Question

Anybody familiar with using contour integration to find exact sums?

Answer 1

Sounds pretty `complex`

Answer 2

So, I was given a few examples of situations in which we can find exact sums of certain series. For example, I have these, all of which use the contour that is a square that encloses all the integers up to n, -n, in, and -in, but not including n+1, -n-1, i(n+1), -i(n+1):

\[\sum_{n=-\infty}^{+\infty}f(n)\]

This uses the closed contour integral (not sure how to make the counterclockwise closed contour integral thingy) w

\[\int\limits_{}^{}f(z) \pi \cot(\pi z)dz\]

Then there is:

\[\sum_{n=-\infty}^{+\infty}(-1)^{n}f(n)\]

This uses the closed contour integral

\[\int\limits_{}^{}f(z) \pi \csc(\pi z)dz\]

I know there is one for

\[\sum_{n=-\infty}^{+\infty}f(n) \sin (\alpha n)\] (same with \(\cos(\alpha n)\) )

But I was unable to get this one to work. I tried using

\[\int\limits_{}^{}f(z) \pi e^{i(\alpha - \pi)z} \csc(\pi z)dz\]

just didn't work.

Also, some of these take advantage of the summand being an even function, which allows you to get a sum from 0 to infinity instead. But if the function is an odd function, I wouldnt know how to compensate for that.

So essentially my question is:

1. If anyone knows the correct contour integral for summands with sine and cosine and

2. If anyone knows how to handle summands that are odd functions (or maybe have no symmetry)

3. If there are other specific contour integrals that can be used in other situations that I haven't mentioned

Answer 3

@ganeshie8

Answer 4

Can you post the exact sum you're trying to find?