Integral to Infinity of Bessel Function of First Kind order Zero

Theorem

$\ds \int_0^\infty \map {J_0} t \rd t = 1$

where $J_0$ denotes the Bessel function of the first kind of order $0$.

Proof

Using the technique of Evaluation of Integral using Laplace Transform:

\(\ds \int_0^\infty e^{-s t} \map {J_0} t \rd t\)

\(=\)

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

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

\(\ds \leadsto \ \ \)

\(\ds \int_0^\infty \map {J_0} t \rd t\)

\(=\)

\(\ds 1\)

letting $s \to 0^+$

$\blacksquare$

Sources

• 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Solved Problems: Evaluation of Integrals: $46 \ \text{(a)}$