Lista de Exercícios Resolvidos - Séries de Fourier e Funções Periódicas

Lista 11 Séries de Fourier extraídos do livro de Boyce e DiPrima Problems In each of Problems 1 through 8 determine whether the given function is periodic If so find its fundamental period 1 sin5x 2 cos2πx 3 sinh2x 4 sinπx L 5 tanπx 6 x² 7 fx 0 2n 1 x 2n 1 2n x 2n 1 n 0 1 2 8 fx 1ⁿ 2n 1 x 2n 1 2n x 2n 1 n 0 1 2 9 If fx x for L x L and if fx 2L fx find a formula for fx in the interval L x 2L in the interval 3L x 2L 10 If fx x 1 1 x 0 x 0 x 1 and if fx 2 fx find a formula for fx in the interval 1 x 2 in the interval 8 x 9 11 If fx L x for 0 x 2L and if fx 2L fx find a formula for fx in the interval L x 0 12 Verify equations 6 and 7 in this section by direct integration In each of Problems 13 through 18 a Sketch the graph of the given function for three periods b Find the Fourier series for the given function 13 fx x L x L fx 2L fx 14 fx 1 L x 0 0 0 x L fx 2L fx 15 fx x π x 0 0 0 x π fx 2π fx 16 fx x 1 1 x 0 1 x 0 x 1 fx 2 fx 17 fx x L L x 0 L 0 x L fx 2L fx 18 fx 0 2 x 1 x 1 x 1 0 1 x 2 fx 4 fx In each of Problems 19 through 24 a Sketch the graph of the given function for three periods b Find the Fourier series for the given function c Plot the partial sum sₘx versus x for m 5 10 and 20 d Describe how the Fourier series seems to be converging 19 fx 1 2 x 0 1 0 x 2 fx 4 fx 20 fx x 1 x 1 fx 2 fx 21 fx x²2 2 x 2 fx 4 fx 22 fx x 2 2 x 0 2 2x 0 x 2 fx 4 fx 23 fx 12 x 2 x 0 12x 12 x² 0 x 2 fx 4 fx 24 fx 0 3 x 0 x²3 x 0 x 3 fx 6 fx 25 Consider the function f defined in Problem 21 and let eₘx fx sₘx a Plot eₘx versus x for 0 x 2 for several values of m b Find the smallest value of m for which eₘx 001 for all x 26 Consider the function f defined in Problem 24 and let eₘx fx sₘx a Plot eₘx versus x for 0 x 3 for several values of m b Find the smallest value of m for which eₘx 01 for all x 27 Suppose that g is an integrable periodic function with period T a If 0 a T show that ₀ᵀ gx dx ₐᵃᵀ gx dx Hint Show first that ₀ᵃ gx dx ₜᵃ gx dx Then in the second integral consider the change of variable s x T b Show that for any value of a not necessarily in 0 a T ₀ᵀ gx dx ₐᵃᵀ gx dx c Show that for any values of a and b ₐᵃᵀ gx dx ᵦᵇᵀ gx dx 28 If f is differentiable and is periodic with period T show that f is also periodic with period T Determine whether Fx ₀ˣ ft dt is always periodic