Latex相关用法
本章主要介绍了latex相关的用法
Latex相关用法
样式 LaTex
[ $\begin{bmatrix}$ ：代表左侧方括号
] $\end{bmatrix}$ ：代表右侧方括号
$\cdots$：代表水平省略号
$\vdots$：代表竖直省略号
$\ddots$：代表对角省略号
\(\cdots\)：代表水平省略号
\(\vdots\)：代表竖直省略号
\(\ddots\)：代表对角省略号
1.字母和符号
1.1 数学中字母样式
${AaBbCc}$ 斜体，大部分数学符号、表达式
$ \mathrm{AaBbCc}$ 正体、公式中的单位或文字
$ \mathbf{AaBbCc}$ 粗体、向量、矩阵
$ \boldsymbol{AaBbCc}$ 粗体、斜体、向量、矩阵
$ \mathtt{AaBbCc}$ 等宽字体、常用于代码
$\mathcal{ABCDEF}$ 花体，用于表示数学中的集合、代数结构、算子
$\mathbb{CRQZN}$ 黑板粗体，常用来表达各种集合
$\text{Aa Bb Cc}$ 用来写公式中的文字
$\mathrm{d}x$ ISO规定导数符号d为正体
$\operatorname{T}$ 运算符
相关样式展示
- \({AaBbCc}\) 斜体，大部分数学符号、表达式
- $ \mathrm{AaBbCc}$ 正体、公式中的单位或文字
- $ \mathbf{AaBbCc}$ 粗体、向量、矩阵
- $ \boldsymbol{AaBbCc}$ 粗体、斜体、向量、矩阵
- $ \mathtt{AaBbCc}$ 等宽字体、常用于代码
- \(\mathcal{ABCDEF}\) 花体，用于表示数学中的集合、代数结构、算子
- \(\mathbb{CRQZN}\) 黑板粗体，常用来表达各种集合
- \(\text{Aa Bb Cc}\) 用来写公式中的文字
- \(\mathrm{d}x\) ISO规定导数符号d为正体
- \(\operatorname{T}\) 运算符
1.2 数学中字母标记
$x'$ x prime x的导数
$x^{\prime}$ x prime x的导数
$x'’$ x double prime x的二次导数
$\overrightarrow{AB}$ a vector pointing from A to B 向量AB
$\underline{x}$ x underline x下
$\hat{x}$ x hat x帽
$\bar{x}$ x bar x把
$\dot{x}$ x dot x点
$\tilde{x}$ x tilde x波浪
$x_i$ x subscript i, x sub i xi
$x^i$ x to the x的i次方
$\ddot{x}$ x double dot x双点
$x^*$ x star, x super asterisk x星
$x\dagger$ x dagger **转置+共轭**的复合操作
$x\ddagger$ x double dagger **转置+共轭**的复合操作
${\color{red}x}$ red x 红x
相关样式展示
- \(x'\) x的导数
- \(x^{\prime}\) x的导数
- \(x'’\) x的二次导数
- \(\overrightarrow{AB}\) 向量AB
- \(\underline{x}\) x下
- \(\hat{x}\) x帽
- \(\bar{x}\) x把
- \(\dot{x}\) x点
- \(\tilde{x}\) x波浪
- \(x_i\) xi
- \(x^i\) x的i次方
- \(\ddot{x}\) x双点
- \(x^*\) x星
- \(x\dagger\) 转置+共轭的复合操作
- \(x\ddagger\) 转置+共轭的复合操作
- \({\color{red}x}\) 红x
1.3 希腊字母
小写 LaTex 大写 LaTex 英文拼写 发音
α $\alpha$ Α $A$ alpha /ˈælfə/
β $\beta$ Β $B$ beta /ˈbeɪtə/
γ $\gamma$ Γ $\Gamma$ gamma /ˈɡæmə/
δ $\delta$ Δ $\Delta$ delta /ˈdeltə/
ε $\epsilon$ Ε $E$ epsilon /ˈepsɪlɑːn/
ζ $\zeta$ Ζ $Z$ zeta /ˈziːtə/
η $\eta$ Η $H$ eta /ˈiːtə/
θ $\theta$ Θ $\Theta$ theta /ˈθiːtə/
ι $\iota$ Ι $I$ iota /aɪˈoʊtə/
κ $\kappa$ Κ $K$ kappa /ˈkæpə/
λ $\lambda$ Λ $\Lambda$ lambda /ˈlæmdə/
μ $\mu$ Μ $M$ mu /mjuː/
ν $\nu$ Ν $N$ nu /njuː/
ξ $\xi$ Ξ $\Xi$ xi /ksaɪ/ 或 /zaɪ/ 或 /ɡzaɪ/
ο $\omicron$ Ο $O$ omicron /ˈɑːməkrɑːn/
π $\pi$ Π $\Pi$ pi /paɪ/
ρ $\rho$ Ρ $P$ rho /roʊ/
