Gamma function
Main definition

The notation Γ (z) {\displaystyle \Gamma (z)} is due to Legendre. If the real part of the complex number z is strictly positive (ℜ (z) > 0 {\displaystyle \Re (z)>0}), then the integral Γ (z) = ∫ 0 ∞ t z − 1 e − t d t {\displaystyle \Gamma (z)=\int _{0}^{\infty }t^{z-1}e^{-t}\,dt} converges absolutely, and is known as the Euler integral of the second kind. (Euler's integral of the first kind is the beta function.) Using integration by parts, one sees that:

Γ (z + 1) = ∫ 0 ∞ t z e − t d t = 0 ∞ + ∫ 0 ∞ z t z − 1 e − t d t = lim t → ∞ (− t z e − t) − (− 0 z e − 0) + z ∫ 0 ∞ t z − 1 e − t d t. {\displaystyle {\begin{aligned}\Gamma (z+1)&=\int _{0}^{\infty }t^{z}e^{-t}\,dt\\&={\Bigl }_{0}^{\infty }+\int _{0}^{\infty }zt^{z-1}e^{-t}\,dt\\&=\lim _{t\to \infty }\left(-t^{z}e^{-t}\right)-\left(-0^{z}e^{-0}\right)+z\int _{0}^{\infty }t^{z-1}e^{-t}\,dt.\end{aligned}}}

Recognizing that − t z e − t → 0 {\displaystyle -t^{z}e^{-t}\to 0} as t → ∞, {\displaystyle t\to \infty,} Γ (z + 1) = z ∫ 0 ∞ t z − 1 e − t d t = z Γ (z). {\displaystyle {\begin{aligned}\Gamma (z+1)&=z\int _{0}^{\infty }t^{z-1}e^{-t}\,dt\\&=z\Gamma (z).\end{aligned}}}

We can calculate Γ (1) {\displaystyle \Gamma (1)}: Γ (1) = ∫ 0 ∞ t 1 − 1 e − t d t = ∫ 0 ∞ e − t d t = 1. {\displaystyle {\begin{aligned}\Gamma (1)&=\int _{0}^{\infty }t^{1-1}e^{-t}\,dt\\&=\int _{0}^{\infty }e^{-t}\,dt\\&=1.\end{aligned}}}

Thus we can show that Γ (n) = (n − 1) ! {\displaystyle \Gamma (n)=(n-1)!} for any positive integer n by induction. Specifically, the base case is that Γ (1) = 1 = 0 ! {\displaystyle \Gamma (1)=1=0!}, and the induction step is that Γ (n + 1) = n Γ (n) = n (n − 1) ! = n !. {\displaystyle \Gamma (n+1)=n\Gamma (n)=n(n-1)!=n!.}

The identity Γ (z) = Γ (z + 1) z {\textstyle \Gamma (z)={\frac {\Gamma (z+1)}{z}}} can be used (or, yielding the same result, analytic continuation can be used) to uniquely extend the integral formulation for Γ (z) {\displaystyle \Gamma (z)} to a meromorphic function defined for all complex numbers z, except integers less than or equal to zero. It is this extended version that is commonly referred to as the gamma function.
change_case:english_lowercase