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EllipticNome

Nome function q(k)

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

EllipticNome(k)

Parameters

k

-

expression denoting a complex number

Description

  

Given the Modulus k, 0<k, entering the definition of Elliptic integrals and JacobiPQ functions,

FunctionAdvisor(definition, EllipticF)[1];

EllipticFz&comma;k=0z1_&alpha;12+1k2_&alpha;12+1&DifferentialD;_&alpha;1

(1)

FunctionAdvisor(definition, JacobiSN)[1];

JacobiSNz&comma;k=sinJacobiAMz&comma;k

(2)

FunctionAdvisor(definition, JacobiAM);

z=JacobiAM0z11k2sinθ2&DifferentialD;θ&comma;k&comma;z::32&comma;32

(3)
  

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

FunctionAdvisor(definition, JacobiTheta1)[1];

JacobiTheta1z&comma;q=_k1=02q_k1+122sinz2_k1+1−1_k1

(4)
  

Alternatively, given the Nome q, q<1, it is possible to compute the corresponding Modulus k, 0<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 );

EllipticNomek=&ExponentialE;πEllipticCKkEllipticKk&comma;with no restrictions on k

(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;EllipticNomek1

EllipticNomek=&ExponentialE;πEllipticCKkEllipticKk

(7)

evalfk&equals;12|k&equals;12

0.01797238701=0.01797238701

(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

Elliptic integrals

EllipticModulus

FunctionAdvisor

InverseJacobiPQ functions

Jacobi Theta functions

JacobiPQ functions