Laplace transform of the Bessel function J0
= Laplace transform of the Bessel function J0
{c}
{title2=$\mathcal L[J_0](p)=(1+p^2)^{-1/2}$}
For real $p>0$, the <Laplace transform> $G$ of the order-zero <Bessel function of the first kind> satisfies $(1+p^2)G'+pG=0$. This follows by transforming $xJ_0''+J_0'+xJ_0=0$, using $\mathcal L[xg]=-\partial_p\mathcal L[g]$ and $J_0(0)=1$, $J_0'(0)=0$. Thus $G=C/\sqrt{1+p^2}$; the <initial value theorem for Laplace transforms> gives $pG(p)\to1$ and fixes $C=1$.