Quiz+1+Summer+2011

1.

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

Solution: math 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 math hence math z=1,-1,-1-2i,1-2i math

2. Is the set math \{z: z \bar z - 2 \geq 0 \mbox{ and } \mbox{Im } z \neq 0\} math 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 math math with the horizontal rays math \mbox{Re }z\geq \sqrt 2,\mbox{ Im }z=0 math and math \mbox{Re }z\leq -\sqrt 2,\mbox{ Im }z=0 math The boundary is closed.
 * z|=\sqrt 2

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

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