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==Definition in terms of the Jacobi theta functions==
===Jacobi theta function description===
Equivalently, Jacobi's elliptic functions can be defined in terms of the [[theta function]]s. Let
Equivalently, Jacobi's elliptic functions can be defined in terms of his [[theta function]]s. If we abbreviate <math>\vartheta_{00}(0;q)</math> as <math>\vartheta_{00}(q)</math>, and <math>\vartheta_{01}(0;q), \vartheta_{10}(0;q), \vartheta_{11}(0;q)</math> respectively as <math>\vartheta_{01}(q), \vartheta_{10}(q), \vartheta_{11}(q)</math> (the ''theta constants'') then the [[Theta function|theta function elliptic modulus]] ''k'' is <math>k=\biggl\{{\vartheta_{10}[q(k)] \over \vartheta_{00}[q(k)]}\biggr\}^2</math>. We define the [[nome (mathematics)|nome]] as <math>q = \exp (\pi i \tau)</math> in relation to the period ratio. We have
:<math>\theta_1(z|\tau)=\displaystyle\sum_{n=-\infty}^\infty (-1)^{n-\frac12}e^{(2n+1)iz+\pi i\tau\left(n+\frac12\right)^2},</math>
: <math id="def.theta">
:<math>\theta_2(z|\tau)=\displaystyle\sum_{n=-\infty}^\infty e^{(2n+1)iz+\pi i\tau \left(n+\frac12\right)^2},</math>
\begin{align}
:<math>\theta_3(z|\tau)=\displaystyle\sum_{n=-\infty}^\infty e^{2niz+\pi i\tau n^2},</math>
\operatorname{sn}(u; k) & = -\frac{\vartheta_{00}(q) \,\vartheta_{11}(\zeta;q)}{\vartheta_{10}(q) \,\vartheta_{01}(\zeta;q)} \\[7pt]
:<math>\theta_4(z|\tau)=\displaystyle\sum_{n=-\infty}^\infty (-1)^n e^{2niz+\pi i\tau n^2}</math>
\operatorname{cn}(u; k) & = \frac{\vartheta_{01}(q) \,\vartheta_{10}(\zeta;q)}{\vartheta_{10}(q) \,\vartheta_{01}(\zeta;q)} \\[7pt]
and <math>\theta_2(\tau)=\theta_2(0|\tau)</math>, <math>\theta_3(\tau)=\theta_3(0|\tau)</math>, <math>\theta_4(\tau)=\theta_4(0|\tau)</math>. Then with <math>K=K(m)</math>, <math>K'=K(1-m)</math>, <math>\zeta=\pi u/(2K)</math> and <math>\tau=iK'/K</math>,
\operatorname{dn}(u; k) & = \frac{\vartheta_{01}(q) \,\vartheta_{00}(\zeta;q)}{\vartheta_{00}(q) \,\vartheta_{01}(\zeta;q)}
\end{align}
</math>
:<math>\begin{align}\operatorname{sn}(u,m)&=\frac{\theta_3(\tau)\theta_1(\zeta|\tau)}{\theta_2(\tau)\theta_4(\zeta|\tau)},\\
where <math>\zeta=\pi u/(2K)</math>.
\operatorname{cn}(u,m)&=\frac{\theta_4(\tau)\theta_2(\zeta|\tau)}{\theta_2(\tau)\theta_4(z|\tau)},\\
\operatorname{dn}(u,m)&=\frac{\theta_4(\tau)\theta_3(\zeta|\tau)}{\theta_3(\tau)\theta_4(\zeta|\tau)}.\end{align}</math>
[[E. T. Whittaker|Edmund Whittaker]] and [[G. N. Watson|George Watson]] defined the [[theta function|Jacobi theta functions]] this way in their textbook ''[[A Course of Modern Analysis]]'':<ref>Whittaker and Watson (1990) pp. 469–470</ref>
:<math>\vartheta_{00}(v;w) = \prod_{n = 1}^\infty (1-w^{2n})[1+2\cos(2v)w^{2n-1}+w^{4n-2}]</math>
:<math>\vartheta_{01}(v;w) = \prod_{n = 1}^\infty (1-w^{2n})[1-2\cos(2v)w^{2n-1}+w^{4n-2}]</math>
:<math>\vartheta_{10}(v;w) = 2 w^{1/4}\cos(v)\prod_{n = 1}^\infty (1-w^{2n})[1+2\cos(2v)w^{2n}+w^{4n}]</math>
:<math>\vartheta_{11}(v;w) = -2 w^{1/4}\sin(v)\prod_{n = 1}^\infty (1-w^{2n})[1-2\cos(2v)w^{2n}+w^{4n}]</math>
The Jacobi zn function can be expressed by theta functions as well:
:<math>\begin{align}\operatorname{zn}(u;k,m)&=\frac{\pi}{2K}\frac{\vartheta_theta_{014}'(\zeta;q|\tau)}{\vartheta_theta_{014}(\zeta;q|\tau)}\\ &=\frac{\pi}{2K}\frac{\vartheta_theta_{003}'(\zeta;q|\tau)}{\vartheta_theta_{003}(\zeta;q|\tau)}+k^2\frac{\operatorname{sn}(u;k,m)\operatorname{cn}(u;k,m)}{\operatorname{dn}(u;k,m)}\\
&=\frac{\pi}{2K}\frac{\vartheta_theta_{102}'(\zeta;q|\tau)}{\vartheta_theta_{102}(\zeta;q|\tau)}+\frac{\operatorname{dn}(u;k,m)\operatorname{sn}(u;k,m)}{\operatorname{cn}(u;k,m)}\\
&=\frac{\pi}{2K}\frac{\vartheta_theta_{111}'(\zeta;q|\tau)}{\vartheta_theta_{111}(\zeta;q|\tau)}-\frac{\operatorname{cn}(u;k,m)\operatorname{dn}(u;k,m)}{\operatorname{sn}(u;k,m)}\end{align}</math>
where <math>'</math> denotes the partial derivative with respect to the first variable.
