Closed form: $\int_0^\pi \left( \frac{2 + 2\cos(x) -\cos((k-\frac{1}{2})x) -2\cos((k+\frac{1}{2})x) - cos((k+\frac{3}{2})x)}{1-\cos(2x)}\right)dx $

Question

Find a closed form for the following definite integral:

$$ I =\int_0^\pi \left( \frac{2 + 2\cos (x) - \cos((k-\frac{1}{2})x) - 2\cos ((k+\frac{1}{2})x) - \cos((k+\frac{3}{2})x)}{1-\cos(2x)}\right) \mathrm{d}x, $$ where $k \in \mathbb{N}_{>0}$.

This is related to a question posted by prof. Igor Rivin in a comment to another question of mine. It has been already proven that $I \in \mathbb{Q}$, but finding a simple closed form is harder.

@ Nico's first guess was that it cannot get significantly simpler than $$ I= \frac{ -2 (-1)^k + (1 + 2k)\Big(\pi + 2(-1)^k \Phi\Big(-1, 1, \frac{3+2k}{2}\Big)\Big)}{2}, $$ and @ Ron Gordon seemed to agree. Still, I hope that a simpler closed-form will be found. — Dal, Nov 01 '14 at 11:58

Claude Leibovici · Answer 1 · 2014-11-01T13:17:50.047

4

Using a CAS, a general formula is obtained. Reworking the results it simplifies to $$I_k=(-1)^k\Big(1+\frac{2k+1}{4} A_k\Big)$$ with $$A_k=-\psi ^{(0)}\left(\frac{1}{4}-\frac{k}{2}\right)+\psi ^{(0)}\left(\frac{3}{4}-\frac{k}{2}\right)+\psi ^{(0)}\left(\frac{k}{2}+\frac{1}{4}\right)-\psi ^{(0)}\left(\frac{k}{2}+\frac{3}{4}\right)$$

which, at least to me, is incredibly beautiful.

edited Nov 01 '14 at 13:17

answered Nov 01 '14 at 13:12

Claude Leibovici

260,315

Thanks for your answer, which is quite beautiful. But -- as far as I can see -- it is not really simpler than the result already proposed. – Dal Nov 01 '14 at 20:04

Closed form: $\int_0^\pi \left( \frac{2 + 2\cos(x) -\cos((k-\frac{1}{2})x) -2\cos((k+\frac{1}{2})x) - cos((k+\frac{3}{2})x)}{1-\cos(2x)}\right)dx $

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