Integral Representation of Bessel Function of the First Kind

Theorem

Let $\map {J_n} x$ denote the Bessel function of the first kind of order $n$.

Order Zero

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

Integer Order

Let $n \in \Z$ be an integer.

Then:

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

Non-Integer Order

Let $n \in \Z$ be an integer.

Then:

$\ds \map {J_n} x = \dfrac {x^n} {2^n \sqrt \pi \map \Gamma {n + \frac 1 2} } \int_0^\pi \map \cos {x \sin \theta} \cos^{2 n} \theta \rd \theta$