title: Markdown 数学公式使用总结
date: 2023-02-15 10:08:12
tags:
- markdown
categories: - 其他实践
cover: https://cover.png
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文章目录
1. 语法
1、行内公式
用 $...$
表示行内公式,如 O_{1 - \left(\vert a \vert \over 2 \right)}
: O 1 − ( ∣ a ∣ 2 ) O_{1 - \left(\vert a \vert \over 2 \right)} O1−(2∣a∣)
2、多行公式
用 $$...$$
表示多行公式,如 \sum_{i=0}^\infty x^2 = x_1 + x_2 + x_3 + ... + x_n
∑ i = 0 ∞ x 2 = x 1 + x 2 + x 3 + . . . + x n \sum_{i=0}^\infty x^2 = x_1 + x_2 + x_3 + ... + x_n i=0∑∞x2=x1+x2+x3+...+xn
2. 上标、下标、分组
上标和下标
- 上标:
^
,如x^y
: x y x^y xy - 下标:
_
,如a_i
: a i a_i ai
分组
符号:{}
,例:C_n^2
: C n 2 C_n^2 Cn2、C_{n^2}
: C n 2 C_{n^2} Cn2
3. 希腊字母
公式 | 小写 | 公式 | 大写 |
---|---|---|---|
\alpha |
α \alpha α | \Alpha |
A \Alpha A |
\beta |
β \beta β | \Beta |
B \Beta B |
\delta |
δ \delta δ | \Delta |
Δ \Delta Δ |
\epsilon |
ϵ \epsilon ϵ | \Epsilon |
E \Epsilon E |
\zeta |
ζ \zeta ζ | \Zeta |
Z \Zeta Z |
\eta |
η \eta η | \Eta |
H \Eta H |
\theta |
θ \theta θ | \Theta |
Θ \Theta Θ |
\iota |
ι \iota ι | \Iota |
I \Iota I |
\kappa |
κ \kappa κ | \Kappa |
K \Kappa K |
\lambda |
λ \lambda λ | \Lambda |
Λ \Lambda Λ |
\mu |
μ \mu μ | \Mu |
M \Mu M |
\nu |
ν \nu ν | \Nu |
N \Nu N |
\xi |
ξ \xi ξ | \Xi |
Ξ \Xi Ξ |
\pi |
π \pi π | \Pi |
Π \Pi Π |
\rho |
ρ \rho ρ | \Rho |
P \Rho P |
\sigma |
σ \sigma σ | \Sigma |
Σ \Sigma Σ |
\tau |
κ \kappa κ | \Tau |
T \Tau T |
\phi |
ϕ \phi ϕ | \Phi |
Φ \Phi Φ |
\psi |
ψ \psi ψ | \Psi |
Ψ \Psi Ψ |
\omega |
ω \omega ω | \omega |
Ω \Omega Ω |
\omicron |
ο \omicron ο | \Omicron |
O \Omicron O |
\gamma |
γ \gamma γ | \Gamma |
Γ \Gamma Γ |
斜体字母加上前缀 var
,\varDelta
: Δ \varDelta Δ
4. 算数运算
公式 | 符号 |
---|---|
\times |
× \times × |
\div |
÷ \div ÷ |
\cdot |
⋅ \cdot ⋅ |
< |
< < < |
> |
> > > |
\ll |
≪ \ll ≪ |
\gg |
≫ \gg ≫ |
\lll |
⋘ \lll ⋘ |
\pm |
± \pm ± |
\le 、\leq |
≤ \le ≤ 、 ≤ \leq ≤ |
\ge 、\geq |
≥ \ge ≥ 、 ≥ \geq ≥ |
\mp |
∓ \mp ∓ |
\leqq |
≦ \leqq ≦ |
\geqq |
≧ \geqq ≧ |
\neq |
≠ \neq = |
\leqslant |
⩽ \leqslant ⩽ |
\geqslant |
⩾ \geqslant ⩾ |
\approx |
≈ \approx ≈ |
5. 集合
