Mathtest: Unterschied zwischen den Versionen

Aktuelle Version vom 28. Juni 2018, 22:32 Uhr

$

 \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.

Mathtest: Unterschied zwischen den Versionen

Aktuelle Version vom 28. Juni 2018, 22:32 Uhr

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@@ Zeile 1: / Zeile 1: @@
-<math>
+<!-- some LaTeX macros we want to use: -->
-   \operatorname{erfc}(x) =
+$
-   \frac{2}{\sqrt{\pi}} \int_x^{\infty} e^{-t^2}\,dt =
+   \newcommand{\Re}{\mathrm{Re}\,}
-   \frac{e^{-x^2}}{x\sqrt{\pi}}\sum_{n=0}^\infty (-1)^n \frac{(2n)!}{n!(2x)^{2n}}
+   \newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)}
-  </math>
+$
+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.
+[[Category:Wiki/en]]