Extra Solution 2.

See text and/or instructor's solution manual.

Answer.   .  

Solution.   Find the Fourier Series   ,

by computing the coefficients with Euler's formulae:  

(12.2)        ,  

        and  

(12.3)        .  

First, calculate   .  

                      

Then

                      

Second, calculate   .  

                      

Alternately,    is an even function so that     is an odd function, and   

                        for all    .

Then Theorem 12.3 shows that

                    ,    

where the coefficients can be computed with the special formula     

                    .  

Now calculate  

                      

Get The Answer.

Therefore,

                    .  

We are done.   

Now use the fact that      and simplify the answer.  

Therefore,

                    .  

We are really done.   

Aside.  We can calculate a few terms in these series to verify that they are the same  ( up to   ).  

We are really really done.   

Aside.  We can let Mathematica double check our work.

Aside.  The Maple commands are similar  


  

                                                              


  

                                                              


  

                                                              

The partial lists of coefficients    and    are:

For illustration, we can sum the first few terms in these series  ( up to   ).  

Aside.  We can compute a few terms of the Fourier series of    using Mathematica's built in procedure  FourierTrigSeries.

The graph is a little more interesting if we use  .

                    .  

We are really really really done.   

Aside.  There are at least two solutions to this problem.  

We can use Mathematica to sum the infinite series.  

However, this sum might not be as familiar as those studied in calculus.

The last two expansions involve only the    and    functions.

Aside.  The Maple commands are similar  


  

                                                              


  

                                                               

We are really really really really done.   

Aside.  We can graph these functions to verify they are correct.  

                    .  

                    .  

Aside.  This expansion of    is related to the one in Example 12.1 where we saw that

                    .  

It is easy to use differentiation and verify that  .

Remark 1.  The above formula for    involves the special function    and is not covered in this course.

Remark 2.  A simplification can be obtained by using Landen's Inversion Formula:   .  

After some tedious steps

                    ,  

can be expressed as

                    .  

                    .  

We are really really really 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  .

               The harmonic function  ,  with  .  

               A contour graph of the harmonic function  .  

This solution is complements of the authors.

(c) 2010 John H. Mathews, Russell W. Howell