Solution for Midterm Review Problems1
(1) Which of the following functions f (z) are entire and which are
not? You must justify your answer. Also nd the complex
derivative f (z) of f (z) if f (z) is entire. Here z = x + yi with
x = Re(z) and y = Im(z).
(a
Math 311 - Spring 2014
Assignment # 1
Completion Date: Wednesday May 7, 2014
Question 1. [p 5, #2]
Show that
(a) Re (iz) = Im z;
(b) Im (iz) = Re z.
Question 2. [p 8, #1 (b)]
Reduce the quantity
5i
to a real number.
(1 i)(2 i)(3 i)
Question 3. [p 8, #1 (c
Solutions for Math 311 Assignment #4
1
(1) Show that if f (z) is continuous at z0 , then
(a) f (z) is continuous at z0 ;
(b) f (z) is continuous at z 0 ;
(c) f (z) is continuous at z 0 .
Proof. Let g(z) = z.
Since g(z) is continuous everywhere and f (z) i
Solutions for Math 311 Assignment #4
(1) Let a function f be analytic everywhere in a domain D. Prove
that if f (z) is real-valued for all z in D, then f (z) must be
constant throughout D.
Proof. Let f (z) = u(x, y) + iv(x, y) where u(x, y) = Re(f (z)
and
Solutions for Math 311 Assignment #2
(1) Find the principal argument and exponential form of
i
(a) z =
;
1
+ i
(b) z = 3 + i;
(c) z = 2 i.
Answer.
(a) Arg(z) = /4 and z = ( 2/2) exp(i/4)
(b) Arg(z) = /6 and z = 2 exp(i/6)
(c) Arg(z) = tan1 (1/2) and z = 5
Math 311 (2015) - Assignment #1
(Due on Sept 11, Friday)
1 + 2i 2 i
+
in the form a + bi.
3 4i
5i
(b) Show that Re (iz) = Im z and Im (iz) = Re z for any complex number z.
1. (a) Express
2. Use mathematical induction to show that the following binomial fo
Lecture 3:
Moduli and Conjugates
Dan Sloughter
Furman University
Mathematics 39
March 10, 2004
3.1
The modulus of a complex number
Denition 3.1. For z = x + iy C, we call
x2 + y 2
|z| =
the modulus, or absolute value, of z.
Note that if z = x + iy is real
Lecture 4:
Polar Coordinates
Dan Sloughter
Furman University
Mathematics 39
March 12, 2004
4.1
Polar coordinates
Recall: If (x, y) is a point in the plane, (x, y) = (0, 0), r is the distance from
(x, y) to the origin, and is the angle between the x-axis a
Math 311Q1, Winter 2017, Assignment #1
Due date: (before 5:00pm) Friday, January 20, 2017
If not specified, report your answers of complex numbers in rectangular coordinates.
1. Compute
4
(a) z0 :=
2i +
(b)
6i1
(1+i)(2i)
and find Re(iz0 ), Im(iz0 ).
6i
.
Total points = 50.
1. [10 points] Find the radius of convergence R, and the point p the
series is centered at for the following:
X
[1 3 5 (2n + 1)]2
n=1
22n (2n)!
(z 1 + 2i)n .
X
1
(z + 5i)n .
n
n
n=1
2. [10 points] Let
1
, p = 1 + i.
1+z
Without doing an
Lecture 5:
Roots of Complex Numbers
Dan Sloughter
Furman University
Mathematics 39
March 14, 2004
5.1
Roots
Suppose z0 is a complex number and, for some positive integer n, z is an nth
root of z0 ; that is, z n = z0 . Now if z = rei and z0 = r0 ei0 , then
Lecture 1:
Complex Numbers
Dan Sloughter
Furman University
Mathematics 39
March 9, 2004
1.1
Basic denitions
Denition 1.1. Given z1 = (x1 , y1 ) and z2 = (x2 , y2 ), we dene the sum of
z1 and z2 by
z1 + z2 = (x1 + x2 , y1 + y2 )
and the product of z1 and z
Math 311 (2015) - Assignment #2
(Due on Sept 18, Friday)
1. Write z in polar form and nd the principal argument Arg z when
(a)
(c)
z = (2 + 2i)(1 i),
z = (1 + 3i)10 ,
(b) z =
5i
,
2+i
(d) z = ( 3 i)15 .
2. Find all satisfying 0 < 2 and |ei 1| = 2 in two w
Lecture 6:
Some Topology
Dan Sloughter
Furman University
Mathematics 39
March 16, 2004
6.1
Topological terminology
Denition 6.1. Given a complex number z0 and a real number
call the set
cfw_z C : |z z0 | <
an
> 0, we
neighborhood of z0 and the set
cfw_z
Due date: Wednesday Nov. 23rd at 4:10.
Total points: 50
1. [20 points] For the following functions, determine the nature of the
singularities (removeable, pole, or essential) and calculate the residues.
For pole singularities, determine the multiplicity a
ASSIGNMENT 4. DUE DATE: WEDNESDAY
OCTOBER 5 @ 4:10 PM IN ASSIGNMENT BOX ON
3RD FLOOR OF CAB
1. [15 points] Let f (z) = + i be an entire function satisfying
A + B + C = 0,
for all z C, where A, B, C R and (A, B) 6= (0, 0). Explain why
f C is a constant fun
Math 311 Spring 2014
Theory of Functions of a Complex Variable
The Field of Complex Numbers: C
Department of Mathematical and Statistical Sciences
University of Alberta
If C is the set of all complex numbers:
C = cfw_ z = (x, y) | x, y R
with addition a
Math 311 Spring 2014
Theory of Functions of a Complex Variable
Limits in the Euclidean Plane and in the Complex Plane
Department of Mathematical and Statistical Sciences
University of Alberta
In this note we show that limits in the complex plane C are exa
Math 311 Spring 2014
Theory of Functions of a Complex Variable
Differentiation
Department of Mathematical and Statistical Sciences
University of Alberta
In this note we will give necessary and sufficient conditions for a function f (z) of a complex variab
Math 311 Spring 2014
Theory of Functions of a Complex Variable
The Extended Complex Plane
Department of Mathematical and Statistical Sciences
University of Alberta
The Extended Complex Plane
To discuss the situation where a function f (z) becomes infinite
Math 311 Spring 2014
Theory of Functions of a Complex Variable
Department of Mathematical and Statistical Sciences
University of Alberta
Lecture A1: M T W R F 9:00 - 10:10 CSC B - 2
Instructor: I. E. Leonard, 679 CAB
telephone: 492-2388
e-mail: [email protected]
Math 311 Spring 2014
Theory of Functions of a Complex Variable
An Example from Euclidean Geometry
Department of Mathematical and Statistical Sciences
University of Alberta
Example. Show that z1 , z2 , z3 are the vertices of an equilateral triangle, if and
Math 311 Spring 2014
Theory of Functions of a Complex Variable
Topics for Final Examination
Department of Mathematical and Statistical Sciences
University of Alberta
The final examination may include questions from all of the course material covered durin
Math 311 Spring 2014
Theory of Functions of a Complex Variable
Topics for Midterm Examination
Department of Mathematical and Statistical Sciences
University of Alberta
The midterm examination may include questions from all of the course material covered t
ASSIGNMENT 3. DUE DATE: WEDNESDAY
SEPTEMBER 28 @ 4:10 PM IN ASSIGNMENT BOX
ON 3RD FLOOR OF CAB
Questions 1, 2, 3 are worth 10 points and question 4 is worth 20
points, making a total of 50 points for this assignment.
1. (i) (5 pts) Write down the equation
1. Evaluate
Z
z 2 e1/z cosh(1/z)dz,
C
where C is any simple-closed curve, oriented counterclockwise, and
containing 0 in its interior.
2. Using the formula in Fact 9.19, p. 80 of your text, compute the
residues at each singularity in the complex plane C,
ASSIGNMENT 2. DUE DATE: WEDNESDAY
SEPTEMBER 21ST @ 4:10 PM IN ASSIGNMENT BOX
ON 3RD FLOOR OF CAB
Each question is worth 10 points, making a total of 50
points.
Please avoid using your calculator to give decimal point answers!
1. Using the binomial theorem