I am struggling with a complex double integral with multiple branch cuts. Even the single variable complex integral I find quite complicated due to the branch cut and the special functions involved in. Here is the simplified version of what I am after.

 F[z_] := D[- Log[EllipticTheta[1, Pi z, E^( - Pi * t)]], {z,  2}]  Integrate[F[z]^{-u}, {z, 0, 1}] 

Where t,u taking real values with t > 0 and u > 0.

It will be great to have an analytic answer (even for particular values of u, say integers to avoid some cuts). But I would be even happy with a numerical integration. I think the main obstruction that I am facing here is how to deal with the cuts in Mathematica.

I am struggling with a complex double integral with multiple branch cuts. Even the single variable complex integral I find quite complicated due to the branch cut and the special functions involved in. Here is the simplified version of what I am after.

F[z_] := D[- Log[EllipticTheta[1, Pi z, E^(- Pi * t)]], {z, 2}] Integrate[F[z]^{-u}, {z, 0, 1}] 

Where $t,u \in \mathbb R$ and $t,u > 0$.

It will be great to have an analytic answer (even for particular values of $u$, say integers to avoid some cuts). But I would be even happy with a numerical integration. I think the main obstruction that I am facing here is how to deal with the cuts in Mathematica.