TITLE: Visualization of complex functions
QUESTION [7 upvotes]: How can I find the image of a rectangle 

$$a<\Re(z)<b \\c<\Im(z)<d$$

under the function $z^3$ , I tried breaking up $z$ into $x+iy$ and cubing it. I got the real and imaginary parts but how can I plot the function under given domain.

REPLY [2 votes]: Let $\phi_i(t) : [0,1] \to \mathbb{C}, \quad i = 1..4$ be defined by
$$\begin{align*}\phi_1(t) & = (b-a)t + a + ci\\
\phi_2(t) & = b + ((d-c)t + c)i \\
\phi_3(t) & = (a-b)t + b + di \\
\phi_4(t) & = a + ((c-d)t + d)i
\end{align*}$$
Then $\phi: [0,4] \to \mathbb{C}$ defined by $\phi(t-k) = \phi_{k+1}(t)$ for suitable $k\in\{0,1,2,3\}$ is a parameterisation of the boundary of your rectangle.

Can you figure out the rest?