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

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

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

Then the Laplace transform of $\map {J_0} {a t}$ is given as:

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

Proof

\(\ds \laptrans {\map {J_0} t}\)

\(=\)

\(\ds \dfrac 1 {\sqrt {s^2 + 1} }\)

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

\(\ds \leadsto \ \ \)

\(\ds \laptrans {\map {J_0} {a t} }\)

\(=\)

\(\ds \dfrac 1 a \dfrac 1 {\sqrt {\paren {s / a}^2 + 1} }\)

Laplace Transform of Constant Multiple

\(\ds \)

\(=\)

\(\ds \dfrac 1 {\sqrt {s^2 + a^2} }\)

simplifying

$\blacksquare$

Sources

• 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Laplace Transforms of Special Functions: $2$
• 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Solved Problems: Bessel Functions: $34 \ \text{(b)}$