10.2 Fourier Series PROBLEMS

Transcription

1 1. Fourier Series 547 PROBEMS In each of Problems 1 through 1 either solve the given boundar value problem or else show that it has no solution =, () =, (π) = 1. + =, () = 1, (π) = 3. + =, () =, () = 4. + =, () = 1, () = 5. + =, () =, (π) = 6. + =, () =, (π) = = cos, () =, (π) = = sin, () =, (π) = = cos, () =, (π) = = cos, () =, (π) = In each of Problems 11 through 16 find the eigenvalues and eigenfunctions of the given boundar value problem. Assume that all eigenvalues are real λ =, () =, (π) = 1. + λ =, () =, (π) = λ =, () =, (π) = λ =, () =, () = λ =, () =, () = 16. λ =, () =, () = 17. In this problem we outline a proof that the eigenvalues of the boundar value problem (18), (19) are real. (a) Write the solution of Eq. (18) as = k 1 ep(iµ) + k ep( iµ), where λ = µ, and impose the boundar conditions (19). Show that nontrivial solutions eist if and onl if ep(iµπ) ep( iµπ) =. (b) et µ = ν + iσ and use Euler s relation ep(iνπ) = cos(νπ) + i sin(νπ) to determine the real and imaginar parts of Eq. (i). (c) B considering the equations found in part (b), show that σ = ; hence µ is real and so is λ. Show also that ν = n, where n is an integer. (i) 1. Fourier Series ater in this chapter ou will find that ou can solve man important problems involving partial differential equations provided that ou can epress a given function as an infinite sum of sines and/or cosines. In this and the following two sections we eplain in detail how this can be done. These trigonometric series are called Fourier series 1 ; 1 Fourier series are named for Joseph Fourier, who made the first sstematic use, although not a completel rigorous investigation, of them in 187 and 1811 in his papers on heat conduction. According to Riemann, when Fourier presented his first paper to the Paris Academ in 187, stating that an arbitrar function could be epressed as a series of the form (1), the mathematician agrange was so surprised that he denied the possibilit in the most definite terms. Although it turned out that Fourier s claim of generalit was somewhat too strong, his results inspired a flood of important research that has continued to the present da. See Grattan-Guinness or Carslaw [Historical Introduction] for a detailed histor of Fourier series.

2 548 Chapter 1. Partial Differential Equations and Fourier Series the are somewhat analogous to Talor series in that both tpes of series provide a means of epressing quite complicated functions in terms of certain familiar elementar functions. We begin with a series of the form a + a m cos mπ + b m m=1 mπ sin. (1) On the set of points where the series (1) converges, it defines a function f, whose value at each point is the sum of the series for that value of. In this case the series (1) is said to be the Fourier series for f. Our immediate goals are to determine what functions can be represented as a sum of a Fourier series and to find some means of computing the coefficients in the series corresponding to a given function. The first term in the series (1) is written as a / rather than simpl as a to simplif a formula for the coefficients that we derive below. Besides their association with the method of separation of variables and partial differential equations, Fourier series are also useful in various other was, such as in the analsis of mechanical or electrical sstems acted on b periodic eternal forces. Periodicit of the Sine and Cosine Functions. To discuss Fourier series it is necessar to develop certain properties of the trigonometric functions sin(mπ /) and cos(mπ /), where m is a positive integer. The first is their periodic character. A function f is said to be periodic with period T > if the domain of f contains + T whenever it contains, and if f ( + T ) = f () () for ever value of. An eample of a periodic function is shown in Figure It follows immediatel from the definition that if T is a period of f, then T is also a period, and so indeed is an integral multiple of T. The smallest value of T for which Eq. () holds is called the fundamental period of f. In this connection it should be noted that a constant ma be thought of as a periodic function with an arbitrar period, but no fundamental period. T FIGURE 1..1 T A periodic function.

3 1. Fourier Series 549 If f and g are an two periodic functions with common period T, then their product f g and an linear combination c 1 f + c g are also periodic with period T. To prove the latter statement, let F() = c 1 f () + c g(); then for an F( + T ) = c 1 f ( + T ) + c g( + T ) = c 1 f () + c g() = F(). (3) Moreover, it can be shown that the sum of an finite number, or even the sum of a convergent infinite series, of functions of period T is also periodic with period T. In particular, the functions sin(mπ /) and cos(mπ /), m = 1,, 3,..., are periodic with fundamental period T = /m. To see this, recall that sin and cos have fundamental period π, and that sin α and cos α have fundamental period π/α. If we choose α = mπ/, then the period T of sin(mπ /) and cos(mπ/) is given b T = π /mπ = /m. Note also that, since ever positive integral multiple of a period is also a period, each of the functions sin(mπ /) and cos(mπ/) has the common period. Orthogonalit of the Sine and Cosine Functions. To describe a second essential propert of the functions sin(mπ /) and cos(mπ/) we generalize the concept of orthogonalit of vectors (see Section 7.). The standard inner product (u, v) of two real-valued functions u and v on the interval α β is defined b (u, v) = β α u()v() d. (4) The functions u and v are said to be orthogonal on α β if their inner product is zero, that is, if β α u()v() d =. (5) A set of functions is said to be mutuall orthogonal if each distinct pair of functions in the set is orthogonal. The functions sin(mπ /) and cos(mπ/), m = 1,,..., form a mutuall orthogonal set of functions on the interval. In fact, the satisf the following orthogonalit relations: cos mπ cos mπ sin mπ nπ, m = n, cos d =, m = n; nπ sin nπ sin d = (6) d =, all m, n; (7), m = n,, m = n. These results can be obtained b direct integration. For eample, to derive Eq. (8), note that sin mπ nπ sin d = 1 (m n)π (m + n)π cos cos d = 1 sin[(m n)π /] sin[(m + n)π /] π m n m + n =, (8)

