MAT389 Fall 2013, Problem Set 12
Rouch
es theorem
12.1 Determine the number of zeroes of the following polynomials inside the unit circle:
(i) z 6
5z 4 + z 3
(ii) 2z 4
2z,
2z 3 + 2z 2
2z + 9.
(i) For all z in the unit circle we have
|z 6 + z 3
2z| |z|6 +
MATH 389 FALL 2011, TEST 1 SOLUTIONS
UNIVERSITY OF TORONTO
(1) Write each of the following in the form x + iy
ei 4
(a)
3+i
Sol:
i
e 4
2 + i 2 (1 + i)(3 i)
=
= 2
3+i
2(3 + i)
20
2
2
2
=
(3 + 1 + i(3 1) =
(4 + 2i) =
(2 + i)
20
20
10
(b) (1
3i)50
Sol: Let w
Math 389 Fall 2011, Test 1, Oct. 18
No aids allowed.
In most questions there is a nal answer, a number or a formula.
In these cases please circle your nal answer, if you arrived at one.
There are six questions in total. Total points = 105
(1) (15 points)
MAT389 Fall 2014, Problem Set 4
Functions
4.1 For each of the functions dened below, describe the domain of denition that is understood:
(i) f (z) =
1
,
z2 + 1
1
(ii) f (z) = Arg ,
z
(iii) f (z) =
z
,
z+z
(iv) f (z) =
1
.
1 |z|2
(i) The function is well-d
MAT389 Fall 2014, Problem Set 5
Holomorphic functions
5.1 For each of the functions below, determine the largest domain over which they are
holomorphic.
(i) f (z) =
eiz
,
z 2 2z + 1
(ii) f (z) = log |z| + i Arg z,
(iii) f (z) = (z 3 1).
z
Note: the functi
MAT389 Fall 2014, Problem Set 3
More Mbius transformations
o
Recall that we denote by D the unit disc |z| < 1, and by H the upper half-plane Im z > 0.
3.1 Show that any Mbius transformation of the form
o
T (z) = ei
z z0
,
z z0
R,
Im z0 > 0
sends the real
MAT389 Fall 2014, Problem Set 2
More geometry
2.1 Show that every line in C can be expressed in the form
z + z + = 0
for some C and R. Why is the condition = 0 necessary?
Hint: recall that every line in C can be expressed as the set of solutions of a line
MAT389 Fall 2014, Midterm 1
Oct 8, 2014
Please justify your reasoning. Answers without an explanation will not be given any credit.
The maximum total mark in the exam is 15 points.
Do not spend too much time on any particular problem. If you get stuck
Test 2 Solutions
(1) Letting w = z + 2i we have
f (z) =
1
1
=
.
w
2
(2i w)
4(1 2i )2
By dierentiating the geometric series we have
1
=
(1 w)2
so
1
=
w
4(1 2i )2
n=1
nw n ,
n=1
n w n
=
4 (2i)n
n=1
n
4n+2 in
(z + 2i)n
(2) Starting with the geometric series,
MAT389 Fall 2014, Problem Set 1
Basic operations
1.1 Express the following complex numbers in the form rei :
(iv) 3 i,
(i) i3 ,
(ii) 1 i,
(iii) 2(1 + i),
(v) 2 2 3 i.
(i) i3 = i = e3i/2
(ii) 1 i = 2ei/4
(iii) 2(1 + i) = 2ei/4
(iv) 3 i = 2ei/6
(v) 2 2 3 i
MAT389 Fall 2014, Midterm 2
Nov 12, 2014
Please justify your reasoning. Answers without an explanation will not be given any credit.
The maximum total mark in the exam is 20 points.
Do not spend too much time on any particular problem. If you get stuck in
MAT389 Fall 2013, Problem Set 8
Integrals of complex-valued functions of a real variable
8.1 In rst-year calculus courses, integrals of the form
b
b
ex sin x dx
ex cos x dx,
a
a
are typically computed by applying integration by parts twice. Notice that th
MAT389 Fall 2013, Problem Set 5
Conformal transformations
5.1 Determine the angle of rotation at the point z = 2 + i of the transformation f (z) = z 2 and
illustrate it for some particular curve. Show that the scale factor of the transformation at
that po
MAT389 Fall 2013, Problem Set 7
Complex exponentials
7.1 Find all the possible values of xi for x 2 R .
Hint: consider the cases x < 0 and x > 0 separately.
