\[\int_{0}^{\infty}t\exp\left(-p^{2}t^{2}\right)I_{0}\left(at\right)K_{0}\left(%
at\right)\,\mathrm{d}t=\frac{1}{4p^{2}}\exp\left(\frac{a^{2}}{2p^{2}}\right)K_%
{0}\left(\frac{a^{2}}{2p^{2}}\right),\]