1.

Find all complex numbers z satisfying

Do not use trigonometric or exponential functions in your answer.

Solution:

hence


2.
Is the set

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

with the horizontal rays

and

The boundary is closed.

3.
Sketch the preimage of the set

under the map f(z)=z^2+1+i. Is it connected?

Solution:
The preimage is the union of the two sets

and