Graphics Reference
In-Depth Information
The basic question here is whether the Fourier series for a function f actually con-
verges to f. Substituting the formula for the a
n
into equation (21.8) we can interchange
the integration with the summation when r < 1, since the series converges absolutely
there, and so we get
•
1
2
p
Â
(
)
()
=
Ú
nn t
i
q
-
()
ur
,
q
r e
ftdt
.
(21.14)
p
-
p
n
=-•
But the standard formula for geometric series gives us
•
1
Â
(
)
nn t
i
q
-
re
=
(
)
i
q
-
t
1
-
re
n
=
0
and
-•
--
i
(
q
t
)
re
re
Â
(
)
-
nn t
i
q
-
re
=
,
(
)
--
i
q
t
1
-
n
=-
1
so that (21.14) becomes
2
1
2
1
-
r
p
()
=
Ú
()
ur
,
q
f t dt
for
r
<
1
.
(21.15)
p
(
)
+
2
p
12
-
r
cos
q
-
t
r
Definition.
The function
2
1
2
1
12
-
r
()
=
Pr
,
f
p
2
-
r
cos
f
+
r
is called the
Poisson kernel
.
We are finally ready to give an answer to the Dirichlet problem that we have been
studying and it is Fourier series that provide that answer.
21.5.3
Theorem.
(1) (Existence) If f(q) is a continuous periodic function of period 2p and if u(r,q)
is the function defined by equation (21.14), then
()
=
()
lim
ur
,
qq
f
,
r
Æ
1
and the convergence is uniform in q.
(2) (Uniqueness) Let f(q) be a continuous periodic function of period 2p. If v(r,q)
is a function satisfying equations (21.5)-(21.7) and which converges uniformly in q to
f(q) as r increases to 1, then v is the function u(r,q) defined by equation (21.15).
Proof.
See [Seel66].
