数学公式备忘

2024-03-17

4 min read

math

用 Sympy 求解是不错的选择

微积分

导数

\[\begin{aligned} (\operatorname{tg} x)^{\prime} & =\sec ^2 x & (\arcsin x)^{\prime} & =\frac{1}{\sqrt{1-x^2}} \\ (\operatorname{ctg} x)^{\prime} & =-\csc ^2 x & (\arccos x)^{\prime} & =-\frac{1}{\sqrt{1-x^2}} \\ (\sec x)^{\prime} & =\sec x \cdot \operatorname{tg} x & (\operatorname{arctg} x)^{\prime} & =\frac{1}{1+x^2} \\ (\csc x)^{\prime} & =-\csc x \cdot \operatorname{ctg} x & (\operatorname{arcctg} x)^{\prime} & =-\frac{1}{1+x^2} \\ \left(a^x\right)^{\prime} & =a^x \ln a & & \end{aligned} \]

运算法则（导数表）：

\[\begin{aligned} \frac{\mathrm{d}}{\mathrm{d} x}(f+g) & =\frac{\mathrm{d} f}{\mathrm{~d} x}+\frac{\mathrm{d} g}{\mathrm{~d} x} & \frac{\mathrm{d}}{\mathrm{d} x}[f(g(x))] & =\frac{\mathrm{d} f}{\mathrm{~d} g} \cdot \frac{\mathrm{d} g}{\mathrm{~d} x}=f^{\prime}(g(x)) \cdot g^{\prime}(x) \\ \frac{\mathrm{d}}{\mathrm{d} x}(c \cdot f) & =c \cdot \frac{\mathrm{d} f}{\mathrm{~d} x} & \frac{\mathrm{d}^n}{\mathrm{~d} x^n} f(x) g(x) & =\sum_{i=0}^n\left(\begin{array}{c} n \\ i \end{array}\right) f^{(n-i)}(x) g^{(i)}(x) \\ \frac{\mathrm{d}}{\mathrm{d} x}(f \cdot g) & =f \cdot \frac{\mathrm{d} g}{\mathrm{~d} x}+g \cdot \frac{\mathrm{d} f}{\mathrm{~d} x} & \frac{\mathrm{d}}{\mathrm{d} x}\left(\frac{1}{f}\right) & =-\frac{f^{\prime}}{f^2} \\ \frac{\mathrm{d}}{\mathrm{d} x}\left(\frac{f}{g}\right) & =\frac{-f \cdot \frac{\mathrm{d} g}{d x}+g \cdot \frac{\mathrm{d} f}{\mathrm{~d} x}}{g^2} & & \end{aligned} \]

积分

参考：常用积分表 / 积分表 /

\[\begin{array}{ll} \int k \mathrm{~d} x=k x+C(k \text { 是常数 }), & \int \frac{\mathrm{d} x}{\sqrt{1-x^2}}=\arcsin x+C, \\ \int x^{\prime \prime} \mathrm{d} x=\frac{x^{\mu+1}}{\mu+1}+C(\mu \neq-1), & \int a^x \mathrm{~d} x=\frac{a^x}{\ln a}+C, \\ \int \frac{\mathrm{d} x}{x}=\ln |x|+C, & \int \cos x \mathrm{~d} x=\sin x+C, \\ \int \frac{\mathrm{d} x}{1+x^2}=\arctan x+C, & \int \sin x \mathrm{~d} x=-\cos x+C, \end{array} \]

运算法则：

\[\begin{aligned} \int c \cdot f(x) \mathrm{d} x & =c \cdot \int f(x) \mathrm{d} x \\ \int(f(x) \pm g(x)) \mathrm{d} x & =\int f(x) \mathrm{d} x \pm \int g(x) \mathrm{d} x \\ \int u d v & =u v-\int v d u \end{aligned} \]

泰勒级数

\[f(a)+\frac{f^{\prime}(a)}{1 !}(x-a)+\frac{f^{\prime \prime}(a)}{2 !}(x-a)^2+\frac{f^{\prime \prime \prime}(a)}{3 !}(x-a)^3+\cdots=\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n !}(x-a)^n \]

