MATH 3007A Course Outline
1 of 3
https:/culearn.carleton.ca/moodle/pluginfile.php/1037080/mod_resource.
MATH 3007A - Functions of a Complex Variable
Fall 2014
Instructor
Dr. S. Melkonian (4279 HP, 520-2600 ext. 2126)
E-mail
[email protected]
Web S
MATH 3007A - Functions of a Complex Variable
Test 4 Solutions
November 24, 2014
[Marks]
1
dz, where is a parametrization of the square with vertices at 1, i, 1
z
and i, traversed once, counterclockwise.
Solution:
1
f(z) = is analytic on A = C \ cfw_0 and
MATH 3007A - Functions of a Complex Variable
Test 2 Solutions
October 20, 2014
[Marks]
[4]
1. Evaluate the following limits:
(a) lim i
z0
(b) lim
z 2
1
z2
=
sin(z) sin( )
d
sin(z) 1
2
= lim
=
sin(z)
z
z 2
z 2
dz
2
z=
2
= cos( ) = 1
2
[2]
2. Determine a
MATH 3007A - Functions of a Complex Variable
Test 1 Solutions
September 29, 2014
[Marks]
[4]
1. Let z = 2 3i. Determine Re(z), Im(z), z, and |z|.
Solution:
Re(z) = 2, Im(z) = 3, z = 2 + 3i, |z| = 13.
[2]
2. Express
[6]
3+i
in the form a + bi, a, b R.
1 2i
MATH 3007
D. Amundsen
Tutorial 7
Nov. 29, 2013
f (x)dx is computed using a semi-circle of radius R in the up-
1. Suppose the real integral
per half plane. State a criteria on the function f so that it is assured that the contribution
from the circular arc
MATH 3007
D. Amundsen
Nov. 15, 2013
Tutorial 6
1. Consider the function f (z) =
2
z 7 (z 2
+ 3)
(a) Find the Laurent series for f (z) centred at z = 0 and convergent on 0 < |z| < r2 .
What is the outer radius r2 ?
(b) Find the Laurent series for f (z) cen
MATH 3007A - Functions of a Complex Variable
Test 3 Solutions
November 10, 2014
[Marks]
[2]
1. Determine all points z C at which f(z) = ez + z is dierentiable.
Solution:
f(z) = ez + z = h(z, z) hz = 1 = 0 f is not dierentiable at any z C.
[2]
2. Let : [1,
MATH 3007A - Functions of a Complex Variable
Solution Set 10
ez
dz, where (t) = eit, 0 t 2.
z2
e2z
dz, where (t) = 2eit , 0 t 2.
(z + i)3
cos(3z)
dz, where (t) = 2e3it , 0 t 2.
(z 1)4
z 3(z
1. Evaluate
Solution:
f(z) = ez is analytic on A = C and is homot
MATH 3007A - Functions of a Complex Variable
Solution Set 11
1. Determine the Laurent series of f about z0 on the given region, the principal part and
the residue of f at z0 , and the nature of the singularity z0.
sin(z)
, z0 = 0, 0 < |z| <
(a) f(z) =
z2
MATH 3007A - Functions of a Complex Variable
Solution Set 12
1. Evaluate
Solution:
1
dx.
x4 + 1
1
, R > 1, 1 (t) = t, R t R, R (t) = Reit , 0 t ,
4+1
z
and = 1 R . f is analytic on A = C except for singularities at the fourth
roots of 1, and is homotopic
MATH 3007A - Functions of a Complex Variable
Solution Set 8
1. Determine whether or not 1 and 2 are homotopic in A.
(a) A = C, 1 (t) = eit , 0 t , 2 (t) = eit , 0 t . Yes.
(b) A = C \ cfw_2, 1 (t) = eit, 0 t , 2 (t) = eit , 0 t . Yes.
(c) A = C \ cfw_0, 1
MATH 3007A - Functions of a Complex Variable
Solution Set 9
2
1. Does f(z) = ez have an analytic antiderivative F (z) on C? Justify your answer.
Solution:
Yes, by the antiderivative theorem, because f is analytic on the simply-connected
domain C.
2. Let A
MATH 3007A - Functions of a Complex Variable
Solution Set 4
1. Determine the value of (1 i)i if
(a) arg(z) [0, 2)
Solution:
1
1 i = 2( 2
(1 i)i = e
1
2
i log(1i)
i) =
=e
2 ei/4 =
i[ln 2+7i/4]
7i/4
2e
log(1 i) = ln 2 +
= e7/4 ei ln
2
7i
4
.
(b) arg(z)
MATH 3007A - Functions of a Complex Variable
Solution Set 6
1. Determine whether the curve is smooth, piecewise smooth, or discontinuous.
(a) (t) = t + i|t|, 2 t 2, is piecewise smooth.
(b) (t) = t + i sin(t), 0 t 6, is smooth.
(c) (t) =
(1 + i)t,
t + 2i,
MATH 3007A - Functions of a Complex Variable
Solution Set 5
1. Determine the open set A C on which f is analytic and compute its derivative.
(a) f(z) = z 3 z
Solution:
A = C, f (z) = 3z 2 1.
1
(b) f(z) = 2
z
Solution:
A = C \ cfw_0, f (z) =
2
.
z3
(c) f(
MATH 3007A - Functions of a Complex Variable
Solution Set 3
1. Let z = 2 + 2 3 i.
(a) Find all polar forms rei of z.
Solution:
|z| = 4 z = 4( 1 + 23 i) = 4ei/3+2ik , k Z.
2
(b) Find arg(z) if arg(z) [0, 2), and express z as z = |z|ei arg(z) .
Solution:
ar
MATH 3007
D. Amundsen
Tutorial 5
Oct. 25, 2013
1. Determine if the following sequences zn for n = 0, 1, 2, . converge.
in2 + 4in 3
n2 2i
n
(b) zn = ein
n+i
(c) zn = (1)n ein
(a) zn =
2. Determine if the following series converge absolutely.
(a)
n=1
(b)
n=