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

${\displaystyle \Gamma (z)=(z-1)!\,}$

${\displaystyle \Gamma (z+1)=z.\Gamma (z)=z.(z-1)!\,}$

${\displaystyle \Gamma (t)=\lim _{n\to \infty }{\frac {n!\;n^{t}}{t\;(t+1)\cdots (t+n)}}={\frac {1}{t}}\prod _{n=1}^{\infty }{\frac {\left(1+{\frac {1}{n}}\right)^{t}}{1+{\frac {t}{n}}}}}$

${\displaystyle \Gamma (t)=\lim _{n\to \infty }{\frac {n!\;n^{t}}{t\;(t+1)\cdots (t+n)}}={\frac {1}{t}}\prod _{n=1}^{\infty }{\frac {\left(1+{\frac {1}{n}}\right)^{t}}{1+{\frac {t}{n}}}}}$