公式 | 符号 |
---|---|
\complement |
∁ \complement ∁ |
\in |
∈ \in ∈ |
\notin |
∉ \notin ∈/ |
/subset |
⊂ \subset ⊂ |
\subseteq |
⊆ \subseteq ⊆ |
\subsetneq |
⊊ \subsetneq ⊊ |
\cap |
∩ \cap ∩ |
\cup |
∪ \cup ∪ |
\varnothing |
∅ \varnothing ∅ |
\emptyset |
∅ \emptyset ∅ |
6. 逻辑运算
公式 | 符号 |
---|---|
\land 、\wedge |
∧ \land ∧ 、 ∧ \wedge ∧ |
\lor 、\vee |
∨ \lor ∨ 、 ∨ \vee ∨ |
\lnot 、\neg |
¬ \lnot ¬ 、 ¬ \neg ¬ |
\forall |
∀ \forall ∀ |
\exists |
∃ \exists ∃ |
\bot |
⊥ \bot ⊥ |
/top |
⊤ \top ⊤ |
\vdash |
⊢ \vdash ⊢ |
\Vdash |
⊩ \Vdash ⊩ |
\vDash |
⊨ \vDash ⊨ |
\models |
⊨ \models ⊨ |
7.括号
公式 | 符号 |
---|---|
(x) |
( x ) (x) (x) |
\Bigg(x \Bigg) |
( x ) \Bigg(x \Bigg) (x) |
\bigg(x \bigg) |
( x ) \bigg(x \bigg) (x) |
\Big(x \Big) |
( x ) \Big( x \Big) (x) |
\big(x \big) |
( x ) \big(x \big) (x) |
\{x \} 、\lbrace x \rbrace |
{ x } \{x \} { x} 、 { x } \lbrace x \rbrace { x} |
[x] |
[ x ] [x] [x] |
\vert x \vert |
∣ x ∣ \vert x \vert ∣x∣ |
\Vert x \Vert |
∥ x ∥ \Vert x \Vert ∥x∥ |
\langle x \rangle |
⟨ x ⟩ \langle x \rangle ⟨x⟩ |
\lceil x \rceil |
⌈ x ⌉ \lceil x \rceil ⌈x⌉ |
\lfloor x \rfloor |
⌊ x ⌋ \lfloor x \rfloor ⌊x⌋ |
二项式系数:\dbinom nr |
( n r ) \dbinom nr (rn) |
\binom nr 、n \choose r |
( n r ) \binom nr (rn) 、 ( n r ) n \choose r (rn) |
8. 空格
公式 | 符号 |
---|---|
无视空格:a b |
a b a b ab |
a \qquad b |
a b a \qquad b ab |
a \quad b |
a b a \quad b ab |
a \ b |
a b a \ b a b |
a \, b |
a b a \, b ab |
a \; b |
a b a \; b ab |
a \! b |
a b a \! b ab |
9. 求和、积分、微分
公式 | 符号 |
---|---|
\sum_1^n |
∑ 1 n \sum_1^n ∑1n |
\sum_{i=0}^\infty x^2 |
∑ i = 0 ∞ x 2 \sum_{i=0}^\infty x^2 ∑i=0∞x2 |
\int |
∫ \int ∫ |
\iint |
∬ \iint ∬ |
\iiint |
∭ \iiint ∭ |
\oint |
∮ \oint ∮ |
\prod |
∏ \prod ∏ |
\coprod |
∐ \coprod ∐ |
\bigcap |
⋂ \bigcap ⋂ |
\bigcup |
⋃ \bigcup ⋃ |
\bigvee |
⋁ \bigvee ⋁ |
\bigwedge |
⋀ \bigwedge ⋀ |
\infty |
∞ \infty ∞ |
\nabla |
∇ \nabla ∇ |
\partial x |
∂ x \partial x ∂x |
\mathrm{d} x |
d x \mathrm{d} x dx |
\dot x |
x ˙ \dot x x˙ |
\ddot x |
x ¨ \ddot x x¨ |
10. 分式、根号、函数
公式 | 符号 |
---|---|
\frac 12 |
