Markdown Cheat Sheet

June 27, 2019 · View on GitHub

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数学相关LaTeX表达

前言

后面介绍的内容是LaTeX\LaTeX排版的数学符号的内容，不止一次有人强调中文版的Wikipedia并不是英文版内容的翻译，并不是。可以对比下参考部分的两个页面，我觉得中文页面做的不错，这两个页面里面关于垂直，貌似有那么点不同。

表 1: 数学模式重音符

示例代码示例代码示例代码示例代码
a^\hat {a}\hat{a}aˇ\check{a}\check{a}a~\tilde{a}\tilde{a}aˊ\acute{a}\acute{a}
**aˋ\grave{a} **\grave{a}**a˙\dot {a} **\dot{a}**aˉ\bar{a} **\bar{a}a¨\ddot a\ddot{a}
**a⃗\vec {a} **\vec{a}**A^\widehat{A} **\widehat{A}A~\widetilde{A}\widetilde{A}a˘\breve a\breve{a}
A^\hat{A}\hat{A}A~\tilde{A}\tilde{A}

表2: 小写希腊字母

示例代码示例代码示例代码示例代码
α\alpha\alphaθ\theta\thetaυ\upsilon\upsilonoo o
β\beta\betaϑ\vartheta\varthetaπ\pi\piϕ\phi\phi
γ\gamma\gammaι\iota\iotaϖ\varpi\varpiφ\varphi\varphi
δ\delta\deltaκ\kappa\kappaρ\rho\rhoχ\chi\chi
ϵ\epsilon\epsilonλ\lambda\lambdaϱ\varrho\varrhoψ\psi\psi
ε\varepsilon\varepsilonμ\mu\muσ\sigma\sigmaω\omega\omega
ζ\zeta\zetaν\nu\nuς\varsigma\varsigma∇\nabla\nabla
η\eta\etaξ\xi\xiτ\tau\tau

表 3: 大写希腊字母

示例代码示例代码示例代码示例代码
Γ\Gamma\GammaΛ\Lambda\LambdaΣ\mitΣ\Sigma\mit\Sigma\Sigma\mit\SigmaΨ\Psi\Psi
Δ\Delta\DeltaΞ\Xi\XiΥ\Upsilon\UpsilonΩ\mitΩ\Omega\mit\Omega\Omega\mit\Omega
Θ\Theta\ThetaΠ\Pi\PiΦ\Phi\Phi

表 4: 数学字母

示例代码
ABCdefxyzXYZ123\mathbf {ABCdefxyzXYZ123}\mathbf {ABCdefxyzXYZ123}
ABCdefxyzXYZ123\mathrm {ABCdefxyzXYZ123}\mathrm {ABCdefxyzXYZ123}
ABCdefxyzXYZ123\mathit {ABCdefxyzXYZ123}\mathit {ABCdefxyzXYZ123}
ABCdefxyzXYZ123\mathcal {ABCdefxyzXYZ123}\mathcal {ABCdefxyzXYZ123}
ABCdefxyzXYZ123\mathscr {ABCdefxyzXYZ123}\mathscr {ABCdefxyzXYZ123}
ABCdefxyzXYZ123\mathfrak {ABCdefxyzXYZ123}\mathfrak {ABCdefxyzXYZ123}
ABCdefxyzXYZ123\mathbb {ABCdefxyzXYZ123}\mathbb {ABCdefxyzXYZ123}
ABCdefxyzXYZ123\boldsymbol{ABCdefxyzXYZ123}\boldsymbol{ABCdefxyzXYZ123}

表 5: 运算符与函数

示例代码示例代码示例代码
∑\sum\sum∏\prod\prodx⋅yx\cdot{y}x\cdot{y}
⋃\bigcup\bigcup⨁\bigoplus\bigoplusx×yx\times {y}x\times {y}
⋁\bigvee\bigvee⋂\bigcap\bigcap∥w∥\left\|w\right\|**\left\
⋀\bigwedge\bigwedge⨄\biguplus\biguplus∭\iiint\iiint
⨂\bigotimes\bigotimes∮\oint\oint∬\iint\iint
∫x dx\int x\,{\rm d}x\int x,{\rm d}x⨆\bigsqcup\bigsqcup⟮⟯\lgroup \rgroup\lgroup \rgroup
∐\coprod\coprod⨀\bigodot\bigodot∂\partial\partial
det⁡\det\detmax⁡\max\maxmin⁡\min\min
log⁡\log\log

表 6: 常用箭头

示例代码示例代码示例代码
←\leftarrow\leftarrow→\rightarrow\rightarrow↔\leftrightarrow\leftrightarrow
⟵\longleftarrow\longleftarrow⟶\longrightarrow\longrightarrow⟷\longleftrightarrow\longleftrightarrow
⇐\Leftarrow\Leftarrow⇒\Rightarrow\Rightarrow⇔\Leftrightarrow\Leftrightarrow
⟸\Longleftarrow\Longleftarrow⟹\Longrightarrow\Longrightarrow⟺\Longleftrightarrow\Longleftrightarrow
↑\uparrow\uparrow↓\downarrow\downarrow↕\updownarrow\updownarrow

