I got this idea from an integral on Wikipedia
This integral is quite difficult to solve with standard techniques from elementary calculus. Instead, the usual approach is to rewrite it as
, then evaluate it as a contour integral by using the following contour that surrounds the pole
The point of this approach is that it leaves the other pole
outside the nice semicircle contour no matter the value of a, which lets you convert the problem into a "simple" calculation of limits (e.g. by using the residue theorem). Compared to integration, the calculation of limits is a relatively algorithmic process that usually works without any "art".
So I figured, why not add more poles to the lower half-plane? After all, this shouldn't affect the contour, and you should still be able to use the residue theorem on a single pole. In other words, you should, at least in principle, still be able to find exact solutions to such integrals by calculating a finite number of limits, even if they might be somewhat... messy.
Well, as far as I can tell, they are. I played around a bit with this one:
has a single pole
in the upper half-plane, and its remaining
poles in the lower half-plane.
Apparently I failed to evaluate any such integrals (except
) manually, but Mathematica returned some... interesting results for n=1 and n=2 after half an hour or so:
Have you seen any other particularly hideous integrals?