Conversions of parameters of elliptic functions

Conversions of parameters of elliptic functions#

Elliptic nome \(q(k)\) (DAMath)#

math53.elliptic_nome(z)#

Returns the elliptic nome \(q(k)\) as a function of the modulus \(|k| < 1\):. See also Wikipedia [1471], MathWorld [1075], Ehrhardt [309] (3.2.8).

\[q(k)=\exp \left(-\pi \frac{K'(k)}{K(k)} \right)\]

An example in Python

>>> from xlcalcnet import ereal
>>> ereal.EllipticNome(0.5)
ereal('5.2359877559829887307E-1')
>>> ereal.EllipticNome('0.51')
ereal('5.3518479027559984754E-1')

Elliptic modulus \(k(q)\) (DAMath)#

math53.elliptic_modulus(q)#

Returns the elliptic modulus k. as a function of the nome \(|q|\) le 1. See also MathWorld [1074], Ehrhardt [309] (3.2.9).The modulus \(k\) is often used as argument of elliptic integrals and Jacobi elliptic functions, the nome \(q\) is used with Jacobi theta functions. \(k(q)\) is explicitly given by

\[k(q) = \frac{\theta_2^2(q)}{\theta_3^2(q)}.\]

An example in Python

>>> from xlcalcnet import ereal
>>> ereal.EllipticModulus(0.5)
ereal('5.2359877559829887307E-1')
>>> ereal.EllipticModulus('0.51')
ereal('5.3518479027559984754E-1')

Jacobi amplitude, \(\mathrm{am}(x, k)\) (DAMath)#

math53.jacobi_amplitude(x, k)#

Returns the Jacobi amplitude function \(\mathrm{am}(x, k)\) for a given modulus \(k\). This is the inverse function of Legendre’s elliptic function of the first kind: \(\mathrm{am}(F(x, k), k) = x\). When \(|k| < 1, \mathrm{am}(x, k)\) is a monotone quasi-periodic function NIST [502], 22.16.2: \(\mathrm{am}(x + 2K(k), k) = \mathrm{am}(x, k) + \pi\), with the special case \(\mathrm{am}(x, 0) = x\). When \(|k| > 1, \mathrm{am}(x, k)\) is periodic with period \(4K(1/k)/k\), and if \(|k| = 1\), then it is equal to the Gudermannian function \(\mathrm{am}(x, \pm1) = \mathrm{gd}(x)\).

See also MathWorld [1080], NIST [502], Ehrhardt [309] (3.2.10).

An example in Python

>>> from xlcalcnet import ereal
>>> ereal.JacobiAmplitude(0.8, 0.5)
ereal('5.2359877559829887307E-1')
>>> ereal.JacobiAmplitude(0.8, '0.51')
ereal('5.3518479027559984754E-1')
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