We have
xi = ei log x = ei(Log |x|+i arg x) = e
If x > 0, arg(x) = 2k for k 2 Z, so
xi = e
2k i Lo
MAT389 Fall 2013, Problem Set 10
Taylor series
10.1 What is the largest disc on which the Taylor series about z = 0 for f (z) = tanh z converges
absolutely?
The hyperbolic tangent is defined as the quotient of two entire functions, tanh z = sinh z/ cosh z
MAT389 Fall 2013, Problem Set 11
Series
11.1 Given a power series in negative powers of z,
S=
X
bn
,
zn
n=0
show that one of two things happen:
a) the series converges nowhere, or
b) there exists an R > 0 such that the series converges:
(i) converges abso
MAT389 Fall 2013, Problem Set 9
9.1 Let R be a closed bounded region, and let f be a function that is continuous on R and
holomorphic everywhere in the interior of R. Assume that f (z) 6= 0 for any z R. Show
that |f (z)| achieves a minimum value in R whic
MAT389 Fall 2013, Problem Set 1
1.1 Express the following complex numbers in the form rei :
(i) i3 ,
(ii) 1 i,
(iii) 2(1 + i),
(iv) 3 i,
(i)
(ii)
(iii)
(iv)
(v)
(v) 2 2 3i.
i3 = i = e3i/2
1 i = 2ei/4
2(1 + i) = 2ei/4
3 i = 2ei/6
2 2 3i = 4ei/3
1.2 Express
MAT389 Fall 2016, Problem Set 8
The Cauchy-Goursat theorem
8.1 For each of the closed contours in the figure, compute the integral
I
dz
,
2+1
z
C
knowing that, for sufficiently small values of , we have
I
I
dz
dz
= ,
= .
2
2
C (i) z + 1
C (i) z + 1
Cauchy
MAT389 Fall 2013, Midterm 1
Oct 8, 2013
Please justify your reasoning. Answers without an explanation will not be given any credit.
1. [2pt] Find all solutions to each of the following equations, and plot them:
(i)
(ii)
(iii)
(iv)
[0.5pt]
[0.5pt]
[0.5pt]
MAT389 Fall 2013, Problem Set 4
Wirtinger derivatives
4.1 Use the Cauchy-Riemann equations as expressed using the Wirtinger operator / z to nd
out where each of the functions below is dierentiable. Find the corresponding derivatives
using /z.
(i) f (z) =
MAT389 Fall 2013, Problem Set 3
Functions
3.1 For each of the functions dened below, describe the domain of denition that is understood:
(i) f (z) =
z2
1
,
+1
1
(ii) f (z) = Arg ,
z
(iii) f (z) =
z
,
z+z
(iv) f (z) =
1
.
1 |z|2
(i) The function is well-de
MAT389 Fall 2013, Problem Set 6
The exponential
6.1 Suppose that a function f (z) = u(x, y) + iv(x, y) satises the following two conditions:
(1) f (x + i0) = ex , and
(2) f is entire, with derivative f (z) = f (z).
Follow the steps below to show that f (z
MAT389 Fall 2013, Problem Set 2
Curves in C
2.1 Show that every line in C can be expressed in the form
z + z + = 0
for some C and R. Why is the condition = 0 necessary?
Hint: recall that every line in C can be expressed as the set of solutions of a linear
Math 389 Fall 2012, Test 1, Oct. 16
Solutions
(1) (a) The given number is
(b) 1 +
1
1
1
11
1 + 2i
=
(1 + 2i)(8 6i) =
(4 22i) =
i
8 + 6i
100
100
25 50
3i = 2ei/3 so the given number is
251 ei51/3 = 251 ei17 = 251 .
(2) (a) 1 + i = 2ei/4 so one cube roots