Gamma Function
Definition
The function defined as follows $\Gamma : (0, \infty) \to \mathbb{R}$ is called the Gamma function. $$ \Gamma (x) := \int_{0}^{\infty} t^{x-1} e^{-t} dt $$
Description
Focusing on the integral in the equation above, it is also referred to as Euler’s integral. The Gamma function is famous as an exceedingly important function not just in pure mathematics but also in physics, statistics, etc. It possesses a plethora of interesting properties, but the most representative one is the concept of generalizing factorials to real numbers.
Theorem
The Gamma function as a generalization of factorials
For a natural number $n \in \mathbb{N}$, $\Gamma (n) = (n-1)!$ holds.
Proof
Strategy: It is sufficient to show that the Gamma function appears in the form of a factorial to address its generalization.
By the definition of the Gamma function $$ \Gamma (n) = \int_{0}^{\infty} t^{n-1} e^{-t} dt $$
Case 1. $n=1$ $$ \Gamma (1) = \int_{0}^{\infty} e^{-t} dt = 1 $$ This can be understood as the same meaning as $0! = 1$.
Case 2. $n>1$
By partial integration $$ \begin{align*} \Gamma (n) =& \int_{0}^{\infty} t^{n-1} e^{-t} dt \\ =& \left[ -t^{n-1} e^{-t} \right] _{0} ^{\infty} - \int_{0}^{\infty} -(n-1)t^{n-2} e^{-t} dt \\ =& (n-1) \int_{0}^{\infty} t^{n-2} e^{-t} dt \\ =& (n-1) \Gamma (n-1) \end{align*} $$
Summarizing for both cases $$ \begin{align*} \Gamma (n) =& (n-1) \cdot (n-2) \cdots 2\cdot\Gamma (2) \\ =& (n-1) \cdot (n-2) \cdots 2\cdot 1\cdot \Gamma (1) \\ =& (n-1)! \end{align*} $$
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