# Is there a closed-form expression for this trigonometric Cauchy Principal Value-type integral?

Consider the following definite integral, $I(n; \theta)$.

$$I(n; \theta) = \int_{0}^{\pi} \frac{\cos(n\phi)}{\cos\phi-\cos\theta} d\phi \quad \text{where } n \in N$$

When $0 < \theta < \pi$, the integral does not converge, but its Cauchy Principal Value exists.

My question is: Is there a closed-form expression for the PV of this integral, i.e. for the expression

$$J(n; \theta) \equiv I(n; \theta)|_{0 < \theta < \pi} = PV \int_{0}^{\pi} \frac{\cos(n\phi)}{\cos\phi-\cos\theta} d\phi \quad \text{where } n \in N$$

Mathematica outputs something complicated involving the MeijerG function. However, I know that for specific values of $n$, the integral is quite tractable.

For instance, when $n=1$, $J(1; \theta) = \pi$.

I considered a couple of contour integration procedures but wasn't really able to make any headway.

Thanks.

• Have you tried using the prosthaphaeresis formulas ? – Lucian Nov 29 '14 at 19:03
• @Lucian Thanks, it does give me something like a reduction formula. Let me try to work through this. – user_of_math Nov 30 '14 at 5:54