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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+ | |||
+ | <syntaxhighlight lang="fortran"> | ||
+ | PROGRAM Truc | ||
+ | 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