参考：泰勒公式 / ¹ / Wiki /

\[\begin{aligned} \ln (1+x) & =\sum_{k=1}^{\infty}(-1)^{k+1} \frac{x^k}{k}(-1 \lt x \leq 1) & \ln (1-x) & =-\sum_{k=1}^{\infty} \frac{x^k}{k}(-1 \leq x \lt 1) \\ \frac{1}{1-x} & =\sum_{n=0}^{\infty} x^n & \frac{1}{1+x} & =\sum_{n=0}^{\infty}(-1)^n x^n \\ e^x & =\sum_{k=0}^{\infty} \frac{x^k}{k !} & \sin x & =\sum_{n=0}^{\infty}(-1)^n \frac{x^{2 n+1}}{(2 n+1) !} \end{aligned} \]

复变函数与积分变换

拉普拉斯变换

参考：Wiki / ² /

\[\begin{gathered}\mathscr{L}[f(t)]=F(s)=\int_0^{\infty} f(t) \mathrm{e}^{-t} \mathrm{~d} t \\ \mathscr{L}^{-1}[F(s)]=f(t)=\frac{1}{2 \pi \mathrm{j}} \int_{\sigma-\mathrm{j} \infty}^{\sigma+\mathrm{j} \infty} F(s) \mathrm{e}^{s t} \mathrm{~d} s\end{gathered} \]

运算法则

\[\begin{array}{|c|c|} \hline \text { Time domain } & s \text { domain } \\ \hline a f(t)+b g(t) & a F(s)+b G(s) \\ \hline t f(t) & -F^{\prime}(s) \\ \hline t^n f(t) & (-1)^n F^{(n)}(s) \\ \hline f^{\prime}(t) & s F(s)-f\left(0^{-}\right) \\ \hline f^{\prime \prime}(t) & s^2 F(s)-s f\left(0^{-}\right)-f^{\prime}\left(0^{-}\right) \\ \hline f^{(n)}(t) & s^n F(s)-\sum_{k=1}^n s^{n-k} f^{(k-1)}\left(0^{-}\right) \\ \hline \frac{1}{t} f(t) & \int_s^{\infty} F(\sigma) d \sigma \\ \hline \int_0^t f(\tau) d \tau=(u * f)(t) & \frac{1}{s} F(s) \\ \hline e^{a t} f(t) & F(s-a) \\ \hline f(t-a) u(t-a) & e^{-a s} F(s) \\ \hline f(t) u(t-a) & e^{-a s} \mathcal{L}\{f(t+a)\} \\ \hline f(a t) & \frac{1}{a} F\left(\frac{s}{a}\right) \\ \hline f(t) g(t) & \frac{1}{2 \pi i} \lim _{T \rightarrow \infty} \int_{c-i T}^{c+i T} F(\sigma) G(s-\sigma) d \sigma \\ \hline(f * g)(t)=\int_0^t f(\tau) g(t-\tau) d \tau & F(s) \cdot G(s) \\ \hline(f * g)(t)=\int_0^T f(\tau) g(t-\tau) d \tau & F(s) \cdot G(s) \\ \hline f^*(t) & F^*\left(s^*\right) \\ \hline(f \star g)(t)=\int_0^{\infty} f(\tau)^* g(t+\tau) d \tau & F^*\left(-s^*\right) \cdot G(s) \\ \hline f(t) & \frac{1}{1-e^{-T s}} \int_0^T e^{-s t} f(t) d t \\ \hline \begin{array}{l} f_P(t)=\sum_{n=0}^{\infty} f(t-T n) \\ f_P(t)=\sum_{n=0}^{\infty}(-1)^n f(t-T n) \end{array} & \begin{array}{l} F_P(s)=\frac{1}{1-e^{-T s}} F(s) \\ F_P(s)=\frac{1}{1+e^{-T s}} F(s) \end{array} \\ \hline \end{array} \]

