Explore Solution 12.2.
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
12.2.a. The function
, and
the approximations
, and
.
We can sum up
and
see that it approximates
.
Figure
12.2.b. The function
, and
the approximation
.
We are done.
Aside. We can sum
the infinite series
and
see if it agrees with
.
Don't be worried, we can plot
to
see what it looks like.
Figure
12.2.c.
is
the periodic extension of
.
We are really done.
We can compare this solution with the other one that we stated.
We can sum the infinite series
and
see if it agrees with
.
That's interesting, it is different from the previous formula.
Don't be worried, we can plot
to
see what it looks like.
Figure
12.2.d.
is
another form of the the periodic extension
of
.
Remark. The above
formulas for
involve
the special functions
and
and
are not covered in this course.
We will mention a little more about these functions in the detailed
solutions. You should not worry about these functions.
Aside. Mathematica
can simplify some of the
expressions.
This form of the answer involves the more
familiar
function.
We are really 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 12.2.e. The
harmonic function
, with
.

Figure 12.2.f. A
contour graph of the harmonic function
.
This solution is complements of the authors.
(c) 2010 John H. Mathews, Russell W. Howell