Hi, does anyone happen to know how to prove this equality.

Let $\displaystyle \displaystyle F(s) = \mathcal{L}\{f(t)\}$, then $\displaystyle \displaystyle\mathcal{L}\left\{\int_0^{\infty}\fra c {t^u \cdot f(u)}{\Gamma (u + 1)}du\right\} = \frac {F(\ln s)}{s\ln s}$