Midterm+Summer+2011

1.

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

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

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

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

[|Midterm.pdf]

4. a) Let

math f(z)=

\begin{cases}

\frac{|z|^2-1}{z-1} & \text{ if $z\neq 1$ }

\\

2 & \text{ if $z=1$ }

\end{cases}

math Determine whether the function f(z) is continuous at the point z=1 (explain why or why not). b) Let math g(z)=z^2-|z|^2 math 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 math \{u+iv|u^2+v^2=1\} math Show that f is constant.

Solutions:

[|midterm (sol).pdf]