σ $\sigma$ Σ $\Sigma$ sigma /ˈsɪɡmə/
τ $\tau$ Τ $T$ tau /taʊ/
υ $\upsilon$ Υ $Y$ upsilon /ˈʊpsɪlɑːn/
φ $\phi$ Φ $\Phi$ phi /faɪ/
χ $\chi$ Χ $X$ chi /kaɪ/
ψ $\psi$ Ψ $\Psi$ psi /saɪ/
ω $\omega$ Ω $\Omega$ omega /oʊˈmeɡə/
1.4 希腊字母，变量
LaTeX写法
$\vartheta$
$\varrho$
$\varkappa$
$\varphi$
$\varpi$
$\varepsilon$
$\varsigma$
相关样式展示
- \(\vartheta\)
- \(\varrho\)
- \(\varkappa\)
- \(\varphi\)
- \(\varpi\)
- \(\varepsilon\)
- \(\varsigma\)
1.5 常用符号
LaTex 英文读法 中文表达
$\times$ multiplies, times 乘
$\div$ divided by 除以
$\otimes$ tensor product 张量积
$($ left parenthesis 左圆括号
$)$ right parenthesis 右圆括号
$[$ left square bracket 左方括号
$]$ right square bracket 右方括号
$\{$ left brace 左大括号
$\}$ right brace 右大括号
$\pm$ plus or minus 正负号
$\mp$ minus or plus 负正号
$<$ less than 小于
$\leq$ less than or equal to 小于等于
$\ll$ much less than 远小于
$>$ greater than 大于号
$\geq$ greater than or equal to 大于等于
$\gg$ much greater than 远大于
$=$ equals, is equal to 等于
$\equiv$ is identical to 完全相等
$\approx$ is approximately equal to 约等于
$\propto$ proportional to 正比于
$\partial$ partial derivative 偏导
$\nabla$ del, nabla 梯度算子
$\infty$ infinity 无穷
$\neq$ does not equal 不等于
$\parallel$ parallel 平行
$\perp$ perpendicular to 垂直
$\angle$ angle 角度
$\triangle$ triangle 三角形
$\square$ square 正方形
$\sim$ similar 相似
$\exists$ there exists 存在
$\forall$ for all 任意
$\subset$ is proper subset of 真子集
$\subseteq$ is subset of 子集
$\varnothing$ empty set 空集
$\supset$ is proper superset of 真超集
$\supseteq$ is superset of 超集
$\cap$ intersection 交集
$\cup$ union 并集
$\in$ is member of 属于
$\notin$ is not member of 不属于
$\N$ set of natural numbers 自然数集合
$\Z$ set of integers 整数集合
$\rightarrow$ arrow to the right 向右箭头
$\leftarrow$ arrow to the left 向左箭头
$\mapsto$ maps to 映射
$\implies$ implies 推出
$\uparrow$ upward arrow 向上箭头
$\Uparrow$ upward arrow 向上箭头
$\downarrow$ downward arrow 向下箭头
$\Downarrow$ downward arrow 向下箭头
$\therefore$ therefore sign 所以
$\because$ because sign 因为
$\star$ asterisk, star, pointer 星号
$!$ exclamation mark 叹号，阶乘
$| x |$ x absolute value of x 绝对值
$\lfloor x \rfloor$ the floor of x 向下取整
$\lceil x \rceil$ the ceiling of x 向上取整
$x!$ x factorial 阶乘
相关样式展示
- \(\times\) 乘
- \(\div\) 除以
- \(\otimes\) 张量积
- \((\) 左圆括号
- \()\) 右圆括号
- \([\) 左方括号
- \(]\) 右方括号
- \(\{\) 左大括号
