Gamma function

The Gamma funciton is defined as

$\Gamma (z)=\int _{0}^{\infty }t^{z-1}e^{-t}dt$

Using integration by parts,

$\Gamma (z)=\int _{0}^{\infty }t^{z-1}d(e^{-t}dt)$

$=\left[t^{z-1}(-e^{-t})\right]_{0}^{\infty }-\int _{0}^{\infty }(z-1)t^{(z-2)}(-e^{-t})dt$

$=(z-1)\int _{0}^{\infty }t^{(z-1)-1}e^{-t}dt$

$=(z-1)\Gamma (z-1).$

Thus,

$\Gamma (z)=(z-1)\Gamma (z-1)$

Using this, and the the fact that $\Gamma (0)=1$, then we can get the factorial. For $n$ a positive integer,

$\Gamma (n)=(n-1)!.$