1 基本公式

1.1 上下标

$$
A_1^2
\\
B_{12}
\\
2^{x^2+y}
$$

$$\begin{gathered} A_1^2 \\ {B}_{12} \\ {2}^{x^2+y} \end{gathered}$$

1.2 分数

$$
\frac{x}{1+x^2}
\\
\frac{\frac{1}{2}+x}{y}
\\
\tfrac{a}{b}
\frac{a}{b}
$$

$$\begin{gathered} \frac{x}{1+x^2} \\ \frac{\frac{1}{2}+x}{y} \\ \frac{a}{b}\frac{a}{b} \end{gathered}$$

1.3 开根号

$$
\sqrt{x}
\sqrt[3]{x}
$$

$$\sqrt{x}\sqrt[3]{x}$$

1.4 指数

$$
\exp \\
e \\
$$

$$\begin{gathered} \exp \\ e \end{gathered}$$

1.5 对数

$$
\log \\
\lg \\
\ln
$$

$$\begin{gathered} \log \\ \lg \\ \ln \end{gathered}$$

1.6 三角函数

$$
\bot \\
\angle \\
30^\circ \\
\sin \\
\cos \\
\tan \\
\cot \\
\sec \\
\csc 
$$

$$\begin{gathered} \perp \\ \angle \\ 30^{\circ } \\ \sin \\ \cos \\ \tan \\ \cot \\ \sec \\ \csc \end{gathered}$$

1.7 集合

$$
\emptyset	\\
\in			\\
\notin		\\
\supset		\\
\supseteq	\\
\bigcap		\\
\bigcup		\\
\bigvee		\\
\bigwedge	\\
\ni
$$

$$\begin{gathered} \varnothing \\ \in \\ \notin \\ \supset \\ \supseteq \\ \bigcap \\ \bigcup \\ \bigvee \\ \bigwedge \\ \ni \end{gathered}$$

1.8 组合数

$$
\binom{n}{k}
\tbinom{n}{k}
$$

$$\left(\genfrac{}{}{0ex}{}{n}{k}\right)\left(\genfrac{}{}{0ex}{}{n}{k}\right)$$

1.9 取模

$$
x \pmod a
\\
2\mod{x}
$$

$$\begin{gathered} x(\mathrm{mod}a) \\ 2\mathrm{mod}x \end{gathered}$$

1.10 极限

$$
\lim_{n \rightarrow +\infty} \frac{1}{n}
$$

$$\underset{n\to +\infty }{\lim }\frac{1}{n}$$

1.11 导数

$$
a'
a''
a^{\prime}
$$

$$a^{\prime }{a}^{\prime \prime }{a}^{\prime }$$

1.12 积分

$$
\int_{1}^{2}
\intop_{2}^{1}
\oint
\smallint
\\
\iint
\oiint
\iiint
\oiiint
$$

$$\begin{gathered} {\int }_1^2\underset{2}{\overset{1}{\int }}\oint \int \\ \iint \iint \iiint \iiint \end{gathered}$$

1.13 微分

$$
\nabla
\\
\partial x
\\
\mathrm{d}x
\\
\dot x
\\
\ddot y
\\
\Delta
$$

$$\begin{gathered} \nabla \\ \partial x \\ \mathrm{d}x \\ \dot{x} \\ \ddot{y} \\ \Delta \end{gathered}$$

1.14 累积/累乘/极限

$$
\sum_{i=1}^{k}
\displaystyle\sum_{i=1}^n
\textstyle\sum_{i=1}^n
\\
\prod_{i=1}^{k}
\displaystyle\prod_{i=1}^n
\textstyle\prod_{i=1}^n
\\
\lim_{k \to \infty}
\lim\limits_{k \to \infty}
\lim\nolimits_{k \to \infty}
$$

$$\begin{gathered} \sum _{i=1}^k\sum _{i=1}^n{\sum }_{i=1}^n \\ {\prod }_{i=1}^k\prod _{i=1}^n{\prod }_{i=1}^n \\ {\lim }_{k\to \infty }\underset{k\to \infty }{\lim }{\lim }_{k\to \infty } \end{gathered}$$