表 7: 其他常用符号

示例代码示例代码示例代码
∴\therefore\therefore∵\because\becausemin⁡f∈H\min \limits_{f \in {H}}\min \limits_{f \in {H}}
⩽\leqslant\leqslant⩾\geqslant\geqslantC≡1\cal {C} \equiv 1\equiv
≈\thickapprox\thickapprox∼∼\thicksim \sim\thicksim \sim(AB)\left(\frac{A}{B}\right)\left(\frac{A}{B}\right)
≠\neq\neq∈\in\in=^\hat{=}\hat{=}
±\pm\pma\sqrt{a}\sqrt{a}≥≤\geq \leq\geq \leq
⊥\perp \perp∠\angle\angle∝\varpropto\varpropto
∞\infty\inftyg′g^\primeg^\prime∀\forall\forall
∃\exist\exist⊥\bot\bot⊤\top\top

注意**\bot和\perp的区别，垂直是\perp**

表8: 使用字体

示例代码备注
ABCdefxyzXYZ123\rm {ABCdefxyzXYZ123}\rm {ABCdefXYZ123}罗马体
ABCdefxyzXYZ123\it{ABCdefxyzXYZ123}\it{ABCdefXYZ123}意大利体
ABCdefxyzXYZ123\bf{ABCdefxyzXYZ123}\bf{ABCdefXYZ123}正粗体，黑体
ABCdefxyzXYZ123\cal {ABCdefxyzXYZ123}\cal {ABCdefXYZ123}花体
ABCdefXYZ123\sf{ABCdefXYZ123}\sf{ABCdefXYZ123}等线体
\mitABCdefxyzXYZ123\mit{ABCdefxyzXYZ123}\mit{ABCdefXYZ123}数字斜体
ABCdefxyzXYZ123\tt{ABCdefxyzXYZ123}\tt{ABCdefXYZ123}打印机字体

表9: 分段函数与公式对齐

分段函数

示例

f(x,y)={1x与y满足某一事实0否则f(x,y) = \begin{cases} 1 & x与y满足某一事实\\ 0 & 否则 \end{cases} 
# 代码
f(x,y) = \begin{cases}
1 & x与y满足某一事实\\
0 & 否则
\end{cases}
L(w)=∑i=1N[yilog⁡π(xi)+(1−yi)log⁡(1−π(xi))]=∑i=1N[yilog⁡π(xi)1−π(xi)+log⁡(1−π(xi))]=∑i=1N[yi(w⋅xi)−log⁡(1+exp⁡(w⋅xi)]\begin{aligned} L(w)&=\sum\limits^{N}_{i=1}[y_i\log\pi(x_i)+(1-y_i)\log(1-\pi(x_i))]\\&=\sum\limits^{N}_{i=1}[y_i\log{\frac{\pi(x_i)}{1-\pi(x_i)}}+\log(1-\pi(x_i))]\\&=\sum\limits^{N}_{i=1}[y_i(w\cdot x_i)-\log(1+\exp(w\cdot{x_i})] \end{aligned}

对齐控制

# 代码
# 通过\begin{aligned}\end{aligned}控制对齐, 使用&表示对齐点.
\begin{aligned}
L(w)&=\sum\limits^{N}_{i=1}[y_i\log\pi(x_i)+(1-y_i)\log(1-\pi(x_i))]\\
&=\sum\limits^{N}_{i=1}[y_i\log{\frac{\pi(x_i)}{1-\pi(x_i)}}+\log(1-\pi(x_i))]\\
&=\sum\limits^{N}_{i=1}[y_i(w\cdot x_i)-\log(1+\exp(w\cdot{x_i})]
\end{aligned}

另外注意到前面的分段函数自动变好了，但是上面多行对齐的公式没有自动编号，如果需要自动编号，外面嵌入equation

L(w)=∑i=1N[yilog⁡π(xi)+(1−yi)log⁡(1−π(xi))]=∑i=1N[yilog⁡π(xi)1−π(xi)+log⁡(1−π(xi))]=∑i=1N[yi(w⋅xi)−log⁡(1+exp⁡(w⋅xi)]\begin{equation} \begin{aligned} L(w)&=\sum\limits^{N}_{i=1}[y_i\log\pi(x_i)+(1-y_i)\log(1-\pi(x_i))]\\ &=\sum\limits^{N}_{i=1}[y_i\log{\frac{\pi(x_i)}{1-\pi(x_i)}}+\log(1-\pi(x_i))]\\ &=\sum\limits^{N}_{i=1}[y_i(w\cdot x_i)-\log(1+\exp(w\cdot{x_i})] \end{aligned} \end{equation}

代码如下

\begin{equation}
\begin{aligned}
L(w)&=\sum\limits^{N}_{i=1}[y_i\log\pi(x_i)+(1-y_i)\log(1-\pi(x_i))]\\&=\sum\limits^{N}_{i=1}[y_i\log{\frac{\pi(x_i)}{1-\pi(x_i)}}+\log(1-\pi(x_i))]\\&=\sum\limits^{N}_{i=1}[y_i(w\cdot x_i)-\log(1+\exp(w\cdot{x_i})]
\end{aligned}
\end{equation}

