4. cosmt. 6. e tan 1rt 10. cos 2 t

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4 20 Chapter 1 Fourier Series of Periodic Functions 3. sin3t s. sinh2t 7. lsint l 9. t 2 4. cosmt 6. e 1 8. tan 1rt 10. cos 2 t 1.1B Sketch two or more periods of the following functions. 1. f(t) = t2, -1[ < t '.5c 1r, f(t + 21r) = f(t).2.f(t)=lt- 11, -1r'.5,t<1r, f(t+21r)=f(t) 3.f(t) = e 1, -1 < t '.5, 1, f(t +2) =f(t) 4.f(t)=l-2l tl, -1'.5,t<l, f(t+2)=f(t) 1.1c i Find a formula for the functionf(t) in Figure 1.3a that is valid (a) in the interval (1r,21r) and (b) in the interval (-2ir,-1r)..2. Find a single formula for the functionf(t) in Figure 1.3b that is valid in the intervals (-41r, -1r) and (1r, 21r). 1.1D Suppose thatf(t) has period L. What is the corresponding period off(at) where a is a positiv cqnstant? 1.1E Show that if a function f has p,eriod' L, then for n = 1, 2, 3,... 1.f(t + nl) = f(t) 2. fn(t) f(nt) also has period L. 1.1F If/and g are functions with period L, show that af,f + g, and/ g also have period L. (a is a constant.) 1.1G 1. If f is differentiable on R. with period L, how can we see graphically that tl;ie derivative/' also has period L? 2. Give a simple example of a periodic function f on R. for which F(x) = f o x f(t) dt is not periodic. 1.1H 1. Iff(x) = cos1rx and g(x) = cos../21rx, is the sum/+ g periodic? 2. If/ 1 is a periodic function with period L 1 and.fi is a periodic function with period L 2, under what conditions on L 1 and L 2 is the sum /1 + Ji a periodic function? Finite Representation; Orthogonality For n = 0, 1, 2,..., the trigonometric functions cos nt and sin nt have period 21r. Therefore, theirfinitelinear combinatiqns, including such expressions as 1 + cos t, sin 3t - ficos 17t, or V3 cos 2t -4 sin 3t - jcos4t, will also hav period 21r, ancl in this section, we will briefly consider such functions. Each must have the f onn N f(t) = + )c n cosnt+s n sinnt) (2) n=i for some choice of real coefficients c 0, c 1,..., c N, s 1,..., s N, where N is finite. (c 0 /2 is

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6 .2.2 Chapter 1 Fourier Series of Periodic Functions Also, 1 J" C Proof: For (4'), see Problem 1.2G. 2 N - J 2 (t)dt = + I)c,;+s,;). 7T -,r 2 n=i Remarks: The coefficient formulas (4) are credited to Euler (1777) and Fourier (1807) who derived them in a related context. These formulas are important in several respects. First, they establish our asserti.on that a given f has at most one representation of the form (2). For example, the right side of the identity sin 2 t =! -! cos 2t cannot be replaced by any other linear combination of cosnt and sinnt. Second, they sliow that for each N, the 2N + 1 functions 1, cost, cos2t,...,cosnt, sint,...,sinni are linearly independent. According to (4), their only linear combination f which is identically zero is that for which all coefficients are zero. Third, for each given!, the coefficients can be determined independently and they. characterize f If both heart-action functions of Figure 1.1 admit finite representation in the form (2), then they can be compared through their respective coefficients. Fourth, for each!, the values of the coefficients determined from (4) do not change with N. Therefore, f can be represented in form (2) only if the number of its coefficients with nonzero values is fini.te. &ample 1: (4') Figure 1.3b presents the graph of the periodic extension off(t) = It I, It I $ 71". Can this/be expressed in the form (2) for some finite integer N? Consider the e n integrals in (4). c 0 =.!.J" f(t) dt =.!. { J 0 -tdt + J" tdt} = 71". 7T -,r 7T -,r 0 However, since sin n7r = 0 and cos n1t = (-lf, then when n 1: 1 J" 1 { Jo 11t } e n = - f(t)cosntdt = - (-t)cosntdt+ tcosntdt π -ir 7T -ir o =.!_ {. (-t) &l n;i lo + J o sinnt dt + t &l n;i I " -1" sinnt dt}. 7T n o -ir -ir n n o o o n (5a) so that =.!_ { -cosnt, 0 + I"} = 3_(-lt-1, 7T n 2 -ir n 2 o 7T n 2 n = 1, 2,..., 0, n = 2, 4,... 7T n 2 7rn 2 1, n odd Cn = 2 (-1 t - 1 = _ { (5b)

