Including mathematical formulae into AniWiki
- Syntax
- Example
  - Stronger form of the Central Limit Theorem
  - Taylor series

Including mathematical formulae into AniWiki

Based on the math plugin, I created a method for AniWiki to generate mathematical formulae, which is fairly similar to TeX/LaTeX.

Syntax

Dokuwiki Syntax:

$...text style mathematical formula...$
$$...display style mathematical formula...$$

I don't think I still need to explain what are textstyle and displaystyle in LaTeX. See the examples below.

Example

Let's try the familiar CLT formula here.

Stronger form of the Central Limit Theorem

Let $\normalsize X_{1}, X_{2}, \ldots$ be a sequence of iid random variables with $\normalsize E(X_{i})=\mu$ and $\normalsize 0<Var(X_i)=\sigma^2<\infty$ . Define $\normalsize \bar{X}_n=(1/n)\sum_{i=1}^n X_i$ . Let $\normalsize G_n(x)$ denote the cdf of $\normalsize \sqrt{n}(\bar{X}_n-\mu)/\sigma$ . Then, for any $\normalsize x$ , $\normalsize -\infty<x<\infty$ ,

$\lim_{n\rightarrow\infty}G_n(x)=\int_{-\infty}^x \frac{1}{\sqrt{2\pi}}e^{-y^2/2}dy$

that is, $\normalsize \sqrt{n}(\bar{X}_n-\mu)/\sigma$ has a limiting standard normal distribution.

Let $X_{1}, X_{2}, \ldots$ be a sequence of iid random variables with $E(X_{i})=\mu$ 
and $0<Var(X_i)=\sigma^2<\infty$. Define $\bar{X}_n=(1/n)\sum_{i=1}^n X_i$. 
Let $G_n(x)$ denote the cdf of $\sqrt{n}(\bar{X}_n-\mu)/\sigma$. 
Then, for any $x$, $-\infty<x<\infty$,

$$\lim_{n\rightarrow\infty}G_n(x)=\int_{-\infty}^x \frac{1}{\sqrt{2\pi}}e^{-y^2/2}dy$$

that is, $\sqrt{n}(\bar{X}_n-\mu)/\sigma$ has a limiting standard normal distribution.

Taylor series

This formula is familiar for us:

$f(x) = f(a) + \frac{f'(a)}{1!}(x - a) + \frac{f^{(2)}(a)}{2!}(x - a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x - a)^n + R_n(x)$

$$f(x) = f(a) + \frac{f'(a)}{1!}(x - a) + \frac{f^{(2)}(a)}{2!}(x - a)^2 
       + \cdots + \frac{f^{(n)}(a)}{n!}(x - a)^n + R_n(x)$$