# Quiz 1 Summer 2011

1.

Find all complex numbers z satisfying
$(z+i)^4=-4$
Do not use trigonometric or exponential functions in your answer.

Solution:
$z+i=(4 (\cos(\pi i) + i \sin(\pi i)) )^{1/4}= \sqrt 2 (\cos (\pi i/4+2\pi i k / 4)+ i \sin (\pi i/4+2\pi i k / 4))=1+i,-1+i,-1-i,1-i$
hence
$z=1,-1,-1-2i,1-2i$

2.
Is the set
$\{z: z \bar z - 2 \geq 0 \mbox{ and } \mbox{Im } z \neq 0\}$
open, closed or neither open nor closed? Is it connected? Sketch its boundary and state whether it is open, closed or neither open nor closed.

Solution:
The set is neither open nor closed. It is not connected. The boundary is the union of the circle
$|z|=\sqrt 2$
with the horizontal rays
$\mbox{Re }z\geq \sqrt 2,\mbox{ Im }z=0$
and
$\mbox{Re }z\leq -\sqrt 2,\mbox{ Im }z=0$
The boundary is closed.

3.
Sketch the preimage of the set
$\{z: \mbox{Im }z\geq 1, \mbox{Re }z \geq 1, |z - (1+i)|\leq 2\}$
under the map f(z)=z^2+1+i. Is it connected?

Solution:
The preimage is the union of the two sets
$\{z| |z|\leq\sqrt 2,0\leq\mbox{arg z} \leq \pi/4\}$
and
$\{z| |z|\leq\sqrt 2,\pi\leq\mbox{arg z} \leq \pi+\pi/4\}$