TITLE: How to integrate $\int_1^\infty e^{-\alpha x}J_0(\beta\sqrt{x^2-1})\mathrm{d}x \,$?
QUESTION [6 upvotes]: This integral is from (6.616.2) in Gradshteyn and Ryzhik.
$$
\int_1^\infty  e^{-\alpha x}J_0(\beta\sqrt{x^2-1})\mathrm{d}x \,=\frac{1}{\sqrt{\alpha^2+\beta^2}}e^{-\sqrt{\alpha^2+\beta^2}}
$$
I want to know how to do this integral and the restriction of $\alpha$ and $\beta$. The integral table doesn't mention it. 
I doubt the results must have some restriction on $\alpha$ and $\beta$, because:

it seems that when $\mathrm{Re}\,\alpha <0$, the integral diverges. 
also when $\alpha$ is purely imaginary, and for $\beta$ real, the result should be complex conjugate when $\alpha$ takes conjugate purely imaginary pairs, $\pm i$ for example, however the results given depends only on $\alpha^2$.
I also found by Mathematica numerical integration that when $\alpha$ is purely imaginary, the integration also seems troublesome.

Edit:
The answer by @Fabian give a general condition that when $\mathrm{Re}\alpha>\mathrm{Im}\beta$, the integral converges. However, what about $\mathrm{Re}\alpha=\mathrm{Im}\beta$. For the simplest case when $\alpha$ is purely imaginary and $\beta$ is real, Mathematica can give the sensible result when $|\alpha|>|\beta|$, while seems diverges when $|\alpha|<|\beta|$:
\[Alpha]=-3I ;\[Beta]=2;
NIntegrate[Exp[-\[Alpha] x]BesselJ[0,\[Beta] Sqrt[x^2-1]],{x,1,Infinity}]
f[a_,b_]:=Exp[-Sqrt[a^2+b^2]]/Sqrt[a^2+b^2]
f[\[Alpha],\[Beta]]//N
-0.351845-0.276053 I
-0.351845+0.276053 I

REPLY [5 votes]: For the purpose of evaluation, I'm just going to consider the case where $\alpha>0$ and $\beta \in \mathbb{R}$.
From the answers here, we know that $$\int_{0}^{\infty} \exp \left(-ax^{2}-\frac{b}{x^{2}} \right) \, dx = \frac{1}{2} \sqrt{\frac{\pi}{a}} e^{-2 \sqrt{ab}}, \quad a, b>0. \tag{1}$$
Using this formula, we get $$ \begin{align} \int_{1}^{\infty} e^{- \alpha x} J_{0}(\beta \sqrt{x^{2}-1}) \, dx &= \int_{0}^{\infty} \frac{u  e^{-\alpha \sqrt{1+u^{2}}} J_{0}(\beta u)}{\sqrt{1+u^{2}}}  \, du  \\ &= \frac{2}{\sqrt{\pi}}\int_{0}^{\infty} u J_{0}(\beta u) \int_{0}^{\infty}\exp \left(-(1+u^{2})t^{2} -\frac{\alpha^{2}}{4t^{2}}\right) \, dt \, du \\ &= \frac{2}{\sqrt{\pi}} \int_{0}^{\infty} \exp \left(- t^{2}- \frac{\alpha^{2}}{4t^{2}} \right)\int_{0}^{\infty} uJ_{0}(\beta u) \exp \left(-t^{2}u^{2} \right) \, du \, dt, \end{align}$$
where
$$ \begin{align}\int_{0}^{\infty} u J_{0}(\beta u) e^{-t^{2}u^{2}} \, du &=  \frac{1}{2}\int_{0}^{\infty} J_{0}(\beta \sqrt{w}) e^{-t^{2}w} \, dw \\ &= \frac{1}{2} \int_{0}^{\infty} e^{-t^{2}w} \sum_{k=0}^{\infty} \frac{(-1)^{k}}{(k!)^{2}} \left(\frac{\beta^{2}w}{4} \right)^{k} \, dw\\ &= \frac{1}{2}\sum_{k=0}^{\infty} \frac{(-1)^{k}}{(k!)^{2}} \left(\frac{\beta^{2}}{4} \right)^{k} \int_{0}^{\infty} e^{-t^{2}w} w^{k} \, dw\\ &= \frac{1}{2t^{2}} \sum_{k=0}^{\infty} \frac{(-1)^{k}}{k!} \left(\frac{\beta^{2}}{4t^{2}} \right)^{k}  \\  &= \frac{1}{2t^{2}} \, \exp \left(-\frac{\beta^{2}}{4t^{2}} \right). \end{align} $$
Therefore, using $(1)$ again, we have $$\begin{align} \int_{1}^{\infty} e^{- \alpha x} J_{0}(\beta \sqrt{x^{2}-1}) \, dx  &=  \frac{2 }{\sqrt{\pi}} \int_{0}^{\infty} \exp \left(- t^{2}- \frac{\alpha^{2}}{4t^{2}} \right) \frac{1}{2t^{2}} \exp \left(- \frac{\beta^{2}}{4t^{2}}\right) \, dt \\ &=\frac{1}{\sqrt{\pi}} \int_{0}^{\infty} \frac{1}{t^{2}} \, \exp \left(-t^{2} - \frac{\alpha^{2}+\beta^{2}}{4} \frac{1}{t^{2}}\right) \, dt \\ &= \frac{1}{\sqrt{\pi}} \int_{0}^{\infty} \exp \left( - \frac{1}{v^{2}}-\frac{\alpha^{2}+\beta^{2}}{4} v^{2} \right) \, dv \\ &= \frac{1}{\sqrt{\pi}} \frac{1}{2} \sqrt{\frac{4 \pi}{\alpha^{2}+\beta^{2}}} \,  e^{-\sqrt{\alpha^{2}+\beta^{2}}} \\&= \frac{1}{\sqrt{\alpha^{2}+\beta^{2}}} \,  e^{-\sqrt{\alpha^{2}+\beta^{2}}}.  \end{align} $$

The generalization
$$\small  \int_{0}^{\infty} J_{\mu}(b t) \, \frac{K_{\nu}(a\sqrt{t^{2}+z^{2}})}{(t^{2}+z^{2})^{\nu/2}} \, t^{\mu+1} \, dt = \frac{b^{\mu}}{a^{\nu}} \left(\frac{\sqrt{a^{2}+b^{2}}}{z} \right)^{\nu-\mu-1}K_{\nu-\mu-1} \left(z \sqrt{a^{2}+b^{2}} \right) $$
is the second entry in section 47 of chapter 13 of the textbook A Treatise on the Theory of Bessel Functions.