Explore Solution 12.1.
Aside. We can let Mathematica explore this example.
For illustration, we can sum the first few terms in these
series ( up to
).
Aside. We can
compute the Fourier series of
using
Mathematica's built in procedure FourierTrigSeries.
A graph of
is
given below.
Figure
12.3. The function
, and
the approximations
,
, and
.
We can sum up
and
see that it approximates
.
Figure
12.3.1. The function
, and
the approximation
.
Higher degree approximations are possible.
Figure
12.3.2. The function
, and
the approximation
.
Since
is
a point of discontinuity of
, we
know that
is
not defined at
, and
,
where
and
denote
the left-hand and right-hand limits,
respectively.
Convergence is not uniform on the closed
interval
, and
the overshooting of
is referred to as "Gibbs
phenomenon."
We are done.
Aside. We can sum
the infinite series
and
see if it agrees with
We can plot
to
see what it looks like.
Figure
12.3.3. The
function
, and
the approximations
,
, and
.
Therefore,
is
the periodic extension of
.
We are really done.
Aside. Let us
announce that in Section
12.2 we interpret the Fourier
Series
as boundary values on the circle
,
and construct the harmonic function
inside the unit disk
with
.

Figure
12.3.4. The harmonic
function
, with
.

Figure
12.3.5. A contour graph of the harmonic
function
.
This solution is complements of the authors.
(c) 2010 John H. Mathews, Russell W. Howell