Section 2.5:
2. z 2 has a zero of order 2 at 0 and sin z has zeros of order 1 at k , k Z. Hence sin z has a removable singularity at 0 which is a zero of order 2 - 1 = 1 and poles of order 1 at k , k Z \ cfw_0. (see the example from Poles in details10.pdf
MAT334, Summer 2009
Quiz #1
Last Name
First Name
Student #
Problem 1
Problem 2
Total
1. Use de Moivre's theorem to write
3i
We have that 3 i = 2 and arctan
lies in the fourth quadrant). So
3i
6
= 2 cos
+ i sin
6
6
6
in the form x + iy.
1/ 3 = arctan 1/
MAT334S
Assignment 2: Solutions
Problem 1. Plugging z = z0 into p(z ) = z n + an1 z n1 + + a1 z + a0 , we
have
p(z0 ) = (z0 )n + an1 (z0 )n1 + + a1 (z0 ) + a0 .
Using the properties of the complex conjugation; z1 z2 = z1 z2 , z1 +z2 = z1 + z2
and ai = ai
Week 1
Section 1.1:
4.
Re
1
z
=
x
, Im
x2 + y 2
1
z
=
y
x2 + y 2
Re(iz ) = Re(ix + i2 y ) = Re(y + ix) = y = Imz, . . .
10. If z = 0 there's nothing to prove. If z = 0 then w = 0/z = 0.
18. Since 1 z = 0, the identity is equivalent to
(1 z ) 1 + z + z 2 +
MAT334S
Assignment 1: Solutions
Problem 1.
cos 4 + i sin 4 = (cos + i sin )4 =
cos4 + 4i cos3 sin 6 cos2 sin2 4i cos sin3 + sin4 =
cos4 6 cos2 sin2 + sin4 + i(4 cos3 sin 4 cos sin3 ).
Comparing real and imaginary parts we get
cos 4 = cos4 6 cos2 sin2 + si
UNIVERSITY OF TORONTO
The Faculty of Arts and Science
FINAL EXAMINATIONS, APRIL 2013
MAT334H1 S
Complex Variables
Duration 3 hours
Instructors: M.-D. Choi and A. Shao
INSTRUCTIONS
o No calculators and no aids are allowed
0 Try as many questions as you can
MAT334S Assignment 1
due Friday January 18 in your tutorial
Problem 1. Use de Moivre's formula
(cos + i sin )n = cos n + i sin n
to express
cos 4
and
sin 4
in terms of
Problem 2. Compute
cos
1i 3
Problem 3. Show that if
Re z1 > 0
and
sin .
50
and
.
Re z2
MAT334F
Term test 1: Solutions
Let f (z) = x2y + x2 + i(xy2 2x + y2), where z = x + iy. Find
all points where f has a complex derivative. Evaluate f (z) at such points.
At which points is f analytic?
The function f (z) can be written in the form f (z) = u
MAT334S Assignment 3
due Friday March 1 in your tutorial
Problem 1.
Find all values of
z C,
such that
ez = 3i.
Problem 2.
Show that for any two nonzero complex numbers
z1
and
z2 ,
Log (z1 z2 ) = Log z1 + Log z2 + 2N i,
where
N
has one of the three values
University of Toronto Mississauga
MAT334F Term Test 1
Thursday October 18 - 9:10am to 11:00am
Name:
Student Number:
Let f (z ) = x2 y + x2 + i(xy 2 2x + y 2 ), where z = x + iy . Find
all points where f has a complex derivative. Evaluate f (z ) at such po
Math334 - Jan 24
Last time
Last time we talked about complex functions. Let f be a function from C to C. Then we can write
f (z) = f (x + iy) = u(x, y) + iv(x, y)
, i.e. in terms of two real-valued functions of two real variables.
We say that a complex f
Math334 - Jan 10
1
Review
Recall last week we talked about the set of complex numbers
C = cfw_z = x + iy | x, y R
such that i2 = 1. x and y are the real part and imaginary part of z respectively.
p We can view x + iy as
a vector (x, y) R2 . Its length is
Math334 - Jan 17
1
Last time
Last time we talked about different properties of a subset in C. This includes open sets, closed sets, bounded
sets, compact sets, connceted sets and convex sets.
