Integral Representation of Bessel Function of the First Kind/Integer Order

Theorem
Let $\map {J_n} x$ denote the Bessel function of the first kind of order $n$. Let $n \in \Z$ be an integer.

Then:
 * $\ds \map {J_n} x = \dfrac 1 \pi \int_0^\pi \map \cos {n \theta - x \sin \theta} \rd \theta$

Proof
Take $C$ to be the unit circle.

Write $t = e^{-i \theta}$ and let $\theta$ be decreasing from $\pi$ to $-\pi$, so that we integrate along $C$ anticlockwise.

Then: