Example 12.2.  The function  ,  extended periodically by the equation  ,  has the Fourier series expansion  

        .  

Explore Solution 12.2.

Applying Theorem 12.3 we see that the Fourier series for     involves only the cosine terms,  ,  and

    .  

We can use Mathematica to construct the coefficients .

This is easy to simplify and obtain

    .

We can obtain this with the indefinite integral

This is easy to simplify and obtain

    .

Thus, we have obtained 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 5 terms and see that it agrees with  .
Then we can sum the infinite series    and see if it agrees with  .  

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

Therefore,   is the periodic extension of  .

We can compare this solution with the other one that we stated.

Therefore,   is another form of the the periodic extension of  .

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