Difference between revisions of "Useful Examples"
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at the end of the paper. | at the end of the paper. | ||
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* HTML like syntax works! e.g. comments with | * HTML like syntax works! e.g. comments with | ||
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Syntax Highlighting: [Version:SyntaxHighlighting] | Syntax Highlighting: [Version:SyntaxHighlighting] | ||
Revision as of 16:11, 6 April 2022
To Edit Sidebar (as admin only!!!)
Go to MediaWiki:Sidebar
To write some LateX
See e.g. LMDZPedia And check out this page in "edit" mode! $$\newcommand{\Re}{\mathrm{Re}\,} \newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)}$$
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.
Miscellaneous =
- HTML like syntax works! e.g. comments with
<!-- This is a comment in the page --> which you see because between "pre" tags
- Some special features:
Link to the internal page with all special features: Special:Version Syntax Highlighting: [Version:SyntaxHighlighting]
