INTRODUCTION

In preceding chapters, Fourier series were used to represent a function f defined on a finite interval (−p, p) or (0, L). When f and f′ are piecewise continuous on such an interval, a Fourier series represents the function on the interval and converges to the periodic extension of f outside the interval. In this way we are justified in saying that Fourier series are associated only with periodic functions. We shall now derive, in a nonrigorous fashion, a means of representing certain kinds of nonperiodic functions that are defined on either an infinite interval (−, ) or a semi-infinite interval (0, ).

From Fourier Series to Fourier Integral

Suppose a function f is defined on (−p, p). If we use the integral definitions of the coefficients (9), (10), and (11) of Section 12.2 in (8) of that section, then the Fourier series of f on the interval is

(1)

If we let then (1) becomes

(2)

We now expand the interval (−p, p) by letting p → . Since p → implies that Δα → 0, the limit of (2) has the form which is suggestive of the definition of the integral Thus if f(t) dt exists, the limit of the first term in (2) is zero and the limit of the sum becomes

(3)

The result given in (3) is called the Fourier integral of f on the interval (−, ). As the following summary shows, the basic structure of the Fourier integral is reminiscent of that of a Fourier series.

Convergence of a Fourier Integral

Sufficient conditions under which a Fourier integral converges to f(x) are similar to, but slightly more restrictive than, the conditions for a Fourier series.

Let f and f′ be piecewise continuous on every finite interval, and let f be absolutely integrable on (−, ).^* Then the Fourier integral of f on the interval converges to f(x) at a point of continuity. At a point of discontinuity, the Fourier integral will converge to the average

where f(x+) and f(x−) denote the limit of f at x from the right and from the left, respectively.

EXAMPLE 1 Fourier Integral Representation

Find the Fourier integral representation of the piecewise-continuous function

SOLUTION

The function, whose graph is shown in FIGURE 15.3.1, satisfies the hypotheses of Theorem 15.3.1. Hence from (5) and (6) we have at once

Substituting these coefficients into (4) then gives

When we use trigonometric identities, the last integral simplifies to

(7) ≡

The Fourier integral can be used to evaluate integrals. For example, at x = 1 it follows from Theorem 15.3.1 that (7) converges to f(1). From this we find

The latter result is worthy of special note since it cannot be obtained in the “usual” manner; the integrand (sin x)/x does not possess an antiderivative that is an elementary function.

Cosine and Sine Integrals

When f is an even function on the interval (−, ), then the product f(x) cos αx is also an even function, whereas f(x) sin αx is an odd function. As a consequence of property (g) of Theorem 12.3.1, B(α) = 0, and so (4) becomes

Here we have also used property (f) of Theorem 12.3.1 to write

Similarly, when f is an odd function on (−, ) the products f(x) cos αx and f(x) sin αx are odd and even functions, respectively. Therefore A(α) = 0 and

We summarize in the following definition.

EXAMPLE 2 Cosine Integral Representation

Find the Fourier integral representation of the function

SOLUTION

It is apparent from FIGURE 15.3.2 that f is an even function. Hence we represent f by the Fourier cosine integral (8). From (9) we obtain

and so (12) ≡

The integrals (8) and (10) can be used when f is neither odd nor even and defined only on the half-line (0, ). In this case (8) represents f on the interval (0, ) and its even (but not periodic) extension to (−, 0), whereas (10) represents f on (0, ) and its odd extension to the interval (−, 0). The next example illustrates this concept.

EXAMPLE 3 Cosine and Sine Integral Representations

Represent f(x) = e^−x, x > 0 (a) by a cosine integral; (b) by a sine integral.

SOLUTION

The graph of the function is given in FIGURE 15.3.3.

(a) Using integration by parts, we find

Therefore from (8) the cosine integral of f is

(13)

(b) Similarly, we have

From (10) the sine integral of f is then

(14)

FIGURE 15.3.4 shows the graphs of the functions and their extensions represented by the two integrals. ≡

Complex Form

The Fourier integral (4) also possesses an equivalent complex form, or exponential form, that is analogous to the complex form of a Fourier series (see Section 12.4). If (5) and (6) are substituted into (4), then

(15)

(16)

(17)

We note that (15) follows from the fact that the integrand is an even function of α. In (16) we have simply added zero to the integrand,

because the integrand is an odd function of α. The integral in (17) can be expressed as

(18)

where (19)

This latter form of the Fourier integral will be put to use in the next section when we return to the solution of boundary-value problems.

Use of Computers

The convergence of a Fourier integral can be examined in a manner that is similar to graphing partial sums of a Fourier series. To illustrate, let’s use the results in parts (a) and (b) of Example 3. By definition of an improper integral, the Fourier cosine integral representation of f(x) = e^−x, x > 0 in (13) can be written as f(x) = limF_b(x), where

and x is treated as a parameter. Similarly, the Fourier sine integral representation of f(x) = e^−x, x > 0 in (14) can be written as f(x) = limG_b(x), where

Because the Fourier integrals (13) and (14) converge, the graphs of the partial integrals F_b(x) and G_b(x) for a specified value of b > 0 will be an approximation to the graph of f and its even and odd extensions shown in Figure 15.3.4(a) and 15.3.4(b), respectively. The graphs of F_b(x) and G_b(x) for b = 20 given in FIGURE 15.3.5 were obtained using Mathematica and its NIntegrate application. See Problem 21 in Exercises 15.3.

15.3 Exercises Answers to selected odd-numbered problems begin on page ANS-38.

In Problems 1–6, find the Fourier integral representation of the given function.

In Problems 7–12, represent the given function by an appropriate cosine or sine integral.

In Problems 13–16, find the cosine and sine integral representations of the given function.

13. f(x) = e^−kx, k > 0, x > 0
14. f(x) = e^−x − e^−3x, x > 0
15. f(x) = xe^−2x, x > 0
16. f(x) = e^−x cos x, x > 0

In Problems 17 and 18, solve the given integral equation for the function f.

17. 
18. 
19. 1. Use (7) to show that

       [Hint: α is a dummy variable of integration.]

    2. Show in general that, for
20. Use the complex form (19) to find the Fourier integral representation of Show that the result is the same as that obtained from (8) and (9).

Computer Lab Assignment

21. While the integral (12) can be graphed in the same manner discussed on page 791 to obtain Figure 15.3.5, it can also be expressed in terms of a special function that is built into a CAS.
    1. Use a trigonometric identity to show that an alternative form of the Fourier integral representation (12) of the function f in Example 2 (with a = 1) is

       

    2. As a consequence of part (a), where

       

       Show that the last integral can be written as

       

       where Si(x) is the sine integral function. See Problem 47 in Exercises 2.3.

    3. Use a CAS and the sine integral form obtained in part (b) to graph F_b(x) on the interval [−3, 3] for b = 4, 6, and 15. Then graph F_b(x) for larger values of b > 0.

* This means that the integral converges.