MathJax basic tutorial and quick reference

1.行内公式：\Gamma(n) = (n-1)!\quad\forall n\in\mathbb N
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2.块级公式x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}

3.x^{y^z}=(1+e^x)^{-2xy^w} \;\;\;\; \times - + \div

4.\sideset{^1_2}{^3_4}\bigotimes

5.方法1：\frac{分子}{分母}$ 方法2：$分子 \over 分母

6.\{[z-(1+\frac23x)y]\div 4\}

7.(1+\frac23x)

8.\frac{du}{dx} | _{x=0}

9.\sqrt {x^3}
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10.\sqrt[3]{\frac {x}{y}}

11.f(x_1,x_2,...,x_n)=x_1^2+x_2^2+,...,+x_n^2

12.\vec{a} \;\cdot \;\vec{b}=0

13.利用\;或\空格表示空格{a}\ \;>\ \;{b}

14.\sum_{i=0}^n \frac{1}{i^2}
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15.\prod_{i=0}^n \frac{1}{i^2}

16.\int_0^1 x^2 {\rm d}{\rm x} \; \oint \iint \iiint \lim

17.\forall x>1

18.\overline{a+b+c+d}

19.\overbrace{a+\underbrace{b+c}_{1.0}+d}^{2.0}

20.\uparrow \; \downarrow \; \rightarrow \; \leftarrow \;\longrightarrow \; \longleftarrow \; \Longrightarrow \; \Longleftarrow
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21.

\left[ \begin{matrix}
    1 & x & x^2 \\
    1 & y & y^2 \\
    1 & z & z^2 \\
    \end{matrix}
    \right]

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22.\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}

23.\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}

24. \frac{\partial z}{\partial y}\frac{\partial y}{\partial x}

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25.\frac{\partial^2z}{\partial x^2}

26.循环神经网络GRU（Gated Recurrent Unit）单元

    z_t=\sigma(W_t\cdot[h_{t-1},x_t])\tag 1
     r_t=\sigma(W_r\cdot[r_t*h_{t-1},x_t])\tag2
     h'=tanh(W\cdot[r_t*h_{t-1},x_t])\tag3
     h_t=(1-z_t)*h_{t-1}+ z_t *h'\tag4

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27.

 \left[
\begin{array}{cc|c}
  1&2&3\\
  4&5&6
\end{array}
\right]

  \begin{pmatrix}
    a & b\\
    c & d\\
  \hline
    1 & 0\\
    0 & 1
  \end{pmatrix}

\bigl( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \bigr)

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28.

\begin{align}
\sqrt{37} & = \sqrt{\frac{73^2-1}{12^2}} \\
 & = \sqrt{\frac{73^2}{12^2}\cdot\frac{73^2-1}{73^2}} \\ 
 & = \sqrt{\frac{73^2}{12^2}}\sqrt{\frac{73^2-1}{73^2}} \\
 & = \frac{73}{12}\sqrt{1 - \frac{1}{73^2}} \\ 
 & \approx \frac{73}{12}\left(1 - \frac{1}{2\cdot73^2}\right)
\end{align}

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29.

 f(n) =
\begin{cases}
n/2,  & \text{if $n$ is even} \\
3n+1, & \text{if $n$ is odd}
\end{cases}

\left.
\begin{array}{l}
\text{if $n$ is even:}&n/2\\
\text{if $n$ is odd:}&3n+1
\end{array}
\right\}
=f(n)

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30.

\begin{array}{c|lcr}
n & \text{Left} & \text{Center} & \text{Right} \\
\hline
1 & 0.24 & 1 & 125 \\
2 & -1 & 189 & -8 \\
3 & -20 & 2000 & 1+10i
\end{array}

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31.

\left\{ 
\begin{array}{c}
a_1x+b_1y+c_1z=d_1 \\ 
a_2x+b_2y+c_2z=d_2 \\ 
a_3x+b_3y+c_3z=d_3
\end{array}
\right.

\left\{
\begin{aligned} 
a_1x+b_1y+c_1z &=d_1+e_1 \\ 
a_2x+b_2y&=d_2 \\ 
a_3x+b_3y+c_3z &=d_3 
\end{aligned} 
\right.

\begin{cases}
a_1x+b_1y+c_1z=\frac{p_1}{q_1} \\
a_2x+b_2y+c_2z=\frac{p_2}{q_2} \\
a_3x+b_3y+c_3z=\frac{p_3}{q_3}
\end{cases}

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32.

\require{AMScd}
\begin{CD}
    A @>a>> B\\
    @V b V V= @VV c V\\
    C @>>d> D
\end{CD}

\require{AMScd}
\begin{CD}
      RCOHR'SO_3Na @>{\text{Hydrolysis,$\Delta, Dil.HCl$}}>> (RCOR')+NaCl+SO_2+ H_2O 
\end{CD}

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33.

\bbox[yellow]
{
e^x=\lim_{n\to\infty} \left( 1+\frac{x}{n} \right)^n
\qquad (1)
}

 \bbox[5px,border:2px solid red]
{
e^x=\lim_{n\to\infty} \left( 1+\frac{x}{n} \right)^n
\qquad (2) 
}

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34.

\begin{align}
   v + w & = 0  &&\text{Given} \tag 1\\
   -w & = -w + 0 && \text{additive identity} \tag 2\\
   -w + 0 & = -w + (v + w) && \text{equations $(1)$ and $(2)$}
\end{align}

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References:
https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference
https://blog.csdn.net/lanxuezaipiao/article/details/44341645

MathJax basic tutorial and quick reference

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