Difference between revisions of "HowToEnterMathSymbols"
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</syntaxhighlight> |
</syntaxhighlight> |
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(Which comes from a preprint of ''Jon M. Borwein, et. al. Some arithmetic properties of short random walk integrals.'') |
(Which comes from a preprint of ''Jon M. Borwein, et. al. Some arithmetic properties of short random walk integrals.'') |
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| − | |||
| + | |||
| − | + | This renders as http://www.cs.kuleuven.be/~dirkn/Extension_MathJax/MathJaxExample.png. |
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<!-- some LaTeX macros we want to use: --> |
<!-- some LaTeX macros we want to use: --> |
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$ |
$ |
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| − | + | We consider, for various values of $s$, the $n$-dimensional integral |
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| − | + | \begin{align} |
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| − | + | \label{def:Wns} |
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| − | + | W_n (s) |
|
| − | + | &:= |
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| − | + | \int_{[0, 1]^n} |
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| − | + | \left| \sum_{k = 1}^n \mathrm{e}^{2 \pi \mathrm{i} \, x_k} \right|^s \mathrm{d}\boldsymbol{x} |
|
| − | + | \end{align} |
|
| − | + | which occurs in the theory of uniform random walk integrals in the plane, |
|
| − | + | where at each step a unit-step is taken in a random direction. As such, |
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| − | + | the integral \eqref{def:Wns} expresses the $s$-th moment of the distance |
|
| − | + | to the origin after $n$ steps. |
|
| − | + | ||
| − | + | By experimentation and some sketchy arguments we quickly conjectured and |
|
| − | + | strongly believed that, for $k$ a nonnegative integer |
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| − | + | \begin{align} |
|
| − | + | \label{eq:W3k} |
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| − | + | W_3(k) &= \Re \, \pFq32{\frac12, -\frac k2, -\frac k2}{1, 1}{4}. |
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| − | + | \end{align} |
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| − | + | Appropriately defined, \eqref{eq:W3k} also holds for negative odd integers. |
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| − | + | The reason for \eqref{eq:W3k} was long a mystery, but it will be explained |
|
| − | + | at the end of the paper. |
|
This documentation comes from the [http://www.mediawiki.org/wiki/Extension:MathJax MathJax Extension page]. Additional documentation on using MathJax can be found at www.mathjax.org. |
This documentation comes from the [http://www.mediawiki.org/wiki/Extension:MathJax MathJax Extension page]. Additional documentation on using MathJax can be found at www.mathjax.org. |
||
Revision as of 09:56, 24 July 2012
We use the MathJax Extension by Dirk Nuyens. This extension enables MathJax (http://www.mathjax.org/) which is a Javascript library written by Davide Cervone.
Usage
The following math environments are defined for inline style math:
$...$(can be turned off, even per page),\(...\)and<math>...</math>.
And the following math environments are defined for display style math:
$$...$$(can be turned off, even per page),\[...\],\begin{...}...\end{...}and:<math>...</math>.
MathJax produces nice and scalable mathematics, see their website (http://www.mathjax.org/) for a demonstration. This extension also enables the usage of \label{} and \eqref{} tags with automatic formula numbering. If needed you can still hand label by using \tag{}.
This extension allows for typical LaTeX math integration. For example: <syntaxhighlight lang="latex"> $
\newcommand{\Re}{\mathrm{Re}\,}
\newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)}
$
We consider, for various values of $s$, the $n$-dimensional integral \begin{align}
\label{def:Wns}
W_n (s)
&:=
\int_{[0, 1]^n}
\left| \sum_{k = 1}^n \mathrm{e}^{2 \pi \mathrm{i} \, x_k} \right|^s \mathrm{d}\boldsymbol{x}
\end{align} which occurs in the theory of uniform random walk integrals in the plane, where at each step a unit-step is taken in a random direction. As such, the integral \eqref{def:Wns} expresses the $s$-th moment of the distance to the origin after $n$ steps.
By experimentation and some sketchy arguments we quickly conjectured and strongly believed that, for $k$ a nonnegative integer \begin{align}
\label{eq:W3k}
W_3(k) &= \Re \, \pFq32{\frac12, -\frac k2, -\frac k2}{1, 1}{4}.
\end{align} Appropriately defined, \eqref{eq:W3k} also holds for negative odd integers. The reason for \eqref{eq:W3k} was long a mystery, but it will be explained at the end of the paper. </syntaxhighlight> (Which comes from a preprint of Jon M. Borwein, et. al. Some arithmetic properties of short random walk integrals.)
This renders as http://www.cs.kuleuven.be/~dirkn/Extension_MathJax/MathJaxExample.png.
$
\newcommand{\Re}{\mathrm{Re}\,}
\newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)}
$
We consider, for various values of $s$, the $n$-dimensional integral \begin{align}
\label{def:Wns}
W_n (s)
&:=
\int_{[0, 1]^n}
\left| \sum_{k = 1}^n \mathrm{e}^{2 \pi \mathrm{i} \, x_k} \right|^s \mathrm{d}\boldsymbol{x}
\end{align} which occurs in the theory of uniform random walk integrals in the plane, where at each step a unit-step is taken in a random direction. As such, the integral \eqref{def:Wns} expresses the $s$-th moment of the distance to the origin after $n$ steps.
By experimentation and some sketchy arguments we quickly conjectured and strongly believed that, for $k$ a nonnegative integer \begin{align}
\label{eq:W3k}
W_3(k) &= \Re \, \pFq32{\frac12, -\frac k2, -\frac k2}{1, 1}{4}.
\end{align} Appropriately defined, \eqref{eq:W3k} also holds for negative odd integers. The reason for \eqref{eq:W3k} was long a mystery, but it will be explained at the end of the paper.
This documentation comes from the MathJax Extension page. Additional documentation on using MathJax can be found at www.mathjax.org.
