Math 381, Complex variables and transforms
Assignment 1, due in the tutorial session, week of September 19, 2016
1. Compute the modulus of
(1 i)n
(2 + 2i)n
,
where n is a positive integer.
2. Find all the values of (1 + i)2/3 .
3. Prove that for all z1 ,
McGill University
Department of Mathematics and Statistics
MATH381 Complex Variables and Transforms
Fall 2008
Assignment 2: The assignment consists of
the WebWork assignment elemderiv due October 8 at 11:55pm, and
the written assignment below. This assi
McGill University
Department of Mathematics and Statistics
MATH 381 Complex Variables and Transforms
Fall 2008
All closed contours are positively oriented, unless explicitly mentioned otherwise.
z 2 dz
1. Let (t) := t + it2 , 1 t 2. Compute the line integ
McGill University
Department of Mathematics and Statistics
MATH381 Complex Variables and Transforms
Fall 2006
1. (a) u(x, y) = 2xy + 3x 2y, ux = 2y + 3, uxx = 0, uy = 2x 2, uyy = 0. Thus
uxx + uyy 0 and u is harmonic everywhere.
(b)
ux = 2y + 3 = vy v = y
McGill University
Department of Mathematics and Statistics
MATH381 Complex Variables and Transforms
Fall 2008
5.6.4 By equation 5.3-8 Log
1
=
1z
|z| < 1, and
Log 1 +
1
z1
n=1
zn
for all |z| < 1. Thus Log(1 z) =
n
= Log 1
=
n=1
=
(1)n+1
n
1
z1
=
n=1
1
n
McGill University
Department of Mathematics and Statistics
MATH381 Complex Variables and Transforms
Fall 2008
1. Evaluate the following integrals:
(c)
|z|=3
sin z
dz
z2
|z|=2
cosh z
dz
(z 3)(z 1)
1
(d)
2i
(a)
(b)
1
2i
|z1|=2
|z|=2
1
dz
(z + 2)(z i)2
sin(2
McGill University
Department of Mathematics and Statistics
MATH381 Complex Variables and Transforms
Fall 2008
Optional Assignment 6: This assignment will NOT be graded! However, it is
recommended that you attempt all the problems to practice for the nal
e
Math 381
Assignment(6)
Due date:Friday, March 27.
Hand in the rst part and the underlined problems in part two; the others
are for more practice.
1
Solve the following dierence equations.
an+1 an = n, a0 = 1, a1 = 1
an+1 an = n2 , a0 = 0, a1 = 0
2
Page 3
MATH381
Sample Midterm Exam
10/28/06
GIVE DETAILED AND COMPLETE SOLUTIONS. FULLY SIMPLIFY YOUR
ANSWERS.
1. Let u(x, y) = 2xy + 3x 2y.
(a) Show that u is harmonic on the entire complex plane.
(b) Find all harmonic conjugates of u.
(c) Let f (z) be an entir
McGill University
Department of Mathematics and Statistics
MATH381 Complex Variables and Transforms
Fall 2008
Assignment 4: This assignment has no WebWork component i.e. it consists solely of the
written questions below. It is due Monday, November 10 at 1
MCGILL UNIVERSITY
FACULTY OF ENGINEERING
FINAL EXAMINATION
MATH 381
COMPLEX VARIABLES AND TRANSFORMS
Examiner: Professor D. Sussman Date: Wednesday December 14, 2005
Associate Examiner: Dr. Benoit Charbonneau Time: 2:00 PM- 5:00 PM
MW
INSTRUCTIONS
Please
_._._._
McGill University April 2012
Faculty of Engineering Final examination
Complex variables and transforms
Math 381
Monday, April 23, 2012
Time: 6pm _ 9pm
/Z/f") \Tff
sociate Examiner: Prof. J. Toth
I Student name (last, rst) Student number (
McGill University April 2012
Faculty of Engineering Final examination
Complex variables and transforms
Math 381
Monday, April 23, 2012
Time: 6pm - 9pm
KMW) JZTeh .
sociate Examiner: Prof. J. Toth
I Student name (last, rst) | Student number (McGi
McGILL UNIVERSITY
FACULTY OF ENGINEERING
FINAL EXAMINATION
MATH 381
COMPLEX VARIABLES AND TRANSFORMS
Examiner: Dr. Axel Hundemer M34 Date: Monday December 11, 2006
Associate Examiner: Professor N.Sancho Alelt? Time 2:00 PM- 5:00 PM
INSTRUCTIONS
1. Please
McGill University
Department of Mathematics and Statistics
MATH381 Complex Variables and Transforms
Fall 2008
1. (a) Where does the function f (z) = z 2 + (x 1)2 + i(y 1)2 have a derivative?
