MATH 3410 COMPLEX VARIABLES
ASSIGNMENT 1, FEBRUARY 11, 2014
1.
√
√
a) Show that the points 1, −1/2 + i 3/2, and −1/2 − i 3/2 are the vertices
of an equilateral triangle.
b) Write each of these complex numbers in polar form:
√
√
√
√ 2
−3 + 3 3i
− 10(1 + i)
√
i)
, ii)
, iii)
6 − 2i .
2 + 2i
2 3 + 2i
2. Use complex numbers to compute the following integral:
π/4
Z
(cos θ)2 (sin θ)4 dθ.
0
3.
4.
a) Show that for any complex number z 6= 1 and any positive integer n we
have
zn − 1
.
1 + z + z 2 + · · · + z n−1 =
z−1
b) Use a) and De Moivre’s formula to establish the following identity:
1 sin n + 12 θ
.
1 + cos θ + cos 2θ + · · · + cos nθ = +
2
2 sin 2θ
a) Find all the solutions of the equation
(2z + 3i − 1)7 = z 7 .
b) Prove that if z0 is a solution to the equation
(2z − 1)2014 = (2z + 1)2014
then Re(z0 ) = 0.
5. Let z be a point of the set S. Prove that if z is not an interior point of S, then
z must be a boundary point of S.
6. The Joukowski mapping is defined by
1
1
z+
.
J(z) =
2
z
Show that
a) J maps the unit circle {z ∈ C : |z| = 1} onto the real interval [−1, 1].
1
2
MATH 3410 COMPLEX VARIABLES ASSIGNMENT 1, FEBRUARY 11, 2014
b) J maps the circle {z ∈ C : |z| = r} (where r > 0 and r 6= 1) onto the
ellipse
u2
v2
+
2
2 = 1.
1
1
1
1
r
+
r
−
2
r
2
r
7. Use the definition of the limit of a sequence of complex numbers to do the
following:
a) Prove that if |z| < 1, then the sequence {z n } converges to 0.
b) Let {zn } be a sequence of complex numbers such that limn→∞ zn = 0 and
{ωn } be a bounded sequence. Prove that
lim ωn zn = 0.
n→∞
8. Define the following function
(
f (z) =
z5
|z|4
0
if z 6= 0
if z = 0.
Prove that the real and imaginary parts of f satisfy the Cauchy-Riemann equations at z = 0 but that f is not differentiable at z = 0.
9. Prove that if f (z) and |f (z)| are both analytic functions in a domain D, then
f (z) must be constant in D.
10.
a) Let
u(x, y) = x3 + y 3 − 3xy 2 − 3x2 y + 2xy + 2x.
Verify that u(x, y) is a harmonic function and find a harmonic conjugate of
u(x, y).
b) Find a harmonic function φ(x, y) such that
φ(x, 0) = 2x3 − 3x + 1.
Due Date: Thursday February 27, at the beginning of class.