\chapter{Gamma function and related}





The \emph{gamma function} was originally studied as a class of integrals, however was eventually found to be an extension of the factorial to real and complex numbers.

Due to the prevalence of factorials in combinatorial formulae, it has since been applied to analytic combinatorics, which in turn promites the gamma function in probability theory, statistics, and physics.


\section{Gamma function}

We now introduce the Gamma function in all its glory.

\begin{definition}[Gamma function]
\[\Gamma (z) = \int^{\infty}_{0} t^{z-1}e^{-t} dt\]
\end{definition}

Of course, the gamma function agrees with the factorial; it is shown by using the product rule to demonstrate the intergal's recursive behaviour in the following way.

\begin{proposition}
	\[\Gamma(1)=1\]
	\[\Gamma(z+1)=z\Gamma(z)\]
\end{proposition}

By inducting on the natural numbers, one proves that the function agrees with the factorial.

\begin{corollary}
\[\Gamma(n+1)=n!\]
\end{corollary}


The gamma function is praised for its applications, however it is also a function of interest from a purely analytical perspective. The reason is because the gamma function is also used as a way to evaluate integrals, or more precisely, suggest methods that may prove effective; if an integral can be manipulated to form of the Gamma function evaluated at some point (or otherwise include the gamma function), this now suggests that one can try to fall back on the gamma function's properties to arrive at a solution.

The recursive formula of the gamma function is one property that can be useful, however there are many such properties that could be employed.


Note that $\Gamma(\frac{1}{2}) = \sqrt{\pi}$, since one can essentially transform it into the Gaussian integral.
\[ \Gamma(\frac{1}{2})=\int^{\infty}_{0} t^{-1/2} e^{-t}dt\]
Now consider the substitution $u=\sqrt{t}$, we now have
\[\Gamma(\frac{1}{2}) = 2\int^{\infty}_{0} e^{-u^2}du \]
which is simply the Gaussian integral!

\begin{proposition}
\[\Gamma (\frac{1}{2}) = \sqrt{\pi} \]
\end{proposition}

\begin{proposition}[Euler's reflection formula]
\[ \Gamma(z)\Gamma(1-z) =\frac{pi}{\sin(\pi z)}\]
\end{proposition}

Among special functions, there is a running theme of "reflection formulae". To prove this result, we require some knowledge on the growth of $\Gamma$.

\begin{lemma}
\[|\Gamma(z)| \leq |\Gamma(\Re(z))|\]
\end{lemma}
This simply follows from the triangle inequality on the integral defining the gamma function.



\begin{proposition}
\[\Gamma (z) \Gamma (z+\frac{1}{2}) = 2^{1-2z}\sqrt{\pi} \Gamma (2z) \]
\end{proposition}


Complex anaysis is a pathway to many abilities some consider to be unnatural; and this is definitely apparent in the study of the gamma function. 

Since $\frac{1}{\Gamma(z)}$ is entire, it has a Hadamard product; reciprocating this product leads to the following alternative definition of the gamma function.

\begin{proposition}
\[\Gamma(z) = \frac{e^{-\gamma z}}{z} \prod^{\infty}_{n=1} [\frac{1}{1+z/n}e^{z/n}] \]
\end{proposition}


\section{Beta function}


The factorial function notably appears in the binomial coefficient, one may ask if there is an analogue of the binomial coefficient with the factorials replaced with Beta functions.

\[\Beta(z,w) = \int^{1}_{0} t^{z-1}(1-t)^{w-1}\]