Hilariously hideous integrals

Demosthenes · 11:19 AM

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

$\reverse \int_{0}^{\pi}\cos(mx)\cos(nx)\,\mathrm{d}x = \frac{\pi}{2}\delta_{mn}$

into

$\reverse \int_{0}^{\pi/2}\ln|\sin(mx)|\cdot\ln|\sin(nx)|\,\mathrm{d}x = \frac{\pi^3}{24}\cdot\frac{\gcd^2(m,n)}{mn} + \frac{\pi\ln^2(2)}{2}$

look promising for a number of identities for orthogonal polynomials. For example, the Legendre polynomials $\reverse P_n(x)$ that I mentioned in another thread satisfy

$\reverse \int_{-1}^{1} P_m(x)P_n(x)\,\mathrm{d}x = \frac{2}{2n+1}\delta_{mn}.$

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

$\reverse \int_{0}^{1}(-\ln x)^{-\mu}\,\mathrm{d}x = \frac{\pi}{\mathrm{\Gamma}(\mu)\,\sin(\mu\pi)}$

It looks quite a bit like the functional equation for the Riemann zeta function, I think (where I replaced replaced its usual parameter $\reverse s$ with $\reverse 1-\mu$ and did a little algebra):

$\reverse 2^{1-\mu}\frac{\mathrm{\zeta(\mu)}}{\mathrm{\zeta(1-\mu)}}\, = \frac{\pi^{\mu}}{\mathrm{\Gamma(\mu)}\,\sin\left(0 .5(1-\mu)\pi\right)}$

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