MATH 215APROBLEM SET I
JUN LI
Problem set due Wednesday, Oct. 3, 5pm, delivered to oce 383Z.
Textbook:
1.4(Page 24). 7, 8, 12.
2.6. 4, 6, 8, 9, 12.
a. Let be a connected region that is divided into two non-empty open subsets 1
and 2 by a piecewise smooth
Complex Analysis, Autumn 2009
Stanford University course 215A
HW 3: (due by October 22 at 3:30pm either in class or in Jose Pereas mailbox)
Please write-up your own solutions to problems in an organized and neat fashion. If collaborating in the problem so
Solutions Homework 3
Wednesday, October 28 2009
Math 215A
Grading Scheme: In this homework questions 1,2.2, 2.9, 4,5,6 were graded. The
mapping from questions to points worth is as follows : p(1) = 2, p(2.2) = 1, p(2.9) = 1,
p(4) = 2, p(5) = 2 and p(6) =
Solutions Homework 2
Wednesday, October 14 2009
Math 215A
Grading Scheme:In this homework questions 2,4,5,6,9,10 were graded. The mapping
from questions to points worth is as follows : p(2) = 2, p(4) = 1, p(5) = 1, p(6) = 1,
p(9) = 1 and p(10) = 4, for a
Solutions Homework 1
Wednesday, October 7 2009
Math 215A
Grading Scheme: In this homework questions 1,4,6,9,10,11 were graded. The mapping from questions to points worth is as follows : p(1) = 2, p(4) = 2, p(6) = 2,
p(9) = 1, p(10) = 1 and p(11) = 2, for
Solutions Homework 5
Monday, November 9 2009
Math 215A
Grading Scheme: In this homework questions 1.10, 2.3, 2.4,2.5,2.6 were graded. The
mapping from questions to points worth is constant and equal to 2.
1. Solution: Gamelin problems, XV.2
9. (a) There w
Complex Analysis, Autumn 2009
Stanford University course 215A
HW 6: (due by November 12 at 3:30pm either in class or in Jose Pereas mailbox)
Please write-up your own solutions to problems in an organized and neat fashion. If collaborating in the problem s
Solutions Homework 7
Wednesday, November 30 2009
Math 215A
Grading Scheme: In this homework questions 1 and 2 were graded. The mapping
from questions to points worth is constant and equal to 5.
1. Solution: Let 0 = 0 and C = 1/2. First, notice that for |z
Solutions Homework 6
Wednesday, November 18 2009
Math 215A
Grading Scheme: In this homework questions 1, 2, 3, 4 and 5 were graded. The
mapping from questions to points worth is constant and equal to 2.
1. Solution: For each n N, let
un (z ) = maxcfw_u(z
Complex Analysis, Autumn 2009
Stanford University course 215A
HW 2: (due by October 8 at 3:30pm either in class or in Jose Pereas mailbox)
Please write-up your own solutions to problems in an organized and neat fashion. If collaborating in the problem sol
Complex Analysis, Autumn 2009
Stanford University, Department of Mathematics course 215A
HW 1: (due by October 1, 3:30pm, either in class or in Jose Pereas mailbox)
Please write-up your own solutions to problems in an organized and neat fashion. If collab
MATH 215 SOLUTION TWO
2.6-15. Suppose f is a non-vanishing continuous function on D that is holomorphic
in D. Prove that if |f (z )| = 1 whenever |z | = 1 then f is constant.
Proof. By maximum module principle, |f (z )| 1 in D. By considering 1/f , which
MATH 215APROBLEM SET II
JUN LI
Problem set due Wednesday, Oct. 10, 5pm, delivered to oce 383Z.
Textbook:
2.6. 15;
2.7. 4;
3.8. 2, 4, 9*, 12, 13, 14;
3.9. 3;
a. Let f be holomorphic on = cfw_z | 1 < Re z < 10, 1 < Im z < 1.
(1). Sketch the strategy that le
MATH 215 SOLUTION THREE
3.8-10. Show that if a > 0, then
0
log x
dx =
log a.
2
+a
2a
x2
Proof. Fix the branch of a translate of log given by log(i) = 0 on C i(, 0]. The
residue of f (z ) = log(z )/(z 2 + a2 ) atz = ia is given by log(ia)/(2ia) = log(a)/2i
MATH 215APROBLEM SET III
JUN LI
Problem set due Wednesday, Oct. 17, 5pm, delivered to oce 383Z.
Textbook:
3.8. 10, 12, 15, 16 , 17;
3.9. 5*.
a . Show that any meromorphic function on C (= P1 ) is a rational function.
b*. Show that the group of automorphis
MATH 215A SOLUTION FOUR
JUN LI
5.7-1. Prove that if f is holomorphic in the unit disc, bounded and not identically
zero, and z1 , z2 , ., zn , . are its zeroes, (|zk | < 1), then
(1 |zn |) < .
n
n
Proof. By dividing f by z if necessary, we may assume f (0
MATH 215APROBLEM SET VI
JUN LI
Problem set due Wednesday, Oct. 24, 5pm, delivered to oce 383Z.
Textbook:
5.6. 10a, 11, 13, 15 .
5.7. 1, 2*.
a. Show that a bounded harmonic function u on the upper-half plan H such that
limza u(z) 1 for all a R must be boun
MATH 215APROBLEM SET V
JUN LI
Problem set due Wed., Oct. 31, 5pm, delivered to oce 383Z.
Textbook:
6.3, 3, 10, 13, 17.
a. Prove the partial fraction formula
1
2
=
2
sin a n= (n + a)2
by evaluating the coutour integral
cot z
dz .
(z + a)2
b. Prove the inn
MATH215APROBLEM SET VI
JUN LI
This set is due 5PM Wednesday, Nov. 14.
8.5. 5, 11, 16.
a. Assuming Riemann hypothesis and following the techniques developed in Chapter 7, discuss how to derive an optimal estimate of 1 (x).
b. Let be a bounded, connected an
Solutions Homework 4
Monday, November 2 2009
Math 215A
Grading Scheme: In this homework questions 1, 2, 4.6 and 6 were graded. The
mapping from questions to points worth is as follows : p(1) = 2, p(2) = 2, p(4.6) = 2
and p(6) = 4, for a total of 10 points