm{\_\&alpha;1}}^2+1}}\&DifferentialD;\mathrm{\_\&alpha;1}$

{z=sin(phi), _alpha1=sin(_theta1)};     #  -1 <= z <= 1

$\left\{\mathrm{\_\&alpha;1}=\sin \left(\mathrm{\_\&theta;1}\right)\&comma;z=\sin \left(\mathrm{\phi }\right)\right\}$

PDEtools[dchange]((11), (10));

$\mathrm{EllipticF}\left(\sin \left(\mathrm{\phi }\right)\&comma;k\right)={\int }_0^{\arcsin \left(\sin \left(\mathrm{\phi }\right)\right)}\frac{\cos \left(\mathrm{\_\&theta;1}\right)}{\sqrt{-{\sin \left(\mathrm{\_\&theta;1}\right)}^2+1}\sqrt{1-k^2{\sin \left(\mathrm{\_\&theta;1}\right)}^2}}\&DifferentialD;\mathrm{\_\&theta;1}$

simplify((12)) assuming phi in RealRange(-Pi/2, Pi/2);

$\mathrm{EllipticF}\left(\sin \left(\mathrm{\phi }\right)\&comma;k\right)={\int }_0^{\mathrm{\phi }}\frac{1}{\sqrt{1-k^2{\sin \left(\mathrm{\_\&theta;1}\right)}^2}}\&DifferentialD;\mathrm{\_\&theta;1}$

where the right-hand side is actually equal to the trigonometric form $\mathrm{InverseJacobiAM}\left(\mathrm{\phi }\&comma;k\right)$. The general relationship between these two forms and the restriction on the values of the parameters such that the relation is valid are given by

FunctionAdvisor( specialize, InverseJacobiAM, EllipticF);

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

FunctionAdvisor( specialize, EllipticF, InverseJacobiAM);

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

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

$\left[\mathrm{EllipticK}\left(-k\right)=\mathrm{EllipticK}\left(k\right)\&comma;\mathrm{EllipticK}\left(0\right)=\frac{\mathrm{\pi }}{2}\&comma;\mathrm{EllipticK}\left(\mathrm{\infty }\right)=0\&comma;\mathrm{EllipticK}\left(\mathrm{\infty }\mathrm{I}\right)=0\right]$

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

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

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

$\left[\mathrm{EllipticF}\left(z\&comma;k\right)\&comma;z\in \left[\mathrm{-1}\&comma;1\&comma;-\frac{1}{k}\&comma;\frac{1}{k}\&comma;\mathrm{\infty }+\mathrm{\infty }\mathrm{I}\right]\right]$

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

$\left[\mathrm{EllipticK}\left(k\right)\&comma;k\in \left[-\mathrm{\infty }-\mathrm{\infty }\mathrm{I}\&comma;\mathrm{-1}\&comma;1\&comma;\mathrm{\infty }+\mathrm{\infty }\mathrm{I}\right]\right]$

$\mathrm{FunctionAdvisor}\left(\mathrm{branch\_cuts}\&comma;\mathrm{EllipticK}\right)$

$\left[\mathrm{EllipticK}\left(k\right)\&comma;k<\mathrm{-1}\vee 1<k\right]$

$\mathrm{EllipticK}\left(2-\frac{1}{100000}\mathrm{I}\right)$

$\mathrm{EllipticK}\left(2-\frac{\mathrm{I}}{100000}\right)$

$=\mathrm{evalf}\left(\right)$

$\mathrm{EllipticK}\left(2-\frac{\mathrm{I}}{100000}\right)=0.8428783289-1.078252932\mathrm{I}$

$\mathrm{EllipticK}\left(2\right)$

$\mathrm{EllipticK}\left(2\right)$

$=\mathrm{evalf}\left(\right)$

$\mathrm{EllipticK}\left(2\right)=0.8428751774-1.078257824\mathrm{I}$

$\mathrm{EllipticK}\left(2+\frac{1}{100000}\mathrm{I}\right)$

$\mathrm{EllipticK}\left(2+\frac{\mathrm{I}}{100000}\right)$

$=\mathrm{evalf}\left(\right)$

$\mathrm{EllipticK}\left(2+\frac{\mathrm{I}}{100000}\right)=0.8428783289+1.078252932\mathrm{I}$