国足踢进世界杯 / 2025-06-18 18:14:05
基本符号
小写希腊字母
注：部分希腊字母在数学公式中常以变量形式出现，例如
ϵ
\epsilon
ϵ在数学中一般写法为
ε
\varepsilon
ε，
ϕ
\phi
ϕ在数学中通常写作
φ
\varphi
φ
符号语法符号语法符号语法
α
\alpha
α\alpha
β
\beta
β\beta
γ
\gamma
γ\gamma
θ
\theta
θ\theta
ε
\varepsilon
ε\varepsilon
δ
\delta
δ\delta
μ
\mu
μ\mu
ν
\nu
ν\nu
η
\eta
η\eta
ζ
\zeta
ζ\zeta
λ
\lambda
λ\lambda
ψ
\psi
ψ\psi
σ
\sigma
σ\sigma
ξ
\xi
ξ\xi
τ
\tau
τ\tau
ϕ
\phi
ϕ\phi
φ
\varphi
φ\varphi
ρ
\rho
ρ\rho
χ
\chi
χ\chi
ω
\omega
ω\omega
π
\pi
π\pi
大写希腊字母
大写希腊字母通常是小写希腊字母的LATEX语法第一个字母改为大写，见下表
符号语法符号语法符号语法
Σ
\Sigma
Σ\Sigma
Π
\Pi
Π\Pi
Δ
\Delta
Δ\Delta
Γ
\Gamma
Γ\Gamma
Ψ
\Psi
Ψ\Psi
Θ
\Theta
Θ\Theta
Λ
\Lambda
Λ\Lambda
Ω
\Omega
Ω\Omega
Φ
\Phi
Φ\Phi
Ξ
\Xi
Ξ\Xi
常用字体
默认的字体为
A
B
C
d
e
f
ABCdef
ABCdef，也就是\mathnormal{ABCdef}（当然，打公式的时候不需要加上这个\mathnormal，直接打字母就是这个效果）
字体语法字体语法
A
B
C
d
e
f
\mathrm{ABCdef}
ABCdef\mathrm{ABCdef}
A
B
C
d
e
f
\mathbf{ABCdef}
ABCdef\mathbf{ABCdef}
A
B
C
d
e
f
\mathit{ABCdef}
ABCdef\mathit{ABCdef}
A
B
C
d
e
f
\pmb{ABCdef}
ABCdef\pmb{ABCdef}
A
B
C
d
e
f
\mathscr{ABCdef}
ABCdef\mathscr{ABCdef}
A
B
C
d
e
f
\mathcal{ABCdef}
ABCdef\mathcal{ABCdef}
A
B
C
d
e
f
\mathfrak{ABCdef}
ABCdef\mathfrak{ABCdef}
A
B
C
d
e
f
\mathbb{ABCdef}
ABCdef\mathbb{ABCdef}
常见运算符
运算符语法运算符语法运算符语法
+
+
++
−
-
−-
×
\times
×\times
±
\pm
±\pm
⋅
\cdot
⋅\cdot
∗
\ast
∗\ast
∪
\cup
∪\cup
∩
\cap
∩\cap
∘
\circ
∘\circ
∨
\lor
∨\lor或\vee
∧
\land
∧\land或\wedge
¬
\lnot
¬\lnot
⊕
\oplus
⊕\oplus
⊖
\ominus
⊖\ominus
⊗
\otimes
⊗\otimes
⊙
\odot
⊙\odot
⊘
\oslash
⊘\oslash
∙
\bullet
∙\bullet
x
\sqrt{x}
x
\sqrt{x}
x
n
\sqrt[n]{x}
nx
\sqrt[n]{x}
大尺寸运算符
运算符语法运算符语法运算符语法
∑
\sum
∑\sum
∏
\prod
∏\prod
∫
\int
∫\int
⋃
\bigcup
⋃\bigcup
⋂
\bigcap
⋂\bigcap
∮
\oint
∮\oint
⋁
\bigvee
⋁\bigvee
⋀
\bigwedge
⋀\bigwedge
∬
\iint
∬\iint
∐
\coprod
∐\coprod
⨆
\bigsqcup
⨆\bigsqcup
∯
\oiint
∬
\oiint
常见关系符号
符号语法符号语法符号语法
<
<
<<
>
>
>>
=
=
==
≤
\leq
≤\leq
≥
\geq
≥\geq
≠
\neq
=\neq
≪
\ll
≪\ll
≫
\gg
≫\gg
≡
\equiv
≡\equiv
⊂
\subset
⊂\subset
⊃
\supset
⊃\supset
≈
\approx
≈\approx
⊆
\subseteq
⊆\subseteq
⊇
\supseteq
⊇\supseteq
∼
\sim
∼\sim
∈
\in
∈\in
∋
\ni
∋\ni
∝
\propto
∝\propto
⊢
\vdash
⊢\vdash
⊣
\dashv
⊣\dashv
⊨
\models
⊨\models
∣
\mid
∣\mid
∥
\parallel
∥\parallel
⊥
\perp
⊥\perp
∉
\notin
∈/\notin
⋈
\Join
⋈\Join
≁
\nsim
≁\nsim
⊊
\subsetneq
⊊\subsetneq
⊋
\supsetneq
