Show that for all $y > 0$, we have $\Gamma ( y ) = y ^ { - 1 } \int _ { 0 } ^ { + \infty } e ^ { - t } t ^ { y } d t$, then that
$$\Gamma ( y ) = e ^ { - y } y ^ { y } \int _ { - 1 } ^ { + \infty } e ^ { - y \phi ( s ) } d s$$
where $\phi$ is the function defined on $] - 1 , + \infty [$ by $\phi ( s ) = s - \ln ( 1 + s )$.
Recall that $\Gamma : ] 0 , + \infty [ \rightarrow \mathbb { R }$ is defined by $\Gamma ( y ) = \int _ { 0 } ^ { \infty } e ^ { - t } t ^ { y - 1 } d t$.