1.15 方程组

$$
\begin{aligned}
f(x)
&= (x+1)^2\\
&= x^2 + 2x + 1
\end{aligned}
$$

$$\begin{array}{rl}f(x)& =(x+1{)}^2\\ & =x^2+2x+1\end{array}$$

$$
f(x) =
\begin{cases}
a &\text{if b} \\
b &\text{if a} \\
\end{cases}
$$

$$f(x)=\{\begin{array}{ll}a& \text{if b}\\ b& \text{if a}\end{array}$$

$$
\begin{cases}
\begin{aligned}
x + 2y &= 1\\
3x - y &= 5
\end{aligned}
\end{cases}
$$

$$\{\begin{array}{l}\begin{array}{rl}x+2y& =1\\ 3x-y& =5\end{array}\end{array}$$

$$
g(x,y) =
\left\{
\begin{array}{rcl}
\frac{M_g - d}{M_f-b}[f(x,y)-b]+d & & {b \leq f(x,y) \leq M_f} \\
F^*_L & & {S_L \leq 0 < S_M} \\
F^*_R & & {S_M \leq 0 < S_R} \\
F_R   & & {S_R \leq 0}
\end{array}
\right.
$$

$$g(x,y)=\{\begin{array}{rcl}\frac{M_g-d}{M_f-b}[f(x,y)-b]+d& & b\le f(x,y)\le M_f\\ F_L^{\ast }& & S_L\le 0<S_M\\ F_R^{\ast }& & S_M\le 0<S_R\\ F_R& & S_R\le 0\end{array}$$

1.16 矩阵

• 在起始、结束标记处用下列词替换 matrix
• pmatrix ：小括号边框
• bmatrix ：中括号边框
• Bmatrix ：大括号边框
• vmatrix ：单竖线边框
• Vmatrix ：双竖线边框

$$
A =
\begin{matrix}
a & b\\
c & d
\end{matrix}
$$

$$A=\begin{array}{cc}a& b\\ c& d\end{array}$$

$$
B =
\begin{pmatrix}
a & b\\
c & d
\end{pmatrix}
$$

$$B=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$$

$$
B =
\begin{bmatrix}
a & b\\
c & d
\end{bmatrix}
$$

$$B=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$$

$$
B =
\begin{Bmatrix}
a & b\\
c & d
\end{Bmatrix}
$$

$$B=\left\{\begin{array}{cc}a& b\\ c& d\end{array}\right\}$$

$$
C =
\begin{vmatrix}
a & b\\
c & d
\end{vmatrix}
$$

$$C=\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|$$

$$
B =
\begin{Vmatrix}
a & b\\
c & d
\end{Vmatrix}
$$

$$B=\Vert \begin{array}{cc}a& b\\ c& d\end{array}\Vert$$

$$
[A\ b] = 
\begin{bmatrix}
\begin{array}{c c c|c}
a_{11} & a_{12} & a_{13} & b_1\\
a_{21} & a_{22} & a_{23} & b_2\\
a_{31} & a_{32} & a_{33} & b_3\\
\end{array}
\end{bmatrix}
$$

$$[Ab]=\left[\begin{array}{c}\begin{array}{cccc}{a}_{11}& a_{12}& a_{13}& b_1\\ a_{21}& a_{22}& a_{23}& b_2\\ a_{31}& a_{32}& a_{33}& b_3\end{array}\end{array}\right]$$

$$
\begin{array}{c:c:c}
a & b & c \\
\hline
d & e & f \\
\hdashline
g & h & i
\end{array}
$$

$$\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}$$

$$
L_{n\times n} =
\begin{bmatrix}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots &\ddots & \vdots \\
a_{n1} & a_{n2} & \cdots & a_{nn} \\
\end{bmatrix}
$$

$$L_{n\times n}=\left[\begin{array}{cccc}{a}_{11}& a_{12}& \cdots & a_{1n}\\ a_{21}& a_{22}& \cdots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{n1}& a_{n2}& \cdots & a_{nn}\end{array}\right]$$

2 修饰符号

2.1 简单的帽子

$$
\hat{\theta}
\widehat{AB}
\\
\bar{y}
\overline{AB}
\\
\tilde{a}
\widetilde{ac}
\\
\bar{a}
\acute{a}
\check{a}
\grave{a}
\\
\dot{a}
\ddot{a} \\
\vec a	\\
\vec {ab} \\
\overrightarrow{xy}
$$

$$\begin{gathered} \hat{\theta }\widehat{AB} \\ \bar{y}\overline{AB} \\ \tilde{a}\widetilde{ac} \\ \bar{a}\acute{a}\check{a}\grave{a} \\ \dot{a}\ddot{a} \\ \vec{a} \\ \overrightarrow{ab} \\ \overrightarrow{xy} \end{gathered}$$

