Difference between revisions of "Essai Latex"
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<!-- some LaTeX macros we want to use: -->
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<!DOCTYPE html>
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$\newcommand{\Re}{\mathrm{Re}\,}
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<html>
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\newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)}$
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<head>
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<title>MathJax TeX Test Page</title>
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We consider, for various values of $s$, the $n$-dimensional integral
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<script type="text/x-mathjax-config">
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\begin{align}
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   MathJax.Hub.Config({tex2jax: {inlineMath: [['$','$'], ['\\(','\\)']]}});
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   \label{def:Wns}
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</script>
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  W_n (s)
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<script type="text/javascript" async
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  &:=
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  src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS_CHTML">
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  \int_{[0, 1]^n}
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</script>
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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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</head>
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\end{align}
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<body>
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which occurs in the theory of uniform random walk integrals in the plane,
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When $a \ne 0$, there are two solutions to \(ax^2 + bx + c = 0\) and they are
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where at each step a unit-step is taken in a random direction. As such,
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$$x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$$
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the integral \eqref{def:Wns} expresses the $s$-th moment of the distance
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</body>
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to the origin after $n$ steps.
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</html>
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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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Revision as of 14:19, 3 April 2016

<!DOCTYPE html> <html> <head> <title>MathJax TeX Test Page</title> <script type="text/x-mathjax-config"> 

 MathJax.Hub.Config({tex2jax: {inlineMath: [['$','$'], ['\\(','\\)']]}});

</script> <script type="text/javascript" async 

 src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS_CHTML">

</script> </head> <body> When $a \ne 0$, there are two solutions to \(ax^2 + bx + c = 0\) and they are $$x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$$ </body> </html>

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