- \(\}\) 右大括号
- \(\pm\) 正负号
- \(\mp\) 负正号
- \(<\) 小于
- \(\leq\) 小于等于
- \(\ll\) 远小于
- \(>\) 大于号
- \(\geq\) 大于等于
- \(\gg\) 远大于
- \(=\) 等于
- \(\equiv\) 完全相等
- \(\approx\) 约等于
- \(\propto\) 正比于
- \(\partial\) 偏导
- \(\nabla\) 梯度算子
- \(\infty\) 无穷
- \(\neq\) 不等于
- \(\parallel\) 平行
- \(\perp\) 垂直
- \(\angle\) 角度
- \(\triangle\) 三角形
- \(\square\) 正方形
- \(\sim\) 相似
- \(\exists\) 存在
- \(\forall\) 任意
- \(\subset\) 真子集
- \(\subseteq\) 子集
- \(\varnothing\) 空集
- \(\supset\) 真超集
- \(\supseteq\) 超集
- \(\cap\) 交集
- \(\cup\) 并集
- \(\in\) 属于
- \(\notin\) 不属于
- \(\N\) 自然数集合
- \(\Z\) 整数集合
- \(\rightarrow\) 向右箭头
- \(\leftarrow\) 向左箭头
- \(\mapsto\) 映射
- \(\implies\) 推出
- \(\uparrow\) 向上箭头
- \(\Uparrow\) 向上箭头
- \(\downarrow\) 向下箭头
- \(\Downarrow\) 向下箭头
- \(\therefore\) 所以
- \(\because\) 因为
- \(\star\) 星号
- \(!\) 叹号，阶乘
- \(| x |\) 绝对值
- \(\lfloor x \rfloor\) 向下取整
- \(\lceil x \rceil\) 向上取整
- \(x!\) 阶乘
2.用 LaTex 写公式
2.1 多项式的数学表达
LaTex
$x^{2}-y^{2} = \left(x+y\right)\left(x-y\right)$
$a_{n}x^{n} + a_{n-1}x^{n-1} + \dotsb + a_{2}x^{2} + a_{1}x + a_{0}$
$\sum_{k=0}^{n}a_{k}x^{k}$
$ ax^{2} + bx + c = 0\ (a\neq 0) $
相关样式展示
- \(x^{2}-y^{2} = \left(x+y\right)\left(x-y\right)\)
- \(a_{n}x^{n} + a_{n-1}x^{n-1} + \dotsb + a_{2}x^{2} + a_{1}x + a_{0}\)
- \(\sum_{k=0}^{n}a_{k}x^{k}\)
- $ ax^{2} + bx + c = 0\ (a\neq 0) $
2.2 根式的数学表达
LaTex
${\sqrt[{n}]{a^{m}}}=(a^{m})^{1/n}=a^{m/n}=(a^{1/n})^{m}=({\sqrt[ {n}]{a}})^{m}$
$\left({\sqrt {1-x^{2}}}\right)^{2}$
相关样式展示
-
\({\sqrt[{n}]{a^{m}}}=(a^{m})^{1/n}=a^{m/n}=(a^{1/n})^{m}=({\sqrt[ {n}]{a}})^{m}\)
-
\(\left({\sqrt {1-x^{2}}}\right)^{2}\)
2.3 分式的数学表达
Latex
$\frac {1}{x+1} + {\frac {1}{x-1}} = {\frac {2x}{x^{2} - 1}}$
$x_{1, 2} = {\frac {-b\pm {\sqrt {b^{2} - 4ac}}}{2a}}$
相关样式展示
-
\(\frac {1}{x+1} + {\frac {1}{x-1}} = {\frac {2x}{x^{2} - 1}}\)
-
\(x_{1, 2} = {\frac {-b\pm {\sqrt {b^{2} - 4ac}}}{2a}}\)
2.4 函数的数学表达
LaTex
$f(x)=ax^{2}+bx+c~~{\text{ with }}~~a,b,c\in \mathbb {R} ,\ a\neq 0$
$f(x_1, x_2) = x_1^2 + x_2^2 + 2x_1x_2$
$\log_{b}(xy)=\log_{b}x+\log_{b}y$
$\ln(xy)=\ln x+\ln y{\text{ for }} x>0 {\text{ and }} y>0$
$f(x)=a\exp \left(-{\frac {(x-b)^{2}}{2c^{2}}}\right)$
相关样式展示
-
\(f(x)=ax^{2}+bx+c~~{\text{ with }}~~a,b,c\in \mathbb {R} ,\ a\neq 0\)
-
\(f(x_1, x_2) = x_1^2 + x_2^2 + 2x_1x_2\)
-