2.2 帽子和袜子

$$
\overleftarrow{AB}
\overrightarrow{AB}
\overleftrightarrow{AB}
\\
\underleftarrow{AB}
\underrightarrow{AB}
\underleftrightarrow{AB}
\\
\overbrace{AB}
\underbrace{AB}
\\
\overline{AB}
\underline{AB}
$$

$$\begin{gathered} \overleftarrow{AB}\overrightarrow{AB}\overleftrightarrow{AB} \\ \underleftarrow{AB}\underrightarrow{AB}\underleftrightarrow{AB} \\ \overbrace{AB}\underbrace{AB} \\ \overline{AB}\underline{AB} \end{gathered}$$

2.3 盒子和帽子

$$
\overbrace{a+b+c}^{\text{note}}
\\
\underbrace{a+b+c}_{\text{note}}
\\
\boxed{\pi=3.14}
\\
\overbrace{a+\underbrace{b+c}_{1.0}+d}^{2.0}
$$

$$\begin{gathered} \stackrel{\text{note}}{\overbrace{a+b+c}} \\ \underset{\text{note}}{\underbrace{a+b+c}} \\ \boxed{\pi =3.14} \\ \stackrel{2.0}{\overbrace{a+\underset{1.0}{\underbrace{b+c}}+d}} \end{gathered}$$

2.4 各种括号

$$
(x+y) \\
[x+y] \\
\{ x+y \} \\
\langle x+y \rangle \\
|x+y| \\
\| x+y \|
$$

( x + y ) [ x + y ] { x + y } ⟨ x + y ⟩ ∣ x + y ∣ ∥ x + y ∥ (x+y) \\ [x+y] \\ \{ x+y \} \\ \langle x+y \rangle \\ |x+y| \\ \| x+y \| (x+y)[x+y]{x+y}⟨x+y⟩∣x+y∣∥x+y∥

$$
( \big( \Big( \bigg( \Bigg( \\
) \big) \Big) \bigg) \Bigg) \\
[ \big[ \Big[ \bigg[ \Bigg[ \\
] \big] \Big] \bigg] \Bigg] \\
\{ \big\{ \Big\{ \bigg\{ \Bigg\{ \\
\} \big\} \Big\} \bigg\} \Bigg\} \\
| \big| \Big| \bigg| \Bigg| \\
\| \big\| \Big\| \bigg\| \Bigg\| \\
\langle \big\langle \Big\langle \bigg\langle \Bigg\langle \\
\rangle \big\rangle \Big\rangle \bigg\rangle \Bigg\rangle \\
\lceil \big\lceil \Big\lceil \bigg\lceil \Bigg\lceil \\
\rceil \big\rceil \Big\rceil \bigg\rceil \Bigg\rceil \\
\lfloor \big\lfloor \Big\lfloor \bigg\lfloor \Bigg\lfloor \\
\rfloor \big\rfloor \Big\rfloor \bigg\rfloor \Bigg\rfloor \\
$$

( ( ( ( ( ) ) ) ) ) [ [ [ [ [ ] ] ] ] ] { { { { { } } } } } ∣ ∣ ∣ ∣ ∣ ∥ ∥ ∥ ∥ ∥ ⟨ ⟨ ⟨ ⟨ ⟨ ⟩ ⟩ ⟩ ⟩ ⟩ ⌈ ⌈ ⌈ ⌈ ⌈ ⌉ ⌉ ⌉ ⌉ ⌉ ⌊ ⌊ ⌊ ⌊ ⌊ ⌋ ⌋ ⌋ ⌋ ⌋ ( \big( \Big( \bigg( \Bigg( \\ ) \big) \Big) \bigg) \Bigg) \\ [ \big[ \Big[ \bigg[ \Bigg[ \\ ] \big] \Big] \bigg] \Bigg] \\ \{ \big\{ \Big\{ \bigg\{ \Bigg\{ \\ \} \big\} \Big\} \bigg\} \Bigg\} \\ | \big| \Big| \bigg| \Bigg| \\ \| \big\| \Big\| \bigg\| \Bigg\| \\ \langle \big\langle \Big\langle \bigg\langle \Bigg\langle \\ \rangle \big\rangle \Big\rangle \bigg\rangle \Bigg\rangle \\ \lceil \big\lceil \Big\lceil \bigg\lceil \Bigg\lceil \\ \rceil \big\rceil \Big\rceil \bigg\rceil \Bigg\rceil \\ \lfloor \big\lfloor \Big\lfloor \bigg\lfloor \Bigg\lfloor \\ \rfloor \big\rfloor \Big\rfloor \bigg\rfloor \Bigg\rfloor \\ ((((()))))[[[[[]]]]]{{{{{}}}}}∣ ​ ​ ​ ​∥ ​ ​ ​ ​⟨⟨⟨⟨⟨⟩⟩⟩⟩⟩⌈⌈⌈⌈⌈⌉⌉⌉⌉⌉⌊⌊⌊⌊⌊⌋⌋⌋⌋⌋

