MATH 621 โ Spring 2005
Homework Set # 5
39) Let fbe analytic and 1:1 on the unit disk โ = {|z|<1}. Let
f(z) =
โ
X
n=0
anznbe the Taylor series expansion of faround the origin.
Show that
area f(โ) = ฯ
โ
X
n=0
n|an|2.
Hint: area f(โ) = Z Zf(โ)
dA. Apply the change of variables formula
for double integrals to transform this to an integral over the unit disk
โ.
40) Let f(z) be an entire function and suppose |f(z)| โค | cos z|for all
zโC. Prove that there exists ฮปโCsuch that f(z) = ฮปcos z.
41) a) Suppose z0is an essential singularity of fand a pole of g.
What can you say about the type of singularity of f.g at z0? What
about f /g?
b) Suppose z0is an essential singularity of fand of g. What can you
say about the type of singularity of f.g at z0?
c) Suppose z0is a pole of order 1 of f, show that z0is an essential
singularity of g=ef
42) Suppose fis holomorphic in the disk {|z|<2}except at the point
z= 1 where it has a pole of order 1. Let
f(z) =
โ
X
n=0
anzn
be the Taylor expansion of faround z= 0. Show that lim
nโโ
anexists.
43) Show that the singular points of the following functions are
poles. Determine the order of each pole and compute the corresponding
residue:
(a) f(z) = 1
z(1 โz2);(b) f(z) = 7z4
(1 + z)2;(c) f(z) = zcot z
(d) f(z) = eiz
z2+a2,areal; (e) f(z) = sinh z
z4(1 โz2)
44) Determine the zeroes and singularities of the following functions
on the extended complex plane (Riemann Sphere) S=Cโช {โ}. In
