Diferenças entre edições de "Utilizador:NunoOliveira"
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Equation \eqref{eq:1} above. |
Equation \eqref{eq:1} above. |
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$\newcommand{\Re}{\mathrm{Re}\,} |
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\newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)}$ |
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We consider, for various values of $s$, the $n$-dimensional integral |
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\begin{align} |
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\label{def:Wns} |
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W_n (s) |
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&:= |
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\int_{[0, 1]^n} |
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\left| \sum_{k = 1}^n \mathrm{e}^{2 \pi \mathrm{i} \, x_k} \right|^s \mathrm{d}\boldsymbol{x} |
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\end{align} |
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which occurs in the theory of uniform random walk integrals in the plane, |
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where at each step a unit-step is taken in a random direction. As such, |
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the integral \eqref{def:Wns} expresses the $s$-th moment of the distance |
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to the origin after $n$ steps. |
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By experimentation and some sketchy arguments we quickly conjectured and |
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strongly believed that, for $k$ a nonnegative integer |
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\begin{align} |
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\label{eq:W3k} |
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W_3(k) &= \Re \, \pFq32{\frac12, -\frac k2, -\frac k2}{1, 1}{4}. |
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\end{align} |
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Appropriately defined, \eqref{eq:W3k} also holds for negative odd integers. |
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The reason for \eqref{eq:W3k} was long a mystery, but it will be explained |
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at the end of the paper. |
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Revisão das 19h31min de 24 de março de 2017
Teste onde é definido por:
Falhou a verificação gramatical (função desconhecida: "\label"): {\displaystyle \begin{align} \dot{x} & = \sigma(y-x) \label{eq:1}\\ \dot{y} & = \rho x - y - xz \\ \dot{z} & = -\beta z + xy \end{align}}
Equation \eqref{eq:1} above.
$\newcommand{\Re}{\mathrm{Re}\,} \newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)}$
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.