Is there a closed-form expression for an integral of the modified Bessel function of the first kind zero-order including the following? $$ \int_{0}^{\infty}x^{-1}e^{-ax}I_{0}(bx^{1/2})dx. $$
where a and b are positive real.
Thanks in advance for both your time and patience.
This integral doesn't converge. An exactly similar integral formula is provided by Gradshteyn and Ryzhik, but it would be aberrant to use it, as the condition $\Re\left(\nu+\mu+\dfrac{1}{2}\right) > 0$ isn't satisfied in your integral.
$$\int_0^\infty x^{\mu-\frac{1}{2}}e^{-\alpha x}I_{2\nu }\left(2\beta\sqrt{x}\right)\mathrm dx=\frac{\Gamma\left(\mu+\nu+\frac{1}{2}\right)}{\Gamma(2\nu+1)}\beta^{-1}e^{\frac{\beta^2}{2\alpha}}\alpha^{-\mu}M_{-\mu,\nu}\left(\frac{\beta^2}{\alpha}\right) \\ \left[\operatorname{Re}\left(\mu+\nu+\frac{1}{2}\right)>0\right]$$