Exercises 1.2
Mihai Halic
Recall
The Circle of Apollonius: The locus of a point C whose distance from a fixed point p is a multiple (0 1) of its
distance from another fixed point q is a circle that is
MATH 334; HOMEWORK # 3
Due October 6, 2009
1. (page 85, # 14) Let P (z ) = A(z z1 ) (z zn ), where A and z1 , ., zn are
complex numbers and A = 0. Show that
P (z )
=
P (z )
n
j =1
1
,
z zj
z = z1 , .,
Math334 - Challenge Problems 2
1
Challenge Problems
Comments: These problems will contribute to the Bonus assigment due the last week of classes. It is encouraged to attempt and work on these problems
Math334 - Exercises 1
1
Practice Exercises
Comments: You are encouraged to work on these problems and solutions will be provided, but they are, well,
not graded. On the other hand, they are occasional
Math334 - Challenge Exercises 1
1
Challenge Problems
Comments: These problems will contribute to the Bonus assigment due the last week of classes. It is encouraged to attempt and work on these problem
Math334 - Practice Exercises 2
1
Other Exercises
Comments: You are encouraged to work on these problems and solutions will be provided, but they are, well,
not graded. On the other hand, they are occa
Exercises 2.3
1)
z
z 1 ( z 2)2 dz
2)
ez
z ( z 3)dz
z 2
3)
e z ( z 3)
z
z 2
f ( z ) e z ( z 3)
1
f (0)
3
z
e
2 i
z 2 z ( z 3)dz 3
z
z 1 ( z 2)2 dz =0
2
0
2
0
d
dz (2 z )
=
2 cos z 1 iz (4 z z 2 1
Exercises 2.4
Short review:
I) e
in
i n2
II) e
ein 1
n 2k (k 0, 1,)
cos(n ) i sin(n )
in 1 n 2k 1 (k 0, 1,)
e
i n
e 2 i (n 4k 1 (k 0, 1,)
cos(n ) i sin(n ) n
2
2 i2
e i (n 4k 1 (k 0, 1,)
b b 2 4a
Elementary formulae
Mihai Halic
Dear students,
I was thinking what can be the most helpful booster for the skills of solving exercises. Many of you are not well
acquainted with some computations in ca
Exercises 1.4
Mihai Halic
Recall: A sequence cfw_ zn is a function form .
>0 N > 0 st n > N zn C < .
Definition of limit of a sequence: lim zn = C or (z n C ) if
n
Let zn = xn + iyn & C = a + ib th
Math334 - Practice Exercises 1
1
Jan 5
1. Let z = 2 + 3i. Find z and |z|.
2. Write
3+4i
1+i
in the form of a + bi.
3. Let z = 2 + 3i and w = 1 i. Find z w
+ zw.
4. *Show that z = z. Thus show that zw
University of Toronto, Faculty of Arts and Science
Midterm Examination, March 17, 2015, 6:10 pm 7:00 pm,
EX300
MAT334HIS Complex Variables
Instructor: Dr. Dinakar Muthiah
Duration 50 minutes
No
University of Toronto
Faculty of Arts and Science
MAT334H1S - Complex Variables
Assignment 1
Due Wednesday January 21, 2015
at the beginning of the lecture
Last name . . . . . . . . . . . . . . . . .
Math334 - Practice Exercises 2
1
Jan 17
1. Prove that |z/w| = |z|/|w| by writing z = a + bi and w = c + di.
2
2. Let f (z) = ez . Express f (z) = f (x + iy) = u(x, y) + iv(x, y).
3. Let f (x + yi) = 3
Math334 - Practice Exercises 6
1. Parametrize the following curves
(a) The circle |z
(b) The circle |z
4
5i| = 3 counterclockwise.
4
5i| = 3 counterclockwise starting from z = 4 + 8i.
(c) The semicirc
Math334 - Practice Exercises 5
1
Logarithms
1. Evaluate the followings
(a) log 2
(b) Log (3 + 2i)
(c) Log ( 3 + 2)
2. While we know as multivalued functions, log z +log w = log(zw), give an example so
Math334 - Practice Exercises 3
1
Jan 24
1. Find f 0 (z).
2
(a) f (z) = ez .
(b) f (z) = z 2 + 3z + 2.
(c) f (z) =
z 2 +1
z 3 +1 .
2. Let f (z) = f (x + iy) = x2 + y 2 x + iy. Write f (z) in terms of z
Exercises for mathematicians, in future courses.
Theorem:
Let C be a simple closed contour. We assume that C and interior (C ) are contained in the domain D.
Suppose f is analytic in D, except at a fi
Exercises 2.6
Observation: For rational functions
P ( x)
we should always check: deg Q deg P 2 .
Q( x)
This is to be sure that the indefinite integral
P( x)
Q( x) dx
is convergent.
x4
z4
dx
We
consid
MATH 334; HOMEWORK # 7
Due November 17, 2009
1. (page 179, # 5) Determine the number of zeros of f (z ) = z 9 + 5z 2 + 3 in the
rst quadrant.
2. (page 180, # 16) Let f, g be analytic on a domain conta