# trigonometric → elliptic in the infinite product representation of θ3(z,q)

If $G(q) = \prod_{n=1}^{\infty} (1-q^{2n})$, then

$\theta_{3}(z,q) = G(q) \prod_{n=1}^{\infty} \left[1 - 2q^{2n-1}\cos(2z) +q^{4n-2}\right]$

Let’s define an analogue of the theta function $\theta_{3}(z,q)$ by replacing that trigonometric cosine $\cos$ with an elliptic $\mathrm{cn}(2z,k)$, where $k$ is the elliptic modulus.

So we’d have:

$A_{3}(z,q;k) = G(q) \prod_{n=1}^{\infty} \left[1 - 2q^{2n-1}\mathrm{cn}(2z;k) +q^{4n-2}\right]$

Here’s a domain colored picture of $A_{3}(z,1/2;1/2)$: