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CHAPTER | ] Fourier Series, Integrals, and Transforms Fourier series (Scc. 11.1) are infinite series designed to represent general periodic functions in terms of simple ones, namely, cosines and sines. They constitute a very important tool, in particular in solving problems that involve ODE and PDE: Tn this chapter wc discuss Fourier series and their engineering use from a practical point of view, in connection with ODESs and with the approximation of periodic functions. Application to PDEs follows in Chap. 12 The theory of Fourier series is complicated, but we shall see that the application of these series is ralher simple. Fourier series are in a certain sense more universal than the familiar Taylor series in calculus becanse many discontinuoas periodie functions of practical interest can be developed in Fourier series but, of course, do not have Taylor series representations. In the last sections (11,7-11.9) we consider Fourier integrals and Fourier transforms, which extend the ideas and techniques of Fourier series to nonperiodic functions and have basic applications to PDES (to be shown in the next chapter. Prerequisite: Flementary integral calculus (needed for Fourier coelficients) Sections that may be omitted in a shorter course; ILA-11.9 References and Answers to Problems: App. 1 Part C, App. 2. 11.1 Fourier Series 478 Fourier series are the basic tool for representing periodic functions, which play an important role in applications. A function f(3) is called a periodie function if f() is defined for all real x (perhaps except at some points, such as x = 7/2, £3a/2, -- - for tan x) and if there is some positive number p, called a period of f(9), such that (ep) fa + p) = fo) for all x. The graph of such a function is obtained by periodic repetition of its graph in any interval of length p (Fig. Familiar periodic functions are the cosine and sine functions. Examples of functions that are not periodic are x, x2, x2, e”, cosh x, and In x, to mention just a few. TE f(x) has period p. it also has the period 2p because (1) implies flw +2p) = fe +p] + p) = fl + p) = SQ9, ete.; thus for any integer n = 1,2, (22) Fl — np) = 0) for all x. SEC. 111 Fourier Series Periodic function Furthermore if f(x) and g(x) have period p, then af(x) + bg(x) with any constants « and b also has the period p. Our problem in the first few sections of this chapter will be the representation of various functions f(x) of period 2 in terms of the simple functions Sd, cosx, sinx, cos2x, sin2x, +", Cos na, Simm, cer. All these functions have the period 27. They form the so-called trigonometrie system. Figure 256 shows lhe first few of them (except for the constant 1, which is períodic with any period). The series to be obtained will be a trigonometrie series, that is, a series of the form dg ta cosx+ by sinx + as cos 2x + by sindx += 4 o ora (4 =00+D (a, cosax + b, sina). e ag Gy, dy. Go, dy," are constants, called the coefficients of the seri ce that ca term has the period 27. Hence if the coefjicienis are such ihai the series converges, ils sum will be a function of period 27. kt can be shown (hat il the series on the left side of (4) converges, then inserting parentheses on the right gives a series that converges and has the same sum as the series om the left. This justifies the equality in (4). Now suppose that f(x) is a given function of period 257 and is such that it can be represented by a (4), that is, (4) converges and, moreover, has the sum f(4). Then, using (he equality sign, we write (5 Ho) = ag + D (a, cosnx + ba sina) na 2m 0 ces x cos 2x cos 3x sn 24 sin 3x Cosine and sine functions having the period 27 SEC. 11.1 Fourier Series £b) The first lkree partial sums of the corresponding Fourier series Fig. 257. Eample 1 Sinee cos (=a) — cos « and cos (0 = 1, this yields, x E; ba — [eos0 cost am) cosa | cosf]= nm cosum. nm Now. cos 7 = =1,cos2m = 1.cos3m' —1, cre: in general. 1 forodd a, cos nm ê 1 2 foradd a, and thus | I-cosnm = for even a, lo iorevena. Henee the Lourier coelficients b,, of onr finelion are dk ak a by — 0, bs ba=0, bs=— EOREM 1 PRO OF CHAP. 11 Fourier Series, Integrals, and Transforms Since the a are zero, the Fourier series of f(x) is 8) * (a 1 Lgn3r4 > sinS 6) E (sima E side sinse . The partial sums are s- * S1= sina, ak (5 Ea ) — (sina + 5 sin3r cte., = 3 Their graphs in Fig. 257 seem to indicate that the series is convergent and has the sum (9). the given function We notice that at x = O and x = 7, the points of discontinuity of f(%), all partial sums have the value zero, the arithmetic mean of the limits —A and & of our function, at these points. Furlhermore, assaming that f(x) is the sum of the series and setting x = 7/2, we have my 4 1 flaj-t- los+ ul— ] + thus 3.5 7 “This is a famous result obtained by Leibniz in 1673 from geomenrio considerations. Tt illustrates that the values of various series with constant terms can be obtained by evaluating Fourier series at specific points. Derivation of the Euler Formulas (6) The key to the Fuler formulas (6) is the orthogonality of (3), a concept basic importance, as follows. Orthogonality of the Trigonometric System (3) | | The trigonometric system (3) is orthogonal on the interval — 1 E x E 7 (hence also on 0 Ex 27 or any other interval of length 277 because of periodicity): that is, | the integral of the product of any two functions in (3) over thai interval is 0, so thai for any integers n and m, (| cosmcosmdr=0 (n 4 m) | | - | - | (9 6 | sinnesinmaár=o (nm) | (e) | sinnecosmedr=0 (n&morn=m). | A DE RE PP PIOR This follows simply by transforming the integrands trigonometrically from products into sums. In (9a) and (9b), by (11) in App. A3.1, 1 1 TD cosmcosmar= — [ cos ln tmrar os [cos (n — mx de E dl 1” ne sim ma de = > [ cos (n — m)x dx — 31 cos (n + mx dx. 484 CHAP. 11 Fourier Series, Integrals, and Transforms Convergence and Sum of a Fourier Series The class of functions that can be represented by Fourier series is surprisingly large and general. Sufficient conditions valid in most applications are as follows. THEOREM 2 | Representation by a Fourier Series Ler f(x) be periodic with period 27 and piecewise continuous (see Sec. 6.1) in the | imerval -7 E x E 7. Furthermore, ler j(%) have a left-hand derivative and a right-hand derivative ar each point of that interval. Then the Fourier series (5) of Ef) [with coeficients (6)] converges. ds sum is f(0), except at points xo where f(%) is discontinuous. There he sum of ihe series is the average of the lefi- and | right-hand limits? of [09 at xo PROOF We prove convergence in Theorem 2. We prove convergence for a continuous function (x) having continuous first and second derivatives. Integrating (6) by parts, we obtain. 1 a => | fojcosnrds= pq f09 sin nx E T codes fre sin nx dx. nm na The first term on the right is zero. Another integration by parts gives — fi cosn]|” = a, f(x) cos nx dx. ra nêm The first term on the right is zero because of the periodicity and continuity of f(x). Since f” is continuous in the interval of integration, wc have Iool=M for an appropriate constant M. Furthermore, |cos nx| = L. It follows that lanl = ] “ro cos nx dr 1 nêm 2The left-hand limit of f(x) at x is defined as the limit of f(x) as « approaches xq from the left and is commonty denoted by f(x — 0). Thus fixo — 0) = Jim, Flo — h) as h-> O through positive values. The right-hand limit is denoted by f(xg + 0) and right-hand limits 8 flo + 0) = Jim, Flxo +) as h => O through positive values Hi-0)=1, fi+0)=1 The Jeft- and right-hand derivatives of f(x) at xp arc defined as the limits of of the function flo fla a . x fx<1 fi) = respectively, as A — O through positive values. Of course if f(x) is continuous at xo, Lhe last term in x/2 both numerators is simply f(x) SEC. 1,1 Fourier Series 485 Similarly, [ba] << 2 M!nê for all n, Hence the absolute value of each term of the Fourier series of f(x) is al most equal to the corresponding term of the series 1 da +20 +++ which is convergent. Hence that Fourier series conver and the proof is complete. (Readers already familiar with uniform convergence will see that. by the Weierstrass test in Sec. 15.5, under our present assumptions the Fourier series converges uniformly, and our derivation of (6) by integrating term by term is then justificd by Theorem 3 of Sec. 15.5.) “The proof of convergence in the case of a piecesise contimous function f(x) and the proof that under the assumptions in the theorem the Fourier series (5) with cocfficients (6) represents f(x) are