Math 380
Bonus problem
Due 4/14
Worth 2 homework points.
1. Let D denote the open unit disk and let @D denote its boundary, the unit circle. Suppose u(eit ) is a
continuous real-valued function on @D. For rei in D, dene
Z
1 r2 2
u(eit )
i
u re =
b
dt:
2
1
Math 380
Solutions to Assignment 3
2.1.13 Let w = z 2 : We use the notation z = x + iy and w = u + iv:
ll
(a) The image of the line x = 1 is given by w = (1 + iy )2 = 1 y 2 + 2yi where y ranges over all real
values. These points in the uv plane are on the
Math 380
Take-home Quiz
Due in class Friday, April 30
Name
Ground rules. You must do this exam in one sitting by yourself. Do not discuss the problems with anyone
between 10 pm April 18 and 10 am April 30. You may use your notes and textbook only.
I
z2
dz
Math380
Vocabulary/Theorem List for Quiz
As mentioned in class Wednesday, the next pop quiz will be Friday, April 9. Be prepared to state
denitions and theorems:
Be able to state the denitions of:
polygonally connected set
simply connected set
domain
Math 380
Spring 2008
San Francisco State University
Title: Introduction to Functions of a Complex Variable
Prerequisites: A grade of C or better in MATH 228 and MATH 325 or the equivalent.
(No knowledge of complex numbers is necessary.)
Instructor: Eric H
Homework 1 (Due on Sept 3)
1. Simplify the following complex numbers to the standard form a + bi.
(i) (1 + 2i)(2 3i) (ii)
2i
1+4i
(iii) (1 3i7 ) + (2 4i9 ) (2 i)2 , (iv)
i
.
(5i)(12i)
2. Solve the following equations for the unknown complex numbers.
(i) z
Homework 2 (Due on Feb 16)
1. Find the roots of the following complex numbers:
(i) the cube roots of i
(ii) the 8th root of 2.
2. Sketch the following sets.
(a) |z 2 + i| 1, (b)|2z + 3| > 4,
(c) Imz > 1, (d) Imz = 1,
(e) 0 argz /4 and z 6= 0, (f) |z 4| |z
Homework 6 (Due on March 17 (Thursday)
Throughout the exercise, z = x + yi.
1. Compute the following numbers
(a) sin(1 i), (b) cosh(1 + i), (c) Im cos(i), (d) Re sinh(i ln 2)
2. Solve the following equations:
(a) tan z = 3i, (b) cosh z = i.
3. Evaluate
th
Sample Test 1
1. (25 points) Compute the following complex numbers.
(a)
1+i
3+4i
(b) (4)1/4
(c) Log(1 i)
(d) |e5i2 |
(e) Imcos(i).
(20 points) 2a. Determine the points for which f (z) = 2y 2 ix2 is analytic.
Explain your answer.
b. Determine if
y
x
i 2
2
Sample Test 2
1. (20 points) Compute the following complex numbers.
(a)
i7
(3i)(1+i)
(b) log(e)
(c) P.V.ii
(d) | cos i|
2. (10 points) Sketch the following regions on the complex plane.
(a). A = cfw_z : |z| > |z + 2|, (b) B = cfw_z : 1 < |z|.
Are they dom
Math 380
Solutions to Sample Exam Problems
1. Find each of the following:
(a) (2 + i)(3 + 4i) = 2 + 11i
p
(b) Arg (1
3i) =
=3
2+
(c) Im( 3 i ) =
i
1
2
(d) cos i = (e1 + e
1
)=2 (or cosh 1)
p
3i) = 2e
(e) the polar form of (1
(f) i1=3 = ei(
=2+2k ) 1=3
= e
Math 380
Spring 2010
Midterm Exam Checklist
Date of Exam: Friday, March 12, 2010
What to bring: Pencil, eraser, calculuator (not really needed). The exam will be closed book and closed
notes.
Complex Arithmetic
mulitplication, addition, division, complex
Math 380
Checklist for Exam 2
Be prepared to do the following basic kinds of calculations:
parametrize directed line segments and sections of circles
set up and evaluate contour integrals
identify whether or not a given domain is simply connected
in a
MATH 380
Final Exam Study Guide
Spring 2010
Time of exam: Friday 21 May, 8:00 sharp to 10:30
What to bring: pencil, calculator, one sheet of notes (double-sided).
Follow the checklists for the midterm exams plus the following items:
nd Taylor and Laurent
1.4.2b 2e3+i
=6
Math 380
Solutions to selected problems from hw 2
p
p
= 2e3 (cos( =6) + i sin( =6) = 2e3 ( 3 + i)=2 = e3 3 + ie3 :
1.4.4b
p
2 2ei =4
2e i 5 = 6
p 13 i=12
=
2e
p i =12
=
2e
:
2 + 2i
p
3+i
1.4.10 Suppose z = x + iy with x
=
0: Then
jez j =
e
Math 380
Solutions to hw 5
3.2.20 Let R = fx + iy : 1 x 1 and 0 y
g and A = w : e 1 jwj e1 and 0 Arg (w)
:
There are three things to show: (1) z 2 R ) ez 2 A, (2) every point in A is the image of some point
in R, and (3) ez is 1-1 on R.
(1) Let z = x + iy
Math 380
Assignment 8 due 4/7/2010
1. Let G be a subset of the complex plane.
(a) Following the language used in the text, what properties must G satisfy in order to qualify as a
domain?
(b) Assuming G is a domain, must it be simply connected? If not, giv
Math 380
Assignment 9 due 4/14
Section 4.6 # 5, 6, 17
Section 5.1 # 11
Section 4.7 # 11
Additional problems
A. In problem 4.7.11, write down the integral for the temperature at the point (1+i)/2. (Don evaluate
t
the integral.)
B. Dene the function u (x; y
Math 380
Hw 8 Solutions
1. Let G
C:
(a) G is a domain if and only if it is both open and connected. This is equivalent to G being both
open and polygonally connected.
(b) If G is a domain, it need not be simply connected. As an example, take G = Cn f0g :