$$
[]
<>
|-2|
\{\}
$$

[ ] < > ∣ − 2 ∣ { } [] <> |-2| \{\} []<>∣−2∣{}

$$
\lgroup x \rgroup \\
\lVert a \rVert \\
\lceil 2.6 \rceil \\
\lfloor 1.2 \rfloor \\
$$

$$\begin{gathered} ⟮x⟯ \\ \Vert a\Vert \\ \lceil 2.6\rceil \\ \lfloor 1.2\rfloor \end{gathered}$$

$$
\ulcorner
\urcorner
\llcorner
\lrcorner
$$

$$┌┐└┘$$

$$
\langle a+b \rangle \\
\lceil a+b \rceil \\
\lfloor a+b \rfloor \\
\lbrace a+b \rbrace \\
\overline{a+b+c+d} \\
\underline{a+b+c+d}
$$

⟨ a + b ⟩ ⌈ a + b ⌉ ⌊ a + b ⌋ { a + b } a + b + c + d ‾ a + b + c + d ‾ \langle a+b \rangle \\ \lceil a+b \rceil \\ \lfloor a+b \rfloor \\ \lbrace a+b \rbrace \\ \overline{a+b+c+d} \\ \underline{a+b+c+d} ⟨a+b⟩⌈a+b⌉⌊a+b⌋{a+b}a+b+c+d​a+b+c+d​

3 特殊字符

3.1 希腊字母

3.2 常见图形

$$
\Box
\square
\blacksquare
\triangle
\triangledown
\blacktriangle
\diamond
\Diamond
\star
\bigstar
\circ
\bullet
\bigcirc
\bigodot
$$

$$\square \square ■△▽▲\diamond ◊\star ★\circ \bullet ◯⨀$$

$$
\diamondsuit
\clubsuit
\heartsuit
\spadesuit
$$

$$♢♣♡♠$$

$$
\angle
\measuredangle
\top
\bot
\infty
$$

$$\angle \measuredangle \top \perp \infty$$

$$
\checkmark
\dagger
\ddagger
\yen
\$
$$

$$✓†‡\yen \$$$

3.3 其他符号

$$
\hbar \\
\imath \\
\jmath \\
\ell \\
\Re \\
\Im \\
\aleph \\
\wp \\
\mho \\
\partial \\
\prime \\
\infty \\
\nabla \\
\triangle \\
\bot \\
\top \\
\angle \\
\surd \\
\flat \\
\natural \\
\sharp \\
$$

$$\begin{gathered} \hslash \\ ı \\ ȷ \\ \ell \\ \Re \\ \Im \\ \aleph \\ \wp \\ \mho \\ \partial \\ \prime \\ \infty \\ \nabla \\ △ \\ \perp \\ \top \\ \angle \\ \surd \\ ♭ \\ ♮ \\ ♯ \end{gathered}$$

$$
\hat{a} \\
\check{a} \\
\tilde{a} \\
\grave{a} \\
\dot{a} \\
\ddot{a} \\
\bar{a} \\
\vec{a} \\
\widehat{a} \\
\acute{a} \\
\breve{a} \\
\widetilde{a} \\
$$

$$\begin{gathered} \hat{a} \\ \check{a} \\ \tilde{a} \\ \grave{a} \\ \dot{a} \\ \ddot{a} \\ \bar{a} \\ \vec{a} \\ \hat{a} \\ \acute{a} \\ \breve{a} \\ \tilde{a} \end{gathered}$$

$$
\mathbb{ABCDEFGHIJKLMNOPQRSTUVWXYZ} \\
\mathbb{abcdefghijklmnopqrstuvwxyz} \\
\mathcal{ABCDEFGHIJKLMNOPQRSTUVWXYZ} \\
\mathcal{abcdefghijklmnopqrstuvwxyz} \\
\mathfrak{ABCDEFGHIJKLMNOPQRSTUVWXYZ} \\
\mathfrak{abcdefghijklmnopqrstuvwxyz} \\
$$

