Chapter 12  Fourier Series and the Laplace Transform

12.4  The Fourier Transform

    In this section we develop the complex Fourier transform of the function  ,  we will use ideas that were introduced in Section 12.1.  If we let be a real-valued function with period , which is piecewise continuous such that also exists and is piecewise continuous, then has the complex Fourier series representation

                         ,
where
                         ,    for all  n.
            
The coefficients are complex numbers.  Previously, we expressed as the real trigonometric series  

(12.19)            .  

Hence a relationship between the coefficients is

                         

We can easily establish these relations.  We start by writing  

(12.20)        

Comparing Equations (12.20) and (12.19), we see that  ,  ,  and  .  

If and are piecewise continuous and have period , then has the complex Fourier series representation

(12.21)            ,
                    where
                         ,    for all  n.

    We've shown how periodic functions are represented by trigonometric series, but many practical problems involve nonperiodic functions.  A representation analogous to a Fourier series for a nonperiodic function is obtained by considering the Fourier series of for    and then taking the limit as  .  The result is known as the Fourier transform of .  

    We start with the nonperiodic function and consider the periodic function with period , where

                         

Then has the complex Fourier series representation

(12.23)            .

    We need to introduce some terminology in order to discuss the terms in Equation (12.23).  First

(12.24)            

is called the frequency.  If t denotes time, then the units for are radians per unit time.  The set of all possible frequencies is called the frequency spectrum, that is,

                         .

Note that, as L increases, the spectrum becomes finer and approaches a continuous spectrum of frequencies.  It is reasonable to expect that the summation in the Fourier series for    will give rise to an integral over .  This result is stated in Theorem 12.9.

Theorem 12.9 (Fourier Transform).  Let be piecewise continuous, and  

                         ,  

for some positive constant  M.  The Fourier transform    of    is defined as  

(12.25)            .  

At points of continuity,    has the integral representation  

                         .  

At a point    of discontinuity of  ,  this integral converges to  .  

The fact that     is transformed into    is commonly expressed by the operator notation  

                         .  

Proof.

Example 12.5.  Show that  .  

Solution.

    Using Equation (12.25), we obtain  

                     
establishing the result.

Explore Solution 12.5.

    Table 12.1 gives some important properties of the Fourier transform.

                    Table 12.1  Properties of the Fourier transform.

Example 12.6.  Show that  .  

Solution.

    Using the result of Example 12.5 and the symmetry property, we obtain

                     

We use the linearity property and multiply each term by and get

                     .

Then rewrite this in the form

                     

establishing the result.

Explore Solution 12.6.