MATH 213A: HOMEWORK 4
DUE THURSDAY, OCT. 18TH
From Ahlfors
p.171 (section
p.173 (section
p.244 (section
p.247 (section
Complex Analysis
4.6.4) 1,2,5
4.6.5) 1,4,5
6.3.2) 1
6.4.1) 2,3,5,6
Let = B1 (0). Find a barrier function at each point of the boundary
o
MATH 213A: HOMEWORK 2
DUE THURSDAY, OCT. 4TH
From Ahlfors
p.123 (section
p.130 (section
p.133 (section
p.136 (section
p.154 (section
Complex Analysis
4.2.3) 2,5
4.3.2) 2,3,4
4.3.3) 1
4.3.4) 1,5,6
4.5.2) 1,2
Also, if f has a removable singularity at some p
MATH 213A: HOMEWORK 3
DUE THURSDAY, OCT. 11TH
From Ahlfors Complex Analysis
p.28 (section 2.1.2) 3
p.161 (section 4.5.3) 3(a),(b),(e),(g),(h)
p.166 (section 4.6.2) 2
1) Compute
1
|z|=1
ez
dz.
1 2z
2) Prove that any harmonic function dened on a simply-conn
MATH 213A: HOMEWORK 5
DUE THURSDAY, OCTOBER 25TH
From Ahlfors Complex Analysis
p.227 (section 5.5.5) 1,2,3
p.232 (section 6.1.1) 1
1a) Let be a simply connected domain, let a , and let
f : B1 (0)
be an injective holomorphic function such that f () = B1 (0
MATH 213A: HOMEWORK 6
DUE THURSDAY, NOV. 1ST
From Ahlfors Complex Analysis
p.238 (section 6.2.2) 2,3,5,6
p.257 (section 6.5.1) 1
1) A holomorphic mapping f : U V is a local bijection on U if for
every z U, there exists an open disk D U centered at z such
MATH 213A: HOMEWORK 7
DUE THURSDAY, NOV. 15TH
From Ahlfors
p.178 (section
p.184 (section
p.186 (section
Complex Analysis
5.1.1) 1,2,5
5.1.2) 2,5
5.1.3) 3,4,5
Prove the lemma needed to nish the proof of Runges little theorem:
If is a simply connected domai
MATH 213A: HOMEWORK 9
DUE THURSDAY, DEC. 6TH
From Ahlfors Complex Analysis
p.200 (section 5.2.4) 1,2
1) Suppose 0 < < 1, 0 < < 1, f is holomorphic on B1 (0),
f (B1 (0) B1 (0), and f (0) = . How many zeroes can f have in
the disk B (0)? What is the answer
MATH 213A: HOMEWORK 8
DUE THURSDAY, NOV. 29TH
From Ahlfors
p.190 (section
p.193 (section
p.198 (section
Complex Analysis
5.2.1) 1,4,5
5.2.2) 2,5
5.2.3) 3,4,5
For N = 1, 2, 3, . . . dene
z2
.
n2
Prove that the ideal generated by cfw_gN in the ring of enti