Given the Modulus k, , entering the definition of Elliptic integrals and JacobiPQ functions,

FunctionAdvisor(definition, EllipticF)[1];

FunctionAdvisor(definition, JacobiSN)[1];

FunctionAdvisor(definition, JacobiAM);

EllipticNome computes the corresponding Nome q, , entering the definition of the related (see below) Jacobi Theta functions, for instance:

FunctionAdvisor(definition, JacobiTheta1)[1];

Alternatively, given the Nome q, , it is possible to compute the corresponding Modulus k, , 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 );

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));

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  and sin(alpha) = k. For example, the Elliptic  function shown in A&S is numerically equal to the Maple  command.