Complex Analysis Prelim
August 23, 2007
Thursday, 9:00am – 12:00pm, MSB 118
Show all your work. Notes and textbooks are not allowed.
1.(a) Find a conformal map from the set S={z:Im z > 0, Re z > 0}onto
the open unit disk Dsuch that 1+iis mapped into 0.
(b) Find all the maps that satisfy (a), and prove that there are no others.
2. State and prove the Fundamental Theorem of Algebra.
3. How many zeros of the polynomial 2z4−z3+ 5z2−10z+ 1 lie in the set
{z: 1 <|z|<3}? (Justify your answer.)
4. Evaluate the integral Z∞
−∞
sin x
x3+xdx using the Residue Theorem (justify
why it can be used).
[Hint: sin x
x, x ∈R, is the imaginary part of an analytic function which is bounded
in the upper half-plane.]
5. Prove that if fis analytic in the open unit disk Dthen
|f(w)−f(0) −wf 0(0)|6|w|2sup
z∈D
|f(z)−f(0) −zf 0(0)|
for all w∈D.
6. Find all the entire functions fthat satisfy sup ¯
¯
¯
¯
zlog |z|
f(z)¯
¯
¯
¯
<∞,where the sup
is over the set {z:z∈C, z 6= 0, f (z)6= 0}, and prove that there are no others.
7. Suppose that Fis a family of analytic functions on the open unit disk such
that f(0) = 1 for all f∈ F. Prove that if {Im f }f∈F is a normal family, then
Fis also a normal family.