Given the Modulus k, $0<\mathrm{\Re }\left(k\right)$, entering the definition of Elliptic integrals and JacobiPQ functions,

FunctionAdvisor(definition, EllipticF)[1];

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

FunctionAdvisor(definition, JacobiSN)[1];

$\mathrm{JacobiSN}\left(z\&comma;k\right)=\sin \left(\mathrm{JacobiAM}\left(z\&comma;k\right)\right)$

FunctionAdvisor(definition, JacobiAM);

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

EllipticNome computes the corresponding Nome q, $\left|q\right|<1$, entering the definition of the related (see below) Jacobi Theta functions, for instance:

FunctionAdvisor(definition, JacobiTheta1)[1];

$\mathrm{JacobiTheta1}\left(z\&comma;q\right)=\sum _{\mathrm{\_k1}=0}^{\mathrm{\infty }}2q^{{\left(\mathrm{\_k1}+\frac{1}{2}\right)}^2}\sin \left(z\left(2\mathrm{\_k1}+1\right)\right){\left(\mathrm{-1}\right)}^{\mathrm{\_k1}}$

Alternatively, given the Nome q, $\left|q\right|<1$, it is possible to compute the corresponding Modulus k, $0<\mathrm{\Re }\left(k\right)$, using EllipticModulus, which is the inverse function of EllipticNome.

EllipticNome is defined in terms of the Complete Elliptic integral of the first kind EllipticK by:

FunctionAdvisor( definition, EllipticNome );

$\left[\mathrm{EllipticNome}\left(k\right)={\&ExponentialE;}^{-\frac{\mathrm{\pi }\mathrm{EllipticCK}\left(k\right)}{\mathrm{EllipticK}\left(k\right)}}\&comma;\mathrm{with\; no\; restrictions\; on}\left(k\right)\right]$

The JacobiPQ functions can be expressed in terms of JacobiTheta functions using EllipticNome

JacobiSN(z,k) = (1/(k^2))^(1/4) * JacobiTheta1(1/2*Pi*z/EllipticK(k),EllipticNome(k)) / JacobiTheta4(1/2*Pi*z/EllipticK(k),EllipticNome(k));

$\mathrm{JacobiSN}\left(z\&comma;k\right)=\frac{{\left(\frac{1}{k^2}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\mathrm{JacobiTheta1}\left(\frac{\mathrm{\pi }z}{2\mathrm{EllipticK}\left(k\right)}\&comma;\mathrm{EllipticNome}\left(k\right)\right)}{\mathrm{JacobiTheta4}\left(\frac{\mathrm{\pi }z}{2\mathrm{EllipticK}\left(k\right)}\&comma;\mathrm{EllipticNome}\left(k\right)\right)}$

Alternative popular notations for elliptic integrals and JacobiPQ functions involve a parameter m or a modular angle alpha, as for instance in the Handbook of Mathematical Functions edited by Abramowitz and Stegun (A&S). These are related to k by $m=k^2$ and sin(alpha) = k. For example, the Elliptic $K\left(m\right)$ function shown in A&S is numerically equal to the Maple $\mathrm{EllipticK}\left(\sqrt{m}\right)$ command.

$\mathrm{FunctionAdvisor}\left(\mathrm{definition}\&comma;\mathrm{EllipticNome}\left(k\right)\right)\left[1\right]$

$\mathrm{EllipticNome}\left(k\right)={\&ExponentialE;}^{-\frac{\mathrm{\pi }\mathrm{EllipticCK}\left(k\right)}{\mathrm{EllipticK}\left(k\right)}}$

$\mathrm{evalf}\left(\mathrm{eval}\left(\&comma;k=\frac{1}{2}\right)\right)$

$0.01797238701=0.01797238701$

$\mathrm{EllipticModulus}\left(\mathrm{EllipticNome}\left(k\right)\right)=k$

$\mathrm{EllipticModulus}\left(\mathrm{EllipticNome}\left(k\right)\right)=k$

$\mathrm{evalf}\left(\mathrm{eval}\left(\&comma;k=2\right)\right)$

$2.=2.$

$\mathrm{EllipticNome}\left(\mathrm{EllipticModulus}\left(q\right)\right)=q$

$\mathrm{EllipticNome}\left(\mathrm{EllipticModulus}\left(q\right)\right)=q$

$\mathrm{evalf}\left(\mathrm{eval}\left(\&comma;q=\frac{1}{2}\right)\right)$

$0.5000000000=0.5000000000$