Definition (Gamma function):
The Gamma function is the unique function that is meromorphic on
and that is given by
![{\displaystyle \Gamma (z):=\int _{0}^{\infty }t^{z-1}e^{-t}dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ad06103b068e33b5dbcad0642744341fdb47213)
whenever
.
Proposition (Gamma function interpolates the factorial):
For
, we have
.
Proof: We use induction on
. The base case is
,
Proposition (existence and uniqueness of the Gamma function):
The integral
![{\displaystyle \int _{0}^{\infty }t^{z-1}e^{-t}dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc2998bda2ae74cbb672ec44ce3520134a861bcc)
converges whenever
, and there exists a unique function
which is meromorphic on
and satisfies
![{\displaystyle \Gamma (z):=\int _{0}^{\infty }t^{z-1}e^{-t}dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ad06103b068e33b5dbcad0642744341fdb47213)
whenever
.
Proof: First, note that the integral
![{\displaystyle \int _{0}^{\infty }t^{z-1}e^{-t}dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc2998bda2ae74cbb672ec44ce3520134a861bcc)
converges for
, 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7600bf6a637c3584d347721099cd0b0176cea094)
where
is sufficiently large. The first integral evaluates to
,
whereas the second integral is less than
.