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10 26 Chapter 1 Fourier Series of Periodic Functions represents f, we denote this relationship as follows: 00 f(t) '""F(t) = + L)cn cos nt + Sn sinnt). n=l Functions f exist for which the formal series F(t) fails to converge at many values of t, but fortunately such functions are rather complicated and do not arise in simple applications. (But see Problem I.SE.) On the other hand, the series can converge at t, with a sum F(t) 1-f(t), and this can occur with elementary functions. Example 2: Whenf(t) = e 1, -1r < t < 1r, then from (8),f has the Fourier coefficients I c 0 = - e I dt = --- J " e" - e-" 2 sinh 1r 7r 7r and, for n = 1,2,..., 1r -,r Cn} = J1r e 1 { C snt }dt = 2 sinh1r( l f { 1 Sn 1r -1r sm nt 1r( I + n -) -n These results can be obtained from partial integrations or with the aid of any integral table. In this case, we see that on the interval (-1r, 1r), t F( )-2. h { 1 oo L( l ) n(cosnt- nsinnt) } e rv t - -sm 7r ' 7r 2 n=l I + n 2 It I < 1r. Now, if e I and F( t) agree on this interval, they cannot agree for t > π since e I is strictly increasing, while F has period 21r. In fact, there arc only a few values oft (such as 0, ±1r) for which it is evident that the last series converges and none for which the sum is easily computed. Moreover, since F(-1r) = F(1r), while e-1r cfa e 1r, it is difficult to believe that this series could represent e ṭ We can appreciate the misgivings of some earlier mathematicians. However, we get some encouragement from the graph of the partial sum F 12 shown in Figure 1.5 because as you see, it does resemble the graph of f(t) = e t on (-1r, 1r), except for the strange oscillations near ±1r. Should those oscillations be there, or is this evidence of computer malfunction? Whenfis a polynomial, the calculation of its Fourier coefficients is facilitated by use of the following result (attributed to Kronecker): If Pm is a polynomial of degree ::; m, and g E C[a", 6], then [ Pm(t)g(t) dt = [pmg - p:,,gl + p::,g2 + + (-1ri;;; ) Gm](t) Ii, (8 1 ) where G is an integral of g, G1 is an integral of G, and so on, and the formula is obtained by successive partial integrations. For example, if f(t) = t 2, It I ::; 1r, then from (8) its nth sine coefficient is

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19 Section 1.4 Symmetry Considerations Which of the functions in Problem l.4a, restricted to the interval (-1r,1r), generates a sine series? a cosine series? 1.4C Find the even and odd parts of each of the functions in Problem l.4a. 1.4D Prove that on a common interval ( a, a) 1. the product of two odd functions is an even function. 2. the product of two even functions is an even function. 3. the product of an even function and an odd function is an odd function. 1.4E Obtain the Fourier series F generated by f defined on (-1r, 1r) when 1. f(t) = t 3 3. f(t) = { -1 1, { t 2 5. f(t) = ' 0, t < 0 t 2: 0 t 2: 0 t<o 2. f(t) = t 4 { 1, It I -5:. 7r /2 4. f(t) = 0, It I > 1r/2 6. j(t) = COS 2 t 1.4F Graph the 211"-periodic extension of each of the functions in Problem 1.4E, assumed defined on (-1r, 1r], and indicate all points of discontinuity. 1.4G Assume each function in Problem l.4e is defined only on (0, 1r). For each: 1. Graph the odd periodic extension and obtain a sine series of period 211". 2. Graph the even periodic extension and obtain a cosine series of period 21r. 1.4H For f(t) = 11" - t, 0 < t < 11", find 1.41 For f(t) = t(11" - t), 0 < t < 11", find 1. a sine series (with period 21r). 1. a sine series (with period 211"). 2. a cosine series (with period 21r). 2. a cosine series (with period 211"). 1.4] 1. Find a sine series for f(t) = cost on O < t < 11". 2. Find a cosine series for g(t) = sin ton O < t < 11". 1.4K Suppose thatf(t) is 211" periodic and also satisfiesf(t + 1r) = -f(t). 1. Show that the Fourier coefficients e n, s n are zero if n is even. 2. Show that when n is odd, then Cn = - 2J " J(t) cos nt dt, 11" 0 2J " S n = - J(t)sinntdt. 11" 0 1.4L 1. If g is integrable on (0, 1r) where g( t) = g( 11" - t), show that g generates a sine series of the form E:=1,3,s n sinnt, with s n = (4/1r) f 0,,.; 2 g(t) sinntdt, n = 1,3,5,... (Hint: Make a substitution to establish that f,,.j 2 g( 1r - t) sin nt dt = (-1 t + ' f0" 1 2 g(t) sin nt dt.) 2. What similar condition on g assumed integrable on (0, 7r) ensures that g generates a cosine series of the form E:=, 3 e n cos nt, with 1r/2 e n = (4/1r) fo g(t) cosnt dt, n = 1, 3, 5,...? ' '

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21 MATHEMATICS A ddressing both physical and mathematical aspects, this self-contained text on boundary value problems is geared toward advanced undergraduates and graduate students in mathematics. Prerequisites include some familiarity with multidimensional calculus and ordinary differential equations. A brief introductory section precedes the two-part treatment, which begins with coverage of basic theory. Topics include Fourier series of periodic functions, linear boundary value problems, diffusion problems related to the heat equation, steadystate problems involving the potential equation, and propagation problems incorporating the wave equation. The second part focuses on extensions, exploring general second-order linear equations and systems, series methods, integral transform methods, and Sturm Liouville problems. Problem sets appear throughout the text, along with a substantial section of answers to selected problems. This corrected edition includes a new Preface and an updated Appendix. Dover corrected and enlarged republication of the work originally published by the PWS Publishing Company, Boston, ISBN-13: ISBN-10: Cover design by John M. Alves $29.95 USA