University of Toronto
Department of Mathematics
MAT 334F Complex Variables
Fall, 1998
Homework 1. Sketch of Solution
Exercise 1: Let z = 1 + i. Then
z 2 2z + 2 = (0 + 2i) (2 + 2i) + 2 = 0.
Let now z = 1 i. Then
z 2 2z + 2 = (0 2i) (2 2i) + 2 = 0.
Exercise
University of Toronto
Complex Variables
Mat 334H1S
3 hours
1. (12 marks) Sketch and describe as accurately as possible the image of
the annulus A = cfw_z |r1 |z | r2 , 0 < r1 < 1 < r2 , under the mapping
z
log z where the log function is dened using the
University of Toronto
Department of Mathematics
MAT 334F Complex Variables
Fall, 1998
Homework 5. Sketch of Solution
Exercise 1, p. 32: (a) The function is dened everywhere except
when the denominator is zero, i.e., z = i.
(b) The function is dened everyw
University of Toronto
Department of Mathematics
MAT 334F Complex Variables
Fall, 1998
Homework 8. Sketch of Solution
Exercise 8, p. 73: We use Equations (11) and (12) from the book.
If z = x + iy ,
(11)
(12)
| sinh z |2 = sinh2 x + sin2 y
| cosh z |2 = si
University of Toronto
Department of Mathematics
MAT 334F Complex Variables
Fall, 1998
Homework 9. Sketch of Solution
Exercise 9, p. 100: The length of the contour C is . By the
triangle inequality, if z is on C , we have |z 2 1| |z |2 1| = 3,
so | z21 1 |
University of Toronto
Department of Mathematics
MAT 334F Complex Variables
Fall, 1998
Homework 8. Sketch of Solution
Exercise 8, p. 73: We use Equations (11) and (12) from the book.
If z = x + iy ,
(11)
(12)
| sinh z |2 = sinh2 x + sin2 y
| cosh z |2 = si
University of Toronto
Department of Mathematics
MAT 334F Complex Variables
Fall, 1998
Homework 7. Sketch of Solution
Exercise 8, p. 67: (a) If x , then |ex+iy | = ex 0, so
ex+iy 0. (b) If x is constant and y , then |ex+iy | = ex and
arg(ex+iy ) , so ex+iy
University of Toronto
Department of Mathematics
MAT 334F Complex Variables
Fall, 1998
Homework 6. Sketch of Solution
Exercise 7, p. 48: We assume that f (0) exists and try to compute
f (0 + z ) f (0)
z 0
z
f (0 + z ) f (0)
= lim
z 0
z
f (0) = lim
(*)
= li
University of Toronto
Department of Mathematics
MAT 334F Complex Variables
Fall, 1998
Homework 5. Sketch of Solution
Exercise 1, p. 32: (a) The function is dened everywhere except
when the denominator is zero, i.e., z = i.
(b) The function is dened everyw
University of Toronto
Department of Mathematics
MAT 334F Complex Variables
Fall, 1998
Homework 3. Sketch of Solution
Exercise 5: Let us rst solve the equation geometrically and then
algebraically.
Geometric solution. The set of points
ei ,
0 < 2 ,
is the
University of Toronto
Department of Mathematics
MAT 334F Complex Variables
Fall, 1998
Homework 2. Sketch of Solution
i
Exercise 2: (b): iz = z = iz .
2 = (2 + 1)2 = (2 i)2 = 3 4i.
(c): (2 + i)
(d): |(2 + 5)( 2 i)| = |(2z + 5)|( 2 i)| = |(2z + 5)| 3.
z
Ex
6.1
SOLUTIONS
Notes: The first half of this section is computational and is easily learned. The second half concerns the
concepts of orthogonality and orthogonal complements, which are essential for later work. Theorem 3 is an important general fact, but