m (Test for the MathJa-extension) |
m (Now it works) |
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The reason for \eqref{eq:W3k} was long a mystery, but it will be explained | 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. | ||

+ | |||

+ | Additional Test | ||

+ | $$ | ||

+ | E=m c^2 | ||

+ | $$ |

## Latest revision as of 23:28, 12 July 2011

$

\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 Test $$ E=m c^2 $$