Laplace Transform of Bessel Function of the First Kind/Proof 1

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:

\(\ds \map {J_n} {a t}\)

\(=\)

\(\ds \dfrac {\paren {a t}^n} {2^n \, \map \Gamma {n + 1} } \paren {1 - \dfrac {\paren {a t}^2} {2 \paren {2 n + 2} } + \dfrac {\paren {a t}^4} {2 \times 4 \paren {2 n + 2} \paren {2 n + 4} } - \cdots}\)

\(\ds \)

\(=\)

\(\ds \sum_{k \mathop = 0}^\infty \dfrac {\paren {-1}^k} {k! \, \map \Gamma {n + k + 1} } \paren {\dfrac {a t} 2}^{n + 2 k}\)

Hence:

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

\(=\)

\(\ds \laptrans {\sum_{k \mathop = 0}^\infty \dfrac {\paren {-1}^k} {k! \, \map \Gamma {n + k + 1} } \paren {\dfrac {a t} 2}^{n + 2 k} }\)

\(\ds \)

\(=\)

\(\ds \sum_{k \mathop = 0}^\infty \dfrac {\paren {-1}^k a^{n + 2 k} } {2^{n + 2 k} k! \, \map \Gamma {n + k + 1} } \laptrans {t^{n + 2 k} }\)

Linear Combination of Laplace Transforms

\(\ds \)

\(=\)

\(\ds \sum_{k \mathop = 0}^\infty \dfrac {\paren {-1}^k a^{n + 2 k} } {2^{n + 2 k} k! \, \map \Gamma {n + k + 1} } \dfrac {\map \Gamma {n + 2 k + 1} } {s^{n + 2 k + 1} }\)

Laplace Transform of Power

\(\ds \)

\(=\)

\(\ds \sum_{k \mathop = 0}^\infty \paren {-1}^k \paren {\dfrac a 2}^{n + 2 k} \dfrac {\map \Gamma {n + 2 k + 1} } {k \, \map \Gamma k \, \map \Gamma {n + k + 1} } \dfrac 1 {s^{n + 2 k + 1} }\)

rearrangement, Gamma Function Extends Factorial

\(\ds \)

\(=\)

\(\ds \sum_{k \mathop = 0}^\infty \paren {-1}^k \paren {\dfrac a 2}^{n + 2 k} \dfrac 1 {k \, \map \Beta {k, n + k + 1} } \dfrac 1 {s^{n + 2 k + 1} }\)

Definition 3 of Beta Function

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