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