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PROBLEMS. Due April 18.
1. If Ω is a simply connected domain and f is a holomorphic function
in Ω, then, for every positive number c, the set of points in Ω where
|f | < c is simply connected.
2. If Ω is a domain not containing 0, then there is a continuous branch of
log z in Ω if and only if 0 and ∞ belong to the same component of the
extended complement of Ω.
3. If Ω is a domain and f is a nonvanishing holomorphic function in Ω,
then there is a branch of log f in Ω if and only if there is a branch of
f 1/n in Ω for every positive integer n
4. If D is the open unit disk and f is holomorphic in a neighborhood
of D, then the number of zeros of f in D equals 1/2π times the net
change experienced by arg(f ) in one counterclockwise circuit around
∂D, provided zeros on ∂D are counted with one-half their multiplicities.
5. If f is holomorphic in a neighborhood of the closure of the annulus
A = {z : r < |z| < R}, of unit modulus on ∂A, and not constant, then
f has at least two zeros in A.
6. Let n ≥ 1 and let {a0 , a1 , . . . , an } be complex numbers such that an 6= 0.
For θ ∈ R, define
f (θ) = a0 + a1 eiθ + a2 e2iθ + · · · + an eniθ
Prove that there exists θ ∈ R such that |f (θ)| > |a0 |.
7. If f is a polynomial on C, then the zeroes of f 0 are contained in the
closed convex hull of the zeroes of f .
8. Let f (z) = z 5 +5z 3 +z 2 +z +1. How many zeros (counting multiplicity)
does f have in the annulus 1 ≤ |z| ≤ 2?