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Version 36 |
| \PMlinkescapeword{arguments} |
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| \PMlinkescapeword{calculate} |
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| \PMlinkescapeword{equivalent} |
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| \PMlinkescapeword{euler} |
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| \PMlinkescapeword{formula} |
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| \PMlinkescapephrase{generated by} |
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| \PMlinkescapeword{natural} |
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| \PMlinkescapeword{satisfies} |
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|
|
| \section*{Introduction} |
\section*{Introduction} |
|
|
| The \emph{gamma function} can be thought of as |
The \emph{gamma function} can be thought of as |
| the natural way to generalize the concept of the factorial |
the natural way to generalize the concept of the factorial |
| to non-integer arguments. |
to non-integer arguments. |
|
|
| \PMlinkname{Leonhard Euler}{EulerLeonhard} came up with a formula for such a generalization in 1729. |
\PMlinkname{Leonhard Euler}{EulerLeonhard} came up with a formula for such a generalization in 1729. |
| At around the same time, |
At around the same time, |
| James Stirling independently arrived at a different formula, |
James Stirling independently arrived at a different formula, |
| but was unable to show that it always converged. |
but was unable to show that it always converged. |
| In 1900, Charles Hermite showed that the formula given by Stirling does work, |
In 1900, Charles Hermite showed that the formula given by Stirling does work, |
| and that it defines the same function as \PMlinkescapetext{Euler's}. |
and that it defines the same function as \PMlinkescapetext{Euler's}. |
|
|
| \section*{Definitions} |
\section*{Definitions} |
|
|
| \PMlinkescapetext{Euler's original formula for the gamma function was} |
\PMlinkescapetext{Euler's original formula for the gamma function was} |
| \[ |
\[ |
| \Gamma(z) = \lim_{n\to\infty}\frac{n^z n!}{\prod_{k=0}^n(z+k)}. |
\Gamma(z) = \lim_{n\to\infty}\frac{n^z n!}{\prod_{k=0}^n(z+k)}. |
| \] |
\] |
| However, it is now more commonly defined by |
However, it is now more commonly defined by |
| \[ |
\[ |
| \Gamma(z) = \int_0^\infty \! e^{-t} t^{z-1} \, dt |
\Gamma(z) = \int_0^\infty \! e^{-t} t^{z-1} \, dt |
| \] |
\] |
| for $z\in\C$ with $\Re(z)>0$, |
for $z\in\C$ with $\Re(z)>0$, |
| and by analytic continuation for the rest of the complex plane, |
and by analytic continuation for the rest of the complex plane, |
| except for the non-positive integers (where it has simple poles). |
except for the non-positive integers (where it has simple poles). |
|
|
| Another equivalent definition is |
Another equivalent definition is |
| \[ |
\[ |
| \Gamma(z) = \frac{e^{-\gamma z}}{z} |
\Gamma(z) = \frac{e^{-\gamma z}}{z} |
| \prod_{n=1}^\infty \left(1 + \frac{z}{n}\right)^{-1} e^{z/n}, |
\prod_{n=1}^\infty \left(1 + \frac{z}{n}\right)^{-1} e^{z/n}, |
| \] |
\] |
| where $\gamma$ is Euler's constant. |