⊋\supsetneq
数学模式重音符
符号语法符号语法符号语法
a
^
\hat{a}
a^\hat{a}
a
ˉ
\bar{a}
aˉ\bar{a}
a
~
\tilde{a}
a~\tilde{a}
a
⃗
\vec{a}
a
\vec{a}
a
˙
\dot{a}
a˙\dot{a}
a
¨
\ddot{a}
a¨\ddot{a}
a
b
c
^
\widehat{abc}
abc
\widehat{abc}
a
b
c
~
\widetilde{abc}
abc
\widetilde{abc}
a
b
c
‾
\overline{abc}
abc\overline{abc}
箭头
如果需要长箭头，只需要在语法前面加上\long，例如\longleftarrow即为
⟵
\longleftarrow
⟵，如果加上\Long则变为双线长箭头，例如\Longleftarrow即为
⟸
\Longleftarrow
⟸
符号语法符号语法符号语法
←
\leftarrow
←\leftarrow
→
\rightarrow
→\rightarrow
↔
\leftrightarrow
↔\leftrightarrow
⇐
\Leftarrow
⇐\Leftarrow
⇒
\Rightarrow
⇒\Rightarrow
⇔
\Leftrightarrow
⇔\Leftrightarrow
↑
\uparrow
↑\uparrow
↓
\downarrow
↓\downarrow
↕
\updownarrow
↕\updownarrow
⇑
\Uparrow
⇑\Uparrow
⇓
\Downarrow
⇓\Downarrow
⇕
\Updownarrow
⇕\Updownarrow
↼
\leftharpoonup
↼\leftharpoonup
↽
\leftharpoondown
↽\leftharpoondown
⇀
\rightharpoonup
⇀\rightharpoonup
⇁
\rightharpoondown
⇁\rightharpoondown
⇌
\rightleftharpoons
⇌\rightleftharpoons
⇋
\leftrightharpoons
⇋\leftrightharpoons
⟺
\iff
⟺\iff
↦
\mapsto
↦\mapsto
括号
括号语法括号语法括号语法
(
)
()
()()
[
]
[]
[][]
{
}
\{\}
{}\{\}
⌊
⌋
\lfloor\rfloor
⌊⌋\lfloor\rfloor
⌈
⌉
\lceil\rceil
⌈⌉\lceil\rceil
⟨
⟩
\langle\rangle
⟨⟩\langle\rangle
大尺寸括号
括号语法括号语法
(
)
\left(\right)
()\left( \right)
[
]
\left[ \right]
[]\left[ \right]
x
1
x
2
…
x
n
⏞
n
\overbrace{x_1x_2\ldots x_n}^{n}
x1x2…xn
n\overbrace{x_1x_2\ldots x_n}^{n}
x
1
x
2
…
x
n
⏟
n
\underbrace{x_1x_2\ldots x_n}_{n}
n
x1x2…xn\underbrace{x_1x_2\ldots x_n}_{n}
注：大尺寸的()和[]是可以根据公式的高度自动调节的，例如
\arg\min_{\theta}
\left[
-\sum_{i=1}^{n}
\left[
\mathbf{y}^{(i)}\ln(h_{\theta}(\mathbf{x}^{(i)})) +
(1-\mathbf{y}^{(i)})\ln(1-h_{\theta}(\mathbf{x}^{(i)}))
\right]
\right]
arg
min
θ
[
−
∑
i
=
1
n
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\arg\min_{\theta} \left[ -\sum_{i=1}^{n} \left[ \mathbf{y}^{(i)}\ln(h_{\theta}(\mathbf{x}^{(i)})) + (1-\mathbf{y}^{(i)})\ln(1-h_{\theta}(\mathbf{x}^{(i)})) \right] \right]
argθmin[−i=1∑n[y(i)ln(hθ(x(i)))+(1−y(i))ln(1−hθ(x(i)))]]
可以看出，括号高度可以框住整个公式
因此在这种大型的公式中，使用大尺寸括号视觉效果更美观
其他常见符号
符号语法符号语法符号语法
∀
\forall
∀\forall
∃
\exist
∃\exist
∠
\angle
∠\angle
∅
\emptyset
∅\emptyset
∂
\partial
∂\partial
∞
\infty
∞\infty
…
\ldots
…\ldots
⋯
\cdots
⋯\cdots
…
\dots
…\dots
⋮
\vdots
⋮\vdots
⋱
\ddots
⋱\ddots
′
\prime
′\prime
∵
\because
∵\because
