What does this Bessel function integral evaluate to?

Question

I'm working with a problem where this integral pops up: $$\int_0^\infty J_\nu(t) \frac{t^{1-\nu}}{\sqrt{t^2 + x^2}} \operatorname{d} t.$$ It is superficially similar to equation 10.22.46 from the DLMF, but the sign of $\nu$ in the exponent is opposite. This can be viewed as a Hankel transform of $\frac{t^{-\nu}}{\sqrt{t^2 + x^2}}$. Because $J_\nu(t)$ is entire there may be some clever way to use the residue theorem. The domain of interest has $\nu \ge -\frac{1}{2}$, with particular interest to integers and half integers. $x$ is likewise real and positive semi-definite.

Answer 1

Here is the Mathematica result: $$ \int_{0}^{\infty} J_{\nu}(t) \frac{t^{1-\nu}}{\sqrt{t^{2} + x^{2}}} \operatorname{d}t = \sqrt{\frac{\pi }{2}} \lvert x \rvert x^{-\nu -\frac{1}{2}} \left(\sqrt{\frac{1}{x^2}} x I_{\nu -\frac{1}{2}}(x)-\pmb{L}_{\nu -\frac{1}{2}}(x)\right) $$ for $$ \Re(\nu )>-\frac{1}{2}\land \Re\left(x^2\right)>0 $$