Product Representations
The Jacobi triple product tells us that for complex numbers w and q with |q| < 1 and w ≠ 0 we have
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
In terms of w and q:
where is the q-Pochhammer symbol and is the q-theta function. Expanding terms out, the Jacobi triple product can also be written
which we may also write as
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
Read more about this topic: Theta Function
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