∴
\therefore
∴\therefore
□
\Box
□\Box
△
\triangle
△\triangle
§
\S
§\S
数学公式写法
上下标
^：上标_：下标
例如:
\sum_{i=1}^{n}X_n表示
∑
i
=
1
n
X
n
\sum_{i=1}^{n}X_n
∑i=1nXn\int_{0}^{\infty}x^2dx表示
∫
0
∞
x
2
d
x
\int_{0}^{\infty}x^2dx
∫0∞x2dx\prod_{i=1}^{n}X_n表示
∏
i
=
1
n
X
n
\prod_{i=1}^{n}X_n
∏i=1nXn
分数
使用\frac{}{}即可，例如\frac{a}{b}表示
a
b
\frac{a}{b}
ba
插入文字
使用\text，例如\text{hello,world!}表示
hello,world!
\text{hello,world!}
hello,world!
常见函数
函数语法函数语法函数语法
log
(
)
\log()
log()\log()
ln
(
)
\ln()
ln()\ln()
lg
(
)
\lg()
lg()\lg()
max
\max
max\max
min
\min
min\min
lim
x
→
∞
\lim_{x \to \infty}
limx→∞\lim_{x \to \infty}
arg
max
c
∈
C
\arg\max_{c \in C}
argmaxc∈C\arg\max_{c \in C}
arg
min
c
∈
C
\arg\min_{c \in C}
argminc∈C\arg\min_{c \in C}
exp
\exp
exp\exp
矩阵、行列式
&表示分隔元素，\\表示换行
A=
\begin{pmatrix}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{pmatrix}
A
=
(
a
11
a
12
a
21
a
22
)
A= \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}
A=(a11a21a12a22)
A=
\begin{bmatrix}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{bmatrix}
A
=
[
a
11
a
12
a
21
a
22
]
A= \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}
A=[a11a21a12a22]
A=
\begin{Bmatrix}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{Bmatrix}
A
=
{
a
11
a
12
a
21
a
22
}
A= \begin{Bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{Bmatrix}
A={a11a21a12a22}
A=
\begin{vmatrix}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{vmatrix}
A
=
∣
a
11
a
12
a
21
a
22
∣
A= \begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix}
A=
a11a21a12a22
A=
\begin{Vmatrix}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{Vmatrix}
A
=
∥
a
11
a
12
a
21
a
22
∥
A= \begin{Vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{Vmatrix}
A=
a11a21a12a22
A=
\begin{matrix}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{matrix}
A
=
a
11
a
12
a
21
a
22
A= \begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix}
A=a11a21a12a22
多行公式对齐
使用\begin{split} \end{split}，在需要对齐的地方添加&符号，注意需要用\\来换行。
例如：
\begin{split}
L(\theta)
&= \arg\max_{\theta}\ln(P_{All})\\
&= \arg\max_{\theta}\ln\prod_{i=1}^{n}
\left[
(h_{\theta}(\mathbf{x}^{(i)}))^{\mathbf{y}^{(i)}}\cdot
(1-h_{\theta}(\mathbf{x}^{(i)}))^{1-\mathbf{y}^{(i)}}
\right]\\
&= \arg\max_{\theta}\sum_{i=1}^{n}
\left[