\(\log_{b}(xy)=\log_{b}x+\log_{b}y\)
-
\(\ln(xy)=\ln x+\ln y{\text{ for }} x>0 {\text{ and }} y>0\)
-
\(f(x)=a\exp \left(-{\frac {(x-b)^{2}}{2c^{2}}}\right)\)
2.5 三角恒等式
LaTex
$\sin ^{2}\theta +\cos ^{2}\theta =1$
$\sin 2\theta =2\sin \theta \cos \theta$
$\sin(\alpha \pm \beta )=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta$
$\tan(\alpha \pm \beta )=\frac {\tan \alpha \pm \tan \beta }{1\mp \tan \alpha \tan \beta }$
相关样式展示
-
\(\sin ^{2}\theta +\cos ^{2}\theta =1\)
-
\(\sin 2\theta =2\sin \theta \cos \theta\)
-
\(\sin(\alpha \pm \beta )=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta\)
-
\(\tan(\alpha \pm \beta )=\frac {\tan \alpha \pm \tan \beta }{1\mp \tan \alpha \tan \beta }\)
2.6 微积分数学表达
LaTex
$\exp(x)=\sum _{k=0}^{\infty }{\frac {x^{k}}{k!}}=1+x+{\frac {x^{2}}{2}}+{\frac {x^{3}}{6}}+{\frac {x^{4}}{24}}+\cdots $
$ \left(\sum _{i=0}^{n}a_{i}\right)\left(\sum _{j=0}^{n}b_{j}\right)=\sum _{i=0}^{n}\sum _{j=0}^{n}a_{i}b_{j}$
$\exp(x) =\lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}$
$\frac {\mathrm{d}}{\mathrm{d}x} \exp(f(x)) =f'(x) \exp(f(x))$
$\int_{a}^{b}f(x) \mathrm {d} x$
$\int _{-\infty }^{\infty }\exp(- x^{2})\mathrm{d}x={\sqrt {\mathrm{\pi} }}$
$\int _{-\infty }^{\infty }\int _{- \infty }^{\infty } \exp \left({- \left(x^{2}+y^{2}\right)} \right) {\mathrm{d}x} {\mathrm{d}y} = \pi$
$\frac {\partial ^{2}f}{\partial x^{2}}=f''_{xx}=\partial _{xx}f=\partial _{x}^{2}f$
${\frac {\partial ^{2}f}{\partial y \partial x}}={\frac {\partial }{\partial y}}\left({\frac {\partial f}{\partial x}}\right)=f''_{xy}$
相关样式展示
- $\exp(x)=\sum _{k=0}^{\infty }{\frac {x^{k}}{k!}}=1+x+{\frac {x^{2}}{2}}+{\frac {x^{3}}{6}}+{\frac {x^{4}}{24}}+\cdots $
- $ \left(\sum {i=0}^{n}a\right)\left(\sum {j=0}^{n}b\right)=\sum {i=0}^{n}\sum {j=0}^{n}ab$
- \(\exp(x) =\lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}\)
- \(\frac {\mathrm{d}}{\mathrm{d}x} \exp(f(x)) =f'(x) \exp(f(x))\)
- \(\int_{a}^{b}f(x) \mathrm {d} x\)
- \(\int _{-\infty }^{\infty }\exp(- x^{2})\mathrm{d}x={\sqrt {\mathrm{\pi} }}\)
- \(\int _{-\infty }^{\infty }\int _{- \infty }^{\infty } \exp \left({- \left(x^{2}+y^{2}\right)} \right) {\mathrm{d}x} {\mathrm{d}y} = \pi\)
- \(\frac {\partial ^{2}f}{\partial x^{2}}=f''_{xx}=\partial _{xx}f=\partial _{x}^{2}f\)
