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I would like to solve the integral $$F_n(\kappa,\theta,\phi)=\int_{-\pi}^{\pi}{\rm e}^{\kappa\cos(x-\theta)}\cos(n\, x-\phi)\,{\rm d}x$$ that appears related to the identity $$I_n(\kappa)=\frac{1}{\pi}\int_{0}^{\pi}{\rm e}^{\kappa\cos(x)}\cos(n\, x)\,{\rm d}x,$$ where $I_n(\kappa)$ is the Modified Bessel Function of the first kind. Any ideas?

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    After elementary trig manipulations it seems that the problem could be solved if the integral $$\int_{-\pi}^{\pi}e^{\kappa \cos(x)}\sin(n\, x){\rm d} x$$ was known.2012-09-13
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    And this integral is zero so I think the problem is solved...2012-09-13
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    Why is the integral zero for any $\kappa$ and $n$?2012-09-13

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