\mathbf{y}^{(i)}\ln(h_{\theta}(\mathbf{x}^{(i)})) +
(1-\mathbf{y}^{(i)})\ln(1-h_{\theta}(\mathbf{x}^{(i)}))
\right]\\
&= \arg\min_{\theta}
\left[
-\sum_{i=1}^{n}
\left[
\mathbf{y}^{(i)}\ln(h_{\theta}(\mathbf{x}^{(i)})) +
(1-\mathbf{y}^{(i)})\ln(1-h_{\theta}(\mathbf{x}^{(i)}))
\right]
\right]\\
&= \arg\min_{\theta}\mathscr{l}(\theta)
\end{split}
L
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=
arg
max
θ
ln
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A
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=
arg
max
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ln
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=
arg
max
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=
arg
min
θ
l
(
θ
)
\begin{split} L(\theta) &= \arg\max_{\theta}\ln(P_{All})\\ &= \arg\max_{\theta}\ln\prod_{i=1}^{n} \left[ (h_{\theta}(\mathbf{x}^{(i)}))^{\mathbf{y}^{(i)}}\cdot (1-h_{\theta}(\mathbf{x}^{(i)}))^{1-\mathbf{y}^{(i)}} \right]\\ &= \arg\max_{\theta}\sum_{i=1}^{n} \left[ \mathbf{y}^{(i)}\ln(h_{\theta}(\mathbf{x}^{(i)})) + (1-\mathbf{y}^{(i)})\ln(1-h_{\theta}(\mathbf{x}^{(i)})) \right]\\ &= \arg\min_{\theta} \left[ -\sum_{i=1}^{n} \left[ \mathbf{y}^{(i)}\ln(h_{\theta}(\mathbf{x}^{(i)})) + (1-\mathbf{y}^{(i)})\ln(1-h_{\theta}(\mathbf{x}^{(i)})) \right] \right]\\ &= \arg\min_{\theta}\mathscr{l}(\theta) \end{split}
L(θ)=argθmaxln(PAll)=argθmaxlni=1∏n[(hθ(x(i)))y(i)⋅(1−hθ(x(i)))1−y(i)]=argθmaxi=1∑n[y(i)ln(hθ(x(i)))+(1−y(i))ln(1−hθ(x(i)))]=argθmin[−i=1∑n[y(i)ln(hθ(x(i)))+(1−y(i))ln(1−hθ(x(i)))]]=argθminl(θ)
上例中，在=前添加了&，因此实现等号对齐；
\begin{split} \end{split}语法默认为右对齐，也就是说如果不在任何地方添加&符号，则公式默认右侧对齐，例如：
\begin{split}
L(\theta)
= \arg\max_{\theta}\ln(P_{All})\\
= \arg\max_{\theta}\ln\prod_{i=1}^{n}
\left[
(h_{\theta}(\mathbf{x}^{(i)}))^{\mathbf{y}^{(i)}}\cdot
(1-h_{\theta}(\mathbf{x}^{(i)}))^{1-\mathbf{y}^{(i)}}
\right]\\
= \arg\max_{\theta}\sum_{i=1}^{n}
\left[
\mathbf{y}^{(i)}\ln(h_{\theta}(\mathbf{x}^{(i)})) +
(1-\mathbf{y}^{(i)})\ln(1-h_{\theta}(\mathbf{x}^{(i)}))
\right]\\
= \arg\min_{\theta}
\left[
-\sum_{i=1}^{n}
\left[
\mathbf{y}^{(i)}\ln(h_{\theta}(\mathbf{x}^{(i)})) +
(1-\mathbf{y}^{(i)})\ln(1-h_{\theta}(\mathbf{x}^{(i)}))
\right]
\right]\\
= \arg\min_{\theta}\mathscr{l}(\theta)
\end{split}
上述LATEX代码没有添加&符号，则公式右对齐：
