Dirichlet Âñ½n {u?ê u?ê Eê/ª 12.1.3 ¼ê Fourier ?êÐm ½n 1 (Dirichlet) ¼ê f(x) ± 2π ±Ï, 1 ◦ XJ¼ê3?Ûk«mþ´Åã1w, K§ Fourier ?ê3 ê¶þÑÂñ, a0 2 + X ∞ n=1 (an cos nx + bn sin nx) = f(x + 0) + f(x − 0) 2 ; 2 ◦ XJ¼ê??ëY, 3?Ûk«mþ´Åã1w, KÙ Fourier ? êÒ3ê¶þýéÂñu f(x). 5, ùp¤¢¼ê f(x) 3k«mþÅã1w´¼êØk: , f(x) ëY këYû f 0 (x), ùk:U´ f(x) ½ f 0 (x) 1 amä:. 1/18 kJ Ik J I £ ¶ '4 òÑ
Dirichlet Âñ½n {u?ê u?ê Eê/ª ~ 1 f(x) = ( 0, −π 6 x < 0 x, 0 6 x < π âc¡~f, k f(x) ∼ π 4 + X ∞ n=1 (−1)n − 1 n2π cos nx + (−1)n+1 n sin nx . (1) dÂñ5½n, 3«m (−π, π) ¥þªA ª. x = 0, π 4 + X ∞ n=1 (−1)n − 1 n2π = 0, =, X ∞ n=1 1 (2n − 1)2 = π 2 8 . 2/18 kJ Ik J I £ ¶ '4 òÑ
Dirichlet Âñ½n {u?ê u?ê Eê/ª ?Ú, k X ∞ n=1 1 n2 = X ∞ n=1 1 (2n − 1)2 + X ∞ n=1 1 (2n) 2 = π 2 8 + 1 4 X ∞ n=1 1 n2 . u´k X ∞ n=1 1 n2 = π 2 6 . qÏ X ∞ n=1 (−1)n−1 n2 ýéÂñ, ¤± X ∞ n=1 (−1)n−1 n2 = X ∞ n=1 1 (2n − 1)2 − X ∞ n=1 1 (2n) 2 = π 2 8 − 1 4 X ∞ n=1 1 n2 = π 2 8 − 1 4 · π 2 6 = π 2 12 . 3/18 kJ Ik J I £ ¶ '4 òÑ
Dirichlet Âñ½n {u?ê u?ê Eê/ª 3 (1) ¥ x = π 2 π 4 + X ∞ n=1 (−1)n − 1 n2π cos nπ 2 + (−1)n−1 n sin nπ 2 = π 2 , Ï X ∞ n=1 1 2n − 1 sin (2n − 1)π 2 = π 4 , = X ∞ n=1 (−1)n−1 2n − 1 = π 4 . 4/18 kJ Ik J I £ ¶ '4 òÑ
Dirichlet Âñ½n {u?ê u?ê Eê/ª f(x) ´ [−π, π] þó¼ê, f(x) ± 2π ±Ïòÿ (−∞, +∞) ´ó¼ê, d, bn = 1 π Z π −π f(x) sin nx dx = 0 (n = 1, 2, · · ·). Ïd f(x) Fourier ?ê a0 2 + X ∞ n=1 an cos nx, Ù¥ an = 2 π Z π 0 f(x) cos nx dx. ù«/ªn?ê¡{u?ê. 5/18 kJ Ik J I £ ¶ '4 òÑ
Dirichlet Âñ½n {u?ê u?ê Eê/ª Ón, f(x) ´ [−π, π] þÛ¼ê, f(x) ± 2π ±Ïòÿ (−∞, +∞) ¦¤Û¼ê, d, an = 1 π Z π −π f(x) cos nx dx = 0 (n = 0, 1, 2, · · ·). Ïd f(x) Fourier ?ê X ∞ n=1 bn sin nx, Ù¥ bn = 2 π Z π 0 f(x) sin nx dx. ù«/ªn?ê¡u?ê. 6/18 kJ Ik J I £ ¶ '4 òÑ
Dirichlet Âñ½n {u?ê u?ê Eê/ª ~ 2 3 [−π, π]þ, f(x) = |x|, §òÿ¤þ± 2π ± Ï±Ï¼ê, ´ëY. Ïdâ Fourier XêOúª bn = 0, a0 = 1 π Z π −π f(x)dx = 2 π Z π 0 xdx = π, an = 1 π Z π −π f(x) cos nxdx = 2 π Z π 0 x cos nxdx = 2 n2π [(−1)n − 1] = − 4 (2k−1)2π , n = 2k − 1. 0, n = 2k; ¤±§Ðmª¹{u¼ê, Ðmª´Âñ. Ïd |x| = π 2 − 4 π X ∞ k=1 1 (2k − 1)2 cos(2k − 1)x, (−∞ < x < +∞). 7/18 kJ Ik J I £ ¶ '4 òÑ
Dirichlet Âñ½n {u?ê u?ê Eê/ª ~ 3 ±Ï 2π 縼ê f(x) = ( 1 2 (π − x), 0 < x 6 2π f(x − 2nπ), 2nπ < x 6 2(n + 1)π, n ´ê Fourier ?ê f(x) ∼ X ∞ n=1 sin nx n . ¤± 1 2 (π − x) = X ∞ n=1 sin nx n , (0 < x < 2π). 8/18 kJ Ik J I £ ¶ '4 òÑ
Dirichlet Âñ½n {u?ê u?ê Eê/ª ~ 4 ¦¼ê f(x) = cos αx (0 < α < 1) 3«m [−π, π] þ Fourier ? êÐm. ) du cos αx ´ó¼ê, § Fourier ?ê{u?ê. a0 = 2 π Z π 0 cos αx dx = 2 sin απ πα , an = 2 π Z π 0 cos αx cos nx dx = sin απ π (−1)n 1 α + n + 1 α − n (n = 1, 2, · · ·). u´ |x| 6 π , k cos αx = sin απ π " 1 α + X ∞ n=1 (−1)n 1 α + n + 1 α − n cos nx# . - x = 0, 1 α + X ∞ n=1 (−1)n 1 α + n + 1 α − n = π sin απ . 9/18 kJ Ik J I £ ¶ '4 òÑ
Dirichlet Âñ½n {u?ê u?ê Eê/ª k«mþ Fourier ?ê 1) e f(x) 3 [−`, `) þk½Â, K± 2` ±Ïòÿ (−∞, +∞) þ, f(x) Fourier ?ê f(x) ∼ a0 2 + X ∞ n=1 an cos nπ ` x + bn sin nπ ` x , Ù¥ an = 1 ` Z ` −` f(x) cos nπ ` x dx, (n = 0, 1, · · ·) bn = 1 ` Z ` −` f(x) sin nπ ` x dx, (n = 1, · · ·). 5 ¼ê g(t) = f( `t π ) Fourier ?ê. 10/18 kJ Ik J I £ ¶ '4 òÑ