markdown数学公式

tony

	算式	markdown
分数，平方	\frac{7x+5}{1+y^2}	\frac{7x+5}{1+y²}
下标	z=z_l	z=z_l
省略号	\cdots	\cdots
开根号	\sqrt{2};\sqrt[n]{3}	\sqrt{2};\sqrt[n]{3}
矢量	\vec{a} \cdot \vec{b}=0	\vec{a} \cdot \vec{b}=0
微积分
	\int ^2_3 x^2 {\rm d}x	\int ²_3 x² {\rm d}x
	\iint	\iint
	\iiint	\iiint
	\oint	\oint
	\mathrm{d}	\mathrm{d}
	\prime	\prime
	\lim	\lim
	\infty	\infty
	\partial	\partial
	\sum	\suml
	\int ^2_3 x^2 {\rm d}x	\int ²_3 x² {\rm d}xl
极限	\lim_{n\rightarrow+\infty} n	\lim_{n\rightarrow+\infty} nl
累加	\sum \frac{1}{i^2}	\sum \frac{1}{i²}l
累乘	\prod \frac{1}{i^2}	\prod \frac{1}{i²}l

矩阵
\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{matrix} \tag{1}

带括号的矩阵
\left[ \begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{matrix} \right]\tag{2}

对数函数

算式	markdown
\ln15	\ln15
\log_2^{10}	\log_2^{10}
\lg7	\lg7

运算符号

运算符	markdown
±\pm	\pm
×	imes
÷	\div
∑	\sum
∏ \prod	\prod
≠	\neq
≤	\leq
≥	\geq

大写	markdown	小写	markdown
A	A	α	\alpha
B	B	β	\beta
Γ	\Gamma	γ	\gamma
Δ	\Delta	δ	\delta
E	E	ϵ	\epsilon
ε	\varepsilon
Z	Z	ζ	\zeta
H	H	η	\eta
Θ	\Theta	θ	heta
I	I	ι	\iota
K	K	κ	\kappa
Λ	\Lambda	λ	\lambda
M	M	μ	\mu
N	N	ν	\nu
Ξ	\Xi	ξ	\xi
O	O	ο	\omicron
Π	\Pi	π	\pi
P	P	ρ	\rho
Σ	\Sigma	σ	\sigma

示例

\begin{aligned} a_0&=\frac{1}{\pi}\int\limits_{-\pi}^{\pi}f(x)\,\mathrm{d}x\\[6pt] a_n&=\frac{1}{\pi}\int\limits_{-\pi}^{\pi}f(x)\cos nx\,\mathrm{d}x=\\ &=\frac{1}{\pi}\int\limits_{-\pi}^{\pi}x^2\cos nx\,\mathrm{d}x\\[6pt] b_n&=\frac{1}{\pi}\int\limits_{-\pi}^{\pi}f(x)\sin nx\,\mathrm{d}x=\\ &=\frac{1}{\pi}\int\limits_{-\pi}^{\pi}x^2\sin nx\,\mathrm{d}x \\[6pt] \end{aligned}

分段函数

f(x) = \left\{ \begin{array}{lr} x^2 & : x < 0\\ x^3 & : x \ge 0 \end{array} \right.

u(x) = \begin{cases} \exp{x} & \text{if } x \geq 0 \\ 1 & \text{if } x < 0 \end{cases}

方程组

\left\{ \begin{array}{c} a_1x+b_1y+c_1z=d_1 \\ a_2x+b_2y+c_2z=d_2 \\ a_3x+b_3y+c_3z=d_3 \end{array} \right.

线性模型

h(\theta) = \sum_{j = 0} ^n \theta_j x_j

均方误差
J(\theta) = \frac{1}{2m}\sum_{i = 0} ^m(y^i - h_\theta (x^i))^2

批量梯度下降
\frac{\partial J(\theta)}{\partial\theta_j}=-\frac1m\sum_{i=0}^m(y^i-h_\theta(x^i))x^i_j

推导过程
\begin{aligned} \frac{\partial J(\theta)}{\partial\theta_j} & = -\frac1m\sum_{i=0}^m(y^i-h_\theta(x^i)) \frac{\partial}{\partial\theta_j}(y^i-h_\theta(x^i)) \\ & = -\frac1m\sum_{i=0}^m(y^i-h_\theta(x^i)) \frac{\partial}{\partial\theta_j}(\sum_{j=0}^n\theta_jx_j^i-y^i) \\ & = -\frac1m\sum_{i=0}^m(y^i-h_\theta(x^i))x^i_j \end{aligned}

case环境的使用
a = \begin{cases} \int x\, \mathrm{d} x\\ b^2 \end{cases}

带方框的等式
\begin{aligned} \boxed{x^2+y^2 = z^2} \end{aligned}

最大（最小）操作符
\begin{gathered} \operatorname{arg\,max}_a f(a) = \operatorname*{arg\,max}_b f(b) \\ \operatorname{arg\,min}_c f(c) = \operatorname*{arg\,min}_d f(d) \end{gathered}

求极限
\begin{aligned} \lim_{a\to \infty} \tfrac{1}{a} \end{aligned}
\begin{aligned} \lim\nolimits_{a\to \infty} \tfrac{1}{a} \end{aligned}

求积分
\begin{aligned} \int_a^b x^2 \mathrm{d} x \end{aligned}

\begin{aligned} \int\limits_a^b x^2 \mathrm{d} x \end{aligned}

求累加
\begin{aligned} \sum_{n=1}\nolimits' C_n \end{aligned}

求累乘
\prod_{{ \begin{gathered} 1\le i \le n\\ 1\le j \le m \end{gathered} }} M_{i,j}

开根号
\sqrt x * \sqrt[3] x * \sqrt[-1] x

省略号的使用
{1+2+3+\ldots+n}