1 2 \frac 12 21 |
强制分式为显示模式:\dfrac 12 |
1 2 \dfrac 12 21 |
强制分式为文本模式:\tfrac 12 |
1 2 \tfrac 12 21 |
用于连续分式:\cfrac 12 |
1 2 \cfrac 12 21 |
\frac {x^2}{1+x} |
x 2 1 + x \frac {x^2}{1+x} 1+xx2 |
用于复杂分式:x^2 \over 1+x |
x 2 1 + x x^2 \over 1+x 1+xx2 |
\sqrt {x^3} |
x 3 \sqrt {x^3} x3 |
\sqrt[3] {\frac xy} |
x y 3 \sqrt[3] {\frac xy} 3yx |
\sin x |
sin x \sin x sinx |
\lim_{x \to 0} |
lim x → 0 \lim_{x \to 0} limx→0 |
\varinjlim |
l i m → \varinjlim lim |
\varprojlim |
l i m ← \varprojlim lim |
\varliminf |
l i m ‾ \varliminf lim |
\varlimsup |
l i m ‾ \varlimsup lim |
11. 箭头符号
公式 | 符号 |
---|---|
\rightarrow |
→ \rightarrow → |
\leftarrow |
← \leftarrow ← |
\Rightarrow |
⇒ \Rightarrow ⇒ |
\Leftarrow |
⇐ \Leftarrow ⇐ |
\longrightarrow |
⟶ \longrightarrow ⟶ |
\longleftarrow |
⟵ \longleftarrow ⟵ |
\Leftrightarrow |
⇔ \Leftrightarrow ⇔ |
\leftrightarrow |
↔ \leftrightarrow ↔ |
\Longleftrightarrow |
⟺ \Longleftrightarrow ⟺ |
\longleftrightarrow |
⟷ \longleftrightarrow ⟷ |
\xrightarrow[y>0] {x+y} |
→ y > 0 x + y \xrightarrow[y>0] {x+y} x+yy>0 |
a \to b |
a → b a \to b a→b |
a \gets b |
a ← b a \gets b a←b |
a \implies b |
a ⟹ b a \implies b a⟹b |
a \impliedby b |
a ⟸ b a \impliedby b a⟸b |
\lim_{x \to 0} |
lim x → 0 \lim_{x \to 0} limx→0 |
\longmapsto |
⟼ \longmapsto ⟼ |
\hookleftarrow |
↩ \hookleftarrow ↩ |
12. 几何和向量
公式 | 符号 |
---|---|
\triangle |
△ \triangle △ |
\Diamond |
◊ \Diamond ◊ |
\Box |
□ \Box □ |
\odot |
⊙ \odot ⊙ |
\angle ABC |
∠ A B C \angle ABC ∠ABC |
30^\circ |
3 0 ∘ 30^\circ 30∘ |
\perp |
⊥ \perp ⊥ |
\sim |
∼ \sim ∼ |
\cong |
≅ \cong ≅ |
\\hat{a} |
a ^ \hat{a} a^ |
\vec{a} |
a ⃗ \vec{a} a |
\overrightarrow{AB} |
A B → \overrightarrow{AB} AB |
\overleftarrow{AB} |
A B ← \overleftarrow{AB} AB |
\overleftrightarrow{AB} |
A B ↔ \overleftrightarrow{AB} AB |
\widehat{e f g} |
e f g ^ \widehat{e f g} efg |
13. 上、下划线和上、下括号
公式 | 符号 |
---|---|
\overline{h i j} |
h i j ‾ \overline{h i j} hij |
\underline{h i j} |
h i j ‾ \underline{h i j} hij |
\underbrace{a+b+\cdots+z}_{10} |
a + b + ⋯ + z ⏟ 10 \underbrace{a+b+\cdots+z}_{10} 10 a+b+⋯+z |
