Chapter 12  Fourier Series and the Laplace Transform

12.1  Fourier Series

Overview

    In this chapter we show how Fourier Series, the Fourier Transform, and the Laplace Transform are related to the study of complex analysis.  We develop the Fourier series representation of a real-valued function of the real variable .  We then discuss complex Fourier series and Fourier transforms.  Finally, we develop the Laplace transform and the complex variable technique for finding its inverse. In this chapter we focus on applying these ideas to solving problems involving real-valued functions, so many of the theorems throughout are stated without proof.

    Let be a real-valued function that is periodic with period  ,  that is

            .

One such function is .   Its graph is obtained by repeating the portion of the graph in any interval of length , as shown in Figure 12.1.

                Figure 12.1.  A function with period .

Exploration.

    Familiar examples of real functions that have period  ,  are and , where n is an integer.  These examples raise the question of whether any periodic function can be represented by a sum of terms involving and , where are real constants.  As we soon demonstrate, the answer to this question is often yes.

Definition 12.1 (Piecewise Continuous). The function is piecewise continuous on the closed interval [a,b], if there exists values with such that U is continuous in each of the open intervals , for and has left-hand and right-hand limits at each of the values , for .  

         We use the symbols    and    for the left-hand and right-hand limit, respectively, of a function as approaches the point .  The graph of a piecewise continuous function is illustrated in Figure 12.2, where the function is

        

The  left-hand and right-hand limits at ,  ,  and    are easily determined:

    At , the left-hand limit is  ,
and the right-hand limit is .

    At , the left-hand limit is  ,
and the right-hand limit is  .

    At , the left-hand limit is  ,
and the right-hand limit is  . 

        Figure 12.2.  A piecewise continuous function over the interval .

Exploration.

Definition 12.2 (Fourier Series). If is periodic with period and is piecewise continuous on, then the Fourier Series for is

(12.1)        ,

where the coefficients   are given by the so-called Euler's formulae:  

(12.2)        ,  

        and  

(12.3)        .  

      We introduced the factor in the constant term    on the right side of Equation (12.1) for convenience so that we can obtain    from the general formula in Equation (12.1) by setting  .  We explain the reasons for this strategy shortly.  Theorem 12.1 deals with convergence of the Fourier series.

Theorem 12.1 (Fourier Expansion). Assume that is the Fourier Series for . If are piecewise continuous on , then is convergent for all .  

The relation holds for all where U is continuous. If is a point of discontinuity of U, then

         ,  

where denote the left-hand and right-hand limits, respectively.  With this understanding, we have the Fourier Series expansion:

         .  

Proof.

Method I. Fourier Series - Trigonometric Polynomials.  Execute the following sells to set up the procedure S[n,t] for finding the Fourier Series - Trigonometric Polynomial of degree n.

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

        .  

Solution.

    Using Equation (12.2) and integrating by parts, we obtain  

    ,  for  .
    
Then using Equation (12.3) we get

    .
    
We compute the coefficient by .

Substituting the coefficients and into Equation (12.1) produces the required solution.  The graphs of and the first three partial sums  ,  ,  and    are shown in Figure 11.3.

        Figure 12.3.  The function , and the approximations ,  ,  and  .

Explore Solution 12.1.

Theorem 12.2.  If    have Fourier series representations, then their sum    has a Fourier series representation, and the Fourier coefficients of  W  are obtained by adding the corresponding coefficients of  U  and  V.

Proof.

Theorem 12.3 (Fourier Cosine Series). Assume that is an even function and has period . If are piecewise continuous, the Fourier series for     involves only the cosine terms,  ,  and we write  

        ,  

where .  

Proof.

Theorem 12.4 (Fourier Sine Series). Assume that is an odd function and has period . If are piecewise continuous, the Fourier series for     involves only the sine terms,  ,  and we write  

        ,  

where  .  

Proof.

Theorem 12.5  (Termwise Integration).  Assume that    has the Fourier series representation  

          .  

Then the integral  of    has a Fourier series representation which can be obtained by termwise integration of the Fourier series of  ,  that is  

        ,  

where we have used the expansion  .  

Proof.

Theorem 12.6  (Termwise Differentiation).  Assume that both    have Fourier series representation and that  

        .  

Then    can be obtained by termwise differentiation of  ,  that is  

        .  

Proof.

Method II.  Fourier Series - Trigonometric Polynomials.  Modify procedure S[n,t] for finding the Fourier Series - Trigonometric Polynomial of degree n to the case of a piecewise continuous function on the subintervals  .

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

        .  

Solution.

    The function    is an even function;  hence we can use Theorem 11.3 to conclude that   for all and that  

    ,  for  .  v
    
We compute the coefficient by

    .

Therefore, we have the found the Fourier series expansion  

        ,
        
we leave it for the reader to show that this is equivalent to

        .  

Using the and Theorem 12.3 produces the required solution. The graphs of and the first two partial sums  ,  
and    are shown in figure below.

        Figure.  The function , and the approximations ,  and  .

 Explore Solution 12.2.