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Homework Assigned on March 31, 2009 due April 7, 2009
Problem 1 (from “Complex Analysis” by Ahlfors, p.83, #6). Suppose that a fractional linear transformation
w =
az + b cz + d
with ad − bc 6 = 0
carries one pair of concentric circles
{ |z − a| = r 1 } , { |z − a| = r 2 }
into another pair of concentric circles
{ |z − b| = s 1 } , { |z − b| = s 2 }.
Prove that the ratios (^) r 1 r 2
and
s 1 s 2
of the radii must be the same.
Problem 2. For any given quadruple of distinct complex numbers z 1 , z 2 , z 3 , z 4 , define its cross-ratio to be (^) z 1 −z 3 z 1 −z 4 z 2 −z 3 z 2 −z 4
Verify that any fractional linear transformation
w =
az + b cz + d
with ad − bc 6 = 0
preserves the cross-ratio. In other words, if the fractional linear transforma- tion maps zj to wj for j = 1, 2 , 3 , 4, then
w 1 −w 3 w 1 −w 4 w 2 −w 3 w 2 −w 4
z 1 −z 3 z 1 −z 4 z 2 −z 3 z 2 −z 4
Problem 3 (from Stein & Shakarchi, p.248, #2). Suppose F (z) is holomor- phic near z = z 0 and F (z 0 ) = F ′^ (z 0 ) = 0, while F ′′^ (z 0 ) 6 = 0. Show that there are two curves Γ 1 and Γ 2 that pass through z 0 , are orthogonal at z 0 , and so that F restricted to Γ 1 is real and has a minimum at z 0 , while F restricted to Γ 2 is also real but has a maximum at z 0.
Hint: Write F (z) = (g(z))^2 for z near z 0 , and consider the mapping z 7 → g(z) and its inverse.
Problem 4 (from Stein & Shakarchi, p.248, #5). Prove that f (z) = −^12
z + (^1) z
is a conformal map from the half-disc { z = x + iy ∈ C
∣ |z|^ <^1 , y >^0
to the upper half-plane H = { y > 0 }.
Hint: The equation f (z) = w reduces to the quadratic equation
z^2 + 2wz + 1 = 0,
which has two distinct roots in C whenever w 6 = ±1. This is certainly the case if w ∈ H.
Problem 5. (a) For 0 ≤ r < 1 and θ ∈ R, use the geometric series to verify that
1 +
∑^ ∞
n=
rneinθ^ +
∑^ ∞
n=
rne−inθ
= Re
1 + reiθ 1 − reiθ
1 − r^2 1 − 2 r cos θ + r^2
(b) For 0 ≤ r < 1 and θ ∈ R, define the Poisson kernel
P (r, θ) =
2 π
1 − r^2 1 − 2 r cos θ + r^2
Use Part(a) to verify that
(i) P (r, θ) ≥ 0 for 0 ≤ r < 1 and θ ∈ R.
(ii)
∫ (^) π
θ=−π
P (r, θ) dθ = 1 for 0 ≤ r < 1.
(iii) lim r→ 1 −
−π≤θ≤π,|θ|>η
P (r, θ) dθ = 0 for any η > 0.
(c) Let v(ϕ) be a continuous function for ϕ ∈ R with period 2π. Let
u(r, θ) =
∫ (^) π
ϕ=−π
P (r, θ − ϕ)v(ϕ)dϕ
for 0 ≤ r < 1 and θ ∈ R. By using the approximate identity argument, show that u(r, θ) approaches v(θ) uniformly in −π ≤ θ ≤ π as r → 1 −.