$$\begin{gathered} \mathbb{ABCDEFGHIJKLMNOPQRSTUVWXYZ} \\ abcdefghij\mathbb{k}lmnopqrstuvwxyz \\ \mathcal{A}\mathcal{BC}\mathcal{D}\mathcal{EFGH}\mathcal{I}\mathcal{J}\mathcal{K}\mathcal{LM}\mathcal{N}\mathcal{O}\mathcal{P}\mathcal{Q}\mathcal{R}\mathcal{S}\mathcal{T}\mathcal{U}\mathcal{V}\mathcal{W}\mathcal{X}\mathcal{Y}\mathcal{Z} \\ abcdefghijklmnopqrstuvwxyz \\ \mathfrak{ABCDEFGHIJKLMNOPQRSTUVWXYZ} \\ \mathfrak{abcdefghijklmnopqrstuvwxyz} \end{gathered}$$

4 运算符

4.1 算术运算符

$$
\times
/
\div
\cdot
\#
\%
$$

$$\times /÷\cdot \#\%+$$

$$
\circ
\ast
\star
\otimes
\oplus
\odot
$$

$$\circ \ast \star \otimes \oplus \odot$$

$$
\pm
\mp
\dotplus
\divideontimes
$$

$$\pm \mp \dotplus \divideontimes$$

4.2 比较运算符

$$
=
= \not
\equiv
\approx
\approxeq
\cong
\sim
\neq
\not=
$$

$$=/\equiv \approx \approxeq \cong \sim \ne \ne$$

$$
<
>
\le
\ge
\gg
\ll
$$

$$<\le \ge \gg \ll$$

$$
\curlyeqprec
\curlyeqsucc
\prec
\succ
\preceq
\succeq
$$

$$⋞⋟\prec \succ ⪯⪰$$

$$
x \leq y \\
x \geq y \\
x \nleq y \\
x \not \leq y \\
x \ngeq y \\
x \not \geq y \\
x \neq y \\
x \approx y \\
x \equiv y
$$

$$\begin{gathered} x\le y \\ x\ge y \\ x\nleqq y \\ x\nleqq y \\ x\ngeqq y \\ x\ngeqq y \\ x\ne y \\ x\approx y \\ x\equiv y \end{gathered}$$

4.3 集合运算符

$$
\in
\owns \not
\subset \not
\supset
\subseteq
\supseteq
\\
\cap
\cup
\land
\lor
\\
\neg
\emptyset
\varnothing
\\
\because
\forall
\exists
\therefore
$$

$$\begin{gathered} \in \ni \not\subset /\supset \subseteq \supseteq \\ \cap \cup \wedge \vee \\ \neg \varnothing \varnothing \\ \because \forall \exists \therefore \end{gathered}$$

$$
\cap
\cup
\land
\lor
\sqcup
\sqcap
$$

$$\cap \cup \wedge \vee \bigsqcup \sqcap$$

5 箭头

$$
\gets
\leftarrow
\to
\rightarrow
\leftrightarrow
\\
\uparrow
\downarrow
\updownarrow
$$

$$\begin{gathered} \leftarrow \leftarrow \to \to \leftrightarrow \\ \uparrow \downarrow \updownarrow \end{gathered}$$

$$
\Leftarrow
\Rightarrow
\Leftrightarrow
\iff
\\
\Uparrow
\Downarrow
\Updownarrow
$$

$$\begin{gathered} \Leftarrow \Rightarrow \iff ⟺ \\ \Uparrow \Downarrow \Updownarrow \end{gathered}$$

$$
\nearrow
\searrow
\swarrow
\nwarrow
$$

$$\nearrow \searrow \swarrow \nwarrow$$

$$
\longleftarrow
\longrightarrow
\longleftrightarrow
\Longleftarrow
\Longrightarrow
\Longleftrightarrow
\longmapsto
$$

$$⟵⟶⟷⟸⟹⟺⟼$$

$$
\xrightarrow{over}
\xrightarrow[over]{}
\xrightarrow[under]{over}
\xleftarrow[]{over}
\xleftarrow[under]{}
\xleftarrow[under]{over}
$$

$$\stackrel{over}{\to }\underset{over}{\to }\underset{under}{\overset{over}{\to }}\stackrel{over}{\leftarrow }\underset{under}{\leftarrow }\underset{under}{\overset{over}{\leftarrow }}$$