where $\gamma$ is Euler's constant. |
|
|
| \section*{Functional equations} |
\section*{Functional equations} |
|
|
| The gamma function satisfies the functional equation |
The gamma function satisfies the functional equation |
| \[ |
\[ |
| \Gamma(z+1) = z \Gamma(z) |
\Gamma(z+1) = z \Gamma(z) |
| \] |
\] |
| except when $z$ is a non-positive integer. |
except when $z$ is a non-positive integer. |
| As $\Gamma(1)=1$, it follows by induction that |
As $\Gamma(1)=1$, it follows by induction that |
| \[ |
\[ |
| \Gamma(n) = (n-1)! |
\Gamma(n) = (n-1)! |
| \] |
\] |
| for positive integer values of $n$. |
for positive integer values of $n$. |
|
|
| Another functional equation satisfied by the gamma function is |
Another functional equation satisfied by the gamma function is |
| \[ |
\[ |
| \Gamma(z) \Gamma(1-z) = \frac{\pi}{\sin \pi z} |
\Gamma(z) \Gamma(1-z) = \frac{\pi}{\sin \pi z} |
| \] |
\] |
| for non-integer values of $z$. |
for non-integer values of $z$. |
|
|
| \section*{Approximate values} |
\section*{Approximate values} |
|
|
| The gamma function for real $z$ looks like this: |
The gamma function for real $z$ looks like this: |
| \begin{center} |
\begin{center} |
| \begin{tabular}{c} |
\begin{tabular}{c} |
| \includegraphics[scale=1]{gammafunc.eps} \\ |
\includegraphics[scale=1]{gammafunc.eps} \\ |
| {\tiny (generated by GNU Octave and gnuplot) } |
{\tiny (generated by GNU Octave and gnuplot) } |
| \end{tabular} |
\end{tabular} |
| \end{center} |
\end{center} |
|
|
| It can be shown that $\Gamma(1/2)=\sqrt{\pi}$. |
It can be shown that $\Gamma(1/2)=\sqrt{\pi}$. |
| Approximate values of $\Gamma(x)$ for some other $x\in(0,1)$ are: |
Approximate values of $\Gamma(x)$ for some other $x\in(0,1)$ are: |
| \[ |
\[ |
| \begin{array}{cc} |
\begin{array}{cc} |
| \Gamma(1/5) \approx 4.5908 & \Gamma(1/4) \approx 3.6256 \\ |
\Gamma(1/5) \approx 4.5908 & \Gamma(1/4) \approx 3.6256 \\ |
| \Gamma(1/3) \approx 2.6789 & \Gamma(2/5) \approx 2.2182 \\ |
\Gamma(1/3) \approx 2.6789 & \Gamma(2/5) \approx 2.2182 \\ |
| \Gamma(3/5) \approx 1.4892 & \Gamma(2/3) \approx 1.3541 \\ |
\Gamma(3/5) \approx 1.4892 & \Gamma(2/3) \approx 1.3541 \\ |
| \Gamma(3/4) \approx 1.2254 & \Gamma(4/5) \approx 1.1642 |
\Gamma(3/4) \approx 1.2254 & \Gamma(4/5) \approx 1.1642 |
| \end{array} |
\end{array} |
| \] |
\] |
|
|
| If the value of $\Gamma(x)$ is known for some $x\in(0,1)$, |
If the value of $\Gamma(x)$ is known for some $x\in(0,1)$, |
| then one may calculate the value of $\Gamma(n+x)$ for any integer $n$ |
then one may calculate the value of $\Gamma(n+x)$ for any integer $n$ |
| by making use of the formula $\Gamma(z+1)=z\Gamma(z)$. |
by making use of the formula $\Gamma(z+1)=z\Gamma(z)$. |
| We have |
We have |
| \begin{eqnarray*} |
\begin{eqnarray*} |
| \Gamma(n+x) & = & (n+x-1)\Gamma(n+x-1) \\ |
\Gamma(n+x) & = & (n+x-1)\Gamma(n+x-1) \\ |
| & = & (n+x-1)(n+x-2)\Gamma(n+x-2) \\ |
& = & (n+x-1)(n+x-2)\Gamma(n+x-2) \\ |
| & \vdots & \\ |
& \vdots & \\ |
| & = & (n+x-1)(n+x-2)\cdots(x)\Gamma(x) |
& = & (n+x-1)(n+x-2)\cdots(x)\Gamma(x) |
| \end{eqnarray*} |
\end{eqnarray*} |
| which is easy to calculate if we know $\Gamma(x)$. |
which is easy to calculate if we know $\Gamma(x)$. |
