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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=_k1=02q_k1+122sinz2_k1+1−1_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=0z1_&alpha;12+1k2_&alpha;12+1&DifferentialD;_&alpha;1

(2)

FunctionAdvisor(definition, JacobiSN)[1];

JacobiSNz&comma;k=sinJacobiAMz&comma;k

(3)

FunctionAdvisor(definition, JacobiAM);

z=JacobiAM0z11k2sinθ2&DifferentialD;θ&comma;k&comma;z::32&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=JacobiTheta20&comma;q2JacobiTheta30&comma;q2&comma;q<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=1k214JacobiTheta1πz2EllipticKk&comma;EllipticNomekJacobiTheta4πz2EllipticKk&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=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=JacobiTheta20&comma;q2JacobiTheta30&comma;q2

(7)

evalfq&equals;12|q&equals;12

0.9999947611=0.9999947617

(8)

EllipticModulusEllipticNomek&equals;k

EllipticModulusEllipticNomek=k

(9)

evalfk&equals;2|k&equals;2

2.=2.

(10)

EllipticNomeEllipticModulusq&equals;q

EllipticNomeEllipticModulusq=q

(11)

evalfq&equals;12|q&equals;12

0.5000000000=0.5000000000

(12)

See Also

EllipticF

EllipticNome

FunctionAdvisor