- \({\frac {\partial ^{2}f}{\partial y \partial x}}={\frac {\partial }{\partial y}}\left({\frac {\partial f}{\partial x}}\right)=f''_{xy}\)
2.7 向量的表达
LaTex
$\mathbf {a} = {\begin{bmatrix} a_{1} \\ a_{2} \\ a_{3} \end{bmatrix}} = [a_{1}\ a_{2}\ a_{3}]^{\operatorname {T} }$
$\left\|\mathbf {a} \right\|=\sqrt {a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}$
$\mathbf {a} \cdot \mathbf {b} = a_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3}$
$\mathbf {a} \cdot \mathbf {b} =\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|\cos \theta $
相关样式展示
-
\(\mathbf {a} = {\begin{bmatrix} a_{1} \\ a_{2} \\ a_{3} \end{bmatrix}} = [a_{1}\ a_{2}\ a_{3}]^{\operatorname {T} }\)
-
\(\left\|\mathbf {a} \right\|=\sqrt {a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}\)
-
\(\mathbf {a} \cdot \mathbf {b} = a_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3}\)
-
$\mathbf {a} \cdot \mathbf {b} =\left|\mathbf {a} \right|\left|\mathbf {b} \right|\cos \theta $
-
\(\|\mathbf {x} \|_{p}=\left(\sum _{i=1}^{n}\left|x_{i}\right|^{p}\right)^{1/p}\)
2.8 矩阵的表达
LaTex
$\mathbf {A} = {\begin{bmatrix} 1 & 2\\ 3 & 4 \\ 5 & 6 \end{bmatrix}}$
$\mathbf {A} ={\begin{bmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\end{bmatrix}}$
$\left(\mathbf {A} +\mathbf {B} \right)^{\operatorname {T} }=\mathbf {A} ^{\operatorname {T} }+\mathbf {B} ^{\operatorname {T} }$
$\left(\mathbf {AB} \right)^{\operatorname {T} }=\mathbf {B} ^{\operatorname {T} }\mathbf {A} ^{\operatorname {T} }$
$ \left(\mathbf {A} ^{\operatorname {T} }\right)^{-1}=\left(\mathbf {A} ^{- 1}\right)^{\operatorname {T} }$
$\mathbf {u} \otimes \mathbf {v} = \mathbf {u} \mathbf {v} ^ {\operatorname {T}} = {\begin{bmatrix}u_{1} \\ u_{2} \\ u_{3} \\ u_{4} \end{bmatrix}} {\begin{bmatrix} v_{1}&v_{2}&v_{3} \end{bmatrix}} = {\begin{bmatrix} u_{1}v_{1} & u_{1}v_{2} & u_{1}v_{3} \\ u_{2}v_{1} & u_{2}v_{2} & u_{2}v_{3} \\ u_{3}v_{1} & u_{3}v_{2} & u_{3}v_{3} \\ u_{4}v_{1} & u_{4}v_{2} & u_{4}v_{3} \end{bmatrix}}$
$\det {\begin{bmatrix} a & b \\ c & d \end{bmatrix}} = ad-bc$
相关样式展示
- \(\mathbf {A} = {\begin{bmatrix} 1 & 2\\ 3 & 4 \\ 5 & 6 \end{bmatrix}}\)
- \(\mathbf {A} ={\begin{bmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\end{bmatrix}}\)
