Using complex analysis for evaluating summations: some general queries

Solution 1:

These are some good questions. Let me try to answer:

1. Yeah, this is hard. The idea is that we need a function that has poles at the integers and is bounded on a contour as the size of the region bounded by the contour becomes infinite. The function $\pi \cot{(\pi z)}$ fits the bill. But I understand the bemusement at this function just magically appearing out of nowhere.

The problem is, it is very difficult to find other functions. For example, the function $\Gamma{(z)}$ has poles at the nonpositive integers, residue $(-1)^n/n!$. But $\Gamma{(z)}$ is unbounded over the increasingly large contours you see in your proofs with $\pi \cot{(\pi z)}$, so it won't work.

2. No real significance, so long as the contours do not intersect poles.

3. Yes, zero is a pole. You can prove that the poles are the only ones with the contour using, e.g., Rouche's Theorem.

4. A contour integral is essentially a line integral, in that one parametrizes each portion of the contour for computation. The idea is that the magnitude of such an integral is bounded as the size of the region enclosed by the contour increases without bound.