Is it possible to take the Laplace transform of a Hankel function with Mathematica?

Question

The following two functions are the fundamental solutions of the wave equation and its Laplace transform (the modified Helmholtz equation) in two dimensions, respectively, when the speed of wave propagation is $1$:

$$ g(x,t) = \frac{H(t - |x|)}{2\pi \sqrt{t^2 - |x|^2}}, $$ $$ G(x,s) = \frac{i}{4}H_0^{(1)}(i s |x|). $$ Here $H$ is the Heavside function and $H_0^{(1)}$ is the Hankel function of the first kind of order $0$.

Question 1: Is it possible to take the Laplace transform of the function $g(x,t)$ in Mathematica and show that it equals $G(x,s)$?

Now the reason I asked Question 1 is because it is the basis for the following question.

The fundamental solution of the damped wave equation is (page 4 here): $$ g_\text{damped}(x,t) = \frac{1}{2\pi}\cosh\bigg(\frac{\alpha}{2}\sqrt{t^2-|x|^2}\bigg)\frac{H(t - |x|)}{2\pi \sqrt{t^2 - |x|^2}} e^{\frac{-\alpha}{2}t}, $$ where $\alpha$ is the damping parameter. When $\alpha = 0$ this reduces to the fundamental solution of the undamped equation $g(x,t)$ above.

Now I 'think' the Laplace transform of this fundamental solution should look something like $$ G_\text{damped}(x,s) = \frac{i}{4} H_0^{(1)}(i s (1-\beta i) |x|), $$ where $\beta$ is some damping parameter analogous to $\alpha$ in the time domain equation. I would like to verify if I am correct about this.

Question 2: Is it possible to take the Laplace transform of the function $g_\text{damped}(x,t)$ in Mathematica so that I can check if the form of the resulting expression looks like $G_\text{damped}(x,s)$?