substantially more complicated; sec. for instance, Reí. [C12]. E EXAMPLE 2 Convergence at a Jump as Indicated in Theorem 2 gular awave in Example “has a jump ata = D. Tts left-hand limit there is —k and its riglt-hand line is & (Fig. 257). Hence the average of these limits is O. The Fourier series (8) of he wave Coes indeed converge to this value when e — O because then all its terms are 0. Similarly for the other jumps. “This is in agreement with Theorem 2. Summary. A Fourier series of a given function f(x) of period 2% is a series of the form (5) with coefficients given by the Euler formulas (6). Theorem 2 gives conditions that are sutficient tor this series to converge and at cach x to have the value f(x), except at discontinuities of f(x). where the series equals the arithmetie mean of the “hand and right-hand limits of f(x) at that point. 1. (Calculus review) Review integration techniques for 6. (Change of scale) IE f() has period p, show hat fax), integrals as they are likely to arise from the Euler a + O, amd J(x/b), b + O, are periodic tunetions of x lormvlas, [or instance, definite integrals of x cos nx, of periods pla and bp, respectively. Give examples. ? sin mx, e 2 cos mr. etc. , [7-12] GRAPHS OF 25-PERIODIC FUNCTIONS 3] FUNDAMENTAL PERIOD Sketch or graph f(%), of period 2m, which for > << 7 Fhe findamentai period is the smallest positive period. Find is given as follows. it for TIO) = 810) 2. cosx, sing cos2r sinZx, cosa singx, Ss) =m— Il 10. J0) — cos 27x, sin27x = ; 27x 2mx ne. fio = [ 3 cosmr, sinns, cos. sin . cos mx ns, cos ; Zan 2mnx (5) — [ cos — sin a k 4. Show that f — const is periodic with any period but FOURIER SERIES as Ho Purglamental peido, 1, Showing the details ol your work, find the Fourier series st and elx) have period p, show that of the given f(x), which is assumed to have (he period 277. he) = aJto) + Dg(4) (a, b, constant) has the period p. Sketch or graph the partial sums up to lat including Thus all functions of period p form 2 vector space. cos 5x and sin 5x SEC. 11.2 Functions of Any Period p = 2L 487 719 2 Functions of Any Period p = 2L The functions considered so far had period 27, for the simplicity of the formulas. Of course, periodic functions in applications will generally have other periods. However, we now show that the transition from period p — 27 to a period 2L is quite simple. The notation p = 2L is practical because L will be the length of a violin string (Sec. 12.2) or the length of a rod im heat conduction (Sec. 12.5), and so on. “The idea is simply to find and use a change of scale that gives from a function g(u) of period 27 a function of period 2L. Now from (5) and (6) in the last section with g(v) instead of f(x) we have the Fourier series «) gw) =a+>D (a, cosnv + b sinnv) je with coelficients => [eva = 57 | sOdo pm 2) an — | e) cosmo do Tl 1 bi=— si dv. no fe sim nv dv We can now write the change of scale as v = kx with k such that the old period v = 27 gives for the new variable x (he new period x = 2L. Thus, 27 = k2L. Hence k = /L and (3) v=kx= ul. This implies dv = (7/L) dx, which upon substitution into (2) cancels 1/27 and 1/7 and gives instead the factors 1/2L and 1/L. Writi (4) s(v) = flo), we Lhus obtain from (1) the Fourier series of the function f(x) of period 2L N Ceci nm o RT (5) f)=aw+> (on cos — x + by sin o ) n=1 iá ê with the Fourier coefficients of f(x) given by the Euler formulas E (a) =) toa E (6) 0d) a — nI- a nTx | too cos TE ar n= 1,200 Ly np Ea nmx hs “00 si =1,2+»: (e) A Ftoosin z dx n=1,2, E 488 EXAMPLE 1 EXAMPLE 2 CHAP. 11 Fourier Series, Integrals, and Transforms Justas in Sec. 11.1, we continue to call (5) with any coefficients a trigonometric series. And we cam integrate from O to 2L or over any other interval of length p = 2L. Periodic Rectangular Wave Find the Fourier series of the function (Fig. 259) O il -2
dr + | ecos dx 21! 2 o 2 = [ = so that the Fourier series has no cosine terms. From (6c), pol [nm bn= 3 | um OS 2 [nm Ea nm |º 2 2 |» nm nm sin —— 2 2 nm 2 [tr ifon=1,3, O ifn=24+ 490 CHAP.11 Fourier Series, Integrals, and Transforms FOURIER SERIES FOR PERIOD p = 2L Find the Fourier series of the function f(3), of period p = 25, and sketch or graph the first three partia! sums. (Show the details of your work.) 1 fgj=1(2
