I'm studying Dirichlet kernels and their related identities; I've stumbled upon a "weighted Dirichlet kernel" so to speak. I'd like to find a closed form for the summation.
To do so, I'm required to integrate something along the lines of \begin{align} \int dz e^{-\alpha z^2-\beta z} \theta_3(z,q) \end{align} (i.e. the indefinite integral)
Usage of special functions is fine (I've already decided to use the elliptic functions so it can't get much worse).
I found a few identities involving integration with respect to $q$, but nothing about integration with respect to $z$. I already know the series representation (this is precisely the representation I'm interested in). I'm wondering if some closed form of the above exists, and what the existing literature, or the people on mathSE can tell me about it. I myself have no idea where to start, and I would certainly be interested in solving the integral in some sort of closed form.