|
|
| (15 intermediate revisions by 2 users not shown) |
| Line 1: |
Line 1: |
| We use the [http://www.mediawiki.org/wiki/Extension:MathJax MathJax Extension] by [http://www.mediawiki.org/wiki/User:Dirk_Nuyens Dirk Nuyens]. This extension enables [http://www.mathjax.org/ MathJax] (http://www.mathjax.org/) which is a Javascript library written by Davide Cervone.
| | #REDIRECT[[Help:Entering mathematics]] |
| | |
| == Usage ==
| |
| | |
| The following math environments are defined for inline style math:
| |
| * <code><nowiki>$...$</nowiki></code> (can be turned off, even per page),
| |
| * <code>\(...\)</code> and
| |
| * <code><math>...</math></code>.
| |
| And the following math environments are defined for display style math:
| |
| * <code><nowiki>$$...$$</nowiki></code> (can be turned off, even per page),
| |
| * <code>\[...\]</code>,
| |
| * <code>\begin{...}...\end{...}</code> and
| |
| * <code>:<math>...</math></code>.
| |
| MathJax produces nice and scalable mathematics, see their website (http://www.mathjax.org/) for a demonstration. This extension also enables the usage of <code>\label{}</code> and <code>\eqref{}</code> tags with automatic formula numbering. If needed you can still hand label by using <code>\tag{}</code>.
| |
| | |
| This extension allows for typical LaTeX math integration.
| |
| For example:
| |
| <syntaxhighlight lang="latex">
| |
| <!-- some LaTeX macros we want to use: -->
| |
| $
| |
| \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.
| |
| | |
| <!-- some LaTeX macros we want to use: -->
| |
| $
| |
| \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 [http://www.mediawiki.org/wiki/Extension:MathJax MathJax Extension page]. Additional documentation on using MathJax can be found at www.mathjax.org.
| |