Laplace Transform of the Bessel function of the first kind, with $n \in \Bbb Z $ and $n \geq 0 $

Question

I'm trying to understand the proof for the general formula for the Laplace Transform of the Bessel function of the first kind, with $n \in \Bbb Z $ and $n \geq 0 $.

I'm following along Proof 2 in the link below, but I don't understand how one can set the initial conditions to $x(0) = 1, x'(0)=0$, as this won't hold for all $n$? Can someone clarify where these conditions have come from. Thanks!

https://proofwiki.org/wiki/Laplace_Transform_of_Bessel_Function_of_the_First_Kind

Answer 1

This is indeed imprecise. The initial conditions actually don't matter as long as they are finite (which is required by the definition of $J_n$), since they get wiped out by the derivatives: $$ \frac{d^2}{ds^2}\mathcal{L}\left[x''\right] = \frac{d^2}{ds^2}\left[s^2\mathcal{L}\left[x\right]- sx(0) - x'(0)\right] = \frac{d^2}{ds^2}\left(s^2\mathcal{L}\left[x\right]\right). $$