4 55 Chapter 1. Partial Differential Equations and Fourier Series as long as m + n and m n are not zero. Since m and n are positive, m + n =. On the other hand, if m n =, then m = n, and the integral must be evaluated in a different wa. In this case sin mπ nπ sin d = = 1 = 1 =. sin mπ d 1 cos mπ sin(mπ /) mπ/ This establishes Eq. (8); Eqs. (6) and (7) can be verified b similar computations. The Euler Fourier Formulas. Now let us suppose that a series of the form (1) converges, and let us call its sum f (): f () = a + a m cos mπ + b mπ m sin. (9) m=1 The coefficients a m and b m can be related to f () as a consequence of the orthogonalit conditions (6), (7), and (8). First multipl Eq. (9) b cos(nπ /), where n is a fied positive integer (n > ), and integrate with respect to from to. Assuming that the integration can be legitimatel carried out term b term, we obtain f () cos nπ d = a + cos nπ b m m=1 d + a m sin mπ m=1 nπ cos d cos mπ nπ cos d d. (1) Keeping in mind that n is fied whereas m ranges over the positive integers, it follows from the orthogonalit relations (6) and (7) that the onl nonzero term on the right side of Eq. (1) is the one for which m = n in the first summation. Hence f () cos nπ d = a n, n = 1,,.... (11) To determine a we can integrate Eq. (9) from to, obtaining f () d = a d + a m cos mπ d + b m sin mπ m=1 m=1 = a, (1) since each integral involving a trigonometric function is zero. Thus a n = 1 f () cos nπ d d, n =, 1,,.... (13) This is a nontrivial assumption, since not all convergent series with variable terms can be so integrated. For the special case of Fourier series, however, term-b-term integration can alwas be justified.

5 1. Fourier Series 551 B writing the constant term in Eq. (9) as a /, it is possible to compute all the a n from Eq. (13). Otherwise, a separate formula would have to be used for a. A similar epression for b n ma be obtained b multipling Eq. (9) b sin(nπ /), integrating termwise from to, and using the orthogonalit relations (7) and (8); thus b n = 1 f () sin nπ d, n = 1,, 3,.... (14) Equations (13) and (14) are known as the Euler Fourier formulas for the coefficients in a Fourier series. Hence, if the series (9) converges to f (), and if the series can be integrated term b term, then the coefficients must be given b Eqs. (13) and (14). Note that Eqs. (13) and (14) are eplicit formulas for a n and b n in terms of f, and that the determination of an particular coefficient is independent of all the other coefficients. Of course, the difficult in evaluating the integrals in Eqs. (13) and (14) depends ver much on the particular function f involved. Note also that the formulas (13) and (14) depend onl on the values of f () in the interval. Since each of the terms in the Fourier series (9) is periodic with period, the series converges for all whenever it converges in, and its sum is also a periodic function with period. Hence f () is determined for all b its values in the interval. It is possible to show (see Problem 7) that if g is periodic with period T, then ever integral of g over an interval of length T has the same value. If we appl this result to the Euler Fourier formulas (13) and (14), it follows that the interval of integration,, can be replaced, if it is more convenient to do so, b an other interval of length. E X A M P E 1 Assume that there is a Fourier series converging to the function f defined b, <, f () =, < ; f ( + 4) = f (). Determine the coefficients in this Fourier series. This function represents a triangular wave (see Figure 1..) and is periodic with period 4. Thus in this case = and the Fourier series has the form f () = a + m=1 a m cos mπ + b m sin mπ (15), (16) where the coefficients are computed from Eqs. (13) and (14) with =. Substituting for f () in Eq. (13) with m =, we have a = 1 d + ( ) 1 For m >, Eq. (13) ields a m = 1 ( ) cos mπ d + 1 d = =. (17) cos mπ d.

6 55 Chapter 1. Partial Differential Equations and Fourier Series FIGURE 1.. Triangular wave. These integrals can be evaluated through integration b parts, with the result that a m = 1 mπ sin cos mπ mπ mπ + 1 mπ sin + cos mπ mπ mπ = 1 + cos mπ + cos mπ mπ mπ mπ mπ = 4 (cos mπ 1), m = 1,,... (mπ) 8/(mπ) =, m odd,, m even. Finall, from Eq. (14) it follows in a similar wa that (18) b m =, m = 1,,.... (19) B substituting the coefficients from Eqs. (17), (18), and (19) in the series (16) we obtain the Fourier series for f : f () = 1 8 π cos π + 1 3π cos π cos 5 + = 1 8 cos(mπ /) π m = 1 8 π m=1,3,5,... cos(n 1)π / (n 1). () E X A M P E et, 3 < < 1, f () = 1, 1 < < 1,, 1 < < 3 and suppose that f ( + 6) = f (); see Figure Find the coefficients in the Fourier series for f. (1)

