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| Title of object: |
gamma function |
| Canonical Name: |
GammaFunction |
| Type: |
Definition |
| Created on: |
2001-11-17 15:28:54 |
| Modified on: |
2006-10-02 03:29:20 |
| Classification: |
msc:33B15, msc:30D30 |
Revision comment (for changes between this and next version):
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xypic}
\def\C{\mathbb{C}}
\def\Re{\operatorname{Re}} |
Content:
\PMlinkescapeword{entire}
\PMlinkescapephrase{generated by}
%\PMlinkescapeword{function}
\PMlinkescapeword{natural}
\PMlinkescapeword{satisfies}
\section*{Introduction}
The gamma function can be thought of as
the natural way to generalize the concept of the factorial
to non-integer arguments.
Leonhard Euler came up with a formula for such a generalization in 1729.
At around the same time,
James Stirling independently arrived at a different formula,
but was unable to show that it always converged.
In 1900, Charles Hermite showed that Stirling's formula does work,
and that it gives the same function as Euler's.
\section*{Definitions}
The \emph{gamma function} is defined by
\[
\Gamma(z) = \int_0^\infty e^{-t} t^{z-1} dt
\]
for $z\in\C$ with $\Re(z)>0$,
and by analytic continuation for the rest of the complex plane,
except for the non-positive integers (where it has simple poles).
The gamma function can equivalently be defined by the formula
\[
\Gamma(z) = \frac{e^{-\gamma z}}{z}
\prod_{n=1}^\infty \left(1 + \frac{z}{n}\right)^{-1} e^{z/n},
\]
where $\gamma$ is Euler's constant.
Euler's original formula was
\[
\Gamma(z) = \lim_{n\to\infty}\frac{n^z n!}{\prod_{k=0}^n(z+k)}.
\]
\section*{Functional equations}
The gamma function satisfies the functional equation
\[
\Gamma(z+1) = z \Gamma(z).
\]
As $\Gamma(1)=1$, it follows by induction that
\[
\Gamma(n) = (n-1)!
\]
for positive integer values of $n$.
Another functional equation satisfied by the gamma function is
\[
\Gamma(z) \Gamma(1-z) = \frac{\pi}{\sin \pi z}.
\]
\section*{Appoximate values}
The gamma function for real $z$ looks like this:
\begin{center}
\begin{tabular}{c}
\includegraphics[scale=1]{gammafunc.eps} \\
{\tiny (generated by GNU Octave and gnuplot) }
\end{tabular}
\end{center}
Approximate values of $\Gamma(z)$ for some $z\in(0,1) are:
\[
\begin{array}{cc}
\Gamma(1/5) \approx 4.5909 & \Gamma(1/4) \approx 3.6256 \\
\Gamma(1/3) \approx 2.6789 & \Gamma(2/5) \approx 2.2182 \\
\Gamma(3/5) \approx 1.4892 & \Gamma(2/3) \approx 1.3541 \\
\Gamma(3/4) \approx 1.2254 & \Gamma(4/5) \approx 1.1642
\end{array}
\]
and the ever-useful $\Gamma(1/2)=\sqrt{\pi}$. These values allow a quick calculation of $\Gamma(n+f)$
where $n$ is a natural number and $f$ is any fractional value for which the gamma function's value is known. Since $\Gamma(z+1)=z\Gamma(z)$, we have
\begin{eqnarray*}
\Gamma(n+f) & = & (n+f-1)\Gamma(n+f-1) \\
& = & (n+f-1)(n+f-2)\Gamma(n+f-2) \\
& \vdots & \\
& = & (n+f-1)(n+f-2)\cdots(f)\Gamma(f)
\end{eqnarray*}
which is easy to calculate if we know $\Gamma(f)$.
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