When modeling physical phenomena by partial diﬀerential equations it is frequently neces-
sary to solve a
boundary value problem
. One of the most famous and important of these is
associated with Laplace’s equation:
Δ
u
=
∂
2
u
∂x
2
+
∂
2
u
∂y
2
= 0
,
where
u
(
x,y
) is deﬁned on a region
R
in the plane. The operator
Δ =
∂
2
2
+
∂
2
2
is called the
Laplacian
and a real-valued function
u
(
) satisfying Δ
u
= 0 is called
harmonic
.
The
Dirichlet problem
for Laplace’s equation is this:
Given a function
f
(
) deﬁned on the boundary of a region
R
, ﬁnd a function
u
(
) deﬁned on
R
that is harmonic in
R
and equal to
f
(
) on the boundary.
Fourier series and convolution combine to solve this problem when
R
is a disk.
As with many problems where circular symmetry is involved, in this case that the functions are
deﬁned on a circular disk, it is helpful to introduce polar coordinates (
r,θ
), with
x
=
r
cos
θ
,
y
=
r
sin
θ
. Writing
U
(
) =
u
(
r
cos
θ,r
sin
θ
), so regarding
u
as a function of
r
and
θ
,
Laplace’s equation becomes
∂
2
U
∂r
2
+
1
r
∂U
+
1
r
2
∂
2
U
∂θ
2
= 0
.
(You need not derive this.)
(a) (5) Let
{
c
n
}
,
n
= 0
,
1
,...
be a bounded sequence of complex numbers, let
r <
1 and
deﬁne
u
(
) by the series
u
(
) = Re
(
c
0
+ 2
∞
X
n
=1
c
n
r
n
e
inθ
)
.
From the assumption that the coeﬃcients are bounded, and comparison with a geometric
series, it can be shown that the series converges, but you need not do this.
Show that
u
(
) is a harmonic function.
[Hint: Superposition in action – the Laplacian of the sum is the sum of the Laplacians,
and the
c
n
comes out, too. Also, the derivative of the real part of something is the real
part of the derivative of the thing.]
(b) (5) Suppose that
f
(
θ
) is a real-valued, continuous, periodic function of period 2
π
and
let
f
(
θ
) =
∞
X
n
=
-∞
c
n
e
inθ
be its Fourier series. Now form the harmonic function
u
(
) as above, with these
coeﬃcients
c
n
. This solves the Dirichlet problem of ﬁnding a harmonic function on the
unit disk
x
2
+
y
2
<
1 with boundary values
f
(
θ
) on the unit circle
x
2
+
y
2
= 1; precisely,
lim
r
→
1
u
(
) =
f
(
θ
)
.