7 1. Fourier Series t FIGURE 1..3 Graph of f () in Eample. Since f has period 6, it follows that = 3 in this problem. Consequentl, the Fourier series for f has the form f () = a + a n cos nπ 3 + b nπ n sin, () 3 where the coefficients a n and b n are given b Eqs. (13) and (14) with = 3. We have Similarl, a n = 1 3 and b n = cos nπ 3 sin nπ 3 a = 1 3 d = 1 nπ 3 d = 1 nπ Thus the Fourier series for f is f () = nπ sin nπ 3 = π 3 f () d = 1 3 sin nπ 3 cos nπ 3 1 cos nπ = nπ sin nπ 3 1 d = 3. (3), n = 1,,..., (4) =, n = 1,,.... (5) cos(π /3) + cos(π/3) cos(4π/3) cos(5π/3) (6) E X A M P E 3 Consider again the function in Eample 1 and its Fourier series (). Investigate the speed with which the series converges. In particular, determine how man terms are needed so that the error is no greater than.1 for all. The mth partial sum in this series, s m () = 1 8 π m cos(n 1)π / (n 1), (7) can be used to approimate the function f. The coefficients diminish as (n 1), so the series converges fairl rapidl. This is borne out b Figure 1..4, where the partial sums for m = 1 and m = are plotted. To investigate the convergence in more detail

8 554 Chapter 1. Partial Differential Equations and Fourier Series we can consider the error e m () = f () s m (). Figure 1..5 shows a plot of e 6 () versus for. Observe that e 6 () is greatest at the points = and = where the graph of f () has corners. It is more difficult for the series to approimate the function near these points, resulting in a larger error there for a given n. Similar graphs are obtained for other values of m. Once ou realize that the maimum error alwas occurs at = or =, ou can obtain a uniform error bound for each m simpl b evaluating e m () at one of these points. For eample, for m = 6 we have e 6 () =.337, so e 6 () <.34 for, and consequentl for all. Table 1..1 shows corresponding data for other values of m; these data are plotted in Figure From this information ou can begin to estimate the number of terms that are needed in the series in order to achieve a given level of accurac in the approimation. For eample, to guarantee that e m ().1 we need to choose m = 1. m = m = FIGURE 1..4 Partial sums in the Fourier series, Eq. (), for the triangular wave. e 6 () FIGURE 1..5 Plot of e 6 () versus for the triangular wave.

9 1. Fourier Series 555 TABE 1..1 Values of the Error e m () for the Triangular Wave m e m () e m () m FIGURE 1..6 Plot of e m () versus m for the triangular wave. In this book Fourier series appear mainl as a means of solving certain problems in partial differential equations. However, such series have much wider application in science and engineering, and in general are valuable tools in the investigation of periodic phenomena. A basic problem is to resolve an incoming signal into its harmonic components, which amounts to constructing its Fourier series representation. In some frequenc ranges the separate terms correspond to different colors or to different audible tones. The magnitude of the coefficient determines the amplitude of each component. This process is referred to as spectral analsis. PROBEMS In each of Problems 1 through 8 determine whether the given function is periodic. If so, find its fundamental period. 1. sin 5. cos π 3. sinh 4. sin π / 5. tan π 6., n 1 < n, 7. f () = n =, ±1, ±,... 1, n < n + 1; ( 1) 8. f () =, n 1 < n, n =, ±1, ±,... 1, n < n + 1;

10 556 Chapter 1. Partial Differential Equations and Fourier Series 9. If f () = for < <, and if f ( + ) = f (), find a formula for f () in the interval < < ; in the interval 3 < <. + 1, 1 < <, 1. If f () = and if f ( + ) = f (), find a formula for f () in, < < 1, the interval 1 < < ; in the interval 8 < < If f () = for < <, and if f ( + ) = f (), find a formula for f () in the interval < <. 1. Verif Eqs. (6) and (7) of the tet b direct integration. In each of Problems 13 through 18: (a) (b) Sketch the graph of the given function for three periods. Find the Fourier series for the given function. 13. f () =, < ; f ( + ) = f () 1, <, 14. f () = f ( + ) = f (), < ;, π <, 15. f () = f ( + π) = f (), < π; + 1, 1 <, 16. f () = f ( + ) = f () 1, < 1; +,, 17. f () = f ( + ) = f (), < < ;, 1, 18. f () =, 1 < < 1, f ( + 4) = f (), 1 < ; In each of Problems 19 through 4: (a) Sketch the graph of the given function for three periods. (b) Find the Fourier series for the given function. (c) Plot s m () versus for m = 5, 1, and. (d) Describe how the Fourier series seems to be converging , <, f () = 1, < ; f ( + 4) = f (). f () =, 1 < 1; f ( + ) = f () 1. f () = /, ; f ( + 4) = f () +, <,. f () = f ( + 4) = f (), < ; 1, <, 3. f () = 1 f ( + 4) = f (), < ;, 3, 4. f () = f ( + 6) = f () (3 ), < < 3; 5. Consider the function f defined in Problem 1 and let e m () = f () s m (). Plot e m () versus for for several values of m. Find the smallest value of m for which e m ().1 for all. 6. Consider the function f defined in Problem 4 and let e m () = f () s m (). Plot e m () versus for 3 for several values of m. Find the smallest value of m for which e m ().1 for all. 7. Suppose that g is an integrable periodic function with period T.

