MATH 300 Problem Set 4 Solutions
In this problem set log z refers to the branch of log z for which < Im log z + 2.
1. Factor the following polynomials into a product of linear terms:
(a) z 2 + 3z 17i
(b) z 3 + z 2 z 1
(c) z 5 1
Solution. (a) By the high s
MATH 300-101, Winter 2012, Term 1, Problem Set 7
In this problem set, Cr (z) denotes the circle of radius r centered at z, traversed in once in counter-clockwise
direction. Cr (z) denotes the same circle, but traversed once in clockwise direction.
1. For
MATH 300-101, Winter 2012, Term 1, Problem Set 5
1. Derive the formulas for the inverse trigonometric functions tan1 (z), cos1 (z), cot1 (z).
2. Find all values of
a) tan1 (1 + i)
b) cos1 (i)
3. Let [z ] denote the principal branch of z . Show that
(i) [z
MATH 300-101, Winter 2012, Term 1, Problem Set 2
1. Find all values of (1 + i)1/2 and (1 i 3)3/4 = [(1 i 3)3 ]1/4
2. Let w be an n-th root of 1 not equal to 1, show that 1 + w + w2 + + wn1 = 0.
3. For complex numbers z, one can dene the trigonometric func
MATH 300-101, Winter 2012, Term 1, Problem Set 3
1. Using the Cauchy-Riemann equations, determine where the following functions are dierentiable
and where they are analytic.
(a) f (z) = z 3
(b) f (z) = ey cos x + iey sin x where x = Re z and y = Im z.
(c)
MATH 300-101, Winter 2012, Term 1, Problem Set 1
1. Calculate the real and imaginary part of each of the following numbers:
(a) (8+i)(12i)
(b) (1 + i)(2 i)(3 + 2i)
(1+2i)2
2. Let Im(z) = 0, prove that Im(z) and Im(1/z) have dierent signs.
3. Let z1 = 1 +
HOMEWORK ASSIGNMENT #1
due in class on Friday, Janyary 13
Student No:
Name (Print):
Note: All homework assignments are due in class one week after being assigned.
They must be on standard 8 1 11 size paper and they must be stapled. Assign2
ments which are
SOLUTIONS TO HOMEWORK ASSIGNMENT # 7
1. Determine the nature of all singularities of the following functions f (z ).
(a) f (z ) = cos 1/z.
1
.
(b) f (z ) = 2
z sin z
z
.
(c) f (z ) = z2
e 1
Solution:
(a) z = 0 is the only singularity. It is an essential s
SOLUTIONS TO HOMEWORK ASSIGNMENT # 5
1. Use Cauchys Integral Theorem to evaluate the following integrals.
z
dz, where C is the positively oriented circle |z 2| = 2.
3
C z +1
z
(b)
dz, where C is the circle |z | = 3, oriented in the clockwise direction.
2
SOLUTIONS TO HOMEWORK ASSIGNMENT # 4
1. Suppose f (z ) is dened for |z z0 | < , where is some positive number. If f (z0 )
show that f (z ) is continuous at z = z0 .
f (z ) f (z0 )
Solution: f (z0 ) means that lim
and equals f (z0 ). Write this in the
z
MATH 300-101, Winter 2012, Term 1, Problem Set 4
In this problem set, log (z) refers to the branch of log where < Im(log (z) + 2.
1. Factor the following polynomials into a product of linear factors
a) z 2 + 3z 15i
b) z 3 + z 2 z 1
c) z 7 1
2. Find the po
MATH 300-101, Winter 2012, Term 1, Problem Set 6
1. Is it always true that Re
f (z) dz =
(Re f (z) dz? Prove this or give a counter example.
2. Let be the circle of radius 1 centered at i + 1. Find the value of
the integral.
1
z
dz without explicitly cal
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MATH 300-101, Winter 2012, Term 1, Problem Set 9
1. Express sin(z) and cos(z) as a power series centered at z = 0.
2. Uniqueness of analytic extensions:
Let f (x), x R be an innitely (real-) dierentiable function dened on R. A function g(z), z D,
dened on
MATH 300-101, Winter 2012, Term 1, Problem Set 10
1. Find the Laurent series of
1
z3
1
cos( 2z ) and calculate
2. Find three dierent Laurent series for f (z) =
C1 (0)
1
z 2 5z+6
1
cos( 2z )
dz.
3
z
all centered at zero.
3. Let f (z) be analytic in the ann
MATH 300-101, Winter 2012, Term 1, Problem Set 8
1. Let f (z) be entire and suppose that f (5) (z) is bounded in the whole plane. Prove that f must be a
polynomial of degree at most 5.
2. Let f (z) be analytic in the annulus 1 |z| 2 and obey |f (z)| 3 on
HOMEWORK ASSIGNMENT # 6
due in class on Friday, March 10
Student No:
Name (Print):
Note: All homework assignments are due in class one week after being assigned.
They must be on standard 8 1 11 size paper and they must be stapled. Assign2
ments which are
SOLUTIONS TOHOMEWORK ASSIGNMENT # 3
1. Find the harmonic function u(x, y) on the region = cfw_z | y > 0, 2 xy 4 that
if xy = 2
satises the boundary conditions u(x, y) =
if xy = 4
Solution: The solution is u = Axy + B, where the constants A, B are chosen
Math 300 Problem Set 7
For section 201, due Thursday, March 8
For section 202, due Wednesday, March 7
1. Evaluate
f (z) dz
(i) where is the counter-clockwise circle of radius 2 centered at the origin and f (z) =
(ii) where is the counter-clockwise circle
Math 300 Problem Set 3
For section 201, due Thursday, January 26
For section 202, due Wednesday, January 25
1. Using the CauchyRiemann equations, determine where the following functions are dierentiable and where they are analytic:
(a) f (z) = z 3
(b) f (
Math 300 Problem Set 2
For section 201, due Thursday, January 19
For section 202, due Wednesday, January 18
NOTE: When sketching a set, you must clearly indicate which points are in the set and which
points are not. For instance, boundary lines or curves