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# Integral of Jacobi Theta function

Let denote one of the Jacobi Theta functions. Prove that

Solution

We have successively,

The sum is evaluated as follows. Consider the function

and integrate it around a square with vertices . The function has poles at every integer with residue as well as at with residues . We also note that as the contour integral of tends to . Thus,

Hence,

and the exercise is complete.

1. Oliver says:

Some more results:

1. .

2.

3.

4.

5.

The derivation is pretty simple! They are left to the reader!

2. John Khan says:

I’m giving the 3rd one a try. We have that

The series is standard. Initiate a contour integration by invoking the kernel and the result follows.