11 1. Fourier Series 557 (a) If a T, show that T Hint: Show first that a g() d = g() d = a+t T a+t a g() d. T in the second integral. (b) Show that for an value of a, not necessaril in a T, T g() d = (c) Show that for an values of a and b, a+t a g() d = g() d. Consider the change of variable s = a+t a b+t b g() d. g() d. 8. If f is differentiable and is periodic with period T, show that f is also periodic with period T. Determine whether F() = f (t) dt is alwas periodic. 9. In this problem we indicate certain similarities between three-dimensional geometric vectors and Fourier series. (a) et v 1, v, and v 3 be a set of mutuall orthogonal vectors in three dimensions and let u be an three-dimensional vector. Show that where u = a 1 v 1 + a v + a 3 v 3, a i = u v i v i v i, i = 1,, 3. (ii) Show that a i can be interpreted as the projection of u in the direction of v i divided b the length of v i. (b) Define the inner product (u, v) b Also let (u, v) = u()v() d. φ n () = cos(nπ /), n =, 1,,... ; ψ n () = sin(nπ /), n = 1,,.... Show that Eq. (1) can be written in the form ( f, φ n ) = a (φ, φ n ) + a m (φ m, φ n ) + b m (ψ m, φ n ). m=1 (c) Use Eq. (v) and the corresponding equation for ( f, ψ n ) together with the orthogonalit relations to show that a n = ( f, φ n ) (φ n, φ n ), n =, 1,,... ; b n = ( f, ψ n ), n = 1,,.... (vi) (ψ n, ψ n ) Note the resemblance between Eqs. (vi) and Eq. (ii). The functions φ n and ψ n pla a role for functions similar to that of the orthogonal vectors v 1, v, and v 3 in three-dimensional m=1 (i) (iii) (iv) (v)

12 558 Chapter 1. Partial Differential Equations and Fourier Series space. The coefficients a n and b n can be interpreted as projections of the function f onto the base functions φ n and ψ n. Observe also that an vector in three dimensions can be epressed as a linear combination of three mutuall orthogonal vectors. In a somewhat similar wa an sufficientl smooth function defined on can be epressed as a linear combination of the mutuall orthogonal functions cos(nπ /) and sin(nπ /), that is, as a Fourier series. 1.3 The Fourier Convergence Theorem In the preceding section we showed that if the Fourier series a + a m cos mπ + b m m=1 mπ sin converges and thereb defines a function f, then f is periodic with period, and the coefficients a m and b m are related to f () b the Euler Fourier formulas: a m = 1 b m = 1 f () cos mπ f () sin mπ (1) d, m =, 1,,... ; () d, m = 1,,.... (3) In this section we adopt a somewhat different point of view. Suppose that a function f is given. If this function is periodic with period and integrable on the interval [, ], then a set of coefficients a m and b m can be computed from Eqs. () and (3), and a series of the form (1) can be formall constructed. The question is whether this series converges for each value of and, if so, whether its sum is f (). Eamples have been discovered showing that the Fourier series corresponding to a function f ma not converge to f (), or ma even diverge. Functions whose Fourier series do not converge to the value of the function at isolated points are easil constructed, and eamples will be presented later in this section. Functions whose Fourier series diverge at one or more points are more pathological, and we will not consider them in this book. To guarantee convergence of a Fourier series to the function from which its coefficients were computed it is essential to place additional conditions on the function. From a practical point of view, such conditions should be broad enough to cover all situations of interest, et simple enough to be easil checked for particular functions. Through the ears several sets of conditions have been devised to serve this purpose. Before stating a convergence theorem for Fourier series, we define a term that appears in the theorem. A function f is said to be piecewise continuous on an interval a b if the interval can be partitioned b a finite number of points a = < 1 < < n = b so that 1. f is continuous on each open subinterval i 1 < < i.. f approaches a finite limit as the endpoints of each subinterval are approached from within the subinterval. The graph of a piecewise continuous function is shown in Figure

13 1.3 The Fourier Convergence Theorem 559 a b FIGURE A piecewise continuous function. The notation f (c+) is used to denote the limit of f () as c from the right; similarl, f (c ) denotes the limit of f () as approaches c from the left. Note that it is not essential that the function even be defined at the partition points i. For eample, in the following theorem we assume that f is piecewise continuous; but certainl f does not eist at those points where f itself is discontinuous. It is also not essential that the interval be closed; it ma also be open, or open at one end and closed at the other. Theorem Suppose that f and f are piecewise continuous on the interval <. Further, suppose that f is defined outside the interval < so that it is periodic with period. Then f has a Fourier series f () = a + m=1 a m cos mπ + b m mπ sin, (4) whose coefficients are given b Eqs. () and (3). The Fourier series converges to f () at all points where f is continuous, and to [ f (+) + f ( )]/ at all points where f is discontinuous. Note that [ f (+) + f ( )]/ is the mean value of the right- and left-hand limits at the point. At an point where f is continuous, f (+) = f ( ) = f (). Thus it is correct to sa that the Fourier series converges to [ f (+) + f ( )]/ at all points. Whenever we sa that a Fourier series converges to a function f, we alwas mean that it converges in this sense. It should be emphasized that the conditions given in this theorem are onl sufficient for the convergence of a Fourier series; the are b no means necessar. Neither are the the most general sufficient conditions that have been discovered. In spite of this, the proof of the theorem is fairl difficult and is not given here. 3 To obtain a better understanding of the content of the theorem it is helpful to consider some classes of functions that fail to satisf the assumed conditions. Functions that are not included in the theorem are primaril those with infinite discontinuities in the interval [, ], such as 1/ as, or ln as. Functions having an infinite number of jump discontinuities in this interval are also ecluded; however, such functions are rarel encountered. 3 Proofs of the convergence of a Fourier series can be found in most books on advanced calculus. See, for eample, Kaplan (Chapter 7) or Buck (Chapter 6).

