MATH 621 – Spring 2005
Homework Set # 3
21) Problem 1, page 64.
22) Problem 2, page 64.
23) Problem 3, page 64.
24) Let P(z) = a0+a1z+···+adzdbe a polynomial of degree d > 0.
Prove that 1
2πi Z{|ξ|=r}
ξd−1|P(ξ)|2dξ =a0¯adr2d
25) Let P(z) = a0+a1z+···+adzdbe a polynomial of degree d > 0.
Prove that there exist positive real constants k,Kand Rsuch that:
k|z|d≤ |P(z)| ≤ K|z|d
for all z∈Csuch that |z|> R.
26) Compute the following line integrals:
a) ZΓ
z2
z−1dz, where Γ is the circle of radius 3 centered at the origin.
b) ZΓ
ez
(z+ 4)(z−1 + i)dz, where Γ is the circle of radius 1 centered
at the origin.
c) ZΓ
z(z+ 3)
(z+i)(z−8)dz, where Γ is the circle of radius 3 centered at the
point 2 + i.
27) a) Let γ(t) = 2eit, 0 ≤t≤π/2. Prove that
ZΓ
dz
z2+ 1
≤π
3
b) Let Γ be the unit circle (i.e. the circle of radius 1 centered at the
origin). Prove that
ZΓ
sin z
z2dz
≤2πe
28) Problem 6, page 65.
