Complex Analysis/Special functions

Definition (Gamma function):

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

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

whenever $\Re z>0$.

Proposition (Gamma function interpolates the factorial):

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

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

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

Proposition (existence and uniqueness of the Gamma function):

The integral

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

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

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

whenever $\Re z>0$.

Proof: First, note that the integral

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

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

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

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

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

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