===Elliptic integral and elliptic nome===
Since the Jacobi functions are defined in terms of the elliptic modulus <math>k(\tau)</math>, we need to invert this and find <math>\tau</math> in terms of <math>k</math>. We start from <math>k' = \sqrt{1-k^2}</math>, the ''complementary modulus''. As a function of <math>\tau</math> it is
:<math>k'(\tau) = \sqrt{1 - k^2} = \biggl\{{\vartheta_{01}[q(k)] \over \vartheta_{00}[q(k)]}\biggr\}^2</math>
Let us define the [[Nome (mathematics)|elliptic nome]] and the [[Elliptic Integral#Complete elliptic integral of the first kind|complete elliptic integral of the first kind]]:
:<math>q(k) = \exp\biggl[-\pi\frac{K(\sqrt{1 - k^2})}{K(k)}\biggr]</math>
These are two identical definitions of the complete elliptic integral of the first kind:
:<math>K(k) = \int_{0}^{\pi/2} \frac{1}{\sqrt{1 - k^2\sin(\varphi)^2}} \partial\varphi</math>
:<math>K(k) = \frac{\pi}{2}\sum_{a = 0}^{\infty} \frac{[(2a)!]^2}{16^{a}(a!)^4} k^{2a}</math>
An identical definition of the nome function can be produced by using a series. Following function has this identity:
:<math>\frac{1 - \sqrt[4]{1 - k^2}}{1 + \sqrt[4]{1 - k^2}} = \frac{\vartheta_{00}[q(k)] - \vartheta_{01}[q(k)]}{\vartheta_{00}[q(k)] + \vartheta_{01}[q(k)]} = \biggl[\sum_{n = 1}^{\infty} 2\,q(k)^{(2n - 1)^2}\biggr] \biggl[1 + \sum_{n = 1}^{\infty} 2\,q(k)^{4n^2}\biggr]^{-1}</math>
Since we may reduce to the case where the imaginary part of <math>\tau</math> is greater than or equal to <math>\sqrt{3}/2</math> (see [[Modular group#Relationship to hyperbolic geometry|Modular group]]), we can assume the absolute value of <math>q</math> is less than or equal to <math>\exp(-\pi\sqrt{3}/2) \approx 0.0658 </math>; for values this small the above series converges very rapidly and easily allows us to find the appropriate value for <math>q</math>. By solving this function after q we get:<ref>{{cite web|access-date=2023-05-28|title=A002103 - OEIS|url=https://oeis.org/A002103}}<!-- auto-translated by Module:CS1 translator --></ref><ref>{{cite web|access-date=2023-05-28|language=en|title=Series Expansion of EllipticNomeQ differs from older Mathematica Version|url=https://mathematica.stackexchange.com/questions/269455/series-expansion-of-ellipticnomeq-differs-from-older-mathematica-version}}<!-- auto-translated by Module:CS1 translator --></ref><ref>{{citation|author=R. B. King, E. R. Canfield|date=1992-08-01|doi=10.1016/0898-1221(92)90210-9|issn=0898-1221|issue=3|pages=13–28|periodical=Computers & Mathematics with Applications|title=Icosahedral symmetry and the quintic equation|volume=24|doi-access=free}}<!-- auto-translated by Module:CS1 translator --></ref>
:<math>q(k) = \sum_{n = 1}^{\infty} \frac{\text{Sw}(n)}{2^{4n - 3}} \biggl(\frac{1 - \sqrt[4]{1 - k^2}}{1 + \sqrt[4]{1 - k^2}}\biggr)^{4n - 3} = k^2\biggl\{\frac{1}{2} + \biggl[\sum_{n = 1}^{\infty} \frac{\text{Sw}(n + 1)}{2^{4n + 1}} k^{2n}\biggr]\biggr\}^4</math>
Where SW(n) is sequence [[OEIS:A002103|A002103]] in the [[On-Line Encyclopedia of Integer Sequences|OEIS]].
==Definition in terms of Neville theta functions==
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