Solutions to Math 312 Midterm Exam October 11, 2013
1 (a). Observe that w3 = ( 3 + i)3 = ( 3)3 + 3( 3)2 i + 3i2 3 + i3 = 8i and so
z
1i
1
i
=
= .
3
w
8i
88
That is, a = b = 1/8.
1 (b). Observe that the point (1/8, 1/8) R2 lies in the third quadrant and ma
Math 312 Fall 2012
Assignment #4
This assignment is due at the beginning of class on Friday, October 12, 2012.
1.
Let z = x + iy . Show that the function
f (z ) = e x
2 y 2
[cos(2xy ) + i sin(2xy )]
is entire, and nd f (z ).
2. Show that if f is analytic
Math 312 Fall 2012
Assignment #7
This assignment is due at the beginning of class on Friday, November 2, 2012.
Note that the Midterm on Wednesday, November 14, 2012, will cover material from Lectures #1
through #21 along with Assignments #1 through #7.
1.
Math 312 Fall 2012
Solutions to Assignment #6
(a) Since D is an annulus and C surrounds the hole (i.e., C is not continuously deformable to a
point), the Cauchy Integral Theorem, Basic Version does not apply. Observe that f (z ) = ez is
continuous in D an
Mathematics 312 (Fall 2012)
Prof. Michael Kozdron
October 26, 2012
Lecture #22: The Cauchy Integral Formula
Recall that the Cauchy Integral Theorem, Basic Version states that if D is a domain and
f (z ) is analytic in D with f (z ) continuous, then
f ( z
University of Regina
Mathematics 312 Complex Analysis I
Lecture Notes
Fall 2012
Michael Kozdron
[email protected]
http:/stat.math.uregina.ca/kozdron
List of Lectures
Lecture #1: Introduction to Complex Variables
Lecture #2: Algebraic Properties
Mathematics 312 (Fall 2013)
Prof. Michael Kozdron
September 13, 2013
Lecture #5: The Complex Exponential Function
Recall that last class we discussed the argument of a complex variable as well as some of the
motivation for its denition.
Denition. Suppose
Mathematics 312 (Fall 2013)
Prof. Michael Kozdron
September 11, 2013
Lecture #4: Polar Form of a Complex Variable
Suppose that z = x + iy is a complex variable. Our goal is to dene the polar form of a
complex variable. We start by describing how our exper
Mathematics 312 (Fall 2013)
Prof. Michael Kozdron
September 9, 2013
Lecture #3: Geometric Properties of C
Recall that if z = a + ib is a complex variable, then the modulus of z is |z| = a2 + b2 which
may be interpreted geometrically as the distance from t
Mathematics 312 (Fall 2013)
Prof. Michael Kozdron
September 20, 2013
Lecture #8: Powers and Roots of Algebraic Equations
Example 8.1. Find all values of z C such that z n = 1 where n is a positive integer.
Solution. Note that any solution will necessarily
Math 312 Fall 2013
Solutions to Assignment #7
1. Since f (z) = 3z 2 5z+i is continuous with antiderivative F (z) = z 3 5z 2 /2+iz, the fundamental
theorem of calculus implies
5
5
13 12 + i i3 i2 + i2
2
2
(3z 2 5z + i) dz =
C
= 3 + 2i.
2. Since f (z) = ez
Mathematics 312 (Fall 2013)
Prof. Michael Kozdron
September 18, 2013
Lecture #7: Applications of Complex Exponentials
Denition. If z = x + iy C, we dene the complex exponential ez as
ez = ex+iy = ex eiy = ex (cos y + i sin y).
Note that
|ez | = |ex eiy |
Math 312 Fall 2013
Solutions to Assignment #8
1. Using the Cauchy integral formula, we nd
sin z
(a)
dz = 2i sin(/2) = 2i,
z
C
2
(b)
C
cos z
dz =
3 + 9z
z
ez
(c)
C
(z + 1)2
C
(cos z)/(z 2 + 9)
cos 0
2
dz = 2i 2
= i,
z
0 +9
9
dz = 2i e(1) = 2ei.
2. Observe
Mathematics 312 (Fall 2013)
Prof. Michael Kozdron
September 6, 2013
Lecture #2: Algebraic Properties of C
Recall that z = a + ib, with i =
1 and a, b R, is a complex variable.
Cartesian Representation (or Geometric Interpretation) of Complex Variables
We
Math 312 Fall 2012
Assignment #3
This assignment is due at the beginning of class on Monday, October 1, 2012.
1.
