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Fourier Series FOURIER SERIES When the French mathematician Joseph Fourier (1768–1830) was trying to solve a prob- lem in heat conduction, he needed to express a function f as an infinite series of sine and cosine functions: ϱ 1 f ͑x͒ ෇ a0 ϩ ͚ ͑an cos nx ϩ bn sin nx͒ n෇1 ෇ a0 ϩ a1 cos x ϩ a2 cos 2x ϩ a3 cos 3x ϩиии ϩ b1 sin x ϩ b2 sin 2x ϩ b3 sin 3x ϩиии Earlier, Daniel Bernoulli and Leonard Euler had used such series while investigating prob- lems concerning vibrating strings and astronomy. The series in Equation 1 is called a trigonometric series or Fourier series and it turns out that expressing a function as a Fourier series is sometimes more advantageous than expanding it as a power series. In particular, astronomical phenomena are usually periodic, as are heartbeats, tides, and vibrating strings, so it makes sense to express them in terms of periodic functions. We start by assuming that the trigonometric series converges and has a continuous func- tion f ͑x͒ as its sum on the interval ͓Ϫ␲, ␲͔, that is, ϱ 2 f ͑x͒ ෇ a0 ϩ ͚ ͑an cos nx ϩ bn sin nx͒ Ϫ␲ ഛ x ഛ ␲ n෇1 Our aim is to find formulas for the coefficients an and bn in terms of f . Recall that for a ͸ n power series f ͑x͒ ෇ cn͑x Ϫ a͒ we found a formula for the coefficients in terms of deriv- ͑n͒ atives: cn ෇ f ͑a͒͞n!. Here we use integrals. If we integrate both sides of Equation 2 and assume that it’s permissible to integrate the series term-by-term, we get ϱ ␲ ␲ ␲ y f ͑x͒ dx ෇ y a0 dx ϩ y ͚ ͑an cos nx ϩ bn sin nx͒ dx Ϫ␲ Ϫ␲ Ϫ␲ n෇1 ϱ ϱ ␲ ␲ ෇ 2␲a0 ϩ ͚ an y cos nx dx ϩ ͚ bn y sin nx dx Ϫ␲ Ϫ␲ n෇1 n෇1 But ␲ ␲ 1 1 y cos nx dx ෇ sin nxͬ ෇ ͓sin n␲ Ϫ sin͑Ϫn␲͔͒ ෇ 0 Ϫ␲ n Ϫ␲ n x␲ ෇ because n is an integer. Similarly,Ϫ␲ sin nx dx 0 . So ␲ y f ͑x͒ dx ෇ 2␲a0 Ϫ␲ Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved. Calculus, Stewart: 1 2 ■ FOURIER SERIES and solving for a0 gives ■■ Notice that a0 is the average value 1 ␲ ͓Ϫ␲ ␲͔ 3 a0 ෇ y f ͑x͒ dx of f over the interval , . 2␲ Ϫ␲ To determine an for n ജ 1 we multiply both sides of Equation 2 by cos mx (where m is an integer and m ജ 1) and integrate term-by-term from Ϫ␲ to ␲: ␲ ϱ ␲ ͑ ͒ ෇ ͫ ϩ ͚ ͑ ϩ ͒ͬ y f x cos mx dx y a0 an cos nx bn sin nx cos mx dx Ϫ␲ Ϫ␲ n෇1 ϱ ϱ ␲ ␲ ␲ 4 ෇ a0 y cos mx dx ϩ ͚ an y cos nx cos mx dx ϩ ͚ bn y sin nx cos mx dx Ϫ␲ Ϫ␲ Ϫ␲ n෇1 n෇1 We’ve seen that the first integral is 0. With the help of Formulas 81, 80, and 64 in the Table of Integrals, it’s not hard to show that ␲ y sin nx cos mx dx ෇ 0 for all n and m Ϫ␲ ␲ 0 for n m y cos nx cos mx dx ෇ ͭ Ϫ␲ ␲ for n ෇ m So the only nonzero term in (4) is am␲ and we get ␲ y f ͑x͒ cos mx dx ෇ am␲ Ϫ␲ Solving for am, and then replacing m by n, we have 1 ␲ 5 an ෇ y f ͑x͒ cos nx dx n ෇ 1, 2, 3, . ␲ Ϫ␲ Similarly, if we multiply both sides of Equation 2 by sin mx and integrate from Ϫ␲ to ␲, we get 1 ␲ 6 bn ෇ y f ͑x͒ sin nx dx n ෇ 1, 2, 3, . ␲ Ϫ␲ We have derived Formulas 3, 5, and 6 assuming f is a continuous function such that Equation 2 holds and for which the term-by-term integration is legitimate. But we can still consider the Fourier series of a wider class of functions: A piecewise continuous function on ͓a, b͔ is continuous except perhaps for a finite number of removable or jump disconti- nuities. (In other words, the function has no infinite discontinuities. See Section 2.5 for a discussion of the different types of discontinuities.) Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved. Calculus, Stewart: FOURIER SERIES ■ 3 7 Definition Let f be a piecewise continuous function on ͓Ϫ␲, ␲͔. Then the Fourier series of f is the series ϱ a0 ϩ ͚ ͑an cos nx ϩ bn sin nx͒ n෇1 where the coefficients an and bn in this series are defined by 1 ␲ a0 ෇ y f ͑x͒ dx 2␲ Ϫ␲ 1 ␲ 1 ␲ an ෇ y f ͑x͒ cos nx dx bn ෇ y f ͑x͒ sin nx dx ␲ Ϫ␲ ␲ Ϫ␲ and are called the Fourier coefficients of f . Notice in Definition 7 that we are not saying f ͑x͒ is equal to its Fourier series. Later we will discuss conditions under which that is actually true. For now we are just saying that associated with any piecewise continuous function f on ͓Ϫ␲, ␲͔ is a certain series called a Fourier series. EXAMPLE 1 Find the Fourier coefficients and Fourier series of the square-wave function f defined by 0 if Ϫ␲ ഛ x Ͻ 0 f ͑x͒ ෇ ͭ and f ͑x ϩ 2␲͒ ෇ f ͑x͒ 1 if 0 ഛ x Ͻ ␲ So f is periodic with period 2␲ and its graph is shown in Figure 1. ■■ Engineers use the square-wave function in describing forces acting on a mechanical system y and electromotive forces in an electric circuit 1 (when a switch is turned on and off repeatedly). Strictly speaking, the graph of f is as shown in Figure 1(a), but it’s often represented as in _π 0 π2π x Figure 1(b), where you can see why it’s called a square wave. (a) y 1 _π 0 π2π x FIGURE 1 Square-wave function (b) SOLUTION Using the formulas for the Fourier coefficients in Definition 7, we have 1 ␲ 1 0 1 ␲ 1 1 a0 ෇ y f ͑x͒ dx ෇ y 0 dx ϩ y 1 dx ෇ 0 ϩ ͑␲͒ ෇ 2␲ Ϫ␲ 2␲ Ϫ␲ 2␲ 0 2␲ 2 Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved. Calculus, Stewart: 4 ■ FOURIER SERIES and, for n ജ 1, 1 ␲ 1 0 1 ␲ an ෇ y f ͑x͒ cos nx dx ෇ y 0 dx ϩ y cos nx dx ␲ Ϫ␲ ␲ Ϫ␲ ␲ 0 ␲ ෇ ϩ 1 sin nxͬ ෇ 1 ͑ ␲ Ϫ ͒ ෇ 0 ␲ ␲ sin n sin 0 0 n 0 n 1 ␲ 1 0 1 ␲ bn ෇ y f ͑x͒ sin nx dx ෇ y 0 dx ϩ y sin x dx ␲ Ϫ␲ ␲ Ϫ␲ ␲ 0 ␲ ෇ Ϫ 1 cos nxͬ ෇ Ϫ 1 ͑ ␲ Ϫ ͒ ␲ ␲ cos n cos 0 n 0 n 0 if n is even ■■ Note that cos n␲ equals 1 if n is even ෇ and Ϫ1 if n is odd. ͭ 2 if n is odd n␲ The Fourier series of f is therefore a0 ϩ a1 cos x ϩ a2 cos 2x ϩ a3 cos 3x ϩиии ϩ b1 sin x ϩ b2 sin 2x ϩ b3 sin 3x ϩиии 1 ෇ ϩ 0 ϩ 0 ϩ 0 ϩиии 2 2 2 2 ϩ sin x ϩ 0 sin 2x ϩ sin 3x ϩ 0 sin 4x ϩ sin 5x ϩиии ␲ 3␲ 5␲ 1 2 2 2 2 ෇ ϩ sin x ϩ sin 3x ϩ sin 5x ϩ sin 7x ϩиии 2 ␲ 3␲ 5␲ 7␲ Since odd integers can be written as n ෇ 2k Ϫ 1, where k is an integer, we can write the Fourier series in sigma notation as 1 ϱ 2 ϩ ͚ sin͑2k Ϫ 1͒x 2 k෇1 ͑2k Ϫ 1͒␲ In Example 1 we found the Fourier series of the square-wave function, but we don’t know yet whether this function is equal to its Fourier series. Let’s investigate this question graphically. Figure 2 shows the graphs of some of the partial sums 1 2 2 2 S ͑x͒ ෇ ϩ sin x ϩ sin 3x ϩиииϩ sin nx n 2 ␲ 3␲ n␲ when n is odd, together with the graph of the square-wave function. Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved. Calculus, Stewart: FOURIER SERIES ■ 5 y y y 1 1 1 S£ S∞ S¡ _π π x _π π x _π π x y y y 1 1 1 S¶ S¡¡ S¡∞ _π π x _π π x _π π x FIGURE 2 Partial sums of the Fourier series for the square-wave function We see that, as n increases,Sn͑x͒ becomes a better approximation to the square-wave function. It appears that the graph of Sn͑x͒ is approaching the graph of f ͑x͒, except where x ෇ 0 or x is an integer multiple of ␲. In other words, it looks as if f is equal to the sum of its Fourier series except at the points where f is discontinuous. The following theorem, which we state without proof, says that this is typical of the Fourier series of piecewise continuous functions. Recall that a piecewise continuous func- tion has only a finite number of jump discontinuities on ͓Ϫ␲, ␲͔. At a number a where f has a jump discontinuity, the one-sided limits exist and we use the notation f ͑aϩ͒ ෇ lim f ͑x͒ f ͑aϪ͒ ෇ lim f ͑x͒ x l aϩ x l aϪ 8 Fourier Convergence Theorem If f is a periodic function with period 2␲ and f and f Ј are piecewise continuous on ͓Ϫ␲, ␲͔, then the Fourier series (7) is convergent.
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