(b) Where is this function analytic? Explain!
(c) Derive a formu
McGILL UNIVERSITY
FACULTY OF ENGINEERING
FINAL EXAMINATION
MATH 381
COMPLEX VARIABLES AND TRANSFORMS
Examiner: Professor W. Jonsson Date: Friday April 21, 2006
Associate Examiner: Professor D. Sussman Time: 2:00 PM 5:00 PM
INSTRUCTIONS
I
Please attempt al
McGill UNIVERSITY
FACULTY OF SCIENCE
FINAL EXAMINATION
MATH .181
COMPLEX VARIABLES AND TRANSFORMS
Examiner: Professor I). Sussman Date: Monday December 13, 2004
Associate Examiner: Professor 1. Klemes Time: 2:00 PM 7 5:00 PM
INSTRUCTIONS
1. Please answe
MATH381
Final Exam Solutions
12/11/06
GIVE DETAILED AND COMPLETE SOLUTIONS. FULLY SIMPLIFY YOUR
ANSWERS.
1. (10 marks) Let u(x, y) be a harmonic function dened on the entire complex plane and
let v(x, y) be a harmonic conjugate of u. Show that u2 v 2 is h
FACULTY OF ENGINEERING
FINAL EXAMINATION
MATHEMATICS MATH381
Complex Variables and Transforms
Examiner: Professor S. W. Drury Date: Monday, 17 December 2007
Associate Examiner: Professor N. Sancho Time: 9: 00 am. 12: 00 noon.
INSTRUCTIONS
Answer all quest
Math 381
Assignment(2)
Due date:Friday, February 6.
Hand in the underlined problems only; the others are for more practice.
Page 70: 7, 9, 13, 18.
Page 76-80: 1, 2, 4, 8, 9, 10, 15, 16, 17, 20, 23.
Page 85-87: 9, 10, 11, 12.
Page 105-107: 1, 15, 19, 2
MATH 381
Assignment(1)
Due date:Wednesday January 28 in the class
Solve the following question from the text book (Complex variable, A. David Wunsch,
3rd Edition)
Page 35-38:1,4, 14, 20, 25, 26, 27.
Page 46-47:6,9, 23,25.
page 53-55:12,17
page 62-63:5
MATH 381- Complex variables and transforms
1. Description: Analytic functions, Cauchy-Riemann equations, simple mappings, Cauchys theorem,
Cauchys integral formula, Taylor and Laurent expansions, residue calculus. Properties of one and two-sided
Fourier a
Math 381, Complex variables and transforms
Assignment 6, due in the tutorial session, week of November 17 , 2014
1. Use contour integration and residues to show that
Z
0
2
d
2
=p
,
a + sin2
a(a + 1)
for a > 0.
2. Use contour integration and residues to e
Math 381, Complex variables and transforms
Assignment 1, due in the tutorial session, week of September 15, 2014
1. Compute the modulus of
(1 i)n
(2 + 2i)n
,
where n is a positive integer.
2. Find all the values of (1 + i)2/3 .
3. Prove that for all z1 ,
Math 381, Complex variables and transforms
Assignment 2, due in the tutorial session, week of September 29, 2014
1. Where does |ez | attain its maximum and minimum values in the region |z 1 i| 2
2. Find all the values of elog(i sinh 1) and state the princ
Math 381, Complex variables and transforms
Assignment 2, due in the tutorial session, week of October 27, 2014
1. Find the Taylor series expansion of
z
(z + 1)(z + 2)
f (z) =
about z = 1 and find the circle within which the series is convergent.
2. Find t
Math 381, Complex variables and transforms
Assignment 5, due in the tutorial session, week of November 10 , 2014
1. Determine the singularities of each of the following functions
sinh z
,
z2 + 2
f (z) =
f (z) =
z1
,
(z 3 1)2
f (z) =
sin(z i/4)
,
e2z i
sho
Math 381, Complex variables and transforms, Assignment 3,
due on the week of October 13, 2014
(in class on Tuesday October 14 for the students registered for the Monday tutorial)
1. Calculate the integral
I
|z|=3/2
cos(z 1)
dz,
z2 z 2
where the contour of