The Gamma funciton is defined as
Γ ( z ) = ∫ 0 ∞ t z − 1 e − t d t {\displaystyle \Gamma (z)=\int _{0}^{\infty }t^{z-1}e^{-t}dt}
Using integration by parts,
Γ ( z ) = ∫ 0 ∞ t z − 1 d ( e − t d t ) {\displaystyle \Gamma (z)=\int _{0}^{\infty }t^{z-1}d(e^{-t}dt)}
Thus, Γ ( z ) = ( z − 1 ) Γ ( z − 1 ) {\displaystyle \Gamma (z)=(z-1)\Gamma (z-1)}
![{\displaystyle =\left[t^{z-1}(-e^{-t})\right]_{0}^{\infty }-\int _{0}^{\infty }(z-1)t^{(z-2)}(-e^{-t})dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce110fa8f5d2a6202768d18ff0b2260509cfa05a)
