I need to calculate the following quantity: $$ \zeta(x;\mu_1,\mu_2,\tau) = _3({x-\mu_1},) \cdot _3({x-\mu_2},) $$ $_3(,)$ being the Jacobi theta function defined as: $$ _3(,)=∑_{=−∞}^∞^{in^2+in2\pi} $$ Note that $\tau$ is the same for both the thetas and is a purely imaginary quantity in the form $\tau = it, \ t \in \mathbb{R}^+$. Under this conditions, $\zeta$ is the product of two Wrapped Normal (WN) distributions of the same variable having the same variance.
Is there a closed form for my expression above?