Help:Editing/Mathematical

LaTeX formulae

SDIY wiki supports embedding mathematical formulas using <m> tags. Click the edit tab above to view the source: <m>\operatorname{erfc}(x) = \frac{2}{\sqrt{\pi}} \int_x^{\infty} e^{-t^2}\,dt = \frac{e^{-x^2}}{x\sqrt{\pi}}\sum_{n=0}^\infty (-1)^n \frac{(2n)!}{n!(2x)^{2n}}</m>

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Use <math> tags, TeX or LaTeX notation, MathML, or AsciiMath notation. See the Extension:MathJax and the MathJax Documentation for more information. Click the edit tab above to view the source:

$

 \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.

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