Laplace Transform of Bessel Function of the First Kind

Theorem
Let $J_n$ denote the Bessel function of the first kind of order $n$.

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


 * $\laptrans {\map {J_n} {a t} } = \dfrac {\paren {\sqrt {s^2 + a^2} - s}^n} {a^n \sqrt {s^2 + a^2} }$

Proof
From Series Expansion of Bessel Function of the First Kind:

Hence: