Complex Analysis Qualifying Examination
August 2008
All problems are worth 10 points.
1) Let
f(z) = 1
z(z−1)(z−2) .
Find the Laurent expansion of fvalid in the annulus {z∈C|1<|z|<2}.
2) Find a one-to-one conformal map of the region {z∈C| |z|<2 and |z−1|>1}
onto the upper half plane.
3) Let fand gbe zero-free functions on the unit disc. If f0(1/n)/f(1/n) =
g0(1/n)/g(1/n) for all n∈N, what can be said about the relation between fand g?
Prove your claim.
4) Let P∞
n=0 anznhave radius of convergence r > 0. Assume that the function
f(z) to which it converges has exactly one singular point z0on {z∈C| |z|=r}, and
that z0is a simple pole. Show that limn→∞(an/an+1) exists and equals z0.
5) Evaluate
Z∞
0
log x
1 + x2dx .
6) Fix the real number α > 1. Show that the equation
sin z=eαz3
has exactly three solutions in the unit disc.
7) a) Show that there exists exactly one germ [f]0of a holomorphic function in a
neighborhood of 0 such that f(0) = 0 and f(z) = z+ (1/2)f(z)2.
b) Show that this germ admits unrestricted analytic continuation in C\ {1
2}.
c) Is there a holomorphic function gon C\ { 1
2}such that [g]0= [f]0? Justify your
answer.
8) In the right-hand half plane, suppose f0(z) = zand for each positive integer n,
fn(z) is the principal branch of zfn−1(z). Is there a neighborhood of the point 1 in
which the family {fn}∞
n=1 is a normal family? Explain why or why not.
9) Prove that the range of the entire function z2+ cos(z) is all of C.
10) Prove that if K={x+iy ∈C| |x| ≤ 1 and |y| ≤ 1}, and u(x, y) is
a harmonic function in a neighborhood of K, then for every ε > 0 there exists a
harmonic polynomial p(x, y) such that max(x,y)∈K|u(x, y )−p(x, y)|< ε.