I'm trying to prove that the Laplace transform of the function
$$
J_0(a\sqrt{x^2+2bx})
$$
is
$$
\frac{1}{\sqrt{p^2+a^2}} \exp\left\{bp- b\sqrt{p^2+a^2} \right\}
$$
as asserted in the eqworld. This formula can also be found in the book "Tables of Integral Transforms" page 207.
My first attempt was to insert $a\sqrt{x^2+bx}$ in the series representation
of the Bessel function
$$
J_0(x) = \sum_{m=0}^\infty \frac{(-1)^m}{(m!)^2}\left(\frac{x}{2}\right)^{2m}
$$
and then integrating term by term. However, the Laplace transform of the function $(x^2+bx)^m$ is not trivial.
My second attempt was to look at a change of variable in the differential equation of $J_0$
$$
t^2J_0''(t)+tJ_0'(t)+t^2J_0(t)=0
$$
but I was not able to find it.
Thanks for any clue.
Best Answer
By the substitution $x\mapsto bx$, the problem is equivalent to finding the Laplace transform of $f_c(x)=J_0(c\sqrt{x^2+x})$ or the inverse Laplace transform of $$ g_c(p)=\frac{1}{\sqrt{p^2+1}}\,\exp\left(\frac{-c}{p+\sqrt{p^2+1}}\right). \tag{1}$$ It is useful to recall that $\mathcal{L}(J_n(x))(p) = \frac{1}{\sqrt{1+p^2}(p+\sqrt{1+p^2})^n}$. It follows that:
$$ \mathcal{L}^{-1}(g_c(p))(x) = \sum_{n\geq 0}\frac{(-c)^n J_n(x)}{n!}=\sum_{n\geq 0}\frac{(-c)^n}{n! 2\pi i^n}\int_{0}^{2\pi}e^{ix\cos\theta}e^{in\theta}\,d\theta \tag{2}$$ and $$\begin{eqnarray*} \mathcal{L}^{-1}(g_c(p))(x) &=& \frac{1}{2\pi}\int_{0}^{2\pi}\exp\left(ix\cos\theta+ice^{i\theta}\right)\,d\theta\\&=&\frac{1}{2\pi}\int_{0}^{2\pi}\exp\left(i(x+c)\cos\theta-c\sin\theta\right)\,d\theta\\&=&I_0\left(\sqrt{c^2-(x+c)^2}\right)=I_0\left(\sqrt{-x(2c+x)}\right). \end{eqnarray*}\tag{3}$$ The last term equals $\color{red}{J_0(\sqrt{x^2+2cx})}$ as wanted.
A particular thing in Mathematics won't ever cease to amaze me: $a=b$ and $b=a$ are the same thing on a semantic level, however to prove $a\mapsto\ldots\mapsto b$ or $b\mapsto\ldots\mapsto a$ may be asymmetrically extremely hard - almost trivial. I do not know for sure which part of this phenomenon is due to the fact that we are used to have the "initial term" on the left and the "final term" on the right of a blank page (in many Asian cultures it is likely the opposite, it might be interesting to investigate "the other side" of this issue, too), which part is due to the fact that we heavily rely on causal links (even when they are $\Leftrightarrow$, in order to achieve a hierarchical, tree-like structure in our minds). In this particular case, I found easier to compute an inverse Laplace transform rather than a direct Laplace transform, and I think we should be trained to "think backwards" more often. That also reminds me of an interesting question I heard once: if both us and Mathematics are built in the image of God, why do we find so difficult to prove many things? Why mathematical truth is not evident in our eyes?