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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.

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  • $\begingroup$ F depends on z and t. Therefore, the Integral depends on t. Therefore, you can not do a numerical integration. And it is quit possible that no analytical solution exists. $\endgroup$ Jul 31, 2023 at 19:08
  • $\begingroup$ I think you should first confirm EllipticTheta is not negative or zero in the range you're integrating. It doesn't look like it is. If so then you're not breaching any branch points or branch cuts of Log when $u$ is an integer. When it's rational, then in general would need to deal with branch point and cuts of root objects but again since not integrating over the orgin would not encounter the Mathematica default branch cut along the negative real axis for root objects as well. $\endgroup$
    – josh
    Aug 1, 2023 at 14:36

2 Answers 2

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Don't see any problem with its numerical integration. For example,

t = Pi; F[z_] := D[-Log[EllipticTheta[1, Pi z, E^(-Pi*t)]], {z, 2}]
Table[NIntegrate[F[z]^-u, {z, 0, 1}], {u, 1, 4,1/2}]

{0.0506606, 0.013688, 0.00384974, 0.00110951, 0.00032505, 0.0000963569, 0.0000288177}

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  • $\begingroup$ t = Pi; u = E; F[z_] := D[-Log[EllipticTheta[1, Pi z, E^(-Pi*t)]], {z, 2}] NIntegrate[F[z]^-u, {z, 0, 1}] results in 0.00064808. $\endgroup$
    – user64494
    Jul 31, 2023 at 19:14
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The elliptic Thetas near the real line are deformations of $sin, cos$ functions with periods of multiples of $\pi$ and exponential growth in imaginary directions

 Manipulate[ComplexPlot3D[EllipticTheta[1, z, q], 
{z, -Pi (1 + I/5), Pi (1 + I/5)}, PlotRange -> All,
Ticks -> {Pi Range[-1, 1], Automatic, Automatic}], 
{{q, 0.8}, 0.01,1}, ControlPlacement -> Top]

enter image description here

The logarithm has of course logarithmic branch cuts along the real line of negative half periods with infintely many copies of $\mathbb C$ with $2\pi i $ distance in the argument. Integrals along the real line make sense only in the intervals of positivity of $sin z$

enter image description here

Structurally, the derivative of $ (Log Sin)'' = Csc^2$ is showing the second derivatives have the form of quadratic rationals of elliptic Thetas, that can be further analyzed via their Fourier series.

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  • $\begingroup$ Doesn't Manipulate[ ComplexPlot[ EllipticTheta[1, z, q]^(-1/2), {z, -Pi (1 + I/5), Pi (1 + I/5)}, PlotRange -> All, Ticks -> {Pi Range[-1, 1], Automatic, Automatic}], {{q, 0.8}, 0.01, 1}, ControlPlacement -> Top] show branch cuts or am I mistaken? $\endgroup$
    – user64494
    Aug 1, 2023 at 14:00
  • $\begingroup$ The theta functions are eintier functions in z. This fact is clear for fast converging Fourier series with coefficients converging to zero faster than any power, which is just a bit slower than a finite fourier sum. They are periodic in the real direction and quasiperiodic in imaginary direction. Their quotients are the meromorphic Jacobi elliptic functions. $\endgroup$
    – Roland F
    Aug 1, 2023 at 19:35

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