- \(\left(\mathbf {A} +\mathbf {B} \right)^{\operatorname {T} }=\mathbf {A} ^{\operatorname {T} }+\mathbf {B} ^{\operatorname {T} }\)
- \(\left(\mathbf {AB} \right)^{\operatorname {T} }=\mathbf {B} ^{\operatorname {T} }\mathbf {A} ^{\operatorname {T} }\)
- $ \left(\mathbf {A} ^{\operatorname {T} }\right)^{-1}=\left(\mathbf {A} ^{- 1}\right)^{\operatorname {T} }$
- \(\mathbf {u} \otimes \mathbf {v} = \mathbf {u} \mathbf {v} ^ {\operatorname {T}} = {\begin{bmatrix}u_{1} \\ u_{2} \\ u_{3} \\ u_{4} \end{bmatrix}} {\begin{bmatrix} v_{1}&v_{2}&v_{3} \end{bmatrix}} = {\begin{bmatrix} u_{1}v_{1} & u_{1}v_{2} & u_{1}v_{3} \\ u_{2}v_{1} & u_{2}v_{2} & u_{2}v_{3} \\ u_{3}v_{1} & u_{3}v_{2} & u_{3}v_{3} \\ u_{4}v_{1} & u_{4}v_{2} & u_{4}v_{3} \end{bmatrix}}\)
- \(\det {\begin{bmatrix} a & b \\ c & d \end{bmatrix}} = ad-bc\)
2.9 概率统计的表达
LaTex
$\Pr(A\vert B)={\frac {\Pr(B\vert A)\Pr(A)}{\Pr(B)}}$
$ f_{X\vert Y=y}(x)={\frac {f_{X,Y}(x,y)}{f_{Y}(y)}}$
$\operatorname {var} (X) = \operatorname {E} \left[X^{2}\right]-\operatorname {E} [X]^{2}$
$\operatorname {var} (aX+bY)=a^{2}\operatorname {var} (X) + b^{2}\operatorname {var} (Y) + 2ab \operatorname {cov} (X,Y)$
$\operatorname {E} [X]=\int _{- \infty }^{\infty }xf_{X}(x) \operatorname {d}x$
$ X\sim N(\mu ,\sigma ^{2})$
$\frac {\exp \left(-{\frac {1}{2}}\left({\mathbf {x} }-{\boldsymbol {\mu }}\right)^{\mathrm {T} }{\boldsymbol {\Sigma }}^{-1}\left({\mathbf {x} }- {\boldsymbol {\mu }}\right)\right)}{\sqrt {(2\pi )^{k}|{\boldsymbol {\Sigma }}|}}$
相关样式展示
- \(\Pr(A\vert B)={\frac {\Pr(B\vert A)\Pr(A)}{\Pr(B)}}\)
- $ f_{X\vert Y=y}(x)={\frac {f_{X,Y}(x,y)}{f_{Y}(y)}}$
- \(\operatorname {var} (X) = \operatorname {E} \left[X^{2}\right]-\operatorname {E} [X]^{2}\)
- \(\operatorname {var} (aX+bY)=a^{2}\operatorname {var} (X) + b^{2}\operatorname {var} (Y) + 2ab \operatorname {cov} (X,Y)\)
- \(\operatorname {E} [X]=\int _{- \infty }^{\infty }xf_{X}(x) \operatorname {d}x\)
- $ X\sim N(\mu ,\sigma ^{2})$
- \(\frac {\exp \left(-{\frac {1}{2}}\left({\mathbf {x} }-{\boldsymbol {\mu }}\right)^{\mathrm {T} }{\boldsymbol {\Sigma }}^{-1}\left({\mathbf {x} }- {\boldsymbol {\mu }}\right)\right)}{\sqrt {(2\pi )^{k}|{\boldsymbol {\Sigma }}|}}\)
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