INTRODUCTION

As we have seen in the preceding two sections, a real function f can be represented by a series of sines and cosines. The functions cos nx, n = 0, 1, 2, … and sin nx, n = 1, 2, … are real-valued functions of a real variable x. The three different real forms of Fourier series given in Definitions 12.2.1 and 12.3.1 will be exceedingly important in Chapters 13 and 14 when we set about to solve linear partial differential equations. However, in certain applications, for example, the analysis of periodic signals in electrical engineering, it is actually more convenient to represent a function f in an infinite series of complex-valued functions of a real variable x such as the exponential functions e^inx, n = 0, 1, 2, …, and where i is the imaginary unit defined by i² = –1. Recall for x a real number, Euler’s formula

e^ix = cos x + i sin x gives e^–ix = cos x – i sin x.(1)

In this section we are going to use the results in (1) to recast the Fourier series in Definition 12.2.1 into a complex form or exponential form. We will see that we can represent a real function by a complex series: a series in which the coefficients are complex numbers. To that end, recall that a complex number is a number z = a + ib, where a and b are real numbers, and i² = –1. The number = a – ib is called the conjugate of z.

Complex Fourier Series

If we first add the two expressions in (1) and solve for cos x and then subtract the two expressions and solve for sin x, we arrive at

(2)

Using (2) to replace cos(nπx/p) and sin(nπx/p) in (8) of Section 12.2, the Fourier series of a function f can be written

(3)

where c₀ = a₀, c_n = (a_n – ib_n), and c_–n = (a_n + ib_n). The symbols a₀, a_n , and b_n are the coefficients (9), (10), and (11) respectively, in Definition 12.2.1. When the function f is real, c_n and c_–n are complex conjugates and can also be written in terms of complex exponential functions:

(4)

(5)

(6)

Since the subscripts of the coefficients and exponents range over the entire set of integers…–3, –2, –1, 0, 1, 2, 3, …, we can write the results in (3), (4), (5), and (6) in a more compact manner by summing over both the negative and nonnegative integers. In other words, we can use one summation and one integral that defines all three coefficients c₀, c_n , and c_–n.

If f satisfies the hypotheses of Theorem 12.2.1, a complex Fourier series converges to f(x) at a point of continuity and to the average

at a point of discontinuity.

EXAMPLE 1 Complex Fourier Series

The series (10) converges to the 2π-periodic extension of f.

You may get the impression that we have just made life more complicated by introducing a complex version of a Fourier series. The reality of the situation is that in areas of engineering, the form (7) given in Definition 12.4.1 is sometimes more useful than that given in (8) of Definition 12.2.1.

Fundamental Frequency

The Fourier series in Definitions 12.2.1 and 12.4.1 define a periodic function and the fundamental period of that function (that is, the periodic extension of f) is T = 2p. Since p = T/2, (8) of Section 12.2 and (7) become, respectively,

(11)

where number ω = 2π/T is called the fundamental angular frequency. In Example 1 the periodic extension of the function has period T = 2π; the fundamental angular frequency is ω = 2π/2π = 1.

Frequency Spectrum

In the study of time-periodic signals, electrical engineers find it informative to examine various spectra of a wave form. If f is periodic and has fundamental period T, the plot of the points (nω, |c_n|), where ω is the fundamental angular frequency and the c_n are the coefficients defined in (8), is called the frequency spectrum of f.

EXAMPLE 2 Frequency Spectrum

In Example 1, ω = 1 so that nω takes on the values 0, ±1, ±2, …. Using | α + iβ | = , we see from (9) that

The following table shows some values of n and corresponding values of c_n.

The graph in FIGURE 12.4.1, lines with arrowheads terminating at the points, is a portion of the frequency spectrum of f.≡

EXAMPLE 3 Frequency Spectrum

Find the frequency spectrum of the periodic square wave or periodic pulse shown in FIGURE 12.4.2. The wave is the periodic extension of the function f :

SOLUTION

Here T = 1 = 2p so p = . Since f is 0 on the intervals (–, –) and (, ), (8) becomes

That is,

Since the last result is not valid at n = 0, we compute that term separately:

The following table shows some of the values of |c_n|, and FIGURE 12.4.3 shows the frequency spectrum of f.

Since the fundamental frequency is ω = 2π/T = 2π, the units nω on the horizontal scale are ±2π, ±4π, ±6π, …. The curved dashed lines were added in Figure 12.4.3 to emphasize the presence of the zero values of |c_n| when n is an even nonzero integer.≡

12.4 Exercises Answers to selected odd-numbered problems begin on page ANS-32.

In Problems 1–6, find the complex Fourier series of f on the given interval.

1. 
2. 
3. 
4. 
5. f(x) = x, 0 < x < 2π
6. f(x) = e^–|x|, –1 < x < 1
7. Find the frequency spectrum of the periodic wave that is the periodic extension of the function f in Problem 1.
8. Find the frequency spectrum of the periodic wave that is the periodic extension of the function f in Problem 3.

In Problems 9 and 10, sketch the given periodic wave. Find the frequency spectrum of f.

9. f(x) = 4 sin x, 0 < x < π; f(x + π) = f(x) [Hint: Use (2).]
10. 
11. 1. Show that a_n = c_n + c_–n and b_n = i(c_n – c_–n).
    2. Use the results in part (a) and the complex Fourier series in Example 1 to obtain the Fourier series expansion of f.
12. The function f in Problem 1 is odd. Use the complex Fourier series to obtain the Fourier sine series expansion of f.