14 56 Chapter 1. Partial Differential Equations and Fourier Series It is noteworth that a Fourier series ma converge to a sum that is not differentiable, or even continuous, in spite of the fact that each term in the series (4) is continuous, and even differentiable infinitel man times. The eample below is an illustration of this, as is Eample in Section 1.. E X A M P E 1 et f () =, < <,, < <, and let f be defined outside this interval so that f ( + ) = f () for all. We will temporaril leave open the definition of f at the points =, ±, ecept that its value must be finite. Find the Fourier series for this function and determine where it converges. (5) 3 3 FIGURE 1.3. Square wave. The equation = f () has the graph shown in Figure 1.3., etended to infinit in both directions. It can be thought of as representing a square wave. The interval [, ] can be partitioned to give the two open subintervals (, ) and (, ). In (, ), f () = and f () =. Clearl, both f and f are continuous and furthermore have limits as from the right and as from the left. The situation in (, ) is similar. Consequentl, both f and f are piecewise continuous on [, ), so f satisfies the conditions of Theorem If the coefficients a m and b m are computed from Eqs. () and (3), the convergence of the resulting Fourier series to f () is assured at all points where f is continuous. Note that the values of a m and b m are the same regardless of the definition of f at its points of discontinuit. This is true because the value of an integral is unaffected b changing the value of the integrand at a finite number of points. From Eq. () a = 1 Similarl, from Eq. (3), f () d = d = ; a m = 1 f () cos mπ d = =, m =. b m = 1 f () sin mπ = (1 cos mπ) mπ, m even; = /mπ, m odd. d = cos mπ sin mπ d d

15 1.3 The Fourier Convergence Theorem 561 Hence f () = + π = + π = + π sin π m=1,3,5,... sin 3π sin(mπ /) m π sin + sin(n 1)π/. (6) n 1 At the points =, ±n, where the function f in the eample is not continuous, all terms in the series after the first vanish and the sum is /. This is the mean value of the limits from the right and left, as it should be. Thus we might as well define f at these points to have the value /. If we choose to define it otherwise, the series still gives the value / at these points, since none of the preceding calculations is altered in an detail; it simpl does not converge to the function at those points unless f is defined to have this value. This illustrates the possibilit that the Fourier series corresponding to a function ma not converge to it at points of discontinuit unless the function is suitabl defined at such points. The manner in which the partial sums s n () = + π sin π (n 1)π sin n 1, n = 1,,... of the Fourier series (6) converge to f () is indicated in Figure 1.3.3, where has been chosen to be one and the graph of s 8 () is plotted. The figure suggests that at points where f is continuous the partial sums do approach f () as n increases. However, in the neighborhood of points of discontinuit, such as = and =, the partial sums do not converge smoothl to the mean value. Instead the tend to overshoot the mark at each end of the jump, as though the cannot quite accommodate themselves to the sharp turn required at this point. This behavior is tpical of Fourier series at points of discontinuit, and is known as the Gibbs 4 phenomenon. 1 n = FIGURE The partial sum s 8 () in the Fourier series, Eq. (6), for the square wave. Additional insight is attained b considering the error e n () = f () s n (). Figure shows a plot of e n () versus for n = 8 and for = 1. The least upper bound of e 8 () is.5 and is approached as and as 1. As n increases, the error 4 The Gibbs phenomenon is named after Josiah Willard Gibbs ( ), who is better known for his work on vector analsis and statistical mechanics. Gibbs was professor of mathematical phsics at Yale, and one of the first American scientists to achieve an international reputation. Gibbs phenomenon is discussed in more detail b Carslaw (Chapter 9).

16 56 Chapter 1. Partial Differential Equations and Fourier Series decreases in the interior of the interval [where f () is continuous] but the least upper bound does not diminish with increasing n. Thus one cannot uniforml reduce the error throughout the interval b increasing the number of terms. Figures and also show that the series in this eample converges more slowl than the one in Eample 1 in Section 1.. This is due to the fact that the coefficients in the series (6) are proportional onl to 1/(n 1). e 8 () FIGURE A plot of the error e 8 () versus for the square wave. PROBEMS In each of Problems 1 through 6 assume that the given function is periodicall etended outside the original interval. (a) (b) Find the Fourier series for the etended function. Sketch the graph of the function to which the series converges for three periods. 1, 1 <, 1. f () = 1, < 1 +, <, 3. f () =, <, π < π/, 5. f () = 1, π/ < π/,, π/ < π. f () =, π <,, < π 4. f () = 1, f () =, 1 <,, < 1 In each of Problems 7 through 1 assume that the given function is periodicall etended outside the original interval. (a) Find the Fourier series for the given function. (b) et e n () = f () s n (). Find the least upper bound or the maimum value (if it eists) of e n () for n = 1,, and 4. (c) If possible, find the smallest n for which e n ().1 for all. 7. f () =, π <,, < π; f ( + π) = f () (see Section 1., Problem 15)

