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## Mandelbrot set
The set of complex numbers c for which the function
$
z_{0} = 0, \qquad z_{n+1} = z_{n}^{2} + c, \quad c \in \mathbb{C}.
$
doesn't diverge
$
M = \left\{\, c \in \mathbb{C} \;\middle|\; \limsup_{n \to \infty} |z_n| \leq 2 \,\right\}.
$
e.g.
$
\begin{align}
z_{0} &= 0 \\
z_{1} &= z_{0}^{2} + c = 0 + c = c \\
z_{2} &= z_{1}^{2} + c = c^2 + c \\
z_{3} &= z_{2}^{2} + c = (c^2 + c)^2 + c \\
\end{align}
$
if
$
\begin{align}
c &= \underbrace{a}_{實部}+\underbrace{b}_{虛部}i \\
c^2 &= (a+bi)^2 \\
&= \underbrace{(a^2-b^2)}_{實部}+\underbrace{2ab}_{虛部}i \\
c^2+c &= (a+bi)^2 + a+bi \\
&= \underbrace{(a^2-b^2+a)}_{實部} + \underbrace{(2ab+b)}_{虛部}i
\end{align}
$
## Julia Set
Recall the Mandelbrot Set:
$
z_{0} = 0, \qquad z_{n+1} = z_{n}^{2} + c, \quad c \in \mathbb{C}.
$
and the C is a fixed complex number
e.g.