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