常见变换

\[\begin{array}{|c|c|c|} \hline \begin{array}{c} \text { Time domain } \\ f(t)=\mathcal{L}^{-1}\{F(s)\} \end{array} & \begin{array}{l} \text { Laplace s-domain } \\ F(s)=\mathcal{L}\{f(t)\} \end{array} & \begin{array}{l} \text { Region of } \\ \text { convergence } \end{array} \\ \hline \delta(t) & 1 & \text { all } s \\ \hline \delta(t-\tau) & e^{-\tau s} & \\ \hline u(t) & \frac{1}{s} & \operatorname{Re}(s)>0 \\ \hline u(t-\tau) & \frac{1}{s} e^{-\tau s} & \operatorname{Re}(s)>0 \\ \hline f(t-\tau) u(t-\tau) & e^{-s \tau} \mathcal{L}\{f(t)\} & \\ \hline u(t)-u(t-\tau) & \frac{1}{s}\left(1-e^{-\tau s}\right) & \operatorname{Re}(s)>0 \\ \hline t \cdot u(t) & \frac{1}{s^2} & \operatorname{Re}(s)>0 \\ \hline t^n \cdot u(t) & \frac{n !}{s^{n+1}} & \begin{array}{c} \operatorname{Re}(s)>0 \\ (n>-1) \end{array} \\ \hline t^q \cdot u(t) & \frac{\Gamma(q+1)}{s^{q+1}} & \begin{array}{c} \operatorname{Re}(s)>0 \\ \operatorname{Re}(q)>-1 \end{array} \\ \hline \sqrt[n]{t} \cdot u(t) & \frac{1}{s^{\frac{1}{n}+1}} \Gamma\left(\frac{1}{n}+1\right) & \operatorname{Re}(s)>0 \\ \hline t^n e^{-\alpha t} \cdot u(t) & \frac{n !}{(s+\alpha)^{n+1}} & \operatorname{Re}(s)>-\alpha \\ \hline(t-\tau)^n e^{-\alpha(t-\tau)} \cdot u(t-\tau) & \frac{n ! \cdot e^{-\tau s}}{(s+\alpha)^{n+1}} & \operatorname{Re}(s)>-\alpha \\ \hline e^{-\alpha t} \cdot u(t) & \frac{1}{s+\alpha} & \operatorname{Re}(s)>-\alpha \\ \hline e^{-\alpha|t|} & \frac{2 \alpha}{\alpha^2-s^2} & -\alpha<\operatorname{Re}(s)<\alpha \\ \hline\left(1-e^{-\alpha t}\right) \cdot u(t) & \frac{\alpha}{s(s+\alpha)} & \operatorname{Re}(s)>0 \\ \hline \sin (\omega t) \cdot u(t) & \frac{\omega}{s^2+\omega^2} & \operatorname{Re}(s)>0 \\ \hline \cos (\omega t) \cdot u(t) & \frac{s}{s^2+\omega^2} & \operatorname{Re}(s)>0 \\ \hline \sinh (\alpha t) \cdot u(t) & \frac{\alpha}{s^2-\alpha^2} & \operatorname{Re}(s)>|\alpha| \\ \hline \cosh (\alpha t) \cdot u(t) & \frac{s}{s^2-\alpha^2} & \operatorname{Re}(s)>|\alpha| \\ \hline e^{-\alpha t} \sin (\omega t) \cdot u(t) & \frac{\omega}{(s+\alpha)^2+\omega^2} & \operatorname{Re}(s)>-\alpha \\ \hline e^{-\alpha t} \cos (\omega t) \cdot u(t) & \frac{s+\alpha}{(s+\alpha)^2+\omega^2} & \operatorname{Re}(s)>-\alpha \\ \hline \ln (t) \cdot u(t) & -\frac{1}{s}[\ln (s)+\gamma] & \operatorname{Re}(s)>0 \\ \hline J_n(\omega t) \cdot u(t) & \frac{\left(\sqrt{s^2+\omega^2}-s\right)^n}{\omega^n \sqrt{s^2+\omega^2}} & \begin{array}{c} \operatorname{Re}(s)>0 \\ (n>-1) \end{array} \\ \hline \operatorname{erf}(t) \cdot u(t) & \frac{1}{s} e^{(1 / 4) s^2}\left(1-\operatorname{erf} \frac{s}{2}\right) & \operatorname{Re}(s)>0 \\ \hline \end{array} \]

