$\newcommand{\on}[1]{\operatorname{#1}}$
In solving a classical fluid mechanics problem involving flow in porous media, I encountered a delicate infinite integral of the form,
$$
\on{f}\left(r,z; \alpha\right) = \int_{0}^{\infty}
\on{K}_{1}\left(\left[k^{2} + \alpha^{2}\right]^{1/2}\,r\right)
\sin\left(kz\right)\,\mathrm{d}k \, ,
$$
wherein $\alpha$ and $r$ are positive real numbers and $z$ is a real number.
- For the specific case where $\alpha = 0$, the integral can be exactly and easily solved as $$ \on{f}\left(r,z; 0\right) = \frac{\pi}{2}\frac{z}{r} \left(r^{2} + z^{2}\right)^{-1/2}\, . $$
- I am curious whether a general expression can be derived for the above integral with arbitrary $\alpha$.
- By replacing $\on{K}_{1}$ with $\on{K}_{0}$ and the sine function with the cosine function, I managed to obtain the following $$ \int_{0}^{\infty}\on{K}_{0}\left(\left[k^{2} + \alpha^{2}\right]^{1/2}\,r\right) \cos\left(kz\right)\,\mathrm{d}k = \frac{\pi}{2} \left(r^{2} + z^{2}\right)^{-1/2} \exp\left(-\alpha\left[r^{2} + z^{2}\right]^{1/2}\right) $$ Could anyone provide hints or guidance on how to approach finding a general expression for the original integral ?.
Thank you very much !.