6 其他

6.1 空间间距

$$
A\!B
\\
AB
\\
A\thinspace B
\\
A\:B
\\
A\ B
\\
A \enspace B
\\
A\quad B
\\
A\qquad B
$$

$$\begin{gathered} AB \\ AB \\ AB \\ AB \\ AB \\ AB \\ AB \\ AB \end{gathered}$$

6.2 字体颜色和大小

$$
\textcolor{blue}{F=ma}
\\
\textcolor{#00ff00}{F=ma}
\\
\textcolor{#ff0000}{F=ma}
\\
\color{blue} one\ line
\\
nothing
$$

$$\begin{gathered} F=ma \\ F=ma \\ F=ma \\ oneline \\ nothing \end{gathered}$$

$$
\colorbox{#00ff00}{F=ma}
\\
\colorbox{aqua}{A}
\\
\fcolorbox{red}{aqua}{A}
$$

$$\begin{gathered} F=ma \\ A \\ \boxed{A} \end{gathered}$$

$$
AB
\Huge AB
\huge AB
\\
AB
\LARGE AB
\Large AB
\large AB
\\
AB
\small AB
\tiny AB
$$

$$\begin{gathered} ABABAB \\ ABABABAB \\ ABABAB \end{gathered}$$

6.3 划掉

$$
\cancel{5}
\bcancel{5}
\xcancel{ABC}
\not =
$$

$$\cancel{5}\bcancel{5}\xcancel{ABC}\ne$$

6.4 省略号

• 横省略号：\cdots
• 竖省略号：\vdots
• 斜省略号：\ddots
• 底省略号: \ldots 效果显示为 $1,2,\dots ,n$

$$
\begin{bmatrix}
{a_{11}}&{a_{12}}&{\cdots}&{a_{1n}} \\
{a_{21}}&{a_{22}}&{\cdots}&{a_{2n}} \\
{\vdots}&{\vdots}&{\ddots}&{\vdots} \\
{a_{m1}}&{a_{m2}}&{\cdots}&{a_{mn}} \\
\end{bmatrix}
$$

$$\left[\begin{array}{cccc}{a}_{11}& a_{12}& \cdots & a_{1n}\\ a_{21}& a_{22}& \cdots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{m1}& a_{m2}& \cdots & a_{mn}\end{array}\right]$$

6.4 对齐

$$
\begin{multline*}
p(x) = 3x^6 + 14x^5y + 590x^4y^2 + 19x^3y^3 \\
-12x^2y^4 - 12xy^5 + 2y^6 - a^3b^3
\end{multline*}
$$

KaTeX parse error: No such environment: multline* at position 8: \begin{̲m̲u̲l̲t̲l̲i̲n̲e̲*̲}̲ p(x) = 3x^6 + …

$$
\begin{align*}
p(x) = 3x^6 + 14x^5y + 590x^4y^2 + 19x^3y^3 \\
-12x^2y^4 - 12xy^5 + 2y^6 - a^3b^3
\end{align*}
$$

$$\begin{array}{r}p(x)=3x^6+14x^{5}y+590x^4{y}^2+19x^3{y}^3\\ -12x^2{y}^4-12x{y}^5+2y^6-a^3{b}^3\end{array}$$

7 宏

$$
\def\Normal#1#2#3
{
\frac{1}{\sqrt{2 \pi} #3} \exp{\left[ -\frac{(#1-#2)^2}{2 #3^2} \right]}
}

f(x) = \Normal(x)(u_1)(\sigma_1) \\
f(y) = \Normal(y)(u_2)(\sigma_2) \\
f(z) = \Normal(z)(u_3)(\sigma_3) \\
$$

$$\begin{gathered} f(x)=\frac{1}{\sqrt{2\pi })(u_1)({\sigma }_1)}\exp \left[-\frac{((-x{)}^2}{2)(u_1)({\sigma }_1{)}^2}\right] \\ f(y)=\frac{1}{\sqrt{2\pi })(u_2)({\sigma }_2)}\exp \left[-\frac{((-y{)}^2}{2)(u_2)({\sigma }_2{)}^2}\right] \\ f(z)=\frac{1}{\sqrt{2\pi })(u_3)({\sigma }_3)}\exp \left[-\frac{((-z{)}^2}{2)(u_3)({\sigma }_3{)}^2}\right] \end{gathered}$$

$$
\def
\EXP
{
e^x = 1 + x + \frac{1}{2!}x^2 + \frac{1}{3!}x^3  + \cdots
}
\EXP
$$

$$e^x=1+x+\frac{1}{2!}{x}^2+\frac{1}{3!}{x}^3+\cdots$$