拉氏变换与积分

参考 / ³，利用拉氏变换定义可证明如下公式。只有反常积分存在且收敛才可套用如下公式：

\[\begin{aligned} &\int_0^{+\infty} f(t) \mathrm{d} t=F(0)\\ &\int_0^{+\infty} t f(t) \mathrm{d} t=-F^{\prime}(0)\\ &\int_0^{+\infty} \frac{f(t)}{t} \mathrm{~d} t=\int_0^{\infty} F(s) \mathrm{d} s \end{aligned} \]

线代&矩阵

参考：线性代数公式定理一览表 /

行列式

代数余子式展开（$A_{i j}=(-1)^{i+j} M_{i j}$）：

\[\begin{aligned} &D=a_{i 1} A_{i 1}+a_{i 2} A_{i 2}+\cdots+a_{i n} A_{i n}=\sum_{k=1}^n a_{i k} A_{i k} \quad(i=1,2, \ldots, n)\\ &D=a_{1 j} A_{1 j}+a_{2 j} A_{2 j}+\cdots+a_{n j} A_{n j}=\sum_{k=1}^n a_{k j} A_{k j} \quad(j=1,2, \ldots, n) \end{aligned} \]

范德蒙德

\[D_n=\left|\begin{array}{cccc} 1 & 1 & \ldots & 1 \\ x_1 & x_2 & \ldots & x_n \\ x_1^2 & x_2^2 & \ldots & x_n^2 \\ \vdots & \vdots & & \vdots \\ x_1^{n-1} & x_2^{n-1} & \ldots & x_n^{n-1} \end{array}\right|=\prod_{1 \leq i \lt j \leq n}\left(x_j-x_i\right) \]

矩阵转置&逆&正交

转置

$(\lambda A)^T=\lambda A^T$
$(A \pm B)^T=A^T \pm B^T$
$(A B)^T=B^T A^T$
设 $A$ 是 $n$ 阶方阵, 则 $\left|A^T\right|=|A|$
$\left(A^T\right)^T=A$

逆

当 $\lambda$ 是非零实数时, $\lambda A$ 也可逆, 且 $(\lambda A)^{-1}=\frac{1}{\lambda} A^{-1}$
$A B$ 可逆, 且 $(A B)^{-1}=B^{-1} A^{-1}$
$A^T$ 可逆, 且 $\left(A^T\right)^{-1}=\left(A^{-1}\right)^T$
$A^{-1}$ 可逆, 且 $\left(A^{-1}\right)^{-1}=A,\left|A^{-1}\right|=\frac{1}{|A|}$
$A^*$ 可逆, 且 $\left(A^*\right)^{-1}=\left(A^{-1}\right)^*=\frac{A}{|A|}, A^*=|A| A^{-1},\left|A^*\right|=|A|^{n-1}$

已知 $A$ 为正交矩阵则

$A^{-1}=A^T$ 也是正交阵;
$|A|= \pm 1$
若 $B$ 也是正交阵, 则 $A B$ 也是正交阵;
$A$ 的行（列）向量组构成 $R^n$ 的一个标准正交基.

