MATH 440/508 - Assignment #3
Due on Friday October 29, 2010
Name —————————————–
Student number —————————
1
Problem 1:
(a) Let α be a complex number and let f be an isomorphism of the
disk {|z − α| ≤ R} with the unit disk such that f (z0 ) = 0. Show that
R(z − z0 )
f (z) = 2
eiθ
R − (z − α)(z¯0 − ᾱ)
for some real number θ.
(b) Let α ∈ [0, 1) be a real number and let Uα be the open set obtained from the unit disk D by deleting the segment [α, 1). Find an
isomorphism of Uα with the upper half disk D+ = {z ∈ C ; |z| <
1 and Im(z) > 0} .
2
Problem 2:
We say that two conformal self-maps f and g of the unit disk D are
conjugate if there is a conformal self-map h of D such that
g = h ◦ f ◦ h−1 . Let f be a conformal self-map of D which is not
the identity map. (a) Show that either f has two fixed points on the
boundary ∂D, counting multiplicity, or f has one fixed point in D. (b)
Show that f has a fixed point in D if and only if it is conjugate to a
unique rotation g(z) = eiφ z. (c) Show that f has two fixed points on
∂D if and only if f is conjugate to g(z) = (z − s)/(1 − sz) for a unique
value of s ∈ (0, 1). (d) Show that any two conformal self-maps of D
with one fixed point on ∂D, of multiplicity two, are conjugate.
3
Problem 3:
For each t > 0, define the kernel function Ct (s) on the real line R by
t
1
,
s ∈ R.
2
π s + t2
Let h(r) be a bounded continuous function on the real line, define a
function h̃(s + it) on the upper half plane H by
Z ∞
Ct (s − r)h(r)d r,
s + it ∈ H.
h̃(s + it) =
Ct (s) =
−∞
(a) Sketch the graph of Ct (s) for a few values of t. (b) Prove that Ct (s)
is a Dirac sequence as t → 0. (c) Prove that h̃(s + it) is a bounded
harmonic function on H and h̃(s + it) → h(r0 ) as s + it → r0 .
4
Problem 4:
Let u be a continuous function on an open set U . Prove that u is
harmonic on U if and only if it satisfies the disk mean value property
This means that for every point z0 ∈ U for all sufficiently small r > 0
we have
ZZ
1
u(z0 ) = 2
u(x, y)dxdy,
πr
D̄(z0 ,r)
where D̄(z0 , r) is the closed disk centered at z0 with radius r.
5
Problem 5:
(a) For t > 0, let
1 −x2 /4t
e
.
4πt
Prove that {Kt } for t → 0 is a Dirac sequence. One calls K the heat
kernel.
K(t, x) = Kt (x) = √
(b) Let D = (∂/∂x)2 − ∂/∂t be the heat operator. Show that DK = 0.
(c) Let f be a piecewise continuous bounded function on R. Let
F (t, x) = (Kt ∗ f )(x). Show that DF = 0 , i.e. F satisfies the heat
equation.