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Hilariously hideous integrals
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Posted 2020-07-23, 06:27 PM
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:



Obviously 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?
"Stephen Wolfram is the creator of Mathematica and is widely regarded as the most important innovator in scientific and technical computing today." - Stephen Wolfram
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Posted 2020-07-24, 10:44 AM in reply to Chruser's post "Hilariously hideous integrals"
Have you seen the following post on Stackexchange: https://math.stackexchange.com/quest...tegral-milking

I think you'll enjoy it.
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Posted 2020-07-25, 10:34 AM in reply to Demosthenes's post starting "Have you seen the following post on..."
Demosthenes said: [Goto]
Have you seen the following post on Stackexchange: https://math.stackexchange.com/quest...tegral-milking

I think you'll enjoy it.

Nice ones! The techniques used to turn



into



look promising for a number of identities for orthogonal polynomials. For example, the Legendre polynomials that I mentioned in another thread satisfy




Also, I like this one from the book "Irresistible Integrals" (page 190) that was mentioned in the Math Underflow thread:



It looks quite a bit like the functional equation for the Riemann zeta function, I think (where I replaced replaced its usual parameter with and did a little algebra):

"Stephen Wolfram is the creator of Mathematica and is widely regarded as the most important innovator in scientific and technical computing today." - Stephen Wolfram

Last edited by Chruser; 2020-07-25 at 11:19 AM.
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