Example 12.1.  The function  ,  extended periodically by the equation  ,  has the Fourier series expansion  

        .  

Explore Solution 12.1.

We can use Mathematica to construct the coefficients .

We can obtain this with the indefinite integral

Thus, we have obtained the desired result

    ,  for  .

We can use Mathematica to construct the coefficients .

We can obtain this with the indefinite integral

This is easy to simplify and obtain the desired result

    .

Now we continue our explorations.

Enter the function u[t], and for illustration construct the Fourier Series - Trigonometric Polynomial of degree n = 9.

We can also get this result using Mathematica's built in procedure FourierTrigSeries.

A graph of    is given below.

The general term for the series is  .  
We can sum up 9 terms and see that it agrees with .
Then we can sum the infinite series    and see if it agrees with  .  

.  

We can plot  S[t]  to see what it looks like.

Therefore,   is the periodic extension of  .

(c) 2006 John H. Mathews, Russell W. Howell