公式编号

关于编号也可以通过行间公式做如下表达

L(w)=∑i=1N[yilog⁡π(xi)+(1−yi)log⁡(1−π(xi))]=∑i=1N[yilog⁡π(xi)1−π(xi)+log⁡(1−π(xi))]=∑i=1N[yi(w⋅xi)−log⁡(1+exp⁡(w⋅xi)]\begin{align} L(w)&=\sum\limits^{N}_{i=1}[y_i\log\pi(x_i)+(1-y_i)\log(1-\pi(x_i))]\\ &=\sum\limits^{N}_{i=1}[y_i\log{\frac{\pi(x_i)}{1-\pi(x_i)}}+\log(1-\pi(x_i))]\nonumber\\ &=\sum\limits^{N}_{i=1}[y_i(w\cdot x_i)-\log(1+\exp(w\cdot{x_i})] \end{align}

代码如下

\begin{align}
L(w)&=\sum\limits^{N}_{i=1}[y_i\log\pi(x_i)+(1-y_i)\log(1-\pi(x_i))]\\
&=\sum\limits^{N}_{i=1}[y_i\log{\frac{\pi(x_i)}{1-\pi(x_i)}}+\log(1-\pi(x_i))]\nonumber\\
&=\sum\limits^{N}_{i=1}[y_i(w\cdot x_i)-\log(1+\exp(w\cdot{x_i})]
\end{align}

以上代码有两点需要注意体会：

  1. align
  2. \nonumber的使用

表X: 矩阵

普通矩阵

M1(x)=[a01a0200],M2(x)=[b11b12b21b22]M3(x)=[c11c12c21c22],M4(x)=[1010]\begin{aligned} M_1(x)= \begin{bmatrix} &a_{01}&a_{02}\\ &0&0 \end{bmatrix} &,M_2(x)= \begin{bmatrix} &b_{11}&b_{12}\\ &b_{21}&b_{22} \end{bmatrix} \\ M_3(x)= \begin{bmatrix} &c_{11}&c_{12}\\ &c_{21}&c_{22} \end{bmatrix} &,M_4(x)= \begin{bmatrix} &1&0\\ &1&0 \end{bmatrix} \end{aligned} 
\begin{aligned}
M_1(x)=
\begin{bmatrix}
&a_{01}&a_{02}\\
&0&0
\end{bmatrix}
&,M_2(x)=
\begin{bmatrix}
&b_{11}&b_{12}\\
&b_{21}&b_{22}
\end{bmatrix}
\\
M_3(x)=
\begin{bmatrix}
&c_{11}&c_{12}\\
&c_{21}&c_{22}
\end{bmatrix}
&,M_4(x)=
\begin{bmatrix}
&1&0\\
&1&0
\end{bmatrix}
\end{aligned}

带省略符号的Matrix

XT=[x11⋯x1N⋮⋱⋮xM1⋯xMN]X^\mathrm T= \left[ \begin{matrix} x_{11} & \cdots & x_{1N} \\ \vdots & \ddots & \vdots \\ x_{M1} & \cdots & x_{MN} \\ \end{matrix} \right] 
% 这里稍微注意下转置符号， 《统计学习方法》中的转置用的是正体的T
% 可以参考 https://zhuanlan.zhihu.com/p/27490955 中关于转置写法的讨论。
X^\mathrm T=
\left[
\begin{matrix}
 x_{11} & \cdots & x_{1N}       \\
 \vdots & \ddots & \vdots 		\\
 x_{M1} & \cdots & x_{MN}       \\
\end{matrix}
\right]

向量

\left[ \begin{array} \\2 \\3 \end{array} \right] 
\left[
\begin{array}
\\2
\\3
\end{array}
\right]
abcde⏞fghij⏟commentklmn‾opqr‾stuv←wxyz→\overbrace{abcde}\underbrace{fghij}_{comment}\overline{klmn}\underline{opqr}\overleftarrow{stuv}\overrightarrow{wxyz} 
\overbrace{abcde}\underbrace{fghij}_{comment}\overline{klmn}\underline{opqr}\overleftarrow{stuv}\overrightarrow{wxyz}

Emoji

表XI: Emoji

:smirk: smirk:smile:smile:laughing:laughing:blush:blush:smiley:smiley:heart_eyes:heart_eyes
😘kissing_heart:wink:wink:kissing:kissing:confused:confused:sweat:sweat:joy:joy
:sob:sob:cry:cry:angry:angry:yum:yum:mask:mask:sunglasses:sunglasses
:heartpulse:heartpulse:alien:alien:cupid:cupid:+1:+1:cn:cn:shit:shit

Refs

  1. Markdown 数学符号速查
  2. Cmd Markdown公式指导手册
  3. Equals_Sign
  4. Emoji
  5. Short Math Guide for LaTeX
  6. List of Mathematical Symbols
  7. 数学公式
  8. Matplotlib Math Text