University of Toronto
MAT334H1F - 2015
Assignment 3 - Day section
Due Wednesday, November 18, 2015 at the beginning of the lecture (10:10 am)
NOTE: Please write your solutions in the same order as the questions, staple the pages together and attach
the qu
MAT334: COMPLEX VARIABLES
HOMEWORK 4: PARTIAL SOLUTIONS
2.1.6. On the plane minus the negative reals (including the origin!), Log is holomorphic, with derivative z 1 . Thus, using the chain rule, we derive
d
d
(Log z)2
[(Log z)3 ] = 3(Log z)2 (Log z) = 3
MAT334: COMPLEX VARIABLES
HOMEWORK 1: PARTIAL SOLUTIONS
1.1.4. If z = x + iy = 0, then
1
x iy
z
z
.
=
= 2 = 2
z
zz
|z|
x + y2
From the above, we get
Re
x
1
= 2
,
z
x + y2
Im
1
y
= 2
.
z
x + y2
Moreover, writing z = x + iy, we also have
Re(iz) = Re(ix y) =
MAT334: COMPLEX VARIABLES
HOMEWORK 2: PARTIAL SOLUTIONS
1.3.14. First, we must show D1 D2 is open. Suppose z is in D1 D2 . Then, z is
in either D1 or D2 . If z D1 , then since D1 is open, there is an open disk B about
z that is contained in D1 . This disk
M A T 334 - M i d t e r m !
I nstructor: G ideon Maschler
D o all your w o r k w i t h n o calculators or other aids. E x p l a i n your answers clearly. I n the
f ollowing, z = X + iy d enotes a c omplex n u m b e r or variable, w i t h
= 1. G ood l uck!
MAT344 Solutions to Term Test # 1
February 15, 2011
1. Let C be the arc of the circle |z | = 5 that lies in the 3rd quadrant. Show
that
5
dz
I :=
z 2 + 16
18
C
Solution: length(C ) = 5/2 and |z 2 + 16| |z 2 | 16 = 9. Combining I
5/2 9 = 5/18.
2. Find the
c omplexworld - M idterm Summer 2011
P age 1 o f 2
g uest
Midterm Summer 2011
PAGE
DISCUSSION
HISTORY
J oin
H elp
S ign In
NOTIFY ME
EDIT
W iki Home
J oin this Wiki
a) Find all c omplex numbers z s atisfying
R ecent Changes
( 2/.?+ 1 ) - ' = 1
M anage Wik
Some solutions to HW 8
Problem 1 (2.4.4). (z 4z + 4)3 = (z 2)6 is zero exactly at 2, and the order of
the zero is 6.
Problem 2 (2.4.6). Log(1 z) for |z| < 1 is zero exactly when z = 0. This follows
from ez = 1 exactly when z = 2ik for k Z. To compute the
Solutions to HW 7
Problem 1 (12).
z2
zn =
= z2
1z
n=0
z n+2
n=0
converges for all |z| < 1.
Problem 2 (14). Note that
Log(1 z) =
n=1
zn
n
converges for all |z| < 1. So to get the rst four (non-zero) terms of the series for
[Log(1 z)]2 , we look at
z2 z3 z
Fall 2012 MAT334 Exam 2
Solutions (LEC0101)
Problem 1. Let the curve be the positively-oriented boundary of the square
with corners at 2 + 2i, 2 + 2i, 2 2i, and 2 2i. Evaluate the following:
z2
ez
dz.
+ 2z 3
Solution. First, the integrand can be factored
Fall 2012 MAT334 Exam 1
Solutions (LEC0101)
Problem 1. Let C denote the bottom half of the positively oriented unit circle
with center i, i.e., counterclockwise from i 1 to i + 1. Evaluate
(z i)3 dz.
C
Solution. The curve C can be parametrized as
(t) = i
MAT344 Solutions to Term Test # 1
February 15, 2011
1. Let C be the arc of the circle |z| = 5 that lies in the 3rd quadrant. Show
that
5
dz
I :=
z 2 + 16
18
C
Solution: length(C) = 5/2 and |z 2 + 16| |z 2 | 16 = 9. Combining I
5/2 9 = 5/18.
2. Find the i
Fall 2012 MAT 334 Exam 1 Solutions
Problem 1. Compute
(z 2)4 dz
where is the left half of the circle of radius 2 centered at 2, oriented from 2+2i to
2-2i.
Solution. We parametrize the half circle by (t) = 2 + 2eit where /2 t 3/2.
Hence
3/2
(z 2)4 dz =
f
Fall 2012 MAT 334 Practice exam 2
You have 50 minutes. Answer 4 of the following 5 questions. If you answer all 5,
your score will be determined by the best 4 solutions you provide.
Problem 1. Show that if f is holomorphic/analytic on a domain D and the i
1.)
We know that z 2 sends the circle |z| = 2 to the circle of radius 4 centered at
zero, so z 2 + 1 sends this circle to the circle of radius 4 centered at 1. So, for
1
|z| = 2, |z 2 + 1| 3 so |z21 3 .
+1|
Thus,
C
dz
sup
z2 + 1
C
1
z2 + 1
cfw_arclength
Fall 2012 MAT 334 Exam 2
You have 50 minutes. Answer 4 of the following 5 questions. If you answer all 5,
your score will be determined by the best 4 solutions you provide.
You must show all your work to receive any credit.
Problem 1. Suppose f = u + iv i
Some solutions to HW 11
Problem 1 (3.2.2). By the Maximum-Modulus Principal, we need only check the
boundary. When z = x 2 is real, the max of |f | is at x = 2, and is equal to 2e2 .
When z = iy, the maximum modulus is 2. When z = 2ei for 0 /2, the
modulu
Some solutions to HW 10
Problem 1 (3.1.2). We use the technique outlined in Example 2 applied to
f (z) = z 4 3z 2 + 3.
When z = x is real and 0 x R, we have f (x) = x4 3x2 + 3 0. This can be
seen by nding the local max and min of f (x) using basic calculu
Some solutions to HW 9
Problem 1 (2.6.2). By the proposition on page 155,
x2
dx = 2i
x4 4x2 + 5
Res
zj H +
P
; zj
Q
where H + is the upper half plane, P is the numerator, and Q is the denominator.
We will do a clever trick to factor Q. Note that 5 = 42 +
MAT334: COMPLEX VARIABLES
HOMEWORK 3: PARTIAL SOLUTIONS
1.5.2.
e
5i
4
= cos
5
5
1
1
+ i sin
= i .
4
4
2
2
1.5.4.
log(i) = ln | i| + i arg(i) = i
+ 2k ,
2
for all integers k.
1.5.6. First, we write
21i = elog 2(1i)
Since log 2 = ln 2 + i2k, for any intege
MAT334: COMPLEX VARIABLES
HOMEWORK 6: PARTIAL SOLUTIONS
2.3.2. The function
ez
z(z 3)
is holomorphic everywhere except at z = 0 and z = 3. The former singularity is
within the circle |z| = 2, while the latter is outside.
Let D be a convex (or simply conne
DEPARTMENT OF MATHEMATICS
University of Toronto
M A T 334F
Midterm Test
Tuesday, November 10, 1998
Instructor: T . B l o o m
Attempt
1.
all questions.
All questions
are of equal value.
F i n d aU complex numbers z s atisfying
s in z = cos z .
2. ( a)
( b)