17 1.3 The Fourier Convergence Theorem , 1 <, 8. f () = f ( + ) = f () (see Section 1., Problem 16) 1, < 1; 9. f () =, 1 < 1; f ( + ) = f () (see Section 1., Problem ) +, <, (see Section 1., 1. f () = f ( + 4) = f (), < ; Problem ), 1 <, 11. f () = f ( + ) = f () (see Problem 6), < 1; 1. f () = 3, 1 < 1; f ( + ) = f () Periodic Forcing Terms. In this chapter we are concerned mainl with the use of Fourier series to solve boundar value problems for certain partial differential equations. However, Fourier series are also useful in man other situations where periodic phenomena occur. Problems 13 through 16 indicate how the can be emploed to solve initial value problems with periodic forcing terms. 13. Find the solution of the initial value problem + ω = sin nt, () =, () =, where n is a positive integer and ω = n. What happens if ω = n? 14. Find the formal solution of the initial value problem + ω = b n sin nt, () =, () =, where ω > is not equal to a positive integer. How is the solution altered if ω = m, where m is a positive integer? 15. Find the formal solution of the initial value problem + ω = f (t), () =, () =, where f is periodic with period π and 1, < t < π; f (t) =, t =, π, π; 1, π < t < π. See Problem Find the formal solution of the initial value problem + ω = f (t), () = 1, () =, where f is periodic with period and 1 t, t < 1; f (t) = 1 + t, 1 t <. See Problem Assuming that show formall that f () = a + 1 a n cos nπ + b n [ f ()] d = a + (an + b n ). nπ sin, (i)

18 564 Chapter 1. Partial Differential Equations and Fourier Series This relation between a function f and its Fourier coefficients is known as Parseval s ( ) equation. Parseval s equation is ver important in the theor of Fourier series and is discussed further in Section Hint: Multipl Eq. (i) b f (), integrate from to, and use the Euler Fourier formulas. 18. This problem indicates a proof of convergence of a Fourier series under conditions more restrictive than those in Theorem (a) If f and f are piecewise continuous on <, and if f is periodic with period, show that na n and nb n are bounded as n. Hint: Use integration b parts. (b) If f is continuous on and periodic with period, and if f and f are piecewise continuous on <, show that n a n and n b n are bounded as n. Use this fact to show that the Fourier series for f converges at each point in. Wh must f be continuous on the closed interval? Hint: Again, use integration b parts. Acceleration of Convergence. In the net problem we show how it is sometimes possible to improve the speed of convergence of a Fourier, or other infinite, series. 19. Suppose that we wish to calculate values of the function g, where g() = (n 1) sin(n 1)π. (i) 1 + (n 1) It is possible to show that this series converges, albeit rather slowl. However, observe that for large n the terms in the series (i) are approimatel equal to [sin(n 1)π ]/(n 1) and that the latter terms are similar to those in the eample in the tet, Eq. (6). (a) Show that [sin(n 1)π ]/(n 1) = (π/)[ f () 1 ], where f is the square wave in the eample with = 1. (b) Subtract Eq. (ii) from Eq. (i) and show that g() = π [ f () 1 ] (ii) sin(n 1)π (n 1)[1 + (n 1) ]. (iii) The series (iii) converges much faster than the series (i) and thus provides a better wa to calculate values of g(). 1.4 Even and Odd Functions Before looking at further eamples of Fourier series it is useful to distinguish two classes of functions for which the Euler Fourier formulas can be simplified. These are even and odd functions, which are characterized geometricall b the propert of smmetr with respect to the -ais and the origin, respectivel (see Figure 1.4.1). Analticall, f is an even function if its domain contains the point whenever it contains the point, and if f ( ) = f () (1)

19 1.4 Even and Odd Functions 565 (a) FIGURE (b) (a) An even function. (b) An odd function. for each in the domain of f. Similarl, f is an odd function if its domain contains whenever it contains, and if f ( ) = f () () for each in the domain of f. Eamples of even functions are 1,, cos n,, and n. The functions, 3, sin n, and n+1 are eamples of odd functions. Note that according to Eq. (), f () must be zero if f is an odd function whose domain contains the origin. Most functions are neither even nor odd, for instance, e. Onl one function, f identicall zero, is both even and odd. Elementar properties of even and odd functions include the following: 1. The sum (difference) and product (quotient) of two even functions are even.. The sum (difference) of two odd functions is odd; the product (quotient) of two odd functions is even. 3. The sum (difference) of an odd function and an even function is neither even nor odd; the product (quotient) of two such functions is odd. 5 The proofs of all these assertions are simple and follow directl from the definitions. For eample, if both f 1 and f are odd, and if g() = f 1 () + f (), then g( ) = f 1 ( ) + f ( ) = f 1 () f () = [ f 1 () + f ()] = g(), (3) so f 1 + f is an odd function also. Similarl, if h() = f 1 () f (), then h( ) = f 1 ( ) f ( ) = [ f 1 ()][ f ()] = f 1 () f () = h(), (4) so that f 1 f is even. Also of importance are the following two integral properties of even and odd functions: 4. If f is an even function, then f () d = f () d. (5) 5 These statements ma need to be modified if either function vanishes identicall.

