Closed-form expression of an infinite sum

[Manchot]

Hey all,

This isn't actually for a homework problem, but I'm trying to find a closed form for the following expression:

S(z) = \sum_{n=1}^\infty \exp(\frac{i}{n}) z^{-n}

(Provided that it converges, of course.) Anyone have any tips, or know of any transcendental functions that might help me out? If you did, that would be great.

 

[jpr0]

There is a method to resum summations using contour integration. I'm not sure how helpful it is with this particular example though.

Have you heard of this method before?

 

[Manchot]

No, I haven't, but I think I have a guess as to what you mean. Do you construct a function whose poles are located at the integers 1 through n, and integrate around them?

 

[jpr0]

Yes, you usually multiply by the function \pi\cot\pi z, which has simple poles of residue 1 at integer values along the real axis. If you pick a contour which encloses the positive real axis, then the integral will give you 2\pi i\times(your summation). If you then deform your contour into one which allows you to evaluate your integral, this will give you your summation in closed form. It doesn't always work though - the integral you have to evaluate could be just as difficult as the original summation. I'll have a closer look at the summation when I get more time.