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 th
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 )]
i
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
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 no
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
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: Introductio
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
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
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
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 int
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
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 e
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
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
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
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 Contou
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 no
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, an
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
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, an
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
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 (re
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
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
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
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 ) =
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)e
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 =