We also mentioned stereographic projection. Points on the spher
Math334 - Jan 12
1
Open sets and Closed sets
In the real line, or R, we have had the concepts of intervals. For example, if we want to denote the set of values
of x such that a < x < b, we will write (a, b). This is an example of an open interval. If we w
Math334 - Jan 19
1
Last time
Last time we talked about complex functions. Let f be a function from C to C. Then we can write
f (z) = f (x + iy) = u(x, y) + iv(x, y)
, i.e. in terms of two real-valued functions of two real variables.
As we would like to ta
Math334 - Jan 26
Last time
Last time, we talked about
another way to describe the CR equations. We let
1
=
i
, and
z
2 x
y
Then the CR equation reduced to
1
=
z
2
+i
x
y
.
f
= 0, where we can treat z and z as independent variables.
z
We also talked a
MAT334S Assignment 2
due Friday February 01 in your tutorial
Problem 1. Let p(z ) = z n + an1 z n1 + + a1 z + a0 be a polynomial of
degree n 1 with a0 , . . . , an1 R. Prove that if p(z0 ) = 0, for some z0 C,
then also p(z0 ) = 0.
Problem 2. Consider the
MAT334S
Practice problems 3
(not to be handed in)
Section 14: 3, 4, 5, 7
Section 18: 1, 3, 5, 7, 9
Additional problem
: Let
the point
z
f
be the map, such that the value
by rotating it counterclockwise about the point
formula for the function
f.
1
1
f (z
Math334 - Practice Exercises 1
1
Jan 5
1. z = 2 3i, |z| =
2.
3+4i
1+i
13
= 3.5i + 0.5i.
3. z w
+ zw = 2.
2
Jan 10
1. arg(2 + 3i) = arctan(1.5) + 2k, k Z and Arg (2 + 3i) = arctan(1.5).
2. It is true that arg(zw) = arg z + arg w. It is not true if arg is
Math334 - Answers to Practice Exercises 3
1
Jan 24
2
1. (a) 2zez .
(b) 2z + 3.
(c)
z 4 3z 2 +2z
.
(z 3 +1)2
2. f (z) = z z z. f 0 (1) = 1.
3. 0.
2
Jan 26
1. Verify the following functions satisfy the Laplace equation, and find their harmonic conjugate.
(a
Math334 - Jan 5
1
Complex Numbers
1.1
Introduction
Let us start with some notations.
N denotes the set of all positive integers.
Z denotes the set of all integers.
R denotes the set of all real numbers.
R2 denotes the set of all tuples of real numbers
MAT334S
Assignment 3: Solutions
Problem 1.
Find all values of z C, such that
ez = 3i.
Solution.
z = log(3i) = ln | 3i| + i arg(3i) = ln 3 i(
+ 2n ),
2
for all integer values of n.
Problem 2.
Show that for any two nonzero complex numbers z1 and z2 ,
Log (z
334 midterm,day section,solutions
1. (7 marks)
Find all solutions to the equation
z 3 = 1000.
Write them in the form a + bi.
In polar form 1000 = 1000ei
So the solutions of z 3 = 1000 are given by 10ei/3 , 10ei , 10e5/3 .
which are
(1) 10(cos(/3) + i sin(
University of Toronto
MAT334H1F - 2015
Assignment 2 - Evening section
Due Tuesday, October 13, 2015 at the beginning of the lecture (6:10 pm)
NOTE: Please write your solutions in the same order as the questions, staple the pages together and attach
the qu
University of Toronto
MAT334H1F - 2015
Assignment 1 - Evening Section
due Tuesday September 29 at the beginning of the lecture (6:00 pm)
NOTE: Please write your solutions in the same order as the questions, staple the pages
together and attach the questio
University of Toronto Department of Mathematics
Practice Problems for MAT334H1F2015
Practice problems are not to be handed in. Their purpose is to help in the understanding
of the material. They are not necessarily related to test questions. Some of them