91Ó°ÊÓ

Sketch the regions associated with the following inequalities. Determine if the region is open, closed, bounded, or compact. (a) \(|z| \leq 1\) (b) \(|2 z+1+i|<4\) (c) \(\operatorname{Re} z \geq 4\) (d) \(|z| \leq|z+1|\) (e) \(0<|2 z-1| \leq 2\)

Sketch the following regions. Determine if they are connected, and what the closure of the region is if they are not closed. (a) \(0<\arg z \leq \pi\) (b) \(0 \leq \arg z<2 \pi\) (c) \(\operatorname{Re} z>0\) and \(\operatorname{Im} z>0\) (d) \(\operatorname{Re}\left(z-z_{0}\right)>0\) and \(\operatorname{Re}\left(z-z_{1}\right)<0\) for two complex numbers \(z_{0}, z_{1}\) (e) \(|z|<\frac{1}{2} \quad\) and \(\quad|2 z-4| \leq 2\)

Show that the functions \(\operatorname{Re} z\) and \(\operatorname{Im} z\) are nowhere differentiable.