20 566 Chapter 1. Partial Differential Equations and Fourier Series 5. If f is an odd function, then f () d =. (6) These properties are intuitivel clear from the interpretation of an integral in terms of area under a curve, and also follow immediatel from the definitions. For eample, if f is even, then f () d = f () d + f () d. etting = s in the first term on the right side, and using Eq. (1), we obtain f () d = f (s) ds + f () d = f () d. The proof of the corresponding propert for odd functions is similar. Even and odd functions are particularl important in applications of Fourier series since their Fourier series have special forms, which occur frequentl in phsical problems. Cosine Series. Suppose that f and f are piecewise continuous on <, and that f is an even periodic function with period. Then it follows from properties 1 and 3 that f () cos(nπ /) is even and f () sin(nπ /) is odd. As a consequence of Eqs. (5) and (6), the Fourier coefficients of f are then given b Thus f has the Fourier series a n = f () cos nπ d, n =, 1,,... ; (7) b n =, n = 1,,.... f () = a + a n cos nπ. In other words, the Fourier series of an even function consists onl of the even trigonometric functions cos(nπ /) and the constant term; it is natural to call such a series a Fourier cosine series. From a computational point of view, observe that onl the coefficients a n, for n =, 1,,..., need to be calculated from the integral formula (7). Each of the b n, for n = 1,,..., is automaticall zero for an even function, and so does not need to be calculated b integration. Sine Series. Suppose that f and f are piecewise continuous on <, and that f is an odd periodic function of period. Then it follows from properties and 3 that f () cos(nπ /) is odd and f () sin(nπ /) is even. In this case the Fourier coefficients of f are a n =, n =, 1,,..., b n = f () sin nπ d, n = 1,,..., (8)

21 1.4 Even and Odd Functions 567 and the Fourier series for f is of the form f () = b n sin nπ. Thus the Fourier series for an odd function consists onl of the odd trigonometric functions sin(nπ /); such a series is called a Fourier sine series. Again observe that onl half of the coefficients need to be calculated b integration, since each a n, for n =, 1,,..., is zero for an odd function. E X A M P E 1 et f () =, < <, and let f ( ) = f () =. et f be defined elsewhere so that it is periodic of period (see Figure 1.4.). The function defined in this manner is known as a sawtooth wave. Find the Fourier series for this function. Since f is an odd function, its Fourier coefficients are, according to Eq. (8), a n =, n =, 1,,... ; b n = = nπ sin nπ d sin nπ nπ = nπ ( 1)n+1, n = 1,,.... Hence the Fourier series for f, the sawtooth wave, is f () = π ( 1) n+1 n cos nπ sin nπ. (9) Observe that the periodic function f is discontinuous at the points ±, ±3,..., as shown in Figure At these points the series (9) converges to the mean value of the left and right limits, namel, zero. The partial sum of the series (9) for n = 9 is shown in Figure The Gibbs phenomenon (mentioned in Section 1.3) again occurs near the points of discontinuit. 3 3 FIGURE 1.4. Sawtooth wave.

22 568 Chapter 1. Partial Differential Equations and Fourier Series n = 9 FIGURE A partial sum in the Fourier series, Eq. (9), for the sawtooth wave. Note that in this eample f ( ) = f () =, as well as f () =. This is required if the function f is to be both odd and periodic with period. When we speak of constructing a sine series for a function defined on, it is understood that, if necessar, we must first redefine the function to be zero at = and =. It is worthwhile to observe that the triangular wave function (Eample 1 of Section 1.) and the sawtooth wave function just considered are identical on the interval <. Therefore, their Fourier series converge to the same function, f () =, on this interval. Thus, if it is required to represent the function f () = on < b a Fourier series, it is possible to do this b either a cosine series or a sine series. In the former case f is etended as an even function into the interval < < and elsewhere periodicall (the triangular wave). In the latter case f is etended into < < as an odd function, and elsewhere periodicall (the sawtooth wave). If f is etended in an other wa, the resulting Fourier series will still converge to in < but will involve both sine and cosine terms. In solving problems in differential equations it is often useful to epand in a Fourier series of period a function f originall defined onl on the interval [, ]. As indicated previousl for the function f () = several alternatives are available. Eplicitl, we can: 1. Define a function g of period so that f (),, g() = f ( ), < <. The function g is thus the even periodic etension of f. Its Fourier series, which is a cosine series, represents f on [, ].. Define a function h of period so that f (), < <, h() =, =,, (11) f ( ), < <. (1) The function h is thus the odd periodic etension of f. Its Fourier series, which is a sine series, also represents f on (, ).

23 1.4 Even and Odd Functions Define a function k of period so that k() = f (),, (1) and let k() be defined for (, ) in an wa consistent with the conditions of Theorem Sometimes it is convenient to define k() to be zero for < <. The Fourier series for k, which involves both sine and cosine terms, also represents f on [, ], regardless of the manner in which k() is defined in (, ). Thus there are infinitel man such series, all of which converge to f () in the original interval. Usuall, the form of the epansion to be used will be dictated (or at least suggested) b the purpose for which it is needed. However, if there is a choice as to the kind of Fourier series to be used, the selection can sometimes be based on the rapidit of convergence. For eample, the cosine series for the triangular wave [Eq. () of Section 1.] converges more rapidl than the sine series for the sawtooth wave [Eq. (9) in this section], although both converge to the same function for <. This is due to the fact that the triangular wave is a smoother function than the sawtooth wave and is therefore easier to approimate. In general, the more continuous derivatives possessed b a function over the entire interval < <, the faster its Fourier series will converge. See Problem 18 of Section 1.3. E X A M P E Suppose that f () = 1, < 1,, 1 <. As indicated previousl, we can represent f either b a cosine series or b a sine series. Sketch the graph of the sum of each of these series for 6 6. In this eample =, so the cosine series for f converges to the even periodic etension of f of period 4, whose graph is sketched in Figure Similarl, the sine series for f converges to the odd periodic etension of f of period 4. The graph of this function is shown in Figure (13) FIGURE Even periodic etension of f () given b Eq. (13) FIGURE Odd periodic etension of f () given b Eq. (13).