LaTeX

$\LaTeX$（ $\LaTeX$ ）示例，更多细节可以参考第三章

希腊字符

LaTex	$\LaTeX$	LaTex	$\LaTeX$	LaTex	$\LaTeX$	LaTex	$\LaTeX$
`\alpha`	$\alpha$	`\nu`	$\nu$	`\digamma`	$\digamma$	`\Pi`	$\Pi$
`\beta`	$\beta$	`\kappa`	$\kappa$	`\varkappa`	$\varkappa$	`\Psi`	$\Psi$
`\gamma`	$\gamma$	`\lambda`	$\lambda$	`\varpi`	$\varpi$	`\varrho`	$\varrho$
`\delta`	$\delta$	`\mu`	$\mu$	`\varsigma`	$\varsigma$	`\Sigma`	$\Sigma$
`\epsilon`	$\epsilon$	`\zeta`	$\zeta$	`\eth`	$\eth$	`\phi`	$\phi$
`\eta`	$\eta$	`\theta`	$\theta$	`\upsilon`	$\upsilon$	`\varphi`	$\varphi$
`\iota`	$\iota$	`\xi`	$\xi$	`\chi`	$\chi$	`\psi`	$\psi$
`\pi`	$\pi$	`\rho`	$\rho$	`\omega`	$\omega$	`\vartheta`	$\vartheta$
`\sigma`	$\sigma$	`\tau`	$\tau$	`\Omega`	$\Omega$	`\hbar`	$\hbar$
`\Gamma`	$\Gamma$	`\Delta`	$\Delta$	`\varepsilon`	$\varepsilon$	`\Upsilon`	$\Upsilon$
`\Lambda`	$\Lambda$	`\Phi`	$\Phi$	`\Xi`	$\Xi$	`\Theta`	$\Theta$

其他符号

LaTex	$\LaTeX$	LaTex	$\LaTeX$	LaTex	$\LaTeX$	LaTex	$\LaTeX$
`\partial`	∂	`\infty`	$\infty$	`\nrightarrow`	$\nrightarrow$	`\mapsto`	$\mapsto$
`\wedge`	∧	`\vee`	$\vee$	`\geq`	$\geq$	`\leq`	$\leq$
`\neg`	$\neg$	`\pm`	$\pm$	`\equiv`	$\equiv$	`\sim`	$\sim$
`\bot`	$\bot$	`\top`	⊤	`\gg`	$\gg$	`\ll`	$\ll$
`\nabla`	$\nabla$	`\varnothing`	∅	`\subset`	⊂	`\subseteq`	$\subseteq$
`\angle`	$\angle$	`\measuredangle`	$\measuredangle$	`\in`	∈	`\notin`	$\notin$
`\surd`	$\surd$	`\forall`	$\forall$	`\mid`	$\mid$	`\propto`	$\propto$
`\exists`	$\exists$	`\nexists`	$\nexists$	`\perp`	$\perp$
`\hookrightarrow`	↪	`\Rightarrow`	⇒	`\vartriangle`	$\vartriangle$
`\int`	$\int$	`\iint`	$\iint$	`\bigcup`	$\bigcup$
`\iiint`	$\iiint$	`\parallel`	$\parallel$	`\bigcap`	$\bigcap$	`a a\ b`	$a a\ b$
`\prod`	$\prod$	`\sum`	$\sum$	`\simeq`	$\simeq$	`a\quad b`	$a\quad b$
`\rightarrow`	$\rightarrow$	`\Leftrightarrow`	⇔	`\iff`	$\iff$	`a\qquad b`	$a\qquad b$

数学标记

LaTex	$\LaTeX$	英文读法	LaTex	$\LaTeX$	英文读法
`x'` / `x^{\prime}`	$x^{\prime}$	$x$ prime	`x_i`	$x_i$	$x$ subscript $\mathrm{i}, x$ sub $\mathrm{i}$
`x''`	$x^{\prime \prime}$	$x$ double prime	`x^i`	$x^i$	$x$ to the $\mathrm{n}, x$ to the nth
`\overrightarrow{AB}`	$\overrightarrow{A B}$	vec pointing from $A$ to $B$	`\ddot{x}`	$\ddot{x}$	$x$ double dot
`\underline{x}`	$\underline{x}$	$x$ underline	`x^*`	$x^*$	$x$ star, $x$ super asterisk
`\hat{x}`	$\hat{x}$	$x$ hat	`x\dagger`	$x \dagger$	$x$ dagger
`\bar{x}`	$\bar{x}$	$x$ bar	`x\ddagger`	$x \ddagger$	$x$ double dagger
`\dot{x}`	$\dot{x}$	$x$ dot	`{\color{red}x}`	${\color{red}x}$	red ${\color{red}x}$
`\tilde{x}`	$\tilde{x}$	$x$ tilde

