EllipticModulus - Maple Programming Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : Special Functions : EllipticModulus

EllipticModulus

Modulus function k(q)

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

EllipticModulus(q)

Parameters

q

-

expression denoting a complex number such that q<1

Description

  

Given the Nome q, q<1, entering the definition of Jacobi Theta functions, for instance

FunctionAdvisor(definition, JacobiTheta1)[1];

JacobiTheta1z&comma;q&equals;_k1&equals;0&infin;2q_k1&plus;122sinz2_k1&plus;11_k1

(1)
  

EllipticModulus computes the corresponding Modulus k, 0<k entering the definition of related elliptic integrals and JacobiPQ elliptic functions.

FunctionAdvisor(definition, EllipticF)[1];

EllipticFz&comma;k&equals;&int;0z1_&alpha;12&plus;1_&alpha;12k2&plus;1&DifferentialD;_&alpha;1

(2)

FunctionAdvisor(definition, JacobiSN)[1];

JacobiSNz&comma;k&equals;sinJacobiAMz&comma;k

(3)

FunctionAdvisor(definition, JacobiAM);

z&equals;JacobiAM&int;0z11k2sin&theta;2&DifferentialD;&theta;&comma;k&comma;z::RealRange32&comma;32

(4)
  

Alternatively, given the Modulus k, 0<k entering Elliptic integrals and JacobiPQ functions, it is possible to compute the corresponding Nome q, q<1, using EllipticNome, which is the inverse function of EllipticModulus.

  

EllipticModulus is defined in terms of JacobiTheta functions by:

FunctionAdvisor( definition, EllipticModulus );

EllipticModulusq&equals;JacobiTheta20&comma;q2JacobiTheta30&comma;q2&comma;Andq<1

(5)
  

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));

JacobiSNz&comma;k&equals;1k21&sol;4JacobiTheta112&pi;zEllipticKk&comma;EllipticNomekJacobiTheta412&pi;zEllipticKk&comma;EllipticNomek

(6)
  

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&equals;k2 and sin(alpha) = k. For example, the Elliptic Km function shown in A&S is numerically equal to the Maple EllipticKm command.

Examples

FunctionAdvisordefinition&comma;EllipticModulusq1

EllipticModulusq&equals;JacobiTheta20&comma;q2JacobiTheta30&comma;q2

(7)

evalfq&equals;12|q&equals;12

0.9999947611&equals;0.9999947617

(8)

EllipticModulusEllipticNomek&equals;k

EllipticModulusEllipticNomek&equals;k

(9)

evalfk&equals;2|k&equals;2

2.&equals;2.

(10)

EllipticNomeEllipticModulusq&equals;q

EllipticNomeEllipticModulusq&equals;q

(11)

evalfq&equals;12|q&equals;12

0.5000000000&equals;0.5000000000

(12)

See Also

EllipticF

EllipticNome

FunctionAdvisor