Laplace Transform of Natural Logarithm/Proof 2

Proof
From Laplace Transform of Power:
 * $\displaystyle \int_0^\infty e^{-s t} t^k \rd t = \dfrac {\map \Gamma {k + 1} } {s^{k + 1} }$

for $k > -1$.

Differentiating $k$:


 * $\displaystyle \int_0^\infty e^{-s t} t^k \ln t \rd t = \dfrac {\map {\Gamma'} {k + 1} - \map \Gamma {k + 1} \ln s} {s^{k + 1} }$

Setting $k = 0$: