Editing Theta function (section)

==Auxiliary functions==

The Jacobi theta function defined above is sometimes considered along with three auxiliary theta functions, in which case it is written with a double 0 subscript:

:<math>\vartheta_{00}(z;\tau) = \vartheta(z;\tau)</math>

The auxiliary (or half-period) functions are defined by

:<math>\begin{align}
\vartheta_{01}(z;\tau)& = \vartheta \left(z+\tfrac12;\tau\right)\\[3pt]
\vartheta_{10}(z;\tau)& = \exp\left(\tfrac14\pi i \tau + \pi i z\right)\vartheta\left(z + \tfrac12\tau;\tau\right)\\[3pt]
\vartheta_{11}(z;\tau)& = \exp\left(\tfrac14\pi i \tau + \pi i\left(z+\tfrac12\right)\right)\vartheta\left(z+\tfrac12\tau + \tfrac12;\tau\right).
\end{align}</math>

This notation follows [[Bernhard Riemann|Riemann]] and [[David Mumford|Mumford]]; [[Carl Gustav Jacobi|Jacobi]]'s original formulation was in terms of the [[nome (mathematics)|nome]] {{math|''q'' {{=}} ''e''<sup>''iπτ''</sup>}} rather than {{mvar|τ}}. In Jacobi's notation the {{mvar|θ}}-functions are written:

:<math>\begin{align}
\theta_1(z;q) &=\theta_1(\pi z,q)= -\vartheta_{11}(z;\tau)\\
\theta_2(z;q) &=\theta_2(\pi z,q)= \vartheta_{10}(z;\tau)\\
\theta_3(z;q) &=\theta_3(\pi z,q)= \vartheta_{00}(z;\tau)\\
\theta_4(z;q) &=\theta_4(\pi z,q)= \vartheta_{01}(z;\tau)
\end{align}</math>

[[File:Jacobi theta 1.png|thumb|Jacobi theta 1]]
[[File:Jacobi theta 2.png|thumb|Jacobi theta 2]]
[[File:Jacobi theta 3.png|thumb|Jacobi theta 3]]
[[File:Jacobi theta 4.png|thumb|Jacobi theta 4]]

The above definitions of the Jacobi theta functions are by no means unique. See [[Jacobi theta functions (notational variations)]] for further discussion.

If we set {{math|''z'' {{=}} 0}} in the above theta functions, we obtain four functions of {{mvar|τ}} only, defined on the upper half-plane. These functions are called ''Theta Nullwert'' functions, based on the German term for ''zero value'' because of the annullation of the left entry in the theta function expression. Alternatively, we obtain four functions of {{mvar|q}} only, defined on the unit disk <math>|q|<1</math>. They are sometimes called [[theta constant]]s:<ref group="note"><math>\theta_1(q)=0</math> for all <math>q\in\mathbb{C}</math> with <math>|q|<1</math>.</ref>

:<math>\begin{align}
\vartheta_{11}(0;\tau)&=-\theta_1(q)=-\sum_{n=-\infty}^\infty (-1)^{n-1/2}q^{(n+1/2)^2} \\
\vartheta_{10}(0;\tau)&=\theta_2(q)=\sum_{n=-\infty}^\infty q^{(n+1/2)^2}\\
\vartheta_{00}(0;\tau)&=\theta_3(q)=\sum_{n=-\infty}^\infty q^{n^2}\\
\vartheta_{01}(0;\tau)&=\theta_4(q)=\sum_{n=-\infty}^\infty (-1)^n q^{n^2}
\end{align}</math>

with the [[Nome (mathematics)|nome]] {{math|''q'' {{=}} ''e''<sup>''iπτ''</sup>}}. 
Observe that <math> \theta_1(q)=0 </math>. 
These can be used to define a variety of [[modular forms]], and to parametrize certain curves; in particular, the '''Jacobi identity''' is

:<math>\theta_2(q)^4 + \theta_4(q)^4 = \theta_3(q)^4</math>

or equivalently,

:<math>\vartheta_{01}(0;\tau)^4 + \vartheta_{10}(0;\tau)^4 = \vartheta_{00}(0;\tau)^4</math>

which is the [[Fermat curve]] of degree four.