Math 534 Homework #2
Autumn 2008
1. (a) Suppose pis a polynomial with all its zeros in the upper half plane H={z: Imz > 0}.
Prove that all of the zeros of p0are contained in H. Hint: Look at the partial fraction expansion of
p0/p.
(b) Use (a) to prove that if pis a polynomial then the zeros of p0are contained in the (closed)
convex hull of the zeros of p. (The closed convex hull is the intersection of all half planes containing
the zeros.)
2. Find the series expansion of
z+ 2i
(z−2)(z2+ 1)
about the point 1.
3. Suppose fis analytic in a connected open set U. If |f(z)|is constant on U, prove that fis
constant on U. Likewise, prove that fis constant if Refis constant.
4. Suppose fis analytic in a connected open set Usuch that for each z∈U, there exists an n
(depending upon z) such that f(n)(z) = 0. Prove fis a polynomial.
5. Suppose fis analytic in Cand |f(z)| ≤ M|z|αfor |z|> R, where αis a non-negative real
number. Prove fis a polynomial of degree ≤α.
7. Prove that if fis non-constant and analytic on all of Cthen f(C) is dense in C.
8. Let fbe analytic in a region Ucontaining the point z= 0. Suppose |f(1/n)|< e−nfor n≥n0.
Prove f(z)≡0.
9. Let fbe analytic in Dand satisfy |f(z)| → 1 as |z| → 1. Prove fis rational.
10. Suppose fhas a power series expansion about 0 which converges in Cand suppose
ZC
|f(x+iy)|dxdy < ∞.
Prove f≡0.
11. Suppose fand gare analytic in Cand |f(z)| ≤ |g(z)|for all z. Prove there exist a constant c
so that f(z) = cg(z) for all z.
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