Difference between revisions of "HowToEnterMathSymbols"
(Fix use of dollar signs, which aren't enabled.) |
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<!-- some LaTeX macros we want to use: --> |
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| − | $ |
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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 |
+ | We consider, for various values of \(s\), the \(n\)-dimensional integral |
\begin{align} |
\begin{align} |
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\label{def:Wns} |
\label{def:Wns} |
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which occurs in the theory of uniform random walk integrals in the plane, |
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, |
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 |
+ | the integral \eqref{def:Wns} expresses the \(s\)-th moment of the distance |
| − | to the origin after |
+ | to the origin after \(n\) steps. |
| − | + | ||
By experimentation and some sketchy arguments we quickly conjectured and |
By experimentation and some sketchy arguments we quickly conjectured and |
||
| − | strongly believed that, for |
+ | strongly believed that, for \(k\) a nonnegative integer |
\begin{align} |
\begin{align} |
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\label{eq:W3k} |
\label{eq:W3k} |
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| Line 80: | Line 80: | ||
\end{align} |
\end{align} |
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Appropriately defined, \eqref{eq:W3k} also holds for negative odd integers. |
Appropriately defined, \eqref{eq:W3k} also holds for negative odd integers. |
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| − | The reason for \eqref{eq:W3k} was |
+ | The reason for \eqref{eq:W3k} was long a mystery, but it will be explained |
at the end of the paper. |
at the end of the paper. |
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Revision as of 10:30, 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:
\(...\)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{}.
Example
Latex Code
<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.)
Rendered text
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.
Additional Information
- This documentation comes from the MathJax Extension page. Additional documentation on using MathJax can be found at www.mathjax.org.
- See our config file for some custom macro definitions that may be helpful.
