| Line 2: | Line 2: | ||
| − | We consider, for various values of | + | We consider, for various values of <math>s</math>, the <math>n</math>-dimensional integral |
| + | |||
\begin{align} | \begin{align} | ||
| − | |||
W_n (s) | W_n (s) | ||
&:= | &:= | ||
| Line 10: | Line 10: | ||
\left| \sum_{k = 1}^n \mathrm{e}^{2 \pi \mathrm{i} \, x_k} \right|^s \mathrm{d}\boldsymbol{x} | \left| \sum_{k = 1}^n \mathrm{e}^{2 \pi \mathrm{i} \, x_k} \right|^s \mathrm{d}\boldsymbol{x} | ||
\end{align} | \end{align} | ||
| + | |||
which occurs in the theory of uniform random walk integrals in the plane, | 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, | where at each step a unit-step is taken in a random direction. As such, | ||
| − | the integral | + | the above integral expresses the <math>s</math>-th moment of the distance |
| − | to the origin after | + | to the origin after <math>n</math> steps. |
By experimentation and some sketchy arguments we quickly conjectured and | By experimentation and some sketchy arguments we quickly conjectured and | ||
| − | strongly believed that, for | + | strongly believed that, for <math>k</math> a nonnegative integer: |
| + | |||
\begin{align} | \begin{align} | ||
\label{eq:W3k} | \label{eq:W3k} | ||
W_3(k) &= \Re \, \pFq32{\frac12, -\frac k2, -\frac k2}{1, 1}{4}. | W_3(k) &= \Re \, \pFq32{\frac12, -\frac k2, -\frac k2}{1, 1}{4}. | ||
\end{align} | \end{align} | ||
| − | Appropriately defined, | + | |
| − | + | Appropriately defined, this equation also holds for negative odd integers. | |
| − | + | ||
Revision as of 14:41, 3 April 2016
\(E=mc^2\)
We consider, for various values of \(s\), the \(n\)-dimensional integral
\begin{align} 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 above integral 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, this equation also holds for negative odd integers.
\(\begin{align}
\nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\ \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\
\nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\
\nabla \cdot \vec{\mathbf{B}} & = 0
\end{align}\)