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
Faculty of Arts and Science
University of Toronto
MAT334: Complex Variables
Term Test II, Winter Term 2017
Duration: 110 minutes
Family Name (PRINT):
(As in your Student ID)
Given Name(s) (PRINT):
(As
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 co
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?
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 con
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
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
Pro
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
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 a
Math334 - Practice Exercises 1
1
Week 1
1. Let z = 2 + 3i. Find z and |z|.
2. Write
3+4i
1+i
in the form of a + bi.
3. Let z = 2 + 3i and w = 1 i. Find z w
+ zw.
4. *Show that z = z. Thus show that z
Math334 - Practice Exercises 2
1
Textbook Exercises
1.2
1.3
1.4
1.5
1-18, 20-25
1-10,15,18,19
1-25, 30-41
1-14, 16-18, 21, 23, 24, 27, 29
2
Other Exercises
Comments: You are encouraged to work on thes
Midterm 1 - Solutions to the Practice problems
Mat 334 Complex Variables
October 11, 2017
1: If a and b are complex numbers such that |a| < 1 and |b| < 1, prove that
ab
< 1.
1 a
b
Solution. We hav
Math334 - Practice Exercises 4
Solutions
2
Limits and Analytic
7. Let f (z) be analytic in a domain D such that Re f (z) = 1 for all z. Show that f is a constant function.
Solution: Write f (z) = u(z)
Instructors: Anup Dixit and Larissa Richards
MAT334: Complex Variables.
Math334 - Practice Exercises 6
Solutions
2
Other Exercises
1. Evaluate the following contour integrals.
Z
(c)
z 1/3 dz for the p
Math334 - Practice Exercises 3
Solutions
2
Logarithms
2. Which of the following statements are true? Why?
(a) eLog z = z Solution: True because eLog z = elog |z|+i Arg z = |z|ei Arg z = z.
(b) Log ez
Instructors: Anup Dixit and Larissa Richards
MAT334: Complex Variables.
Math334 - Practice Exercises 6
Comments: You are encouraged to work on these problems and solutions will be provided, but they a
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 fu
Math334 - Final Exam
There are 8 questions in the final exam. Each question may contain several parts.
You will have 3 hours to complete the final exam.
The exam will NOT be marked using Crowdmark.
Test #3
10
1 Compute
z2
z 1 dz.
z 2
Solution:
10
z2
I
dz. The Cauchy formula yields:
z 1
z 2
f ( k ) ( z0 )
1
k!
2 i
f ( w)
( w z0 ) k 1
z 2
d 9 20
( z ) z 1
9
20
20.19.18.17.16.15.14.13.1
Exam 1
Mihai Halic 07 Jun 16
1) a) Determine the polar decomposition of z = ( 3 + i )
Solution: r = x 2 + y 2
b) Let w = z 8 . Compute
r = 3 + 1 = 2, tan =
1
7
7
7
=
z = 2(cos
+ i sin ) .
6
6
6
3
MAT334 Fall 2016 - Midterm 1 Solution
Instructors: Askold Khovanskii and Dmitry Faifman
1. Consider the sets A = cfw_z : |z 3i| 1 and B = cfw_w : |w 4| 1.
(a) (2 points) Sketch the sets A and B.
(b) (
Exam
MAT334 Complex Variables Spring 2016
Christopher J. Adkins
Solutions
Question 1
(a) Write the number z in standard form (i.e. a + ib where a, b R)
z=
(1 + i)5
7 2i
(b) Determine the radius of con
1.
(a) (8 pts) Find the set D of all points z C for which f (z) = |z|2 + iz is
differentiable (in the complex sense). Justify your answer. Draw a square
around the asnwer D =.
Solution Write f (z) = x
Math334 - Practice Exercises 4
1
Jan 31 and Feb 2
1. By using sin z =
eiz eiz
,
2i
etc, show the followings
(a) sin iz = i sinh z
(b) sinh iz = i sin z
(c) cos iz = cosh z
(d) cosh iz = cos z
2. By us
Faculty of Arts and Science
University of Toronto
MAT334: Complex Variables
Term Test I, Winter Term 2017
Duration: 100 minutes
Solution
Family Name (PRINT):
Given Name(s) (PRINT):
Student Number:
Sig
Math334 - Review
Recall
Final exam - 8 questions, 3 hours.
Show work! Includes saying which singularities are inside the contour, or whether the function is analytic,
etc.
About 90 out of 115 point
Math334 - Mar 28
Last Time
We talked about Taylor Series and Laurent Series.
(Laurent Series) Let f be analytic in the (open) annulus r < |z z0 | < R. Then f can be
expressed in the sum of two series
MAT334: Complex Variables
Fall 2017
Tutorial #2: More Geometry, and Complex Sequences and Series
TA: Jonathan Mostovoy
Note: All supplemental material for tutorials may be found on: http:/mathfin.ca/m
MAT334: Complex Variables
Fall 2017
Tutorial #1: Complex Arithmetic and a bit of Geometry
TA: Jonathan Mostovoy
Note: All supplemental material for tutorials may be found on: http:/mathfin.ca/mostovoy
Math334 - Practice Exercises 2
Solutions
2
Other Exercises
2
2. Let f (z) = ez . Express f (z) = f (x + iy) = u(x, y) + iv(x, y).
Solution: u(x, y) = ex
2
y 2
cos(2xy), v(x, y) = ex
2
y 2
sin(2xy).
3.
Instructors: Larissa Richards and Anup Dixit
MAT334: Complex Variables.
Practice Exercises 5
Comments: You are encouraged to work on these problems and solutions will be provided, but they are, well,
Instructors: Larissa Richards and Anup Dixit
MAT334: Complex Variables.
Practice Exercises 5
Solutions
2
Other Exercises
1. Parametrize the following curves
(b) The circle |z 4 5i| = 3 counterclockwis