By Kenneth Hoffman

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B) Show that f E L~(O, 21r). (c) Is f of bounded variation on (0, 21r)? Is it (a) Verify that L~(O, 21r)? (d) We wish to determine the Fourier series expansion of f: ;o + Lancosnx. f(x) = n=l Compute an, n 2: 1, by noticing that the integral In= 111" cot ~ sinnxdx does not depend on n (compute In- In-l)· (e) Determine the value of ao and prove that 00 L ~ cosnx =-In (2sin ~), x E (0, 21r). 8 Let x be a real parameter and Iet j(t) = f be defined by exeit. (a) What is the period of f? Show that Cn(f) {~n = n!

N as N--+ +oo. Hereis an example. Take f (t) = { +1 if 0 ~ t < 1f' -1 if 1f ~ t < 21f. By writing the exponentials in terms of sines, we have the following approximations for N = 1, 3, 5: 4 . sin3t); 3 1 . 1 . 4 . ;:(smt + "3 sm3t + "5 sm5t). 5. 3. fl(t) = ~sint. N tends to we have following important general result. f as N increases. 4), tends to otherwise, f in L~(O, a) as N--+ +oo. Expressed 32 Lesson 4. 4. fs(t) y = ~(sint+ isin3t). 5. j5(t) = ~(sint + i sin3t + i sin5t). The proof of this theorem requires more background than is available in these early lessons.

1. (d) Show that k 2: 2. (e) Compute the Fourier series of h and use it to determine the Fourier series of h for all k 2: 2. 10 Let f be the 27r-periodic function defined on [-7r,7r) by f(x) = cosh(ax), a > 0. (a) Show that the Fourier seriesoff converges uniformly to (b) Compute the expansion of f f. in a series of cosines. _ [coth(7ra)- 1l'a - 1~ L.... a 2 + n 2 2a a E R\{0}. n=l (d) Justify the term-by-term differentiation of the series for fand show that . n 2 sinh(a7r) ~( 1)n+1 . 7l' sm ax = xE(-7r,7r).