# Midterm Summer 2011

1.

a) Find all complex numbers z satisfying
$(2/\bar{z}+1)^3=1$

Write the answers in the form a+i b
b) Sketch the set of complex numbers satisfying
$|2/\bar{z}+1|^3=1$
Is this set open?
Closed?
Connected?
Does it contain points not in its boundary?

2. a) Find the radius of convergence of
$\sum\limits_{n=0}^{\infty} e^{(1+i)n} z^n$
b) Compute
$\sum\limits_{n=1}^{\infty} n \left(\frac{1+2i}{5}\right)^{n-1}$
or show that the sum diverges.

3.
a) Evaluate the integral
$\int_{|z-3 i|=2} \frac{e^{\pi z/2}}{z^2+4} dz$
where the integration is along the circle |z-3 i|=2 traversed counter-clockwise.
b) Evaluate the integral
$\int_{|z+3 i|=2} \frac{e^{\pi z/2}}{z^2+4} dz$
where the integration is along the circle |z+3 i|=2 traversed counter-clockwise.
c) Evaluate the integral
$\int_{C} \frac{e^{\pi z/2}}{z^2+4} dz$
where C is the curve in the following picture:

Midterm.pdf

4. a) Let

$f(z)= \begin{cases} \frac{|z|^2-1}{z-1} & \text{ if z\neq 1 } \\ 2 & \text{ if z=1 } \end{cases}$
Determine whether the function f(z) is continuous at the point z=1 (explain why or why not).
b) Let
$g(z)=z^2-|z|^2$
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
$\{u+iv|u^2+v^2=1\}$
Show that f is constant.

Solutions:

midterm (sol).pdf