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.  

First, we will introduce the Fourier series for a real-valued function    of the real variable  .  Then we discuss Fourier transforms.  

Finally, we develop the Laplace transform and the complex variable techniques for finding its inverse.

Our goal is to apply these ideas to solving problems, so many of the theorems are stated without proof.

Background

    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  .

    Familiar examples of real functions that have period  ,  are  ,  where  n  is an integer.  

These examples raise the question of whether any periodic function can be represented by a sum of terms involving  ,  

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  ,  if there exists values    

with    such that    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 below, where the function    is

                              

                              Figure 12.2.  A piecewise continuous function    over the interval  .

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   .

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    is continuous.   

If    is a point of discontinuity of  ,  then  

                        ,  

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

                         .  

Proof.

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    .

The coefficient    is obtained with the separate computation  

                    .  

Then using Equation (12.3) we get  

                    ,    for    .

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 12.3.

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

Explore Solution 12.1.

Solution Details 12.1.

Theorem 12.2.   If    have Fourier series representations, then their sum      has a Fourier series representation,

and the Fourier coefficients of    are obtained by adding the corresponding coefficients of  .  

Proof.

Theorem 12.3 (Fourier Cosine Series).   Assume that    is an even function and has period  .  

Here 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  .  

Here 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.

Example 12.2.  The function   ,   extended periodically by the equation   ,   

has the Fourier series expansion  

                    ,  

which can be written in the alternative form  

                    .  

Solution.

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

                    ,    for    .  
    
The coefficient    is obtained with the separate computation  

                    .

Using the    and Theorem 12.3 produces the required solution.

Therefore, we have the found the Fourier series expansion  

                    .  

It is easy to see that    for all  ,  and we can express    in the form   

                    .  

Therefore,  

                    .  

The graphs of      and the first two partial sums   ,   and      are shown below.

                    Figure 12.2.a.  The function  ,  and the approximations  ,  and  .

Explore Solution 12.2.

Solution Details 12.2.

Extra Example 1.   Given     extended periodically by the equation   ,  

find the Fourier series expansion.     

Extra Solution 1.

Extra Example 2.   Given  ,   extended periodically by the equation   ,  

find the Fourier series expansion.     

Extra Solution 2.

Exercises for Section 12.1.  Fourier Series Representation

This material is coordinated with our book Complex Analysis for Mathematics and Engineering.

(c) 2012 John H. Mathews, Russell W. Howell