\overbrace{a+b+\cdots+z}_{10} |
a + b + ⋯ + z ⏞ 10 \overbrace{a+b+\cdots+z}_{10} a+b+⋯+z 10 |
13. 特殊字符
公式 | 符号 |
---|---|
\eth |
ð \eth ð |
\S |
§ \S § |
\% |
% \% % |
/dagger |
† \dagger † |
\ddagger |
‡ \ddagger ‡ |
\ast 、* |
∗ \ast ∗ 、 ∗ * ∗ |
\circ |
∘ \circ ∘ |
\bullet |
∙ \bullet ∙ |
\ldots |
… \ldots … |
\cdots |
⋯ \cdots ⋯ |
\vdots |
⋮ \vdots ⋮ |
\ddots |
⋱ \ddots ⋱ |
\smile |
⌣ \smile ⌣ |
\frown |
⌢ \frown ⌢ |
\wr |
≀ \wr ≀ |
\oplus |
⊕ \oplus ⊕ |
\bigoplus |
⨁ \bigoplus ⨁ |
\otimes |
⊗ \otimes ⊗ |
\bigotimes |
⨂ \bigotimes ⨂ |
\bigodot |
⨀ \bigodot ⨀ |
\boxtimes |
⊠ \boxtimes ⊠ |
\boxplus |
⊞ \boxplus ⊞ |
\triangleleft |
◃ \triangleleft ◃ |
\triangleright |
▹ \triangleright ▹ |
\imath |
ı \imath |
\hbar |
ℏ \hbar ℏ |
\ell |
ℓ \ell ℓ |
\mho |
℧ \mho ℧ |
\Finv |
Ⅎ \Finv Ⅎ |
\Re |
ℜ \Re ℜ |
\Im |
ℑ \Im ℑ |
\wp |
℘ \wp ℘ |
\diamondsuit |
♢ \diamondsuit ♢ |
\heartsuit |
♡ \heartsuit ♡ |
\clubsuit |
♣ \clubsuit ♣ |
\spadesuit |
♠ \spadesuit ♠ |
\Game |
⅁ \Game ⅁ |
\flat |
♭ \flat ♭ |
\natural |
♮ \natural ♮ |
\sharp |
♯ \sharp ♯ |
14. 阵列、矩阵、等式对齐、分段函数、编号
1、阵列
语法: $$\begin{array} … \end{array}$$
,r
右对齐,l
左对齐,c
居中,|
垂直线,\hline
横线,\\
换行,元素之间以 &
间隔
$$
\begin{array} {c|lcr}
n & \text{Left} & \text{Center} & \text{Right} \\
\hline
1 & 0.24 & 1 & 125 \\
2 & -1 & 189 & -8 \\
3 & -20 & 2000 & 1+10i
\end{array}
$$
n Left Center Right 1 0.24 1 125 2 − 1 189 − 8 3 − 20 2000 1 + 10 i \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} n123Left0.24−1−20Center11892000Right125−81+10i
2、矩阵
语法: $$\begin{matrix} … \end{matrix}$$
,每行以 \\
结尾,元素之间以 &
间隔
$$
\begin{matrix}
1 & x & x^2 \\
1 & y & y^2 \\
1 & z & z^2 \\
\end{matrix}
$$
1 x x 2 1 y y 2 1 z z 2 \begin{matrix} 1 & x & x^2 \\ 1 & y & y^2 \\ 1 & z & z^2 \\ \end{matrix} 111xyzx2y2z2
添加括号:
pmatrix
:()bmatrix
:[ ]Bmatrix
:{ }vmatrix
:| |Vmatrix
:‖ ‖
添加省略号:
$$
\begin{pmatrix}
1 & a_1^2 & a_1^2 & \cdots & a_1^2 \\
1 & a_2^2 & a_2^2 & \cdots & a_2^2 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
1 & a_n^2 & a_n^2 & \cdots & a_n^2 \\
