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
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
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
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
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
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
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: 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 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 (
MATH381
Final Exam
12/11/06
GIVE DETAILED AND COMPLETE SOLUTIONS. FULLY SIMPLIFY YOUR
ANSWERS.
1. (10 marks) Let u(x, y) be a harmonic function defined on the entire complex plane and
let v(x, y) be a harmonic conjugate of u. Show that u2 v 2 is harmonic.
Question 1 Find all solutions z C of the equation
1
3
4
z = +i
.
2
2
Solution 1 We write the number z0 :=
1
2
+i
3
2
in polar form. Note that
z0 = cos /3 + i sin /3 = exp(i ).
3
As z0 has magnitude |z0 | = 1, we know that the solutions are on the unit cir
Question 1 Find all solutions z C of the equation
z 2 2z + 1 = i.
and determine their real parts Re z.
Question 2
1. Is the function
f: CC
f (x + iy) := sin x cosh y + i cos x sinh y
holomorphic on C?
2. Find a harmonic conjugate to
u : R2 R
u(x, y) = y 2
MATH 381 Final
Fall 2014
Instructor: Kamran
1. Find all the complex numbers w satisfying the equation
! + ! + 1 = 0
and plot them in the complex plane. Explain why the plot is symmetric with respect
to both t
MATH 381
Final Exam
12/18/08
GIVE DETAILED AND COMPLETE SOLUTIONS. FULLY SIMPLIFY YOUR
ANSWERS. ALL CLOSED CONTOURS ARE POSITIVELY ORIENTED.
1. (10 marks) Let u(x, y) be a harmonic function defined on the entire complex plane and
let v(x, y) be a harmonic
Math 381, Complex variables and transforms
Assignment 6, will not be marked/Solutions will follow
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 evaluate the int
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
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
MATH 381 FALL 2012 PRACTICE FINAL
1. Compute
cosh ez
dz,
z 2 4z + 3
around the (positively oriented) square with corners at z = 3 1, and z = 3 i.
2. Find the radius of convergence of the Taylor series of
1
f (z) = 1/2
,
z 1
expanded about z = 2, where we
Math 381
Assignment(5)
Due date:Wednesday, March 25.
Hand in the rst part and the underlined problems in part two; the others
are for more practice.
1
1. Find the Taylor series for the Logz about the point z0 = 1. Find the
radius of convergence.
2. Find t
Math 381
Assignment(3)
Due date:Wednesday, February 18 .
Hand in the underlined problems only; the others are for more practice.
Page 119-120: 5, 6, 10, 11, 13, 15, 17, 27.
Page 126-128: 2, 3, 5, 8, 9, 11, 12, 13, 14, 15.
Page 132-133: 1, 4, 8, 11, 14,
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
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
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
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
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
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