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A friend of mine found this interesting problem in one of Feynman's books one day and neither of us ever managed to come up with a solution. The question is to show that the following two integrals are equivalent and equal to $ \frac{\left(\pi\ p^{a}\right)}{2\left(a!\right)}$

$$\int_{0}^{\pi\ }e^{p\cos x\ }\sin\left(p\sin\left(x\right)\right)\sin\left(ax\right)dx$$

$$\int_{0}^{\pi\ }e^{p\cos x\ }\cos\left(p\sin\left(x\right)\right)\cos\left(ax\right)dx$$

The fact that the answer contains a factorial suggests to us we might need the Gamma Function but we are not very familiar with how to apply it here. Anyone has a step-by-step solution to this?

Sine of the Time
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Enzo Bestetti
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    One approach: think of e.g. the first integral as (related to) Fourier transforms of $e^{p \cos(x)} \sin(p \sin(x))$. See e.g. this, this, and this. Also this, wherein Feynman's trick is used---maybe that's how he would've wanted it done. – Jakob Streipel Jan 15 '24 at 11:40

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