Theta Function - Product Representations

Product Representations

The Jacobi triple product tells us that for complex numbers w and q with |q| < 1 and w ≠ 0 we have

\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}.

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 and then

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

\vartheta(z; \tau) = \prod_{m=1}^\infty \left( 1 - \exp(2m \pi i \tau)\right) \left( 1 + \exp((2m-1) \pi i \tau + 2 \pi i z)\right) \left( 1 + \exp((2m-1) \pi i \tau -2 \pi i z)\right).

In terms of w and q:

\vartheta(z; \tau) = \prod_{m=1}^\infty \left( 1 - q^{2m})\right) \left( 1 + q^{2m-1}w^2\right) \left( 1 + q^{2m-1}/w^2\right)

where is the q-Pochhammer symbol and is the q-theta function. Expanding terms out, the Jacobi triple product can also be written

\prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 + (w^{2}+w^{-2})q^{2m-1}+q^{4m-2}\right),

which we may also write as

\vartheta(z|q) = \prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 + 2 \cos(2 \pi z)q^{2m-1}+q^{4m-2}\right).

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

\vartheta_{01}(z|q) = \prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 - 2 \cos(2 \pi z)q^{2m-1}+q^{4m-2}\right).
\vartheta_{10}(z|q) = 2 q^{1/4}\cos(\pi z)\prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 + 2 \cos(2 \pi z)q^{2m}+q^{4m}\right).
\vartheta_{11}(z|q) = -2 q^{1/4}\sin(\pi z)\prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 - 2 \cos(2 \pi z)q^{2m}+q^{4m}\right).

Read more about this topic:  Theta Function

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