FINAL EXAM
Due Wednesday, April 22, 4 pm. Please submit to my mailbox.
You are not allowed to talk to anyone about the problems of this exam before
you submit your work. You may use your textbook, but no other texts, notes,
or other aids.
1. Find a one-to-one conformal map of the region {z∈C:|z|<2 and |z−1|>
1}onto the right half-plane.
2. Let fbe a function analytic in the unit disk. The circle with center at the
origin and radius ris mapped by fonto a curve whose length is denoted by
L(r). Prove the inequality L(r)≥2πr|f0(0)|. Is it sharp?
3. If a function f=u+iv is analytic in a region Gand u2=vin Gthen f
is constant in G.
4. Suppose that fis analytic in the annulus A={1<|z|<2}and continuous
on the closure Aof A. If |f(z)| ≤ 3 on |z|= 1, and |f(z)| ≤ 12 on |z|= 2
then |f(z)| ≤ 3|z|2in A.
5. Give examples of power series converging:
(a) at all points of the circle of convergence;
(b) at no points of the circle of convergence;
(c) at all but one point of the circle of convergence.
6. Suppose that fis analytic in Dand that |f(z)| ≤ 1/(1 − |z|) for all z∈D.
How large can |f0(0)|be?
7. Suppose that fis analytic in the disk {z∈C:|z|<2}.
(a) Can f(1/n)=(−1)nfor every positive integer n?
(b) Can f(1/n) = (−1)n/n for every positive integer n?
In each case, either give an example of an analytic function with the given
sequence of values, or prove that no such function exists.