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
feature: false
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)