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| 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]] |
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| == Usage ==
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| The following math environments are defined for inline style math:
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| * <code>\(...\)</code> and
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| * <code><math>...</math></code>.
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| And the following math environments are defined for display style math:
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| * <code><nowiki>$$...$$</nowiki></code> (can be turned off, even per page),
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| * <code>\[...\]</code>,
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| * <code>\begin{...}...\end{...}</code> and
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| * <code>:<math>...</math></code>.
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| 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>.
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| == Example ==
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| === Latex Code ===
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| <syntaxhighlight lang="latex">
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| <!-- some LaTeX macros we want to use: -->
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| \(
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| \newcommand{\Re}{\mathrm{Re}\,}
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| \newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)}
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| \)
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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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| &:=
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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}
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| \end{align}
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| which occurs in the theory of uniform random walk integrals in the plane,
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| 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
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| to the origin after \(n\) steps.
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| By experimentation and some sketchy arguments we quickly conjectured and
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| strongly believed that, for \(k\) a nonnegative integer
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| \begin{align}
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| \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
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| at the end of the paper.
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| </syntaxhighlight>
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| (Which comes from a preprint of ''Jon M. Borwein, et. al. Some arithmetic properties of short random walk integrals.'')
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| === Rendered text ===
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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: -->
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| \(
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| \newcommand{\Re}{\mathrm{Re}\,}
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| \newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)}
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| \)
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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)
| |
| &:=
| |
| \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}
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| \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.
| |
|
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| By experimentation and some sketchy arguments we quickly conjectured and
| |
| strongly believed that, for \(k\) a nonnegative integer
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| \begin{align}
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| \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
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| at the end of the paper.
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| == Additional Information ==
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| * This documentation comes from the [http://www.mediawiki.org/wiki/Extension:MathJax MathJax Extension page]. Additional documentation on using MathJax can be found at [http://www.mathjax.org www.mathjax.org].
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| * See our [http://people.cs.kuleuven.be/~dirk.nuyens/Extension_MathJax/mwMathJaxConfig.js config file] for some custom macro definitions that may be helpful.
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