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Homework answers / question archive / 2) Let (x) = x2 for 0 < x 0) (b) eax (a > 0) (c) X" (m (m = integer) (d) tan x2 (e) (sin(x/b)| (b > 0) (f) x cos ax (a > 0) 2
2) Let (x) = x2 for 0 < x 0) (b) eax (a > 0) (c) X" (m (m = integer) (d) tan x2 (e) (sin(x/b)| (b > 0) (f) x cos ax (a > 0) 2. Show that cos x + cos ax is periodic if a is a rational number. What is its period? 3. Prove property (5) concerning the integrals of even and odd functions. 4. (a) Use (5) to prove that if $(x) is an odd function, its full Fourier series on (-1,1) has only sine terms. (b) Also, if $(x) is an even function, its full Fourier series on (-1,1) has only cosine terms. (Hint: Don't use the series directly. Use the formulas for the coefficients to show that every second coefficient vanishes.) 5. Show that the Fourier sine series on (0, 1) can be derived from the full Fourier series on (-1,1) as follows. Let $(x) be any (continuous) function on (0,1). Let õ(x) be its odd extension. Write the full series for ?(x) on (-1,1). [Assume that its sum is (x).] By Exercise 4, this series has only sine terms. Simply restrict your attention to 0) < x < 1 to get the sine series for $(x). Show that the cosine series on (0, 1) can be derived from the full series on (-1, 1) by using the even extension of a function. 7. Show how the full Fourier series on (-1,1) can be derived from the full series on (-1, 1) by changing variables w = (1/1)x. (This is called a change of scale; it means that one unit along the x axis becomes all units along the w axis.) 10.) (a) Let $(x) be a continuous function on (0, 1). Under what conditions is its odd extension also a continuous function? (b) Let $(x) be a differentiable function on (0,1). Under what conditions is its odd extension also a differentiable function? (c) Same as part (a) for the even extension. (d) Same as part (b) for the even extension. 118 CHAPTER 5 FOURIER SERIES 11. Find the full Fourier series of et on (-1,1) in its real and complex forms. (Hint: It is convenient to find the complex form first.) 12. Repeat Exercise 11 for cosh x. (Hint: Use the preceding result.) 13. Repeat Exercise 11 for sin x. Assume that I is not an integer multiple of 1. (Hint: First find the series for eix). 14. Repeat Exercise 11 for |x|. 15. Without any computation, predict which of the Fourier coefficients of sin x| on the interval (-1, ) must vanish. 16. Use the De Moivre formulas (11) to derive the standard formulas for cos(0 + 0) and sin(0 + 0). 17.) Show that a complex-valued function f(x) is real-valued if and only if its complex Fourier coefficients satisfy on=C-n, where denotes the complex conjugate. L (odd) dx = 0 and La (even) dx = 2 26 (even) dx. (5)
