https://wiki.cosmos.esa.int/planckpla2015/index.php?title=Rod_test&feed=atom&action=historyRod test - Revision history2024-03-28T15:08:20ZRevision history for this page on the wikiMediaWiki 1.31.6https://wiki.cosmos.esa.int/planckpla2015/index.php?title=Rod_test&diff=805&oldid=prevRbailey: Created page with '<!-- 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…'2012-06-14T10:50:06Z<p>Created page with '<!-- 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…'</p>
<p><b>New page</b></p><div><!-- some LaTeX macros we want to use: --><br />
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\newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)}<br />
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<br />
We consider, for various values of $s$, the $n$-dimensional integral<br />
\begin{align}<br />
\label{def:Wns}<br />
W_n (s)<br />
&:= <br />
\int_{[0, 1]^n} <br />
\left| \sum_{k = 1}^n \mathrm{e}^{2 \pi \mathrm{i} \, x_k} \right|^s \mathrm{d}\boldsymbol{x}<br />
\end{align}<br />
which occurs in the theory of uniform random walk integrals in the plane, <br />
where at each step a unit-step is taken in a random direction. As such, <br />
the integral \eqref{def:Wns} expresses the $s$-th moment of the distance <br />
to the origin after $n$ steps.<br />
<br />
By experimentation and some sketchy arguments we quickly conjectured and <br />
strongly believed that, for $k$ a nonnegative integer<br />
\begin{align}<br />
\label{eq:W3k}<br />
W_3(k) &= \Re \, \pFq32{\frac12, -\frac k2, -\frac k2}{1, 1}{4}.<br />
\end{align}<br />
Appropriately defined, \eqref{eq:W3k} also holds for negative odd integers. <br />
The reason for \eqref{eq:W3k} was long a mystery, but it will be explained <br />
at the end of the paper.</div>Rbailey