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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma(0)=1} , then we can get the factorial. For $n$ a positive integer,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma(n) = (n-1)!. }