MATH 621 – Spring 2005
Homework Set # 7
56) Let ωj,j= 1,...,k be the k-th roots of unity. Prove that for any
polynomial P(z) of degree d < k:
P(0) = 1
k
k
X
j=1
P(ωj)
Hint: Compute Z|z|=ρ
P(z)
z(zk−1)dz for ρsufficiently large.
57) a) Let fbe a meromorphic function on the Riemann sphere
S=C∪ {∞}, so a rational function (as shown in class). Prove that
X
P
νPf= 0 ,
where the sum is taken over all points in Sand
νPf=
k, if Pis a zero of order k;
−k, if Pis a pole of order k;
0,otherwise.
b) Let Pi∈S,i= 1,...,r, and let m1,...,mrbe integers such that
Pr
i=1 mi= 0. Prove that there is a meromorphic function fon the
Riemann sphere such that νPif=mifor i= 1,...,r.
58) Find the value of the integral:
ZC
3z3+ 2
(z−1)(z2+ 9) dz
for: a) Cthe circle |z−2|= 2; b) Cthe circle |z|= 4.
59) Compute the integral of the following functions over the circle
|z|= 2:
a) f(z) = 1
(z−3)(1 + 2z)2(1 −3z)3;
b) f(z) = 1
(1 + eπz )2;c) f(z) = cos(1/z)
1 + z4
60) Let Ube a connected open set, and let Dbe an open disk whose
closure is contained in U. Let fbe analytic on Uand not constant.
Assume that the absolute value of |f|is constant on the boundary
of D. Prove that fhas at least one zero in D. (Hint: Consider
g(z) = f(z)−f(z0) with z0∈D)
