Editing Theta function (section)

==Product representations==
The [[Jacobi triple product]] (a special case of the [[Macdonald identities]]) tells us that for complex numbers {{mvar|w}} and {{mvar|q}} with {{math|{{abs|''q''}} < 1}} and {{math|''w'' ≠ 0}} we have
:<math>\prod_{m=1}^\infty 
\left( 1 - q^{2m}\right)
\left( 1 + w^2 q^{2m-1}\right)
\left( 1 + w^{-2}q^{2m-1}\right)
= \sum_{n=-\infty}^\infty  w^{2n}q^{n^2}.
</math>

It can be proven by elementary means, as for instance in Hardy and Wright's ''[[An Introduction to the Theory of Numbers]]''.

If we express the theta function in terms of the nome {{math|''q'' {{=}} ''e''<sup>''πiτ''</sup>}} (noting some authors instead set {{math|''q'' {{=}} ''e''<sup>2''πiτ''</sup>}})  and take {{math|''w'' {{=}} ''e''<sup>''πiz''</sup>}} then

:<math>\vartheta(z; \tau) = \sum_{n=-\infty}^\infty \exp(\pi i \tau n^2) \exp(2\pi i z n) = \sum_{n=-\infty}^\infty w^{2n}q^{n^2}. </math>

We therefore obtain a product formula for the theta function in the form

:<math>\vartheta(z; \tau) = \prod_{m=1}^\infty 
\big( 1 - \exp(2m \pi i \tau)\big)
\Big( 1 + \exp\big((2m-1) \pi i \tau + 2 \pi i z\big)\Big)
\Big( 1 + \exp\big((2m-1) \pi i \tau - 2 \pi i z\big)\Big).
</math>

In terms of {{mvar|w}} and {{mvar|q}}:
:<math>\begin{align}
\vartheta(z; \tau) &= \prod_{m=1}^\infty 
\left( 1 - q^{2m}\right)
\left( 1 + q^{2m-1}w^2\right)
\left( 1 + \frac{q^{2m-1}}{w^2}\right) \\
&= \left(q^2;q^2\right)_\infty\,\left(-w^2q;q^2\right)_\infty\,\left(-\frac{q}{w^2};q^2\right)_\infty \\
&= \left(q^2;q^2\right)_\infty\,\theta\left(-w^2q;q^2\right) \end{align}</math>

where {{math|(&nbsp;&nbsp;;&nbsp;&nbsp;)<sub>∞</sub>}} is the [[q-Pochhammer symbol|{{mvar|q}}-Pochhammer symbol]] and {{math|''θ''(&nbsp;&nbsp;;&nbsp;&nbsp;)}} is the [[q-theta function|{{mvar|q}}-theta function]]. Expanding terms out, the Jacobi triple product can also be written

:<math>\prod_{m=1}^\infty 
\left( 1 - q^{2m}\right)
\Big( 1 + \left(w^2+w^{-2}\right)q^{2m-1}+q^{4m-2}\Big),</math>

which we may also write as

:<math>\vartheta(z\mid q) = \prod_{m=1}^\infty 
\left( 1 - q^{2m}\right)
\left( 1 + 2 \cos(2 \pi z)q^{2m-1}+q^{4m-2}\right).</math>

This form is valid in general but clearly is of particular interest when {{mvar|z}} is real. Similar product formulas for the auxiliary theta functions are

:<math>\begin{align}
\vartheta_{01}(z\mid q) &= \prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 - 2 \cos(2 \pi z)q^{2m-1}+q^{4m-2}\right),\\[3pt]
\vartheta_{10}(z\mid q) &= 2 q^\frac14\cos(\pi z)\prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 + 2 \cos(2 \pi z)q^{2m}+q^{4m}\right),\\[3pt]
\vartheta_{11}(z\mid q) &= -2 q^\frac14\sin(\pi z)\prod_{m=1}^\infty \left( 1 - q^{2m}\right)\left( 1 - 2 \cos(2 \pi z)q^{2m}+q^{4m}\right).
\end{align}</math>
In particular, <math display="block">\lim_{q\to 0}\frac{\vartheta_{10}(z\mid q)}{2 q^{\frac14}} = \cos(\pi z),\quad \lim_{q\to 0}\frac{-\vartheta_{11}(z\mid q)}{2 q^{-\frac14}} = \sin(\pi z)</math>so we may interpret them as one-parameter deformations of the periodic functions <math>\sin, \cos</math>, again validating the interpretation of the theta function as the most general 2 quasi-period function.