How to find $\int_0^{\pi}\frac{\sin n\theta}{\cos\theta-\cos\alpha}d\theta$

Question

I was doing some work in physics and I came up with a definite integral. I tried everything I could but couldn't solve the integral. The integral is $$ \int_0^\pi {\sin\left(n\theta\right)\over \cos\left(\theta\right) - \cos\left(\alpha\right)}\,{\rm d}\theta\,, \qquad\qquad n\ \in\ {\mathbb Z}\,,\qquad 0\ \leq\ \alpha\ \leq\ 2\pi $$

There is singularity at $\alpha=\theta$ which increases its difficulty.

I tried complex analysis but couldn't solve it.please help me with method,also provide an answer with proof if you like, I would appreciate it.

@IvoTerek Everyone knows it's $d\theta$. But I agree he should tell us anyway. — David H, Jun 29 '14 at 06:52
I even erased my early comment, I read the question quite fast and didn't saw that he told us $\alpha $ was constant. Being $\mathrm{d}\theta $ was the only way. Anyway I edited it so that people don't do the same as me now haha. No stress. — Ivo Terek, Jun 29 '14 at 06:56
I'm pretty sure there is a related problem with cosine in the numerator instead of sine that's already been solved. Can anybody find it? — David H, Jun 29 '14 at 07:00
@DavidH, here it is: http://math.stackexchange.com/questions/547684/how-find-this-int-0-pi-frac-cosnx-cosxadx — CuriousGuest, Jun 29 '14 at 08:23
@JibinJoyK - Asking people to give you a free service "ASAP" is considered impolite and rude in many cultures, specifically those which tend to use the term in the first place. — Nathaniel Bubis, Jun 30 '14 at 23:19
@nbubis Sorry I didn't know that,I will keep it in mind in my future posts :-) — TheDragonReborn, Jul 01 '14 at 01:32

JJacquelin · Accepted Answer · 2014-06-29T21:32:46.393

Thanks to complex analysis, it is rather easy to obtain $$ \int_0^{\pi}{\cos\left(n\theta\right)\over \cos\left(\theta\right) - \cos\left(\alpha\right)}\,{\rm d}\theta=\frac{\pi cos(n\alpha)}{sin(\alpha)} $$ By the way, this result is also obtained in attachment, but with a method much more complicated than usual. In fact, this complicated method is proposed for the much more difficult integral :

$$ \int_0^{\pi}{\sin\left(n\theta\right)\over \cos\left(\theta\right) - \cos\left(\alpha\right)}\,{\rm d}\theta\ $$ The closed form obtained involves the Incomplete Beta function in the complex range. Nevertheless, one of the parameters is nul, which makes think that further simplification might be possible (may be involving polylogarithms)

Several numerical checking were done. Of course, for the numerical computation of the integral (noted $I$ in attachement), the Cauchy principal value is considered.

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How to find $\int_0^{\pi}\frac{\sin n\theta}{\cos\theta-\cos\alpha}d\theta$

1 Answers1