SF14

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$\int_0^\infty e^{-st}t^{x-1}\,dt,\,\,\,s>0\,$

Let $p = st, \,\,\, dp = sdt, \,\,\, t=p/s, \,\,\, dt=dp/s\,$

$\int_0^\infty e^{-p} \left(\frac{p}{s}\right)^{x-1} \frac{dp}{s} = \frac{1}{s^x}\int_0^\infty e^{-p} p^{x-1} \,dp\,$

$= \frac{\Gamma(x)}{s^x}\,$

Special Functions

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