Example 12.3. Find
the function
that
is harmonic in the unit disk
and takes on the boundary values
.
Explore Solution 12.3.
Solution using Fourier
series.
Using the result
of Example 12.1 in Section
12.1, we have the Fourier series for
,
.
Using Equation (12.11.1) for the
extended Fourier series solution of the Dirichlet problem, we
obtain
.
An approximation using a partial
sum.
Summing up the first
seven terms we get the approximations
and
.
Figure
12.13.A. The functions
and
.

Another
3-D graph and contour graph of the harmonic
function
.
The "true solution" using the infinite
sum.
Summing up all of the
terms we get the boundary value function
on
the unit circle
,
and the harmonic function
in
the unit disk
.
Aside. The Maple
commands are similar
We can use Mathematica to plot the boundary
function
and
harmonic function
.
Figure
12.13.B. The boundary function
,
and
the harmonic function
.

Another
3-D graph and contour graph of the harmonic
function
.
Remark 1. As we saw in Example 12.1 in Section 12.1, when we sum the infinite series we get solutions involving the logarithm function.
Remark 2. As we saw in Example 12.1 in Section 12.1, the other form of the solution will be similar.
This solution is complements of the authors.
(c) 2010 John H. Mathews, Russell W. Howell