Department of Mechanics: Student's corner: Pružnost a pevnost: Speciální cvičení: Difference between revisions
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We consider, for various values of $s$, the $n$-dimensional integral | We consider, for various values of $s$, the $n$-dimensional integral | ||
<math> | |||
\begin{align} | \begin{align} | ||
\label{def:Wns} | \label{def:Wns} | ||
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\left| \sum_{k = 1}^n \mathrm{e}^{2 \pi \mathrm{i} \, x_k} \right|^s \mathrm{d}\boldsymbol{x} | \left| \sum_{k = 1}^n \mathrm{e}^{2 \pi \mathrm{i} \, x_k} \right|^s \mathrm{d}\boldsymbol{x} | ||
\end{align} | \end{align} | ||
</math> | |||
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, | ||
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, | ||
Latest revision as of 17:13, 14 December 2012
Cvičení
$
\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 [math]\displaystyle{ \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} }[/math] 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.