Assignment 1 - Complex Analysis
MATH 440/508 – M.P. Lamoureux
Due Monday, Sept 21
1. For p > 0, define a curve Γ in the complex plane by the parameterization
φ(t) = {
0+0·i
t=0
.
π
t + t sin( tp ) · i 0 < t ≤ 1
a.
Sketch the curve, showing that it is continuous.
b.
For what values of p is the arc length finite?
Infinite?
c. Find a continuous, bounded function f (z) so that the integral
Z
f (z)dz
Γ
is infinite, for some values of p.
2. Give an example to show that the Mean Value Theorem for Derivatives
does not necessarily hold for complex-valued functions. Specifically, find a
function φ : [a, b] → C that is differentiable on the interval, but there is no
point t∗ that satisfies
φ(b) − φ(a)
φ0 (t∗ ) =
.
b−a
Hint: there are plenty of easy examples, so try to find something simple.
3. With f (z) = z the complex conjugation function, compute explicitly
a. the integral
R
|z|=1 f (z)dz
b. the integral
R
Γ f (z)dz,
around the closed unit circle.
where Γ is the square with corners (±1, ±i).
4. Suppose z, w are two complex numbers so that zw 6= 1. Prove that
w−z
a. |z| < 1 and |w| < 1 implies | 1−wz
| < 1;
w−z
b. |z| = 1 or |w| = 1 implies | 1−wz
| = 1;
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