Math 300 Course Review:
The Final Exam will be held on Monday, December 17th, 8:30 a.m. - 11:00 a.m. in SWNG 222.
Calculators, laptops, notes, cheat sheets, etc. will NOT be allowed.
The Final Exam will cover all material from the course, including the fo
1. a. Write 2ei 4 in rectangular form x + iy.
8
in polar form rei .
b. Write 1i
3+i
2. Define S = cfw_z C : Re (z) > 0, |z| > 1.
a. Sketch S.
b. Sketch the image of the domain S under the function f (z) = Log z.
3. a. For what values of z is the function
Department of Mathematics, University of British
Columbia
MATH 300, Section 202
Midterm
Closed book. Calculators, laptops, notes, cheat sheets, etc. are
NOT allowed.
This exam consists of 5 numbered problems, each of equal value.
Time: 1:00-1:50 pm
Date:
Be sure that this examination has 2 pages including this cover
The University of British Columbia
Sessional Examinations - December 2004
Mathematics 300
Introduction to Complex Variables
Closed book examination
Name Signature
Student Number Instructors
Midterm Solutions
1. Give all possible values of the following in the form x + iy.
(a) (1 i)i
Solution:
(1 i)i = ei log(1i) = ei(Log|1i|+iarg(1i) = ei(Log 2+(i( 4 +2k)
= e 4 2k+iLog 2 = e 4 2k eiLog 2 = e 4 2k (cos Log 2 + i sin Log 2)
= e 4 2k cos Log 2
Solutions:
1. Since f (z) = f (x, y) = u(x, y) + iv(x, y) is differentiable at
(z)
exists and
z = x + iy, we know that the limit limz0 f (z+z)f
z
is equal to f (z). This limit needs to exist for any path along which
z = x + iy approaches 0, and so we can
February 10, 2015 Math 300 Name: Page 2 out of 1}.
IL Express the following in the form of a. "i ib where a and b are reai numbers.
#1 + 3?;
(a) 2 + 37;
(Jig + 1000
1 ____. .
(a) 2
4H: («1+5an W M61; 2; «q
L51) WW 1; WWW.»
m3: 7 (M3~.)(2»"s:) 0
Review worksheet for Midterm 1
Math 300, Section 202, Spring 2015
1. Find the limit of the function f (z) = (z/)2 , if it exists, as z tends to zero. If
z
you think the limit does not exist, explain your reasoning for this conclusion.
Solution. If z 0 alo
March 16, 2015 Math 300 Name: _____M_ Page 2 out of 9
1. (3) Express the principal mine of
.... 37ri
[ (m1 Maj]
in the form (L +11).
(7 points)
n 3112 3111M Adi}
(Wlw'l) z Q, 3( ) (1%)
MW MMEEWA ,2 Arg(w\»¥(§):
3
a ~ m
=1) Lo(~\wx)z 5%; wt, +35 Um
Solutions to Math 300 2012WT1 Solutions
p
1. (a) z = i log( 1 + i) = i log 2 + i( 3 + 2n) , so cos z = (eiz + e
4
iz
)/2 =
3
4
i
+ 4.
p
p
p
( 3+1 i( 3 1)
p 3+i =
p
. Since |z| = 1 and z lies in the fourth quadrant, Log(z) =
2(1+i)
2 2
p
p
i arctan( p3 1
Math 300, Fall 2014, Section 101 Page 1 of 8
Midterm Exam II
November 5, 2014
No books. No notes. No calculators. No electronic devices of any kind.
Name ____________ Student Number
Problem 1. (3 points)
Find all values of ii .
__ Q aZf/M\ _ QL : Q
Math 300 Midterm Review:
The midterm will be held in class on Friday, February 15. Calculators,
laptops, notes, cheat sheets, etc. will NOT be allowed.
The midterm will cover the material from Chapters 1, 2 and 3 of the text
(but not including 1.7, 2.6, 2
SOLUTIONS TO HOMEWORK ASSIGNMENT # 7
1. Determine the nature of all singularities of the following functions f (z ).
(a) f (z ) = cos 1/z.
1
.
(b) f (z ) = 2
z sin z
z
.
(c) f (z ) = z2
e 1
Solution:
(a) z = 0 is the only singularity. It is an essential s
SOLUTIONS TO MIDTERM #1, MATH 300
1. (9 marks) Answer true or false to the following questions by putting either true or
false in the boxes. If the answer is true give a proof, and if the answer is false give a
counter-example.
(a) Log ez = z complex numb
SOLUTIONS TO MIDTERM #2, MATH 300
1. (12 marks) Answer true or false to the following statements. Give valid reasons for all
your answers.
(a) If f (z ) is analytic on a simple closed smooth curve C then
f (z )dz = 0.
C
(b) The function f (z ) = ze1/z has
SOLUTIONS TO HOMEWORK ASSIGNMENT # 5
1. Use Cauchys Integral Theorem to evaluate the following integrals.
z
dz, where C is the positively oriented circle |z 2| = 2.
3
C z +1
z
(b)
dz, where C is the circle |z | = 3, oriented in the clockwise direction.
2
SOLUTIONS TO HOMEWORK ASSIGNMENT #2
1. Find all nth roots of the following complex numbers z. Express your answers in the form
a + b.
(a) n = 3, z = 8.
(b) n = 4, z = 2 + 12.
Solution:
(a) 8 = 8e(3/2+2k) = the 3rd roots are: 2e(3/2+2k)/3 , k = 0, 1, 2.
2
SOLUTIONS TOHOMEWORK ASSIGNMENT # 3
1. Find the harmonic function u(x, y ) on the region = cfw_z | y > 0, 2 xy 4 that
if xy = 2
satises the boundary conditions u(x, y ) =
if xy = 4
Solution: The solution is u = Axy + B, where the constants A, B are chos
SOLUTIONS TO HOMEWORK ASSIGNMENT # 4
1. Suppose f (z ) is dened for |z z0 | < , where is some positive number. If f (z0 )
show that f (z ) is continuous at z = z0 .
f (z ) f (z0 )
Solution: f (z0 ) means that lim
and equals f (z0 ). Write this in the
z
Math 300, Fall 2014, Section 101 Page 1 of 8
Midterm Exam I
October 1, 20124
No books. No notes. No calculators. No electronic devices of any kind.
Name _________ Student Number .
Problem 1. (3 points)
Find all solutions in (C of the equation
22(i+1)