Complex Analysis Prelim
January, 2005
Throughout let D = {z : |z| < 1} and C = complex plane.
1. Let u(x, y) =
y
x2 + y2
. Show that u is harmonic in the punctured plane C\{0} in 2
ways:
A. From the definition of harmonic.
B. By finding a function f , analytic in C\{0} with u = Ref.
2. Construct a conformal map of the region D\{z : |z + 1
2
| ≤ 1
2
} onto the region
C\{z : Re z ≤ 0}.
3. Let α be a real number and consider the integral
∫ ∞
−∞
dx
x2 − 2x + α
. Determine for
what α the integral converges and, in those cases, determine its value. Include the
details of your contour argument.
4. Let f(z) = cos(i z3). Determine Z(f) = {z : f(z) = 0}. Indicate with a picture where
the solutions lie in C.
5. Let p(z) = 3z15 + 4z8 + 6z5 + 19z4 + 3z + 1. Show that p(z) has 4 zeros for |z| < 1
and 11 zeros for 1 < |z| < 2.
6. Let f : D → D be analytic and satisfy f(1
2
) =
1
2
and f ′(
1
2
) = −1. Find an explicit
formula for f .
7. Let f and g be analytic in a nonempty connected open set U and satisfy |f | = |g|
there. What else can you deduce about the relationship between f and g. Justify
your answer.
8. Let f be analytic in D and satisfy |f(z)| ≤
1
1− |z|
there. Show that |f ′(0)| ≤ 4.
9. Let {bn} be a sequence of complex numbers such that lim sup |bn|
1
n = 1. Let F be the
family of function f(z) =
∞∑
n=0
anz
n which are analytic in D and satisfy |an| ≤ |bn|,
n = 0, 1, 2, . . . Prove that F is a compact family in the topology of uniform convergence
on compact sets in D.
10. A. State carefully the Riemann Mapping Theorem.
B. Let f be a conformal map from D onto D satisfying f(0) = 0 and |f ′(0)| = 1.
Using only the Riemann Mapping Theorem show that f(eiθz) = eiθf(z) for every
real number θ.
C. Deduce that there is a real number θ0 such that f(z) = eiθ0z.