L
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arg
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=
arg
min
θ
l
(
θ
)
\begin{split} L(\theta) = \arg\max_{\theta}\ln(P_{All})\\ = \arg\max_{\theta}\ln\prod_{i=1}^{n} \left[ (h_{\theta}(\mathbf{x}^{(i)}))^{\mathbf{y}^{(i)}}\cdot (1-h_{\theta}(\mathbf{x}^{(i)}))^{1-\mathbf{y}^{(i)}} \right]\\ = \arg\max_{\theta}\sum_{i=1}^{n} \left[ \mathbf{y}^{(i)}\ln(h_{\theta}(\mathbf{x}^{(i)})) + (1-\mathbf{y}^{(i)})\ln(1-h_{\theta}(\mathbf{x}^{(i)})) \right]\\ = \arg\min_{\theta} \left[ -\sum_{i=1}^{n} \left[ \mathbf{y}^{(i)}\ln(h_{\theta}(\mathbf{x}^{(i)})) + (1-\mathbf{y}^{(i)})\ln(1-h_{\theta}(\mathbf{x}^{(i)})) \right] \right]\\ = \arg\min_{\theta}\mathscr{l}(\theta) \end{split}
L(θ)=argθmaxln(PAll)=argθmaxlni=1∏n[(hθ(x(i)))y(i)⋅(1−hθ(x(i)))1−y(i)]=argθmaxi=1∑n[y(i)ln(hθ(x(i)))+(1−y(i))ln(1−hθ(x(i)))]=argθmin[−i=1∑n[y(i)ln(hθ(x(i)))+(1−y(i))ln(1−hθ(x(i)))]]=argθminl(θ)
如果希望左对齐，例如
\begin{split}
&\ln h_{\theta}(\mathbf{x}^{(i)})
= \ln\frac{1}{1+e^{-\theta^T \mathbf{x}^{(i)}}}
= -\ln(1+e^{\theta^T \mathbf{x}^{(i)}})\\
&\ln(1-h_{\theta}(\mathbf{x}^{(i)}))
= \ln(1-\frac{1}{1+e^{-\theta^T \mathbf{x}^{(i)}}})
= -\theta^T \mathbf{x}^{(i)}-\ln(1+e^{\theta^T \mathbf{x}^{(i)}})
\end{split}
ln
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\begin{split} &\ln h_{\theta}(\mathbf{x}^{(i)}) = \ln\frac{1}{1+e^{-\theta^T \mathbf{x}^{(i)}}} = -\ln(1+e^{\theta^T \mathbf{x}^{(i)}})\\ &\ln(1-h_{\theta}(\mathbf{x}^{(i)})) = \ln(1-\frac{1}{1+e^{-\theta^T \mathbf{x}^{(i)}}}) = -\theta^T \mathbf{x}^{(i)}-\ln(1+e^{\theta^T \mathbf{x}^{(i)}}) \end{split}
lnhθ(x(i))=ln1+e−θTx(i)1=−ln(1+eθTx(i))ln(1−hθ(x(i)))=ln(1−1+e−θTx(i)1)=−θTx(i)−ln(1+eθTx(i))
除了\begin{split} \end{split}，也可以用\begin{align} \end{align}，用法与split相同，对齐方式也相同；
只有一点不同：采用align环境会默认为每一条公式编号（如下例），split则不会编号。
\begin{align}
&\ln h_{\theta}(\mathbf{x}^{(i)})
= \ln\frac{1}{1+e^{-\theta^T \mathbf{x}^{(i)}}}
= -\ln(1+e^{\theta^T \mathbf{x}^{(i)}})\\
&\ln(1-h_{\theta}(\mathbf{x}^{(i)}))
= \ln(1-\frac{1}{1+e^{-\theta^T \mathbf{x}^{(i)}}})
= -\theta^T \mathbf{x}^{(i)}-\ln(1+e^{\theta^T \mathbf{x}^{(i)}})
\end{align}
