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*[[Áreas temáticas|'''Áreas Temáticas''']]
 
*[[Áreas temáticas|'''Áreas Temáticas''']]
  
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<!-- some LaTeX macros we want to use: -->
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<math xmlns="http://www.w3.org/1998/Math/MathML">
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$
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   <mstyle displaystyle="true">
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   \newcommand{\Re}{\mathrm{Re}\,}
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    <mi>f</mi>
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  \newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)}
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    <mrow>
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$
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      <mo>(</mo>
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      <mi>a</mi>
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We consider, for various values of $s$, the $n$-dimensional integral
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      <mo>)</mo>
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\begin{align}
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    </mrow>
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  \label{def:Wns}
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     <mo>=</mo>
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  W_n (s)
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    <mfrac>
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  &:=
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      <mn>1</mn>
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  \int_{[0, 1]^n}
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      <mrow>
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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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        <mn>2</mn>
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\end{align}
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        <mi>&#x3C0;</mi>
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which occurs in the theory of uniform random walk integrals in the plane,
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        <mi>i</mi>
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where at each step a unit-step is taken in a random direction.  As such,
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      </mrow>
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the integral \eqref{def:Wns} expresses the $s$-th moment of the distance
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    </mfrac>
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to the origin after $n$ steps.
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    <msub>
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      <mo>&#x222E;</mo>
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By experimentation and some sketchy arguments we quickly conjectured and
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      <mrow>
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strongly believed that, for $k$ a nonnegative integer
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        <mi>&#x3B3;</mi>
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\begin{align}
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      </mrow>
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  \label{eq:W3k}
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    </msub>
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  W_3(k) &= \Re \, \pFq32{\frac12, -\frac k2, -\frac k2}{1, 1}{4}.
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    <mfrac>
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\end{align}
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      <mrow>
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Appropriately defined, \eqref{eq:W3k} also holds for negative odd integers.
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        <mi>f</mi>
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The reason for \eqref{eq:W3k} was  long a mystery, but it will be explained
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        <mo>(</mo>
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at the end of the paper.
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        <mi>z</mi>
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        <mo>)</mo>
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      </mrow>
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      <mrow>
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        <mi>z</mi>
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        <mo>&#x2212;</mo>
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        <mi>a</mi>
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      </mrow>
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    </mfrac>
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    <mi>d</mi>
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    <mi>z</mi>
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  </mstyle>
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</math>

Edição das 14h31min de 7 de maio de 2014

[math] \ltmstyle displaystyle="true"\gt \ltmi\gtf\lt/mi\gt \ltmrow\gt \ltmo\gt(\lt/mo\gt \ltmi\gta\lt/mi\gt \ltmo\gt)\lt/mo\gt \lt/mrow\gt \ltmo\gt=\lt/mo\gt \ltmfrac\gt \ltmn\gt1\lt/mn\gt \ltmrow\gt \ltmn\gt2\lt/mn\gt \ltmi\gtπ\lt/mi\gt \ltmi\gti\lt/mi\gt \lt/mrow\gt \lt/mfrac\gt \ltmsub\gt \ltmo\gt∮\lt/mo\gt \ltmrow\gt \ltmi\gtγ\lt/mi\gt \lt/mrow\gt \lt/msub\gt \ltmfrac\gt \ltmrow\gt \ltmi\gtf\lt/mi\gt \ltmo\gt(\lt/mo\gt \ltmi\gtz\lt/mi\gt \ltmo\gt)\lt/mo\gt \lt/mrow\gt \ltmrow\gt \ltmi\gtz\lt/mi\gt \ltmo\gt−\lt/mo\gt \ltmi\gta\lt/mi\gt \lt/mrow\gt \lt/mfrac\gt \ltmi\gtd\lt/mi\gt \ltmi\gtz\lt/mi\gt \lt/mstyle\gt [/math]

Disponível em "http://wikiien.ien.gov.br/mediawiki/index.php?title=Página_principal&oldid=126"