Math 417 Midterm Exam Solutions
Friday, July 11, 2008
1. Find all values of:
(a) log(3 4i)
(b) (2 + 2i)i
Solution:
(a) The modulus of z = 3 4i is r = 5 and the principal argument is = tan1
4
.
3
Therefore, the values of log(3 4i) are
log z = ln r + i( +
MATH 417
Instructor: D. Cabrera
Homework 2
Due June 23
1. Find and sketch the image of the rectangle 0 < x < 1, < y < under the trans2
2
formation w = ez .
Solution: The transformation w = ez can be written as
w = ez = ex cos y + iex sin y
If we dene w =
Math 417 Midterm Exam Solutions
Friday, July 11, 2008
1. (30 pts) Determine all possible series representations of the function:
f (z ) =
1
z (z 2 + 1)
about z = 0 and state their regions of validity.
Solution: The function has singular points at z = 0, z
MATH 417
Instructor: D. Cabrera
Final Exam
Practice Problems
Contour Integrals
1. Evaluate
C
2. Evaluate
C
ez
dz where C is the circle |z | = 2 oriented counterclockwise.
z2 + 1
1
dz where C is the circle |z | = 4 oriented counterclockwise.
(z + 5)(z 1)5
Math 417, Complex Analysis, Final Exam
Thursday, August 8, 2008
YOU MUST SHOW ALL OF YOUR COMPUTATIONS IN THE EXAM BOOKLET TO
RECEIVE FULL CREDIT
1. (30 pts) Determine all possible series representations of the function:
f (z ) =
1
+ 1)
z (z 2
about z = 0
MATH 417
Instructor: D. Cabrera
Homework 7
Due July 28
1. Find all singular points of the given function. For each isolated singular point, classify
the point as being a removable singularity, a pole of order N (specify N ), or an essential
singularity.
1
MATH 417
Instructor: D. Cabrera
Homework 6
Due July 21
1. Find the radius of convergence for each power series below.
(a)
n=2
(b)
n2 (z 3)n
en (z + i)n
n=4
Solution:
(a) Using the Ratio Test we have
cn+1
cn
(n + 1)2 (z 3)n+1
= lim
n
n2 (z 3)n
(n + 1)2
| z
MATH 417
Instructor: D. Cabrera
Homework 5
Due July 16
1. Show that
f (z ) dz = 0
C
where C is the circle |z | = 2 oriented clockwise for each function below:
(a) f (z ) = zez
1
(b) f (z ) = 2
z +9
Solution:
(a) The function f (z ) = zez is entire and C i
MATH 417
Instructor: D. Cabrera
Homework 4
Due July 7
1. Find all values of each expression below.
(a) (1 i)i
(b) cos(1 i)
(c) sin1 (2)
Solution:
(a) Here we use the formula
z c = ec log z
(1 i)i = ei log(1i)
2 and the principal argument is = . Therefore,
MATH 417
Instructor: D. Cabrera
Homework 3
Due June 30
1. Let a function f (z ) = u + iv be dierentiable at z0 .
(a) Use the Chain Rule and the formulas x = r cos and y = r sin to show that
ux = ur cos u
sin
,
r
vx = vr cos v
sin
r
(b) Then use the Cauc
MATH 417
Instructor: D. Cabrera
Homework 1
Due June 18
1. Find the modulus and conjugate of each complex number below.
(a) 2 + i
(b) (2 + i)(4 + 3i)
3i
(c)
2 + 3i
Solution:
(a) For the complex number z = 2 + i, the modulus is
|z | = (2)2 + 12 = 5
and the
Math 417, Complex Analysis, Final Exam
Thursday, August 8, 2008
YOU MUST SHOW ALL OF YOUR COMPUTATIONS IN THE EXAM BOOKLET TO
RECEIVE FULL CREDIT
1. (30 pts) Determine all possible series representations of the function:
f (z ) =
1
+ 1)
z (z 2
about z = 0
Math 417 Midterm Exam Solutions
Friday, July 9, 2010
Solve any 4 of Problems 16 and 1 of Problems 78. Write your solutions in the booklet
provided. If you attempt more than 5 problems, you must clearly indicate which problems
should be graded. Answers wit
MATH 417
Instructor: D. Cabrera
Final Exam
Practice Problems
Contour Integrals
1. Evaluate
C
2. Evaluate
C
ez
dz where C is the circle |z | = 2 oriented counterclockwise.
z2 + 1
1
dz where C is the circle |z | = 4 oriented counterclockwise.
