Cauchy OK Dirichlet O{ Abel O{ ëY5 È©Ò È©Òe¦ §13.3 ¹ëCþ2ÂÈ© 13.3.1 Âñ59ÙO{ ù!·?ÚĹëCþ2ÂÈ©. (½å, ?Øäk áþÈ©. Ù(J±aíäkáe9Ã.¼êÈ©£=× È©¤. b¼ê f(x, u) 3 I = [a, +∞) × [α, β] þëY, eé?¿½ u ∈ [α, β], 2ÂÈ© Z +∞ a f(x, u)dx ÑÂñ, ¡¹ëCþ u 2ÂÈ©. ù«È©(½ «m [α, β] þ ¼ê u ∈ [α, β] 7−→ ϕ(u) = Z +∞ a f(x, u)dx, ·8Ò´ïÄùa¼êëY5!5ÚÈ5. 1/31 kJ Ik J I £ ¶ '4 òÑ
Cauchy OK Dirichlet O{ Abel O{ ëY5 È©Ò È©Òe¦ ½Â 1 XJ3¼ê ϕ(u) ¦é?¿½ê ε, oUé X (> a), b > X , Ø ª Z b a f(x, u)dx − ϕ(u) a), b > X , Ø ª Z +∞ b f(x, u)dx < ε é?¿ u ∈ [α, β] ¤á, K¡2ÂÈ© R +∞ a f(x, u)dx 3 [α, β] þ ñ. ùp [α, β] ±¤m«m½Ã¡«m. 2/31 kJ Ik J I £ ¶ '4 òÑ
Cauchy OK Dirichlet O{ Abel O{ ëY5 È©Ò È©Òe¦ âc½nØJy²e(J. ½n 2 á«mþ¹ëCþ2ÂÈ© R +∞ a f(x, u)dx ´Âñ¿ ©7^´ lim b→+∞ β(b) = 0 Ù¥ β(b) = sup u∈[α,β] Z +∞ b f(x, u)dx ~ 1 áȩ Z +∞ 1 1 u + x2 dx 3 [0, +∞) þÂñ. ù´Ï Z +∞ b 1 u + x2 dx 6 Z +∞ b 1 x2 dx = 1 b → 0, (b → +∞). 3/31 kJ Ik J I £ ¶ '4 òÑ
Cauchy OK Dirichlet O{ Abel O{ ëY5 È©Ò È©Òe¦ ½n 3 (Cauchy OK) È© R +∞ a f(x, u)dx 3«m [α, β] þÂñ ¿©7^´: é?¿½ê ε, o3= ε k'ê B, ¦ b1, b2 > B, ÒkØ ª Z b2 b1 f(x, u)dx < ε é«m [α, β] þ u ¤á. ½n 4 (WeierstrassO{) f(x, u) 3« I = [a, +∞) × [α, β] þëY. XJ33 [a, +∞) þ2Âȼê p(x), ¦éu¿© x ±9 [α, β] þ?¿ u Ñk |f(x, u)| 6 p(x), KÈ© R +∞ a f(x, u)dx 3 [α, β] þÂñ. p(x) ¡§¼ê. 4/31 kJ Ik J I £ ¶ '4 òÑ
Cauchy OK Dirichlet O{ Abel O{ ëY5 È©Ò È©Òe¦ ~ 2 ïÄÈ© Z +∞ 1 sin ux x2 dx 3 −∞ 1 , k sin ux x2 6 1 x2 , È© Z +∞ 1 dx x2 Âñ, d'O{È©'u u 3¢¶þÂñ. 5/31 kJ Ik J I £ ¶ '4 òÑ
Cauchy OK Dirichlet O{ Abel O{ ëY5 È©Ò È©Òe¦ ~ 3 y² α > 0 , È© Z +∞ 0 x sin βx α2 + x2 dx 3 β > β0 > 0 þÂ ñ. y² d©ÜÈ©{, Z +∞ b x sin βx α2 + x2 dx = − x cos βx β(α2 + x2) +∞ b + 1 β Z +∞ b cos βxd x α2 + x2 = b cos bβ β(α2 + b 2) + 1 β Z +∞ b cos βx α2 − x 2 (α2 + x2) 2 dx, d b cos bβ β(α2 + b 2) 6 1 bβ0 9 cos βx β α2 − x 2 (α2 + x2) 2 6 1 β0(α2 + x2) β0 > 0 þÂñ. 6/31 kJ Ik J I £ ¶ '4 òÑ
Cauchy OK Dirichlet O{ Abel O{ ëY5 È©Ò È©Òe¦ ~ 4 0 0 Âñ. y² ? 0 6 u 0, Z +∞ b p e −ux cos x xp+1 dx 6 Z +∞ b p e −bu xp+1 dx 6 e −bu b p . 7/31 kJ Ik J I £ ¶ '4 òÑ
Cauchy OK Dirichlet O{ Abel O{ ëY5 È©Ò È©Òe¦ Z +∞ b e −ux sin x xp dx 6 3 e −bu b p 6 3 b p . ¤±, ? ε > 0, b ¿©, é?¿ 0 6 u 0 Âñ. 8/31 kJ Ik J I £ ¶ '4 òÑ
Cauchy OK Dirichlet O{ Abel O{ ëY5 È©Ò È©Òe¦ ~ 5 y²È© Z +∞ a ue−uxdx 3 0 6 u 0 Âñ. Ï Z +∞ b ue−uxdx = 0, u = 0; e −ub, u > 0, ¤± lim b→+∞ β(b) = lim b→+∞ sup u>0 Z +∞ b ue−uxdx = 1 6= 0, ¤È©ØÂñ. 9/31 kJ Ik J I £ ¶ '4 òÑ
Cauchy OK Dirichlet O{ Abel O{ ëY5 È©Ò È©Òe¦ ~ 6 y²È© Z +∞ 0 sin xu x dx 3 0 0 β(b) = sup u>0 Z +∞ b sin xu x dx = sup u>0 Z +∞ bu sin x x dx > Z +∞ 0 sin x x dx = π 2 6→ 0 (b → +∞), ¤± Z +∞ 0 sin xu x dx 3 0 < u < +∞ þØÂñ. 10/31 kJ Ik J I £ ¶ '4 òÑ