MATH 730: Complex Analysis
Spring 2010
Homework 2, due Thursday, February 18
1. Prove that
f (x + iy ) = ex cos y + iex sin y
is an analytic function (dened on all of C) and compute its derivative.
2. Verify the Cauchy-Riemann equations for z 3 .
3. Show
MATH 730: Complex Analysis
Spring 2010
Homework 1, due Tuesday, February 9
1. Ahlfors, pp.2-3, #1 and #2
2. Ahlfors, p.4, #2 and #3
3. Prove that every non-zero complex number has a multiplicative inverse.
4. Ahlfors, p.9, #4
5. Ahlfors, p.11, #1
6. Ahlfo
MATH 730: Complex Analysis
Spring 2010
Homework 8, solutions/comments
1. Use the principle branch of the logarithm to evaluate
log zdz
log zdz = lim
| z | =r
0
C
where C = cfw_rei : + .
Since F (z ) = z log z z is an anti-derivative, the integral is equal
MATH 730: Complex Analysis
Spring 2010
Homework 7, solutions/comments
1. (Ahlfors, p.130, #4.) Show that a meromorphic function in the extended plane is rational.
Assume f is a meromorphic function in the extended plane.
Step 1. First, we show that f has
MATH 730: Complex Analysis
Spring 2010
Homework 4, due Thursday, March 11
1. Ahlfors, p.88, #1. (Find the xed points in each case, then classify the
Mbius transformation.)
o
2. For each case of the previous problem, describe the orbit S k (0), k =
1, 2, 3
MATH 730: Complex Analysis
Spring 2010
Homework 8, due Tuesday, May 4
1. Use the principle branch of the logarithm to evaluate
log zdz
log zdz = lim
| z | =r
0
C
where C = cfw_rei : + .
2. Compute the Laurent series expansion of f (z ) =
lowing annuli
1
(
MATH 730: Complex Analysis
Spring 2010
Homework 7, due Thursday, April 22
1. Ahlfors, p.130, #4.
2. Ahlfors, p.161, #1.
3. Ahlfors, p.161, #3. (Do as many as you like, but no less than four
integrals)
MATH 730: Complex Analysis
Spring 2010
Homework 6, due Thursday, April 15
1. Find the winding number of the following closed curve at each of the
given 5 points.
1
2
3
4
2. Ahlfors, p.123, #1.
3. Verify Cauchys estimate at the origin (for all n and r ) fo
MATH 730: Complex Analysis
Spring 2010
Final Exam (take-home) - due Tuesday, May 18 at 5pm
1. Compute
1
sin z
dz where is shown below.
(z 1)(z 2)2
2
3
4
5
2. Prove that log |z | does not have a harmonic conjugate dened on C\cfw_0.
3. Choose a single-value