Ph.D. Prelim in Complex Analysis
January 18, 1994
1. Let f be analytic in the unit disk D. Use Cauchy’s integral formula to establish
the power series representation of f in D. Obtain both an integral formula and a
derivative formula for the n-th coefficient.
2. Let Ω be a region and let F = {f : f is analytic in Ω and |f(z)| ≤ 1, ∀z ∈ Ω}. Fix
z0 ∈ Ω and show that ∃ g ∈ F such that Re g′(z0) ≥ Re f ′(z0), ∀f ∈ F .
3. Let f be analytic and nonconstant in a region Ω with µ = Ref and v = Imf f .
(a) Show that |f ′(z)|2 = u2x + u2y = v2x + v2y.
(b) Determine all real numbers a and b such that au2 + bv2 is harmonic in Ω.
4. Let Ω = {z : |z − i| < 1} and H = {z : Im z > 0}. Map H\Ω conformally onto Ω.
5. If p is a polynomial, prove that the series
∞∑
n=0
p(n)zn defines a rational function.
HINT: Note that any linear combination of rational functions is a rational function.
6. (a) Let f be analytic in the unit disk D with
lim
|z|→1−
f(z) = 0 .
Prove f ≡ 0.
(b) Let g be analytic in D. Prove that the statement
lim
|z|→1−
g(z) = ∞
is impossible.
7. Let f be meromorphic in C and bounded outside of some circle. Determine the form
of f as completely as possible.
8. Let Γ = {z : |z| = 1}.
(a) Show that the mapping z
7−→ (z + 1)2 takes Γ onto the cardioid r = 2(1 + cos θ).
Sketch this cardioid.
(b) Let g(w) =
∫
Γ
z(z + 1)
z2 + 2z − w
dz (Γ traversed once counterclockwise). Use the result
of part (a) to sketch a domain containing 0 on which g is analytic.
(c) Determine g(0) and g′(0).