Solution Details 12.2.

See text and/or instructor's solution manual.

Answer.   .  

Alternative 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.   

We have shown that     for all  ,   and it is easy to see that     for all  .   

Furthermore, we can express the odd coefficients    in the form   

                    .  

Therefore,  

                    .  

We are really really done.   

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

We are 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.

                         .  

We are really really done.   

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

We can use Mathematica to sum the infinite series.  

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

The last expansions involve only the    and    functions.

Aside.  The Maple commands are similar  


  

                                                              


  

                                                              


  

                                                              

Aside.  We mention below that Landen's Inversion Formula will permit us to write:  

                    .  

We are really really really done.   

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

                    .  

                    .  

                    .  

Remark 1.  The above formulas for    involve the special functions    and    and are 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

                    .  

                              .

Remark 3.  Do not worry when an unknown function such as or is involved in a computer computation.

There are many new and fascinating facts waiting to be revealed in higher mathematics.

We are 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