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 bounded domain D and continuous on its
closure. Show that 
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  < . Choose < , such that B (z, ) D. let = cfw_ :
 z  = .
(
Let
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
( p)k+1

D(p,r)
=
k!
2rk+1
f ( )d
D(p,r)
k!
M 2r
2r
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 P.
1
(c) Iff has an removable singularity at P, and lim
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
f
Res( ; z = Pi ) =
g
k
i=1
f (Pi )
g (Pi )
If the zer
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 in
D(0, 1). But
f ( )d =
(1 1)d = 0
(b)NO. For example,
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!
and
+
n=1
z n!
2n
Problem 2.
Find all entire functions
Solution of Homework 1
Problem (1.4):
Solution:By denition,
RHS =
=
=
=
=
=
z + w2
(z + w)(z + w)
(z + w)(z + w)
z z + z w + wz + ww
z 2 + w2 + 2Re(z w)
LHS
Similarly,
z + w2 = (z + w)(z + w)
z w2 = (z w)(z w)
We can get (b)
Problem (1.9):
Solut
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  z  > 4
whose derivative is
z
.
(z 1)(z 2)(z 3)
I
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 holomorphic function on C such that f (z )  cos z . Pro