C OMPLEX A NALYSIS M ATH 220A
Final Exam (sample)
Problem 1.
Does there exist a conformal automorphism of the unit disc such that
i
(1/2) = 0 and (0) = 3 ?
Problem 2.
Let f and g be analytic on a boun
Solution of Homework 3
Problem (2.20):
Solution:
F C 0 (D), and F is holomorphic on D. choose z D, since F is holomorphic in z, so > 0, there exist a > 0, we have |F (z ) F ( )| <
when| z | < . Choos
Solution of Homework 4
Problem (3.30)
Solution:
(a)By cauchy integra formula, for arbitary r > 0
k!
k
f (p) =
z k
2i
f ( )
d
( p)k+1
D(p,r)
Let Cr = sup f (z )
D(0,r)
k
f (P )|
z k
k!
|f ( )|
d
2
(|
Solution of Homework 5
Problem (4.3):
Solution:
(a) Iff has an essential singularity at P, then
P.
(b) Iff has an pole at P, then
1
f
1
f
has a essential singularity at
has a removable singularity at
Solution of Homework 6
Problem (4.38):
Solution:
By residue theorem,
1
2i
f (z )
dz =
g (z )
D(P,r)
k
i=1
f
Res( ; z = Pi )
g
If all Pi are simple zeros, we have
f (z )
dz =
g (z )
1
2i
D(P,r)
k
i=1
Solution of Homework 2
Problem (2.25):
Solution:
(a) For example, f (z ) = z z 1 = |z |2 1 is a C 1 function. we know that
f (z )
z = 0 if and only if z = 0. So f is not holomorphic on any open set i
C OMPLEX A NALYSIS M ATH 220A
Midterm Exam
Friday, October 30, 2009 12:00 pm - 1:00 pm
Problem
Points
Student's name:
1
2
3
4
5
Problem 1.
Find the radius of convergence for the series:
+
n=1
z 2n
n!
C OMPLEX A NALYSIS M ATH 220A
Final Exam
Monday, December 7, 2009 1:30 - 3:30 pm
Problem
Points
Student's name:
1
2
3
4
5
Problem 1.
Show that there is a holomorphic function dened in the set
= cfw_z
C OMPLEX A NALYSIS M ATH 220A
Midterm Sample Exam
Problem 1.
Find the largest disk centered at 1 in which the Taylor series for
1
=
1 + z2
ak (z 1)k
will converge.
Problem 2.
Let f (z ) be entire holo