1. ## Complex Mapping

I want to make sure if I'm on the right track with approach I used so far for part (b) of this problem:
"Sketch the region onto which the sector $r\leq1$, $0\leq\theta\leq\frac{\pi}{4}$ is mapped by the transformation:
(a) $w = z^2$
(b) $w = z^3$
(c) $w = z^4$"

I assumed $r\leq1$ is same as $0 \leq r \leq 1$.

(b) $w = z^3 = (re^{i\theta})^3 = r^3e^{i3\theta} = r^3(\cos3\theta + i\sin3\theta) = r^3\cos3\theta+ir^3\sin3\theta$
$u(r,\theta) = r^3\cos3\theta$ and $v(r,\theta) = r^3\sin3\theta$
$\rho = r^3 \Longrightarrow 0^3 \leq \rho \leq 1^3 \Longrightarrow0 \leq \rho \leq 1$
$\phi = 3\theta \Longrightarrow 0 \leq 3\theta \leq \frac{3\pi}{4} \Longrightarrow 0 \leq \phi \leq \frac{3\pi}{4}$
What I plotted is in the attachment. Sounds right?

2. Looks good to me.