1.

a) Find all complex numbers z satisfying


Write the answers in the form a+i b
b) Sketch the set of complex numbers satisfying

Is this set open?
Closed?
Connected?
Does it contain points not in its boundary?



2. a) Find the radius of convergence of

b) Compute

or show that the sum diverges.


3.
a) Evaluate the integral

where the integration is along the circle |z-3 i|=2 traversed counter-clockwise.
b) Evaluate the integral

where the integration is along the circle |z+3 i|=2 traversed counter-clockwise.
c) Evaluate the integral

where C is the curve in the following picture:

Midterm.pdf

4. a) Let


Determine whether the function f(z) is continuous at the point z=1 (explain why or why not).
b) Let

Determine all the points where g'(z) exists. Find g'(z) at each such point.


5. Suppose that the function f is analytic (i.e. complex-differentiable) at all points of the complex plane. Suppose also that the image of f is contained in the unit circle

Show that f is constant.


Solutions:

midterm (sol).pdf