Evaluating a definite integral using Bessel functions I've been examining an integral I remember encountering in complex analysis:
$$
I = \int_0^\pi e^{\cos\theta} \cos\left(n\theta - \sin\theta\right) d\theta
$$
where $n$ is a nonnegative integer. It's not too hard to show that $I = \frac{\pi}{n!}$ by changing $\cos\left(n\theta - \sin\theta\right)$ into $\operatorname{Re}\left[e^{i(n\theta - \sin\theta)}\right]$ and making the substitution $z = e^{i\theta}$ to turn the integral into a contour integral on the complex unit circle.
However, I came across the following representation of the Bessel functions of the first kind on Wikipedia:
$$
J_n(x) = \frac{1}{\pi} \int_0^\pi \cos\left(n\theta - x\sin\theta\right) d\theta
$$
Given the similarity between this formula when $x = 1$ and the initial integral, I'm sure there's some way to evaluate the integral using properties of the Bessel functions. But so far I'm stuck as to figuring it out. I tried expanding $e^{\cos\theta}$ as a series but that got ugly pretty quickly. Any suggestions?
 A: If we compute the Fourier cosine series of $e^{\cos\theta}$ over $[-\pi,\pi]$ we get:
$$ e^{\cos\theta} = I_0(1)+2\sum_{k\geq 1}I_k(1)\cos(k\theta)\tag{1}$$
where $I_k$ is a modified Bessel function of the first kind. On the other hand, by using Briggs formulas and your definition of $J_n(x)$ it follows that:
$$ \frac{1}{2\pi}\int_{-\pi}^{\pi}\cos(n\theta-\sin\theta)\,d\theta = J_n(1),$$
$$ \frac{1}{\pi}\int_{-\pi}^{\pi}\cos(n\theta-\sin\theta)\cos(k\theta)\,d\theta = J_{n+k}(1)+J_{n-k}(1),\tag{2}$$
so we have:
$$ \int_{-\pi}^{\pi}e^{\cos\theta}\cos(n\theta-\sin\theta)\,d\theta =\\2\pi \cdot I_0(1)\,J_n(1)+2\pi\cdot\sum_{k\geq 1}I_k(1)\left(J_{n-k}(1)+J_{n+k}(1)\right)\tag{3}$$
and now we may exploit:
$$ \sum_{n=-\infty}^{+\infty}t^n\,J_n(z) = \exp\left(\frac{z}{2}\left(t-\frac{1}{t}\right)\right) \tag{4}$$
together with $I_n(z)=(-i)^n\,J_n(iz)$ to write $(3)$ as a convolution and deduce:
$$ I = \pi\cdot [z^n]\,e^{z} = \frac{\pi}{n!}\tag{5} $$
in a extremely convoluted and probably useless very nice way.