字母样式

Latex	$\LaTeX$	说明
`{AaBbCc}`	${AaBbCc}$	斜体, 大部分数学符号、表达式
`\mathrm {AaBbCc}`	$\mathrm {AaBbCc}$	正体, 公式中的单位或文字
`\mathbf{AaBbCc}`	$\mathbf{AaBbCc}$	粗体, 向量、矩阵
`\boldsymbol {AaBbCc}`	$\boldsymbol {AaBbCc}$	粗体、斜体, 向量、矩阵
`\mathtt {AaBbCc}`	$\mathtt {AaBbCc}$	等宽字体, 常用于代码
`\mathcal {AaBbCc}`	$\mathcal{A B C D E F}$	花体, 用于表示数学中的集合、代数结构、算子
`\mathbb {AaBbCc}`	$\mathbb {AaBbCc}$	黑板粗体 (blackboard bold)，常用来表达各种集合
`\text {Aa Bb Cc}`	$\text {Aa Bb Cc}$	用来写公式中的文字
`\mathrm{d}x`	$\mathrm{d}x$	ISO 规定导数符号 $\mathrm{d}$ 为正体
`\operatorname{T}`	$\operatorname{T}$	运算符

Almendros, Ines N. Rodriguez . “Table of Integrals, Series, and Products.” (2009). ↩︎
郑君里, 应启珩, and 杨为理. 信号与系统.上册.3版. 高等教育出版社, 2011. ↩︎
李叶舟, and 刘文君. 复变函数及其应用. 北京邮电大学出版社, 2020. ↩︎

LaTex	\(\LaTeX\)	英文读法	LaTex	\(\LaTeX\)	英文读法
`x'` / `x^{\prime}`	\(x^{\prime}\)	\(x\) prime	`x_i`	\(x_i\)	\(x\) subscript \(\mathrm{i}, x\) sub \(\mathrm{i}\)
`x''`	\(x^{\prime \prime}\)	\(x\) double prime	`x^i`	\(x^i\)	\(x\) to the \(\mathrm{n}, x\) to the nth
`\overrightarrow{AB}`	\(\overrightarrow{A B}\)	vec pointing from \(A\) to \(B\)	`\ddot{x}`	\(\ddot{x}\)	\(x\) double dot
`\underline{x}`	\(\underline{x}\)	\(x\) underline	`x^*`	\(x^*\)	\(x\) star, \(x\) super asterisk
`\hat{x}`	\(\hat{x}\)	\(x\) hat	`x\dagger`	\(x \dagger\)	\(x\) dagger
`\bar{x}`	\(\bar{x}\)	\(x\) bar	`x\ddagger`	\(x \ddagger\)	\(x\) double dagger
`\dot{x}`	\(\dot{x}\)	\(x\) dot	`{\color{red}x}`	\({\color{red}x}\)	red \({\color{red}x}\)
`\tilde{x}`	\(\tilde{x}\)	\(x\) tilde

LaTex	\(\LaTeX\)	LaTex	\(\LaTeX\)	LaTex	\(\LaTeX\)	LaTex	\(\LaTeX\)
`\alpha`	\(\alpha\)	`\nu`	\(\nu\)	`\digamma`	\(\digamma\)	`\Pi`	\(\Pi\)
`\beta`	\(\beta\)	`\kappa`	\(\kappa\)	`\varkappa`	\(\varkappa\)	`\Psi`	\(\Psi\)
`\gamma`	\(\gamma\)	`\lambda`	\(\lambda\)	`\varpi`	\(\varpi\)	`\varrho`	\(\varrho\)
`\delta`	\(\delta\)	`\mu`	\(\mu\)	`\varsigma`	\(\varsigma\)	`\Sigma`	\(\Sigma\)
`\epsilon`	\(\epsilon\)	`\zeta`	\(\zeta\)	`\eth`	\(\eth\)	`\phi`	\(\phi\)
`\eta`	\(\eta\)	`\theta`	\(\theta\)	`\upsilon`	\(\upsilon\)	`\varphi`	\(\varphi\)
`\iota`	\(\iota\)	`\xi`	\(\xi\)	`\chi`	\(\chi\)	`\psi`	\(\psi\)
`\pi`	\(\pi\)	`\rho`	\(\rho\)	`\omega`	\(\omega\)	`\vartheta`	\(\vartheta\)
`\sigma`	\(\sigma\)	`\tau`	\(\tau\)	`\Omega`	\(\Omega\)	`\hbar`	\(\hbar\)
`\Gamma`	\(\Gamma\)	`\Delta`	\(\Delta\)	`\varepsilon`	\(\varepsilon\)	`\Upsilon`	\(\Upsilon\)
`\Lambda`	\(\Lambda\)	`\Phi`	\(\Phi\)	`\Xi`	\(\Xi\)	`\Theta`	\(\Theta\)