\end{pmatrix}
$$
( 1 a 1 2 a 1 2 ⋯ a 1 2 1 a 2 2 a 2 2 ⋯ a 2 2 ⋮ ⋮ ⋮ ⋱ ⋮ 1 a n 2 a n 2 ⋯ a n 2 ) \begin{pmatrix} 1 & a_1^2 & a_1^2 & \cdots & a_1^2 \\ 1 & a_2^2 & a_2^2 & \cdots & a_2^2 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & a_n^2 & a_n^2 & \cdots & a_n^2 \\ \end{pmatrix} 11⋮1a12a22⋮an2a12a22⋮an2⋯⋯⋱⋯a12a22⋮an2
水平增广矩阵,使用阵列语法:
$$
\left[
\begin{array}{cc|c}
1&2&3\\
4&5&6
\end{array}
\right]
$$
[ 1 2 3 4 5 6 ] \left[ \begin{array}{cc|c} 1&2&3\\ 4&5&6 \end{array} \right] [142536]
垂直增广矩阵:
$$
\begin{pmatrix}
a & b\\
c & d\\
\hline
1 & 0\\
0 & 1
\end{pmatrix}
$$
( a b c d 1 0 0 1 ) \begin{pmatrix} a & b\\ c & d\\ \hline 1 & 0\\ 0 & 1 \end{pmatrix} ac10bd01
3、等式对齐
语法: \begin{align} … \end{align}
,每行以 \\
结尾,元素之间以 &
间隔
$$
\begin{align}
\sqrt{37} & = \sqrt{\frac{73^2-1}{12^2}} \\
& = \sqrt{\frac{73^2}{12^2}\cdot\frac{73^2-1}{73^2}} \\
& = \sqrt{\frac{73^2}{12^2}}\sqrt{\frac{73^2-1}{73^2}} \\
& = \frac{73}{12}\sqrt{1 - \frac{1}{73^2}} \\
& \approx \frac{73}{12}\left(1 - \frac{1}{2\cdot73^2}\right)
\end{align}
$$
37 = 7 3 2 − 1 1 2 2 = 7 3 2 1 2 2 ⋅ 7 3 2 − 1 7 3 2 = 7 3 2 1 2 2 7 3 2 − 1 7 3 2 = 73 12 1 − 1 7 3 2 ≈ 73 12 ( 1 − 1 2 ⋅ 7 3 2 ) \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} 37=122732−1=122732⋅732732−1=122732732732−1=12731−7321≈1273(1−2⋅7321)
4、分段函数
语法: \begin{cases} … \end{cases}
,每行以 \\
结尾,元素之间以 &
间隔
$$
f(n) =
\begin{cases}
n/2, & \text{if $n$ is even} \\
3n+1, & \text{if $n$ is odd}
\end{cases}
$$
f ( n ) = { n / 2 , if n is even 3 n + 1 , if n is odd f(n) = \begin{cases} n/2, & \text{if $n$ is even} \\ 3n+1, & \text{if $n$ is odd} \end{cases} f(n)={ n/2,3n+1,if n is evenif n is odd
$$
\left.
\begin{array}{l}
\text{if $n$ is even:}&n/2\\
\text{if $n$ is odd:}&3n+1
\end{array}
\right\}
=f(n)
$$
if n is even: n / 2 if n is odd: 3 n + 1 } = f ( n ) \left. \begin{array}{l} \text{if $n$ is even:}&n/2\\ \text{if $n$ is odd:}&3n+1 \end{array} \right\} =f(n) if n is even:if n is odd:n/23n+1}=f(n)
5、编号
语法:\tag
$$
y=x^2 \tag{1.5a}
$$
y = x 2 (1.5a) y=x^2 \tag{1.5a} y=x2(1.5a)