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.

Some more results:

1..2.3.4.5.The derivation is pretty simple! They are left to the reader!

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.