Describe the range of the function f (z ) = 2z 3 for z in the quarter disk
cfw_|z | < 1, 0 < Arg(z ) < /2.
2.
Let S = cfw_z : 1 Im(z ) 2. Dete
Math 312 Fall 2012
Assignment #6
This assignment is due at the beginning of class on Wednesday, October 24, 2012.
Determine whether the Fundamental Theorem of Calculus for Integrals over Closed Contours or the
Cauchy Integral Theorem, Basic Version or bot
Math 312 Fall 2013
Solutions to Assignment #6
(a) Since D is an annulus and C surrounds the hole (i.e., C is not continuously deformable to a
point), the Cauchy Integral Theorem, Basic Version does not apply. Observe that f (z ) = ez is
continuous in D an
Math 312 Midterm Exam October 11, 2013
This exam is worth 50 points.
This exam has 5 problems and 1 numbered page.
You have 50 minutes to complete this exam. Please read all instructions carefully, and check
your answers. Write your solutions in the exam
Solutions to Math 312 Midterm Exam November 8, 2013
1. Since i = ei/2 = ei/2+2ki for any k Z, we see that if e = i, then cfw_ i/2 + 2 ki :
k Z. Hence z cfw_ /2 + 2 k : k Z.
2. Since z (t) = 3ieit , 0 t /2, by denition we have
C
1
dz =
|z |2
/2
0
1
3i
3i
Math 312 Midterm Exam November 8, 2013
This exam is worth 50 points.
This exam has 5 problems and 1 numbered page.
You have 50 minutes to complete this exam. Please read all instructions carefully, and check
your answers. Write your solutions in the exam
Math 312 Fall 2013
Solutions to Assignment #5
1. Since e2+i/4 = e2 ei/4 and since ei/4 is the polar form of 1/ 2 + i/ 2, we conclude that
e2
e2
e2+i/4 = + i,
2
2
i.e., a = b = e2 / 2.
2. Note that
sin(2i) =
ei(2i) ei(2i)
e2 e2
=
=
2i
2i
e2 e2
2
2
i,
and s
Math 312 Fall 2013
Solutions to Assignment #3
1. Consider the quarter disk D = cfw_|z | < 1, 0 < Arg(z ) < /2. If z D, then the polar form of z
is z = rei with 0 r < 1, 0 < < /2. Hence,
f (z ) = f (rei ) = 2r3 e3i = 2r3 ei(3+)
using the fact that 1 = ei .
Math 312 Fall 2013 Final Exam Solutions
2+i
(2 + i)(i + 1)
2i + 2 + i2 + i
3i + 1
13
=
=
=
= i.
21
i1
(i 1)(i + 1)
i
2
22
1
1. (b) Note that 1 + i = 2ei/4 so that Arg(1 + i) = /4. This implies z = log 2 + i.
2
4
1
3
1. (c) We have z = 2ei/3 = 2 [cos(/3) +
University of Regina
Department of Mathematics & Statistics
Final Examination
201230
Mathematics 312
Complex Analysis I
Instructor: Michael Kozdron
This exam has 8 problems and 2 numbered pages.
This exam is worth 150 points. The number of points per prob
University of Regina
Department of Mathematics & Statistics
Final Examination
201330
Mathematics 312
Complex Analysis I
Instructor: Michael Kozdron
This exam has 10 problems and 2 numbered pages.
This exam is worth 150 points. The number of points per pro
Math 312 Fall 2013
Solutions to Assignment #1
1. Note that
3 + 2i
8+i
=
2i
6 2i
47
+i
55
23 11
+i
20 20
=
7
17
+ i.
20 20
2. Let z = a + ib so that Re(z ) = a. Since iz = b + ia we see that Im(iz ) = a = Re(z ) as required.
3. Note that i2 = 1, i3 = i, an
Math 312 Fall 2013
Solutions to Assignment #2
1. Since the polar form of i is i = ei/2 , can write z as
z=
2i
2
2
= e4 ei/2 ei = e4 ei(/21) .
4+i
3e
3
3
Since /2 1 (, ], we conclude that |z | = (2/3)e4 and Arg(z ) = 1 so that the polar
form of z is, in fa
Math 312 Fall 2012 Final Exam Solutions
1. Since ez = 0 for all z C, we can multiply ez + 2ez = 3 by ez and simplify obtain
e2z 3ez + 2 = 0. Notice that e2z 3ez + 2 = (ez 2)(ez 1) and so e2z 3ez + 2 = 0
i either ez 2 = 0 or ez 1 = 0. Consider rst the equa