Laplace Transform of Bessel Function of the First Kind of Order Zero

Theorem
Let $J_0$ denote the Bessel function of the first kind of order $0$.

Then the Laplace transform of $J_0$ is given as:


 * $\laptrans {\map {J_0} {a t} } = \dfrac 1 {\sqrt {s^2 + a^2} }$

Proof
From Bessel Function of the First Kind of Order $0$:

Hence:

Then:

The two expressions match, and the result follows.