Difference between revisions of "Main Page"
| Line 16: | Line 16: | ||
\right) | \right) | ||
\end{equation} | \end{equation} | ||
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. | |||
== First Section == | == First Section == | ||
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Take a look at this beautifully typesetted equation: \begin{equation} p(n) = \mathcal{C}_n\times {}_1\!F_2\left(n+1;\frac{2n+1}{2}+\frac{\Gamma_\sigma}{\Gamma}, n+\frac{\Gamma_\sigma}{\Gamma}; -\frac{P_\sigma} {2\Gamma} \right) \end{equation}
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.
First Section
First Subsection
First subsubsection
<m>x^2</m>
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...tomorrow we will run faster, stretch out our arms farther...
And then one fine morning— So we beat on, boats against the
current, borne back ceaselessly into the past.