"MathJax"의 두 판 사이의 차이
Pythagoras0 (토론 | 기여) |
Pythagoras0 (토론 | 기여) |
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<!-- some LaTeX macros we want to use: --> | <!-- some LaTeX macros we want to use: --> | ||
| − | + | <math> | |
\newcommand{\Re}{\mathrm{Re}\,} | \newcommand{\Re}{\mathrm{Re}\,} | ||
\newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)} | \newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)} | ||
| − | + | </math> | |
| − | 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} | ||
\label{def:Wns} | \label{def:Wns} | ||
| 67번째 줄: | 37번째 줄: | ||
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 \eqref{def:Wns} expresses the | + | the integral \eqref{def:Wns} 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} | ||
| 82번째 줄: | 52번째 줄: | ||
==관련된 항목들== | ==관련된 항목들== | ||
* [[수식표현 안내]] | * [[수식표현 안내]] | ||
| + | [[분류:수식표현]] | ||
2020년 11월 12일 (목) 07:24 기준 최신판
관련링크
문서 구조
- item 1
- definition 1
- item 2
- definition 2-1
- definition 2-2
참조 예
- 맥스웰 방정식의 벡터 해석학적 표현
\[\iint_{S} \mathbf{E}\cdot\,d\mathbf{S} = \frac {Q} {\varepsilon_0} \label{gau}\] \[\int_{C} \mathbf{E}\cdot\,d\mathbf{r} =-\frac{d}{dt}\iint_{S} \mathbf{B}\cdot\,d\mathbf{S}\label{far} \]
- \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.