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문서 구조

item 1
definition 1
item 2
definition 2-1
definition 2-2

참조 예

\[\iint_{S} \mathbf{E}\cdot\,d\mathbf{S} = \frac {Q} {\varepsilon_0} \label{gau}\] \[\int_{C} \mathbf{E}\cdot\,d\mathbf{r} =-\frac{d}{dt}\iint_{S} \mathbf{B}\cdot\,d\mathbf{S}\label{far} \]

  • \ref{gau}를 가우스 법칙이라 한다
  • \ref{far}를 패러데이 법칙이라 한다

newcommand 사용 예

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

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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