LaTex	\(\LaTeX\)	LaTex	\(\LaTeX\)	LaTex	\(\LaTeX\)	LaTex	\(\LaTeX\)
`\partial`	∂	`\infty`	\(\infty\)	`\nrightarrow`	\(\nrightarrow\)	`\mapsto`	\(\mapsto\)
`\wedge`	∧	`\vee`	\(\vee\)	`\geq`	\(\geq\)	`\leq`	\(\leq\)
`\neg`	\(\neg\)	`\pm`	\(\pm\)	`\equiv`	\(\equiv\)	`\sim`	\(\sim\)
`\bot`	\(\bot\)	`\top`	⊤	`\gg`	\(\gg\)	`\ll`	\(\ll\)
`\nabla`	\(\nabla\)	`\varnothing`	∅	`\subset`	⊂	`\subseteq`	\(\subseteq\)
`\angle`	\(\angle\)	`\measuredangle`	\(\measuredangle\)	`\in`	∈	`\notin`	\(\notin\)
`\surd`	\(\surd\)	`\forall`	\(\forall\)	`\mid`	\(\mid\)	`\propto`	\(\propto\)
`\exists`	\(\exists\)	`\nexists`	\(\nexists\)	`\perp`	\(\perp\)
`\hookrightarrow`	↪	`\Rightarrow`	⇒	`\vartriangle`	\(\vartriangle\)
`\int`	\(\int\)	`\iint`	\(\iint\)	`\bigcup`	\(\bigcup\)
`\iiint`	\(\iiint\)	`\parallel`	\(\parallel\)	`\bigcap`	\(\bigcap\)	`a a\ b`	\(a a\ b\)
`\prod`	\(\prod\)	`\sum`	\(\sum\)	`\simeq`	\(\simeq\)	`a\quad b`	\(a\quad b\)
`\rightarrow`	\(\rightarrow\)	`\Leftrightarrow`	⇔	`\iff`	\(\iff\)	`a\qquad b`	\(a\qquad b\)

Latex	\(\LaTeX\)	说明
`{AaBbCc}`	\({AaBbCc}\)	斜体, 大部分数学符号、表达式
`\mathrm {AaBbCc}`	\(\mathrm {AaBbCc}\)	正体, 公式中的单位或文字
`\mathbf{AaBbCc}`	\(\mathbf{AaBbCc}\)	粗体, 向量、矩阵
`\boldsymbol {AaBbCc}`	\(\boldsymbol {AaBbCc}\)	粗体、斜体, 向量、矩阵
`\mathtt {AaBbCc}`	\(\mathtt {AaBbCc}\)	等宽字体, 常用于代码
`\mathcal {AaBbCc}`	\(\mathcal{A B C D E F}\)	花体, 用于表示数学中的集合、代数结构、算子
`\mathbb {AaBbCc}`	\(\mathbb {AaBbCc}\)	黑板粗体 (blackboard bold)，常用来表达各种集合
`\text {Aa Bb Cc}`	\(\text {Aa Bb Cc}\)	用来写公式中的文字
`\mathrm{d}x`	\(\mathrm{d}x\)	ISO 规定导数符号 \(\mathrm{d}\) 为正体
`\operatorname{T}`	\(\operatorname{T}\)	运算符