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\begin{align} &\ln h_{\theta}(\mathbf{x}^{(i)}) = \ln\frac{1}{1+e^{-\theta^T \mathbf{x}^{(i)}}} = -\ln(1+e^{\theta^T \mathbf{x}^{(i)}})\\ &\ln(1-h_{\theta}(\mathbf{x}^{(i)})) = \ln(1-\frac{1}{1+e^{-\theta^T \mathbf{x}^{(i)}}}) = -\theta^T \mathbf{x}^{(i)}-\ln(1+e^{\theta^T \mathbf{x}^{(i)}}) \end{align}
lnhθ(x(i))=ln1+e−θTx(i)1=−ln(1+eθTx(i))ln(1−hθ(x(i)))=ln(1−1+e−θTx(i)1)=−θTx(i)−ln(1+eθTx(i))
但可以在align后加一个*号，则align环境也可以取消公式自动编号，如下： (也就是说align*和split的用法完全相同)
\begin{align*}
&\ln h_{\theta}(\mathbf{x}^{(i)})
= \ln\frac{1}{1+e^{-\theta^T \mathbf{x}^{(i)}}}
= -\ln(1+e^{\theta^T \mathbf{x}^{(i)}})\\
&\ln(1-h_{\theta}(\mathbf{x}^{(i)}))
= \ln(1-\frac{1}{1+e^{-\theta^T \mathbf{x}^{(i)}}})
= -\theta^T \mathbf{x}^{(i)}-\ln(1+e^{\theta^T \mathbf{x}^{(i)}})
\end{align*}
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\begin{align*} &\ln h_{\theta}(\mathbf{x}^{(i)}) = \ln\frac{1}{1+e^{-\theta^T \mathbf{x}^{(i)}}} = -\ln(1+e^{\theta^T \mathbf{x}^{(i)}})\\ &\ln(1-h_{\theta}(\mathbf{x}^{(i)})) = \ln(1-\frac{1}{1+e^{-\theta^T \mathbf{x}^{(i)}}}) = -\theta^T \mathbf{x}^{(i)}-\ln(1+e^{\theta^T \mathbf{x}^{(i)}}) \end{align*}
lnhθ(x(i))=ln1+e−θTx(i)1=−ln(1+eθTx(i))ln(1−hθ(x(i)))=ln(1−1+e−θTx(i)1)=−θTx(i)−ln(1+eθTx(i))
方程组
使用\begin{cases} \end{cases}
例如：
\begin{cases}
\begin{split}
p &= P(y=1|\mathbf{x})=
\frac{1}{1+e^{-\theta^T\mathbf{X}}}\\
1-p &= P(y=0|\mathbf{x})=1-P(y=1|\mathbf{x})=
\frac{1}{1+e^{\theta^T\mathbf{X}}}
\end{split}
\end{cases}
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\begin{cases} \begin{split} p &= P(y=1|\mathbf{x})= \frac{1}{1+e^{-\theta^T\mathbf{X}}}\\ 1-p &= P(y=0|\mathbf{x})=1-P(y=1|\mathbf{x})= \frac{1}{1+e^{\theta^T\mathbf{X}}} \end{split} \end{cases}
⎩
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⎧p1−p=P(y=1∣x)=1+e−θTX1=P(y=0∣x)=1−P(y=1∣x)=1+eθTX1
注意LATEX语法可以嵌套使用，上例即为\begin{cases} \end{cases}下嵌套了begin{split} \end{split}。
也可以将公式和文字结合起来，例如：
\text{Decision Boundary}=
\begin{cases}
1\quad \text{if }\ \hat{y}>0.5\\
0\quad \text{otherwise}
\end{cases}
Decision Boundary
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\text{Decision Boundary}= \begin{cases} 1\quad \text{if}\quad \hat{y}>0.5\\ 0\quad \text{otherwise} \end{cases}
Decision Boundary={1ify^>0.50otherwise 注：\quad表示空格。
原装电瓶和修复电瓶怎么分辨（怎样分辨原厂电瓶和翻新电瓶）“69”究竟是什么意思？一文带你了解其背后的文化与含义