ExempleMaths : Différence entre versions

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The reason for \eqref{eq:W3k} was  long a mystery, but it will be explained  
 
The reason for \eqref{eq:W3k} was  long a mystery, but it will be explained  
 
at the end of the paper.
 
at the end of the paper.
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 +
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<syntaxhighlight lang="fortran">
 +
  PROGRAM Truc
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  IMPLICIT NONE
 +
  DO i = 1, iim
 +
    DO j = 1, jjm
 +
      .....
 +
    END DO
 +
  END DO
 +
  END PROGRAM
 +
</syntaxhighlight>

Version actuelle en date du 6 avril 2022 à 16:16

$$\newcommand{\Re}{\mathrm{Re}\,} \newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)}$$

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.


   PROGRAM Truc
   IMPLICIT NONE
   DO i = 1, iim
     DO j = 1, jjm
       .....
     END DO
   END DO
   END PROGRAM