24 57 Chapter 1. Partial Differential Equations and Fourier Series PROBEMS In each of Problems 1 through 6 determine whether the given function is even, odd, or neither tan 4. sec e In each of Problems 7 through 1 a function f is given on an interval of length. In each case sketch the graphs of the even and odd etensions of f of period., <,, < 1, 7. f () = 8. f () = 1, < 3 1, 1 < 9. f () =, < < 1. f () = 3, < < 4, < 1, 11. f () = 1. f () = 4, < < 1 1, 1 < 13. Prove that an function can be epressed as the sum of two other functions, one of which is even and the other odd. That is, for an function f, whose domain contains whenever it contains, show that there is an even function g and an odd function h such that f () = g() + h(). Hint: Assuming f () = g() + h(), what is f ( )? 14. Find the coefficients in the cosine and sine series described in Eample. In each of Problems 15 through find the required Fourier series for the given function and sketch the graph of the function to which the series converges over three periods , < < 1, f () =, 1 < < ; cosine series, period 4 Compare with Eample 1 and Problem 5 of Section 1.3., < 1, 16. f () = sine series, period 4 1, 1 < ; 17. f () = 1, π; cosine series, period π 18. f () = 1, < < π; sine series, period π, < < π, 19. f () = 1, π < < π, sine series, period 6π, π < < 3π. f () =, < 1; series of period 1 1. f () =, ; cosine series, period Compare with Eample 1 of Section 1... f () =, < < ; sine series, period In each of Problems 3 through 6: (a) Find the required Fourier series for the given function. (b) Sketch the graph of the function to which the series converges for three periods. (c) Plot one or more partial sums of the series. 3., < < π, f () =, π < < π; cosine series, period 4π 4. f () =, π < < ; sine series, period π 5. f () =, < < ; sine series, period 4 6. f () =, < < 4; cosine series, period 8 In each of Problems 7 through 3 a function is given on an interval < <. (a) Sketch the graphs of the even etension g() and the odd etension h() of the given function of period over three periods.

25 1.4 Even and Odd Functions 571 (b) (c) (d) Find the Fourier cosine and sine series for the given function. Plot a few partial sums of each series. For each series investigate the dependence on n of the maimum error on [, ]. 7. f () = 3, < < 3 8. f () = 9. f () = (4 4 3)/4, < < 3. f () = , < < Prove that if f is an odd function, then f () d =., < < 1,, 1 < < 3. Prove properties and 3 of even and odd functions, as stated in the tet. 33. Prove that the derivative of an even function is odd, and that the derivative of an odd function is even. 34. et F() = f (t) dt. Show that if f is even, then F is odd, and that if f is odd, then F is even. 35. From the Fourier series for the square wave in Eample 1 of Section 1.3, show that π 4 = = ( 1) n n + 1. This relation between π and the odd positive integers was discovered b eibniz in From the Fourier series for the triangular wave (Eample 1 of Section 1.), show that 37. Assume that f has a Fourier sine series (a) Show formall that n= π 8 = = 1 (n + 1). f () = n= b n sin(nπ /),. [ f ()] d = bn. Compare this result with that of Problem 17 in Section 1.3. What is the corresponding result if f has a cosine series? (b) Appl the result of part (a) to the series for the sawtooth wave given in Eq. (9), and thereb show that π 6 = = 1 n. This relation was discovered b Euler about More Specialized Fourier Series. et f be a function originall defined on. In this section we have shown that it is possible to represent f either b a sine series or b a cosine series b constructing odd or even periodic etensions of f, respectivel. Problems 38 through 4 concern some other more specialized Fourier series that converge to the given function f on (, ).

26 57 Chapter 1. Partial Differential Equations and Fourier Series 38. et f be etended into (, ] in an arbitrar manner. Then etend the resulting function into (, ) as an odd function and elsewhere as a periodic function of period 4 (see Figure 1.4.6). Show that this function has a Fourier sine series in terms of the functions sin(nπ /), n = 1,, 3,... ; that is, where b n = 1 f () = b n sin(nπ /), f () sin(nπ /) d. This series converges to the original function on (, ). FIGURE Graph of the function in Problem 38. FIGURE Graph of the function in Problem et f first be etended into (, ) so that it is smmetric about = ; that is, so as to satisf f ( ) = f () for <. et the resulting function be etended into (, ) as an odd function and elsewhere (see Figure 1.4.7) as a periodic function of period 4. Show that this function has a Fourier series in terms of the functions sin(π /), sin(3π /), sin(5π /),... ; that is, where f () = b n = b n sin f () sin (n 1)π, (n 1)π This series converges to the original function on (, ]. 4. How should f, originall defined on [, ], be etended so as to obtain a Fourier series involving onl the functions cos(π /), cos(3π /), cos(5π /),...? Refer to Problems 38 and 39. If f () = for, sketch the function to which the Fourier series converges for 4 4. d.