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.


  • 1
    $\begingroup$ Have you tried using the prosthaphaeresis formulas ? $\endgroup$ – Lucian Nov 29 '14 at 19:03
  • $\begingroup$ @Lucian Thanks, it does give me something like a reduction formula. Let me try to work through this. $\endgroup$ – user_of_math Nov 30 '14 at 5:54

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