EllipticNome - Maple Help

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EllipticNome

Nome function q(k)

 Calling Sequence EllipticNome(k)

Parameters

 k - expression denoting a complex number

Description

 Given the Modulus k, $0<\mathrm{Re}\left(k\right)$, entering the definition of Elliptic integrals and JacobiPQ functions,
 ${\mathrm{EllipticF}}{}\left({z}{,}{k}\right){=}{{∫}}_{{0}}^{{z}}\frac{{1}}{\sqrt{{-}{{\mathrm{_α1}}}^{{2}}{+}{1}}{}\sqrt{{-}{{\mathrm{_α1}}}^{{2}}{}{{k}}^{{2}}{+}{1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_α1}}$ (1)
 ${\mathrm{JacobiSN}}{}\left({z}{,}{k}\right){=}{\mathrm{sin}}{}\left({\mathrm{JacobiAM}}{}\left({z}{,}{k}\right)\right)$ (2)
 $\left[{z}{=}{\mathrm{JacobiAM}}{}\left({{∫}}_{{0}}^{{z}}\frac{{1}}{\sqrt{{1}{-}{{k}}^{{2}}{}{{\mathrm{sin}}{}\left({\mathrm{θ}}\right)}^{{2}}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{θ}}{,}{k}\right){,}{z}{::}\left({\mathrm{RealRange}}{}\left({-}\frac{{3}}{{2}}{,}\frac{{3}}{{2}}\right)\right)\right]$ (3)
 EllipticNome computes the corresponding Nome q, $\left|q\right|<1$, entering the definition of the related (see below) Jacobi Theta functions, for instance:
 ${\mathrm{JacobiTheta1}}{}\left({z}{,}{q}\right){=}{\sum }_{{\mathrm{_k1}}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{2}{}{{q}}^{{\left({\mathrm{_k1}}{+}\frac{{1}}{{2}}\right)}^{{2}}}{}{\mathrm{sin}}{}\left({z}{}\left({2}{}{\mathrm{_k1}}{+}{1}\right)\right){}{\left({-}{1}\right)}^{{\mathrm{_k1}}}$ (4)
 Alternatively, given the Nome q, $\left|q\right|<1$, it is possible to compute the corresponding Modulus k, $0<\mathrm{Re}\left(k\right)$, 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:
 $\left[{\mathrm{EllipticNome}}{}\left({k}\right){=}{{ⅇ}}^{{-}\frac{{\mathrm{π}}{}{\mathrm{EllipticCK}}{}\left({k}\right)}{{\mathrm{EllipticK}}{}\left({k}\right)}}{,}{\mathrm{with no restrictions on}}{}\left({k}\right)\right]$ (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));
 ${\mathrm{JacobiSN}}{}\left({z}{,}{k}\right){=}\frac{{\left(\frac{{1}}{{{k}}^{{2}}}\right)}^{{1}{/}{4}}{}{\mathrm{JacobiTheta1}}{}\left(\frac{{1}}{{2}}{}\frac{{\mathrm{π}}{}{z}}{{\mathrm{EllipticK}}{}\left({k}\right)}{,}{\mathrm{EllipticNome}}{}\left({k}\right)\right)}{{\mathrm{JacobiTheta4}}{}\left(\frac{{1}}{{2}}{}\frac{{\mathrm{π}}{}{z}}{{\mathrm{EllipticK}}{}\left({k}\right)}{,}{\mathrm{EllipticNome}}{}\left({k}\right)\right)}$ (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

 > ${\mathrm{FunctionAdvisor}\left(\mathrm{definition},\mathrm{EllipticNome}\left(k\right)\right)}_{1}$
 ${\mathrm{EllipticNome}}{}\left({k}\right){=}{{ⅇ}}^{{-}\frac{{\mathrm{π}}{}{\mathrm{EllipticCK}}{}\left({k}\right)}{{\mathrm{EllipticK}}{}\left({k}\right)}}$ (7)
 > $\mathrm{evalf}\left(\genfrac{}{}{0}{}{\phantom{k=\frac{1}{2}}}{}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{}}{k=\frac{1}{2}}\right)$
 ${0.01797238701}{=}{0.01797238701}$ (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)
 >