Dirichlet È© Laplace È© Fresnel È© §13.4 ¹ëCþÈ©A^ 13.4.1 A2ÂÈ© 1 ◦ Dirichlet È© Z +∞ 0 sin x x dx. d Dirichlet O{ùÈ©Âñ, ØýéÂñ. Ú?ÂñÏf e −ux , ¿Ä¹ëCþÈ© I(u) = Z +∞ 0 e −ux sin x x dx. du u > 0, dÈ©´Âñ. ȼê3« [0, +∞)×[0, +∞) þëY, Ï I(u) Ò3 [0, +∞) þëY. AO3: u = 0 ëY, í lim u→0+ I(u) = I(0) = Z +∞ 0 sin x x dx. 1/15 kJ Ik J I £ ¶ '4 òÑ
Dirichlet È© Laplace È© Fresnel È© ,¡, ò I(u) ûq I 0 (u) = − Z +∞ 0 e −ux sin xdx, Ù¥3È©Òeé u ûÜn5´Ï u > u0 > 0 , È© Z +∞ 0 e −ux sin xdx ´Âñ. d©ÜÈ©{ I 0 (u) = e −ux cos x +∞ 0 + u Z +∞ 0 e −ux cos x dx = −1 + u e −ux sin x +∞ 0 + u Z +∞ 0 e −ux sin x dx = −1 − u 2 I 0 (u), l I 0 (u) = − 1 1 + u2 , 2/15 kJ Ik J I £ ¶ '4 òÑ
Dirichlet È© Laplace È© Fresnel È© ü>È©¦ I(u) = − arctan u + c, u > 0 , ·k |I(u)| = Z +∞ 0 e −ux sin x x dx 6 Z +∞ 0 e −uxdx = 1 u , u → +∞ I(u) → 0, dd½Ñ~ê c = π 2 . k I(u) = π 2 − arctan u, (u > 0). - u ªu"= Z +∞ 0 sin x x dx = I(0) = π 2 . 5 '?ê X ∞ n=1 sin n n = π − 1 2 . 3/15 kJ Ik J I £ ¶ '4 òÑ
Dirichlet È© Laplace È© Fresnel È© 2 ◦ Laplace È© I(β) = Z +∞ 0 cos βx α2 + x2 dx (α > 0, β > 0), J(β) = Z +∞ 0 x sin βx α2 + x2 dx (α > 0, β > 0). Ïé?¿ β > 0 Ú α > 0, k | cos βx| α2 + x2 6 1 α2 + x2 , I(β) 3 β ∈ [0, +∞) þÂñ. , , d Dirichlet O{, J(β) é?¿ β > β0 > 0 Âñ. w, I(β) J(β) Ñ3 β > β0 > 0 þÂñ. dÈ©Òe¦5½n , I(β) é β û3È©Òe?1, ¿k I 0 (β) = Z +∞ 0 ∂ ∂β cos βx α2 + x2 dx = − Z +∞ 0 x sin βx α2 + x2 dx = −J(β). 4/15 kJ Ik J I £ ¶ '4 òÑ
Dirichlet È© Laplace È© Fresnel È© β > 0 , k π 2 = Z +∞ 0 sin βx x dx, u´k I 0 (β) + π 2 = − Z +∞ 0 x sin βx α2 + x2 dx + Z +∞ 0 sin βx x dx = α 2 Z +∞ 0 sin βx x(α2 + x2) dx, þªqé β 3È©Òe¦û, u´qk I 00(β) = α 2 Z +∞ 0 cos βx α2 + x2 dx = α 2 I(β), ù´ ~Xê5©§, ¦Ï) I(β) = c1e αβ + c2e −αβ , 5/15 kJ Ik J I £ ¶ '4 òÑ
Dirichlet È© Laplace È© Fresnel È© Ù¥ c1, c2 ?¿~ê. dué β > 0 k |I(β)| 6 Z +∞ 0 dx α2 + x2 = π 2α , I(β) k., qÏ α > 0, ¤± c1 7L", k I(β) = c2e −αβ . 5¿, d, $Ñ´3 β > 0 be?1. e¡5(½ c2 , duÈ© I(β) 3 β ∈ [0, +∞) þÂñ, I(β) 3 [0, +∞) þëY, AO3 β = 0 ?mëY, u´k c2 = lim β→0+ I(β) = I(0) = Z +∞ 0 dx α2 + x2 = π 2α . I(β) = Z +∞ 0 cos βx α2 + x2 dx = π 2α e −αβ (α > 0, β > 0). 6/15 kJ Ik J I £ ¶ '4 òÑ
Dirichlet È© Laplace È© Fresnel È© , é α > 0, β > 0, k J(β) = −I 0 (β) = π 2 e −αβ , u´ Z +∞ 0 x sin βx α2 + x2 dx = π 2 e −αβ (α > 0, β > 0). 7/15 kJ Ik J I £ ¶ '4 òÑ
Dirichlet È© Laplace È© Fresnel È© ~ 1 ¦È© I(α) = Z +∞ 0 sin2 αx 1 + x2 dx. ) Ï sin2 αx = 1−cos 2αx 2 , ¤± I(α) = 1 2 Z +∞ 0 1 − cos 2αx 1 + x2 dx = 1 2 Z +∞ 0 1 1 + x2 dx − 1 2 Z +∞ 0 cos 2αx 1 + x2 dx = π 4 − 1 2 · π 2 e −2α = π 4 1 − e −2α . 8/15 kJ Ik J I £ ¶ '4 òÑ
Dirichlet È© Laplace È© Fresnel È© 3 ◦ Fresnel È© Z +∞ 0 sin x 2dx, Z +∞ 0 cos x 2dx. È©C Z +∞ 0 sin x 2dx = 1 2 Z +∞ 0 sin t √ t dt, TÈ©´^Âñ. t > 0 , d 1 √ t = 2 √ π Z +∞ 0 e −tu2 du, Z +∞ 0 sin t √ t e −vtdt = 2 √ π Z +∞ 0 e −vt sin tdt Z +∞ 0 e −tu2 du, È©gSÒ Z +∞ 0 sin t √ t e −vtdt = 2 √ π Z +∞ 0 du Z +∞ 0 e −(u 2+v)t sin tdt = 2 √ π Z +∞ 0 du 1 + (u2 + v) 2 . 9/15 kJ Ik J I £ ¶ '4 òÑ
Dirichlet È© Laplace È© Fresnel È© ª Z +∞ 0 sin t √ t e −vtdt = 2 √ π Z +∞ 0 du 1 + (u2 + v) 2 . màÈ©'u v 3 [0, +∞) þÂñ, àÈ©'u v 3 [0, +∞) þÂñ. Ïd v → 0 + , ±3 ªüàÈ©Òe4, =k Z +∞ 0 sin t √ t dt = 2 √ π Z +∞ 0 du 1 + u4 = 2 √ π π 2 √ 2 = r π 2 . ¤± Z +∞ 0 sin x 2dx = r π 8 . aq Z +∞ 0 cos x 2dx = r π 8 . Iyþ¡A©gSÜn5. 10/15 kJ Ik J I £ ¶ '4 òÑ