一个MATLAB 的程序学习

$$
z = \overbrace{
   \underbrace{x}_\text{real} + i
   \underbrace{y}_\text{imaginary}
  }^\text{complex number}
$$

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

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

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

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

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

$$
\frac{\partial J(\theta)}{\partial\theta_j}=-\frac1m\sum_{i=0}^m(y^i-h_\theta(x^i))x^i_j
$$

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

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

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

$$
\sqrt x * \sqrt[3] x * \sqrt[-1] x
$$
$$\dot{\sigma}=\beta*\pi$$