# Complex Analysis/Special functions

Definition (Gamma function):

The Gamma function is the unique function that is meromorphic on ${\displaystyle \mathbb {C} }$ and that is given by

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

whenever ${\displaystyle \Re z>0}$.

Proposition (Gamma function interpolates the factorial):

For ${\displaystyle n\in \mathbb {N} }$, we have

${\displaystyle n!=\int _{0}^{\infty }t^{n}e^{-t}dt}$.

Proof: We use induction on ${\displaystyle n}$. The base case is ${\displaystyle n=1}$, ${\displaystyle \Box }$

Proposition (existence and uniqueness of the Gamma function):

The integral

${\displaystyle \int _{0}^{\infty }t^{z-1}e^{-t}dt}$

converges whenever ${\displaystyle \Re z>0}$, and there exists a unique function ${\displaystyle \Gamma }$ which is meromorphic on ${\displaystyle \mathbb {C} }$ and satisfies

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

whenever ${\displaystyle \Re z>0}$.

Proof: First, note that the integral

${\displaystyle \int _{0}^{\infty }t^{z-1}e^{-t}dt}$

converges for ${\displaystyle \Re z>0}$, because we have the estimate

{\displaystyle {\begin{aligned}\int _{0}^{\infty }|t^{z-1}e^{-t}|dt&=\int _{0}^{\infty }t^{\Re z-1}e^{-t}dt\\&=\int _{0}^{1}t^{\Re z-1}e^{-t}dt+\int _{1}^{\infty }t^{\Re z-1}e^{-t}dt\\&\leq \int _{0}^{1}t^{\Re z-1}dt+\int _{1}^{\infty }t^{k}e^{-t}dt,\end{aligned}}}

where ${\displaystyle k\in \mathbb {N} }$ is sufficiently large. The first integral evaluates to

${\displaystyle \int _{0}^{1}t^{\Re z-1}dt=\left[{\frac {t^{\Re z}}{\Re z}}\right]_{0}^{1}={\frac {1}{\Re z}}}$,

whereas the second integral is less than ${\displaystyle k!}$. ${\displaystyle \Box }$