Properties of "Gamma Function"¶

When you play with probability density function, it's inevitable to deal with "gamma function". Thereby I'd like to share with you fundamental properties of "Gamma Function".
First of all, just in case, before getting into properties of "Gamma function", we should be on the same page on what the "Gamma Function" is. "Gamma Function" is depicted as below.
$$ \Gamma(s) = \int_0^{\infty} x^{s-1}e^{-x}dx\ \ \ \ s.t\ \ \ \ s>0$$

These are crucial properties of "Gamma Function".

1. $ \Gamma(s+1) = s\Gamma(s)$
2. $ \Gamma(n+1) = n!\ $ where $\ \ n$ is natural number
3. $\Gamma(\frac{1}{2}) = \sqrt\pi$

1. $ \Gamma(s+1) = s\Gamma(s)$¶

Let me show you how to derive this property. $$\Gamma(s+1) = \int_0^\infty x^se^{-x}dx $$ So, I'd like to apply "Integration by parts". Just in case, you can check what the Integration by parts is here. However　we have to be familier with $\int e^{-x}dx$ beforehand.
if　let $\varphi(x)$ be $\varphi(x) = -x$, $f(t)$ be $f(t) = e^t$, we can apply "chain rule of Integration" as below.
$$-\int e^{-x}\cdot -1 dx =-\int e^t dt = -e^t = -e^{-x}$$ Let's back on truck! Let us assume $f(x) = x^s$, $g(x) = e^{-x}$, now is the time to apply "Integration by parts".
$$\begin{eqnarray}\Gamma(s+1) &=& \int_0^\infty x^se^{-x}dx \\ &=& [-x^s e^{-x}]_0^\infty + s\int^\infty_0 x^{s-1}e^{-x}dx\end{eqnarray}$$
From above, $\lim_{x \to \infty}x^se^{-x}$ seems to be what we have to deal with. In this case, we can apply "L'Hopital's rule". Just in case you are not familiar with "L'Hopital's rule", you can check here.
In this case, we apply "L'Hopital's rule" until $s-n<0$ as below. $$\begin{eqnarray} \lim_{x \to \infty}x^se^{-x} &=& \lim_{x \to \infty}s(s-1)(s-2) \cdots(s-(n-1))x^{s-n}e^{-x} \end{eqnarray}$$ Then we get the result as following, $$\lim_{x \to \infty}x^se^{-x} = 0$$
Consequently,
$$\begin{eqnarray}\Gamma(s+1) &=& s\int^\infty_0 x^{s-1}e^{-x}dx\end{eqnarray}$$ $$\therefore \Gamma(s+1) = s\Gamma(s)$$

2. $ \Gamma(n+1) = n!$ where $\ n$ is natural number¶

As you may know, "Gamma function" is extention of "factorial". Needless to say, it should equipped this property :)
$$\begin{eqnarray} \Gamma(n+1) &=& n\Gamma(n)\\ &=& n(n-1)(n-2)\cdots2\cdot1\Gamma(1)\end{eqnarray}$$
What we have to rectify is $\Gamma(1)$. You can calculate $\Gamma(1)$ with comparative ease as below.
$$\begin{eqnarray}\Gamma(1) &=& \int^\infty_0 x^0 e^{-x}dx\\ &=&[-e^{-x}]^\infty_0 \\ &=& 0 - (-1) = 1\end{eqnarray}$$ Consequently, $$ \Gamma(n+1) = n!$$