(z + 5)(z 1)5
Math 417 Midterm Exam Solutions
Friday, July 11, 2008
1. (30 pts) Determine all possible series representations of the function:
f (z ) =
1
z (z 2 + 1)
about z = 0 and state their regions of validity.
Solution: The function has singular points at z = 0, z
MATH 417
Instructor: D. Cabrera
Homework 1
Due June 18
1. Find the modulus and conjugate of each complex number below.
(a) 2 + i
(b) (2 + i)(4 + 3i)
3i
(c)
2 + 3i
Solution:
(a) For the complex number z = 2 + i, the modulus is
|z | = (2)2 + 12 = 5
and the
MATH 417
Instructor: D. Cabrera
Homework 2
Due June 23
1. Find and sketch the image of the rectangle 0 < x < 1, < y < under the trans2
2
formation w = ez .
Solution: The transformation w = ez can be written as
w = ez = ex cos y + iex sin y
If we dene w =
MATH 417
Instructor: D. Cabrera
Homework 3
Due June 30
1. Let a function f (z ) = u + iv be dierentiable at z0 .
(a) Use the Chain Rule and the formulas x = r cos and y = r sin to show that
ux = ur cos u
sin
,
r
vx = vr cos v
sin
r
(b) Then use the Cauc
MATH 417
Instructor: D. Cabrera
Homework 4
Due July 7
1. Find all values of each expression below.
(a) (1 i)i
(b) cos(1 i)
(c) sin1 (2)
Solution:
(a) Here we use the formula
z c = ec log z
(1 i)i = ei log(1i)
2 and the principal argument is = . Therefore,
MATH 417
Instructor: D. Cabrera
Homework 6
Due July 21
1. Find the radius of convergence for each power series below.
(a)
n=2
(b)
n2 (z 3)n
en (z + i)n
n=4
Solution:
(a) Using the Ratio Test we have
cn+1
cn
(n + 1)2 (z 3)n+1
= lim
n
n2 (z 3)n
(n + 1)2
| z
MATH 417
Instructor: D. Cabrera
Homework 7
Due July 28
1. Find all singular points of the given function. For each isolated singular point, classify
the point as being a removable singularity, a pole of order N (specify N ), or an essential
singularity.
1
MATH 417
Instructor: D. Cabrera
Homework 8
Due August 4
1. Compute the improper integral
0
cos 2x
dx
(x2 + 1)2
Solution: To evaluate the integral consider the complex integral
C
ei(2z )
dz
(z 2 + 1)2
where C is the union of the contours C1 and CR shown be
Math 417 Midterm Exam Solutions
Friday, July 11, 2008
1. Find all values of:
(a) log(3 4i)
(b) (2 + 2i)i
Solution:
(a) The modulus of z = 3 4i is r = 5 and the principal argument is = tan1
4
.
3
Therefore, the values of log(3 4i) are
log z = ln r + i( +
Math 417 Midterm Exam Solutions
Friday, July 9, 2010
Solve any 4 of Problems 16 and 1 of Problems 78. Write your solutions in the booklet
provided. If you attempt more than 5 problems, you must clearly indicate which problems
should be graded. Answers wit
Math 417, Complex Analysis, Midterm Exam
Friday, July 11, 2008
YOU MUST SHOW ALL OF YOUR COMPUTATIONS IN THE EXAM BOOKLET TO
RECEIVE FULL CREDIT
1. Find all values of:
(a) log(3 4i)
(b) (2 + 2i)i
2. Complete each of the following:
(a) Is |ez | = e|z | ? E
Math 417 - Complex Analysis with Applications
Summer 2010
Welcome to Math 417! This course is an introduction to Complex Analysis. Complex
Analysis is one of the great subjects of modern mathematics and an invaluable tool in
physics and engineering. In th