EllipticModulus - Maple Help
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EllipticModulus

Modulus function k(q)

 Calling Sequence EllipticModulus(q)

Parameters

 q - expression denoting a complex number such that $\left|q\right|<1$

Description

 Given the Nome q, $\left|q\right|<1$, entering the definition of Jacobi Theta functions, for instance
 ${\mathrm{JacobiTheta1}}{}\left({z}{,}{q}\right){=}{\sum }_{{\mathrm{_k1}}{=}{0}}^{{\mathrm{\infty }}}{}{2}{{q}}^{{\left({\mathrm{_k1}}{+}\frac{{1}}{{2}}\right)}^{{2}}}{\mathrm{sin}}{}\left({z}\left({2}{\mathrm{_k1}}{+}{1}\right)\right){\left({-1}\right)}^{{\mathrm{_k1}}}$ (1)
 EllipticModulus computes the corresponding Modulus k, $0<\mathrm{\Re }\left(k\right)$ entering the definition of related elliptic integrals and JacobiPQ elliptic functions.
 ${\mathrm{EllipticF}}{}\left({z}{,}{k}\right){=}{{\int }}_{{0}}^{{z}}\frac{{1}}{\sqrt{{-}{{\mathrm{_α1}}}^{{2}}{+}{1}}\sqrt{{-}{{k}}^{{2}}{{\mathrm{_α1}}}^{{2}}{+}{1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_α1}}$ (2)
 ${\mathrm{JacobiSN}}{}\left({z}{,}{k}\right){=}{\mathrm{sin}}{}\left({\mathrm{JacobiAM}}{}\left({z}{,}{k}\right)\right)$ (3)
 $\left[{z}{=}{\mathrm{JacobiAM}}{}\left({{\int }}_{{0}}^{{z}}\frac{{1}}{\sqrt{{1}{-}{{k}}^{{2}}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}^{{2}}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{\theta }}{,}{k}\right){,}{z}{::}\left[{-}\frac{{3}}{{2}}{,}\frac{{3}}{{2}}\right]\right]$ (4)
 Alternatively, given the Modulus k, $0<\mathrm{\Re }\left(k\right)$ entering Elliptic integrals and JacobiPQ functions, it is possible to compute the corresponding Nome q, $\left|q\right|<1$, using EllipticNome, which is the inverse function of EllipticModulus.
 EllipticModulus is defined in terms of JacobiTheta functions by:
 > FunctionAdvisor( definition, EllipticModulus );
 $\left[{\mathrm{EllipticModulus}}{}\left({q}\right){=}\frac{{{\mathrm{JacobiTheta2}}{}\left({0}{,}{q}\right)}^{{2}}}{{{\mathrm{JacobiTheta3}}{}\left({0}{,}{q}\right)}^{{2}}}{,}\left|{q}\right|{<}{1}\right]$ (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));
 ${\mathrm{JacobiSN}}{}\left({z}{,}{k}\right){=}\frac{{\left(\frac{{1}}{{{k}}^{{2}}}\right)}^{{1}}{{4}}}{\mathrm{JacobiTheta1}}{}\left(\frac{{\mathrm{\pi }}{z}}{{2}{\mathrm{EllipticK}}{}\left({k}\right)}{,}{\mathrm{EllipticNome}}{}\left({k}\right)\right)}{{\mathrm{JacobiTheta4}}{}\left(\frac{{\mathrm{\pi }}{z}}{{2}{\mathrm{EllipticK}}{}\left({k}\right)}{,}{\mathrm{EllipticNome}}{}\left({k}\right)\right)}$ (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 $m={k}^{2}$ and sin(alpha) = k. For example, the Elliptic $K\left(m\right)$ function shown in A&S is numerically equal to the Maple $\mathrm{EllipticK}\left(\sqrt{m}\right)$ command.

Examples

 > ${\mathrm{FunctionAdvisor}\left(\mathrm{definition},\mathrm{EllipticModulus}\left(q\right)\right)}_{1}$
 ${\mathrm{EllipticModulus}}{}\left({q}\right){=}\frac{{{\mathrm{JacobiTheta2}}{}\left({0}{,}{q}\right)}^{{2}}}{{{\mathrm{JacobiTheta3}}{}\left({0}{,}{q}\right)}^{{2}}}$ (7)
 > $\mathrm{evalf}\left(\genfrac{}{}{0}{}{\phantom{q=\frac{1}{2}}}{}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{}}{q=\frac{1}{2}}\right)$
 ${0.9999947611}{=}{0.9999947617}$ (8)
 > $\mathrm{EllipticModulus}\left(\mathrm{EllipticNome}\left(k\right)\right)=k$
 ${\mathrm{EllipticModulus}}{}\left({\mathrm{EllipticNome}}{}\left({k}\right)\right){=}{k}$ (9)
 > $\mathrm{evalf}\left(\genfrac{}{}{0}{}{\phantom{k=2}}{}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{}}{k=2}\right)$
 ${2.}{=}{2.}$ (10)
 > $\mathrm{EllipticNome}\left(\mathrm{EllipticModulus}\left(q\right)\right)=q$
 ${\mathrm{EllipticNome}}{}\left({\mathrm{EllipticModulus}}{}\left({q}\right)\right){=}{q}$ (11)
 > $\mathrm{evalf}\left(\genfrac{}{}{0}{}{\phantom{q=\frac{1}{2}}}{}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{}}{q=\frac{1}{2}}\right)$
 ${0.5000000000}{=}{0.5000000000}$ (12)