Rót gån hardy cho một lớp tích phân liouville các hàm số sơ cấp

❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ◗❯❨ ◆❍❒◆ P❍❸▼ ❚❍➚ ❉➚❯ ❍■➋◆ ❘Ó❚ ●➴◆ ❍❆❘❉❨ ❈❍❖ ▼❐❚ ▲❰P ❚➑❈❍ P❍❹◆ ▲■❖❯❱■▲▲❊ ❈⑩❈ ❍⑨▼ ❙➮ ❙❒ ❈❻P ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❇➻♥❤ ✣à♥❤ ✲ ◆➠♠ ✷✵✷✶ ❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ◗❯❨ ◆❍❒◆ P❍❸▼ ❚❍➚ ❉➚❯ ❍■➋◆ ❘Ó❚ ●➴◆ ❍❆❘❉❨ ❈❍❖ ▼❐❚ ▲❰P ❚➑❈❍ P❍❹◆ ▲■❖❯❱■▲▲❊ ❈⑩❈ ❍⑨▼ ❙➮ ❙❒ ❈❻P ❈❍❯❨➊◆ ◆●⑨◆❍✿ P❍×❒◆● P❍⑩P ❚❖⑩◆ ❙❒ ❈❻P ▼❶ ❙➮✿ ✽ ✹✻ ữớ ữợ ❞➝♥✿ P●❙✳❚❙✳ ❚❍⑩■ ❚❍❯❺◆ ◗❯❆◆● ❇➻♥❤ ✣à♥❤ ✲ ✷✵✷✶ ❉❆◆❍ ▼Ư❈ ❈⑩❈ ❑Þ ❍■➏❯ R ✿ ❱➔♥❤ ✈✐ ♣❤➙♥ Q(x) rữớ tự ợ số ỳ t R(x) rữớ tự ợ số tỹ C(x) rữớ tự ợ số ự (2j − 1)!! = 1.3.5 (2j − 1) (2j)!! = 2.4 (2j) (3j − 2)!!! = 1.4.7 (3j − 2) (3j − 1)!!! = 2.5.8 (3j − 1) ử ỵ ❈⑩❈ ❍⑨▼ ❙➮ ❙❒ ❈❻P ❱⑨ ✣➚◆❍ ▲Þ ▲■❖❯❱■▲▲❊ ✶✳✶ ✶✳✷ ❱➔♥❤ ✈➔ tr÷í♥❣ ✈✐ ♣❤➙♥ ✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✶✳✶ ❈→❝ ❦❤→✐ ♥✐➺♠ ✈➔ t➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✶✳✷ ▼ð rë♥❣ ❧♦❣❛r✐t ✈➔ ♠ð rë♥❣ ♠ô ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ số sỡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✷✳✸ ▼ët sè ✈➼ ❞ö →♣ ❞ö♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷ ❘Ó❚ ●➴◆ ❍❆❘❉❨ ❈❍❖ ▲❰P ❚➑❈❍ P❍❹◆ ▲■❖❯❱■▲▲❊ ✷✳✶ ▼ët sè ❦➳t q✉↔ ❝❤✉➞♥ ❜à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷ ❘ót ❣å♥ ❍❛r❞② ❝❤♦ t➼❝❤ ♣❤➙♥ ▲✐♦✉✈✐❧❧❡ ✷✺ ✷✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✷✳✶ ▼ët sè ❦➳t q✉↔ ✈➲ rót ❣å♥ ❍❛r❞② ✷✳✷✳✷ ❈→❝ ❤➺ q✉↔ ✷✳✷✳✸ ❈→❝ ✈➼ ❞ö ✸ ▼❐❚ ❙➮ ⑩P ❉Ö◆● ✸✽ ✸✳✶ ▼ët sè ❞↕♥❣ t➼❝❤ ♣❤➙♥ ▲✐♦✉✈✐❧❧❡ ✤➦❝ ❜✐➺t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✸✳✷ ❈→❝ t➼❝❤ ♣❤➙♥ ❑✐➸✉ ▲✐♦✉✈✐❧❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✹✻ ✶ ▼Ð ✣❺❯ ❱✐➺❝ ❦➳t ❧✉➟♥ ♠ët t➼❝❤ ♣❤➙♥ ❝õ❛ ♠ët ❤➔♠ ❝➜♣ ❝â ❝á♥ ❧➔ ♠ët ❤➔♠ sè ❝➜♣ ❤❛② ❦❤ỉ♥❣ ❧➔ ♠ët ❝➙✉ ❤ä✐ q✉❛♥ trå♥❣ ✤➣ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ tø t❤í✐ ◆❡✇t♦♥ ✈➔ ▲❡✐❜♥✐③✳ P❤➛♥ ❧ỵ♥ ❞ü❛ tr➯♥ ❝→❝ ❝ỉ♥❣ tr➻♥❤ ❝õ❛ ▲✐♦✉✈✐❧❧❡ ❬✶✵❪✱ ❘✐s❝❤ ❬✶✸❪✱ ✈➔ ❘♦s❡♥t❧✐❝❤t ❬✶✹❪✱ r➜t ♥❤✐➲✉ t✐➳♥ ❜ë ✤➣ ✤↕t ✤÷đ❝ ✈➲ ✈➜♥ ✤➲ ♥➔② tr♦♥❣ s✉èt ❤❛✐ t❤➳ ❦✛ ❬✶✱ ✷✱ ✸✱ ✹✱ ✼✱ ✾✱ ✶✻✱ ✶✼❪✳ ❚✉② ♥❤✐➯♥✱ ❝â ♠ët sè ❧ỵ♣ t➼❝❤ ♣❤➙♥ r➜t ✤÷đ❝ ❤å❝ s✐♥❤✱ s✐♥❤ ✈✐➯♥ q✉❛♥ t➙♠ t➼♥❤ t♦→♥ ♥❤÷♥❣ ✈➝♥ ❝❤÷❛ ❝â ❝➙✉ tr↔ ❧í✐ ❤♦➔♥ t♦➔♥ ✤➛② ✤õ ❝❤♦ ❝➙✉ ❤ä✐ ♥➔②✳ ▼ët ✈➼ ❞ö tr♦♥❣ sè ♥➔② ❧➔ ❧ỵ♣ ❝→❝ t➼❝❤ ♣❤➙♥ ❝â ❞↕♥❣ s xr eax dx, ✈ỵ✐ r, s ❧➔ ❝→❝ sè ♥❣✉②➯♥✳ ợ t ố tữủ ❝ù✉ tr♦♥❣ ♠ët tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t s❛✉ ✤➙② ❝õ❛ ởt ỵ ỵ ✭❚✐➯✉ ❝❤✉➞♥ ▲✐♦✉✈✐❧❧❡ ✤è✐ ✈ỵ✐ t➼❝❤ ♣❤➙♥✱ ✶✽✸✺✮✳ ❤ú✉ t✛ ✈ỵ✐ g ❈❤♦ f, g ❧➔ ❝→❝ ❤➔♠ sè ❦❤→❝ ❤➡♥❣ sè✳ ❑❤✐ ✤â f (x)eg(x) dx ❧➔ ♠ët ❤➔♠ số sỡ tỗ t ởt ❤➔♠ sè ❤ú✉ t✛ R(x)eg(x) , R s❛♦ ❝❤♦ f (x)eg(x) dx = ❤♦➦❝ ♠ët ❝→❝❤ t÷ì♥❣ ✤÷ì♥❣ f (x) = R(x)g (x) + R (x) ❚✐➯✉ ❝❤✉➞♥ ♥➔② t❤÷í♥❣ ✤÷đ❝ sû ❞ư♥❣ ✤➸ ❝❤♦ ❤å❝ s✐♥❤ t➼♥❤ t♦→♥ ✈➔ ♥❤➟♥ ❜✐➳t r➡♥❣ ♠ët sè ❧ỵ♣ ♥❤➜t ✤à♥❤ t➼❝❤ ♣❤➙♥ ❝ê ✤✐➸♥ ❝❤➥♥❣ ❤↕♥ ♥❤÷ ❡r❢(x) =√ π x e−u du, x ▲✐(x) = du , ln u x ❙✐(x) = sin u , u ổ t ữủ t ữợ ❤➔♠ ❝➜♣✳ ❚✉② ♥❤✐➯♥✱ ❜➜t ❝❤➜♣ ✈❛✐ trá t❤✐➳t ②➳✉ ❝õ❛ ♥â tr♦♥❣ ✈✐➺❝ ①→❝ ✤à♥❤ ✤➦❝ t➼♥❤ ✏❦❤æ♥❣ ❝➜♣✑ ❝õ❛ ❝→❝ t➼❝❤ ♣❤➙♥ q✉❛♥ trå♥❣ tr♦♥❣ ❝→❝ ù♥❣ ❞ö♥❣✱ ♠ù❝ ✤ë ❧✐➯♥ q✉❛♥ ❝õ❛ ❦➳t q✉↔ ♥➔② tr t ố t ữủ ợ tr♦♥❣ ♠ët ✈➔✐ ❧ỵ♣ ❝♦♥ ❝õ❛ ❧ỵ♣ t➼❝❤ ♣❤➙♥ ▲✐♦✉✈✐❧❧❡ ❬✶✶✱ ✶✷✱ ✶✺❪✳ ❈❤õ ✤➲ ❝õ❛ ▲✉➟♥ ✈➠♥ ❧✐➯♥ q✉❛♥ ✤➳♥ ❝➙✉ tr↔ ❧í✐ ❝❤♦ ❝➙✉ ❤ä✐ ♥➯✉ tr➯♥ ✤è✐ ✈ỵ✐ ❧ỵ♣ t➼❝❤ ♣❤➙♥ ▲✐♦✉✈✐❧❧❡✳ ✣â ❧➔ ❧ỵ♣ ❝→❝ t➼❝❤ ♣❤➙♥ ❝õ❛ ❤➔♠ sè ❝â ❞↕♥❣ ❧➔ ❝→❝ ❤➔♠ sè ❤ú✉ t✛✱ g f (x)eg(x) dx tr♦♥❣ ✤â f, g ❦❤ỉ♥❣ ❧➔ ❤➔♠ ❤➡♥❣✳ ▼ư❝ t✐➯✉ ❝õ❛ ▲✉➟♥ ✈➠♥ ❧➔ t➟♣ tr✉♥❣ ❣✐↔✐ q✉②➳t ❝→❝ ❜➔✐ t♦→♥ s❛✉✿ ✶✳ ❉ü❛ ✈➔♦ t✐➯✉ ❝❤✉➞♥ ▲✐♦✉✈✐❧❧❡ ♥❣❤✐➯♥ ❝ù✉ ✤÷❛ r❛ ♠ët t❤✉➟t t rút r t ữợ t➼❝❤ ♣❤➙♥ t❤➔♥❤ ❤❛✐ t❤➔♥❤ ♣❤➛♥ ❝ì ❜↔♥ ✏❝ü❝ ✤↕✐✑ ✈➔ ✏❝ü❝ t✐➸✉✑ s❛♦ ❝❤♦ ♣❤➙♥ t➼❝❤ ♥➔② ❝â t❤➸ ự ỵ tt rút r ✤➸ ①→❝ ✤à♥❤ ❧✐➺✉ ❝→❝ t➼❝❤ ♣❤➙♥ ✤â ❝â ♣❤↔✐ ❧➔ ❤➔♠ ❝➜♣ ❤❛② ❦❤ỉ♥❣✳ ✷✳ ◆❣❤✐➯♥ ❝ù✉ ♠ët sè →♣ ❞ö♥❣ ❝õ❛ t❤✉➟t t♦→♥✳ ◆❣♦➔✐ ♣❤➛♥ ▼ð ✤➛✉✱ ❑➳t ❧✉➟♥✱ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✱ ▲✉➟♥ ✈➠♥ ✤÷đ❝ ❝❤✐❛ t❤➔♥❤ ❜❛ ❝❤÷ì♥❣✳ ❈❤÷ì♥❣ ✶ ❞➔♥❤ ❝❤♦ ✈✐➺❝ t➻♠ ❤✐➸✉ ỵ tờ qt tr ởt trữớ ✈➔ ♠ët sè ❤➺ q✉↔ ✤➦❝ ❜✐➺t ❝õ❛ ♥â ✤è✐ ợ số sỡ r ữỡ ú tổ t➟♣ tr✉♥❣ ♥❣❤✐➯♥ ❝ù✉ ✤÷❛ r❛ ♠ët t❤✉➟t t♦→♥ ✤➸ t ữợ t t t ♣❤➛♥ ❝ì ❜↔♥ ✏❝ü❝ ✤↕✐✑ ✈➔ ✏❝ü❝ t✐➸✉✑ s❛♦ ❝❤♦ t õ t ự ỵ t❤✉②➳t rót ❣å♥ ❝õ❛ ❍❛r❞② ✤➸ ①→❝ ✤à♥❤ ❧✐➺✉ ❝→❝ t➼❝❤ ♣❤➙♥ ✤â ❝â ♣❤↔✐ ❧➔ ❤➔♠ ❝➜♣ ❤❛② ❦❤æ♥❣✱ ✈➔ ❦❤✐ ✤➣ ❦❤➥♥❣ ✤à♥❤ t❤➻ ❧✐➺✉ ❝â t❤➸ t➼♥❤ t♦→♥ ✤÷đ❝ ❣✐→ trà ❝❤➼♥❤ ①→❝ ❤❛② ❦❤ỉ♥❣✳ ❈❤÷ì♥❣ ✸ ♥❣❤✐➯♥ ❝ù✉ ♠ët sè →♣ ❞ö♥❣ ❝õ❛ t❤✉➟t t♦→♥ rót ❣å♥ ❍❛r❞② ❝❤♦ ❧ỵ♣ t➼❝❤ ♣❤➙♥ ▲✐♦✉✈✐❧❧❡ ❝→❝ ❤➔♠ sỡ ữủ t ữợ sỹ ữợ ❞➝♥ ❦❤♦❛ ❤å❝ ❝õ❛ t❤➛② P●❙✳ ❚❙✳ ❚❤→✐ ❚❤✉➛♥ ◗✉❛♥❣✱ ❑❤♦❛ ❚♦→♥ ✈➔ ❚❤è♥❣ ❦➯✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ◗✉② ◆❤ì♥✳ ◆❤➙♥ ❞à♣ ♥➔② tæ✐ ①✐♥ ❜➔② tä sü ❦➼♥❤ trå♥❣ ✈➔ ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ ❚❤➛② ✤➣ ❣✐ó♣ ✤ï tæ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✳ ✸ ❚ỉ✐ ❝ơ♥❣ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ✤➳♥ ❇❛♥ ❣✐→♠ ❤✐➺✉ ❚r÷í♥❣ ✣↕✐ ❤å❝ ◗✉② ◆❤ì♥✱ Pỏ t ũ qỵ t ổ ợ Pữỡ ❚♦→♥ ❝➜♣ ❦❤â❛ ✷✷ ✤➣ ❞➔② ❝ỉ♥❣ ❣✐↔♥❣ ❞↕② tr♦♥❣ s✉èt ❦❤â❛ ❤å❝✱ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐ ❝❤♦ tæ✐ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐✳ ◆❤➙♥ ✤➙② tỉ✐ ❝ơ♥❣ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ sü ❤é trñ ✈➲ ♠➦t t✐♥❤ t❤➛♥ ❝õ❛ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧ ✤➣ ❧✉ỉ♥ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ ❣✐ó♣ ✤ï ✤➸ tæ✐ ❤♦➔♥ t❤➔♥❤ tèt ❦❤â❛ ❤å❝ ✈➔ ❧✉➟♥ ✈➠♥ ũ ữủ tỹ ợ sỹ ♥é ❧ü❝ ❝è ❣➢♥❣ ❤➳t sù❝ ❝õ❛ ❜↔♥ t❤➙♥✱ ♥❤÷♥❣ ❞♦ ✤✐➲✉ ❦✐➺♥ t❤í✐ ❣✐❛♥ ❝â ❤↕♥✱ tr➻♥❤ ✤ë ❦✐➳♥ t❤ù❝ ✈➔ ❦✐♥❤ ♥❣❤✐➺♠ ♥❣❤✐➯♥ ❝ù✉ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❧✉➟♥ ✈➠♥ ❦❤â tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ ❚æ✐ r➜t ữủ ỳ õ ỵ qỵ t ổ ❣✐→♦ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✳ ✹ ❈❤÷ì♥❣ ✶ ❈→❝ ❤➔♠ sè sỡ ỵ r ữỡ ú tæ✐ tr➻♥❤ ❜➔② ❤❛✐ ✈➜♥ ✤➲✿ ❱➔♥❤ ✈✐ ♣❤➙♥ ✈➔ ỵ P t t ữỡ ú tổ ự ởt số ỵ ▲✐♦✉✈✐❧❧❡ ❝❤♦ t➼♥❤ ❝➜♣ ❝õ❛ ♠ët sè t➼❝❤ ♣❤➙♥✳ ✶✳✶ ❱➔♥❤ ✈➔ tr÷í♥❣ ✈✐ ♣❤➙♥ ✶✳✶✳✶ ❈→❝ ❦❤→✐ ♥✐➺♠ ✈➔ t➼♥❤ ❝❤➜t ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶ ∂:R→R ✭❬✺❪✮ ✳ ❈❤♦ R ❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ ✈à 1✳ ❚❛ ❣å✐ →♥❤ ①↕ ✤÷đ❝ ❣å✐ ❧➔ →♥❤ ①↕ ✤↕♦ ❤➔♠ ♥➳✉   ∂(a + b) = ∂(a) + ∂(b) , ∀a, b ∈ R   ∂(ab) = ∂(a)b + a∂(b) ▼ët ✈➔♥❤ ✤÷đ❝ tr❛♥❣ ❜à ♠ët ✤↕♦ ❤➔♠ ❝ư t❤➸ ❣å✐ ❧➔ t❛ t❤÷í♥❣ ✈✐➳t ∂(a) = a ❧➔ ✈➔♥❤ ✈✐ ♣❤➙♥✳ ❚r÷í♥❣ ▼➺♥❤ ✤➲ ✶✳✶✳✷ ✭❬✺❪✮ ▼✐➲♥ ♥❣✉②➯♥ R R ✈➔♥❤ ✈✐ ♣❤➙♥✳ ✤÷đ❝ ❣å✐ ❧➔ ♠ët ✤÷đ❝ ❣å✐ ❧➔ ♠ët ♠✐➲♥ ♥❣✉②➯♥ ✈✐ ♣❤➙♥ tr÷í♥❣ ✈✐ ♣❤➙♥ ✳ ❈❤♦ R ❧➔ ♠ët ✈➔♥❤ ✈✐ ♣❤➙♥✳ ❑❤✐ ✤â ✶✮ ✶✬❂✵✳ ✷✮ ∂(n1) = ✈ỵ✐ ♠å✐ n ∈ Z ✸✮ ∂(na) = n∂(a) ✈ỵ✐ ♠å✐ a ∈ R, n ∈ Z ✣➸ t❤✉➟♥ t✐➺♥✱ ♥❣÷í✐ ♥➳✉ R ♥➳✉ ❧➔ ✈➔♥❤ ✈✐ ♣❤➙♥✳ R ✺ ❈❤ù♥❣ ♠✐♥❤✳ ✷✮ ❚❛ ❝â = (1 · 1) = · + · ✶✮ ❚❛ ❝â ♥➯♥ =1 +1 ❉♦ ✤â = 0✳ ∂(n1) = n∂(1) = n0 = ✸✮ ❚❛ ❝❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü ♥❤÷ tr♦♥❣ ✷✮ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ q✉② ♥↕♣ t❤❡♦ ▼➺♥❤ ✤➲ ✶✳✶✳✸ ✭❬✺❪✮ n ✳ ❈❤♦ R ❧➔ ♠ët ✈➔♥❤ ✈✐ ♣❤➙♥✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ a ∈ R t❤➻ (a ) = n nan−1 a ✳ ❈❤ù♥❣ ợ n=2 ự ữỡ ♣❤→♣ q✉② ♥↕♣✳ (a2 ) = (aa) = a a + aa = 2aa ✳ t❤➻ ●✐↔ sû ♠➺♥❤ ✤➲ ✤ó♥❣ ✈ỵ✐ n = k✱ t❛ ❝â ❚❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ ♠➺♥❤ ✤➲ ✤ó♥❣ ✈ỵ✐ (ak+1 ) (ak ) = kak−1 a ✳ n = k + 1✳ ❚❤➟t ✈➟②✱ = (ak a) = (ak ) a + ak a = kak−1 a a + ak a = kak a + ak a = (k + 1)ak a n = k + 1✳ ❱➟② ♠➺♥❤ ✤➲ ✤ó♥❣ ✈ỵ✐ ▼➺♥❤ ✤➲ ✶✳✶✳✹ ✭❬✺❪✮ ✳ ❈❤♦ R ❧➔ ♠ët tr÷í♥❣ ✈✐ ♣❤➙♥✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ a ∈ R \ {0} t❤➻ (a−1 ) = −a−2 a ❈❤ù♥❣ ♠✐♥❤✳ ❉♦ ✤â ❱➻ = = (aa−1 ) = a a−1 + a(a−1 ) ♥➯♥ a(a−1 ) = −a a−1 ✳ (a−1 ) = −a−2 a ✳ ▼➺♥❤ ✤➲ ✶✳✶✳✺ ✳ ❈❤♦ R ❧➔ ♠ët tr÷í♥❣ ✈✐ ♣❤➙♥✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ a ✭❬✺❪✮ ∈ R ✈➔ b ∈ R \ {0} t❤➻ ∂ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â (ab−1 ) = a b−1 + a(b−1 ) = a b−1 + a(−b−2 b ) = (a b − ab )(b−2 )✳ t r ú ỵ a a b − ab = b b2 Q ⊆ R✱ ♠é✐ ♣❤➛♥ tû ❝õ❛ Q ❧➔ ♠ët ❤➡♥❣ sè✳ R ❧➔ ♠ët ♠✐➲♥ ♥❣✉②➯♥ ✈✐ ♣❤➙♥ t❤➻ tr÷í♥❣ ❝→❝ tữỡ F s R ợ t ❧➔ ♠ët tr÷í♥❣ ✈✐ ♣❤➙♥✳ ✻ ❱➼ ❞ư ✶✳✶✳✻✳ ▼å✐ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ R ❝â ✤ì♥ ✈à d(a) = 0, ợ t t tữớ a R, ❧➔ ♠ët ✈➔♥❤ ✈✐ ♣❤➙♥✳ ❱➼ ❞ö ✶✳✶✳✼✳ ❈❤♦ P (x) t❤ỉ♥❣ t❤÷í♥❣✳ ❑❤✐ ✤â ❧➔ ✈➔♥❤ ❝→❝ ✤❛ t❤ù❝ ❝â ❤➺ sè t❤ü❝ ✈ỵ✐ ♣❤➨♣ t♦→♥ ✤↕♦ ❤➔♠ P (x) ❧➔ ♠ët ✈➔♥❤ ✈✐ ♣❤➙♥✳ ❍ì♥ ♥ú❛✱ tr÷í♥❣ ❝→❝ t❤÷ì♥❣ ❝õ❛ ❈❤♦ ♥➯♥ R(x) ❱➼ ❞ư ✶✳✶✳✽✳ P (x) ❧➔ tr÷í♥❣ R(x) ❝→❝ ♣❤➙♥ t❤ù❝ ❝â ❤➺ sè tỹ ởt trữớ ợ t ❤➔♠ t❤ỉ♥❣ t❤÷í♥❣✳ ❱➔♥❤ ❝→❝ ❤➔♠ t❤ü❝ ❦❤↔ ✈✐ ✈ỉ ❤↕♥✱ ✈➔♥❤ ❝→❝ ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ tr➯♥ ♠➦t ♣❤➥♥❣ ♣❤ù❝ ợ t tổ tữớ ✈✐ ♣❤➙♥✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✾ ❧➔ ✈➔♥❤ ❝♦♥ ❝õ❛ ◆❤÷ ✈➟②✱ R ✭❬✺❪✮ ✳S ✤÷đ❝ ❣å✐ ♠ët ✈➔♥❤ ❝♦♥ ✈✐ ♣❤➙♥ ❝õ❛ ✈➔♥❤ ✈✐ ♣❤➙♥ R ♥➳✉ S ✈➔ ✤â♥❣ ❦➼♥ ✈ỵ✐ ♣❤➨♣ t♦→♥ ✤↕♦ ❤➔♠✳ S⊆R ❧➔ ✈➔♥❤ ❝♦♥ ✈✐ ♣❤➙♥ ♥➳✉ ∈ S ✱ a, b ∈ S t❤➻ a−b ∈ S ✈➔ a∈S t❤➻ a ∈ S✳ ▼ët ❝→❝❤ t÷ì♥❣ tü✱ ✐❞❡❛❧ I ❝õ❛ ✈➔♥❤ ✈✐ ♣❤➙♥ R ✤÷đ❝ ❣å✐ ❧➔ ✐❞❡❛❧ ✈✐ ♣❤➙♥ ♥➳✉ I ✤â♥❣ ❦➼♥ ✈ỵ✐ ♣❤➨♣ t♦→♥ ✤↕♦ ❤➔♠✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✵ ✭❬✺❪✮ ✳ ❈❤♦ R, S ❧➔ ❝→❝ ✈➔♥❤ ✈✐ ♣❤➙♥✳ :RS ữủ ởt ỗ ♣❤➙♥ ♥➳✉    ϕ(a + b) = ϕ(a) + ϕ(b)       ϕ(ab) = ϕ(a)ϕ(b)    ϕ(1)       ϕ(a ) ▼➺♥❤ ✤➲ ✶✳✶✳✶✶ =1 = (ϕ(a)) ✳ ❈❤♦ R✱ S ❧➔ ❝→❝ ✈➔♥❤ ✈✐ ♣❤➙♥ ✈➔ : R S ởt ỗ ✈➔♥❤✳ ❑❤✐ ✤â ✶✮ ker ϕ = x ∈ R : ϕ(x) = ❧➔ ✐❞❡❛❧ ❝õ❛ R, ✷✮ ⑩♥❤ ①↕ f : R/ ker f → Imf ❧➔ ♠ët ✤➥♥❣ ❝➜✉ ✈✐ ♣❤➙♥✳ ✸✸ ✈ỵ✐ Q ≡ ❤♦➦❝ ❜➟❝ tè✐ ✤❛ ❜➡♥❣ deg g − 2✱ s❛♦ ❝❤♦ f (x)eg(x) dx = u(x)eg(x) dx + R(x)eg(x) ❍ì♥ ♥ú❛✱ ✈➳ tr→✐ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② ❧➔ ❝➜♣ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ u ≡ 0✱ ✈➔ ♥➳✉ u ≡ t➼❝❤ ♣❤➙♥ ð ✈➳ ♣❤↔✐ ❦❤ỉ♥❣ ❝➜♣ ♥❣❤➽❛ ❧➔ ✈ỵ✐ ❜➜t ❦ý ❤➔♠ ❤ú✉ t✛ v ✈➔ Q t❤ä❛ ♠➣♥ f (x)eg(x) dx = v(x)eg(x) dx + Q(x)eg(x) , ✈ỵ✐ deg v ≥ deg u ❍➺ q✉↔ ✷✳✷✳✾ ✭❬✻❪✮ ✳ ❈❤♦ g ✈➔ P ♥❤÷ tr♦♥❣ ✭✷✳✷✳✻✮✳ õ ợ t ý f Q tỗ t ❞✉② ♥❤➜t ♠ët ❤➔♠ ❤ú✉ t✛ R ✈➔ ❞✉② ♥❤➜t ❤➔♠ ❤ú✉ t✛ u ∈ Ng ❝â ❞↕♥❣ m u(x) = j=1 Aj + x − λj γ ars γrs + Q(x) rs Ng ợ Ng Ng ữủ tr ự ỵ trữớ ủ ✷ ✈➔ Q ≡ ♥➳✉ P ≡ ❤♦➦❝ ❧➔ ♠ët ✤❛ t❤ù❝ ❝â ❜➟❝ tè✐ ✤❛ ❧➔ deg(P ) − 1✱ s❛♦ ❝❤♦ f (x)eg(x) dx = u(x)eg(x) dx + R(x)eg(x) ❍ì♥ ♥ú❛ ✱ ✈➳ tr→✐ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ ❧➔ ❝➜♣ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ u ≡ ✈➔ ♥➳✉ u ≡ 0✱ t➼❝❤ ♣❤➙♥ ð ✈➳ ♣❤↔✐ ❧➔ ❦❤ỉ♥❣ ❝➜♣ ♥❤ä ♥❤➜t t❤❡♦ ♥❣❤➽❛ ✤è✐ ✈ỵ✐ ❜➜t ❦ý ❤➔♠ ❤ú✉ t✛ v ✈➔ ρ t❤ä❛ ♠➣♥ f (x)eg(x) dx = v(x)eg(x) dx + ρ(x)eg(x) , ✈ỵ✐ deg v ≥ deg u✳ ✷✳✷✳✸ ❈→❝ ✈➼ ❞ư ❱➼ ❞ư ✷✳✷✳✶✵ ✭❬✻❪✮ ✳ ❚➻♠ rót ❣å♥ ❍❛r❞② ❝õ❛ f (x)e ✈ỵ✐ ❜➜t ❦ý ✤❛ t❤ù❝ f ✈➔ a ∈ C \ {0} ax2 dx ✸✹ ▲í✐ ❣✐↔✐✳ ❚❛ ❝â g(x) = ax2 ✈➔ P = s♣❛♥{1} + s♣❛♥{ax, ax2 + 1, ax3 + 2x, } ◆➳✉ deg f (x) = n, t❛ ❝â t❤➸ t➻♠ ✤÷đ❝ A0 , A1 , , An ∈ C s❛♦ ❝❤♦ n−1 Aj+1 (axj+1 + jxj−1 ) f (x) = A0 + j=0 ❉♦ ✤â✱ t❛ ❝â f (x)e ax2 dx = A0 e ax2 n−1 dx + Aj+1 (axj+1 + jxj−1 )e ax2 dx j=0 = A0 e ax2 n−1 Aj+1 xj e dx + ax2 j=0 n−1 ❈→❝ ✤❛ t❤ù❝ Aj+1 xj ❝â t❤➸ ①→❝ ✤à♥❤ ❝→❝ ❤➺ sè A0 , A1 , , An u(x) = A0 ✈➔ P (x) = j=0 t tự f trữợ ỡ ỳ t ❜➯♥ tr→✐ ❧➔ ❝➜♣ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ✈➔ ❣✐→ trà ❝õ❛ ♥â ❧➔ ❱➼ ❞ö ✷✳✷✳✶✶ ✭❬✻❪✮ P (x)e ✳ ax2 ✳ ❚➻♠ rót ❣å♥ ❍❛r❞② ❝õ❛ (10x4 − 3x1 )ex dx ❚➼❝❤ ♣❤➙♥ ♥➔② ❝â ❝➜♣ ❦❤ỉ♥❣❄ ▲í✐ ❣✐↔✐✳ ❚❛ ❝â g(x) = x5 , f (x) = 10x4 − 3x + 1, Eg = span{βj (x) = 5xj+4 + jxj−1 : j = 0, 1, 2, }, Ng = span{1, x, x2 , x3 } ❉♦ ✤â✱ f (x) = − 3x + 2β0 (x)✱ rót ❣å♥ ❍❛r❞② t❛ ✤÷đ❝ (10x4 − 3x + 1)ex dx = ❱➻ u(x) = − 3x ❦❤→❝ ✤❛ t❤ù❝ 5 (1 − 3x)ex dx + 2ex ♥➯♥ t➼❝❤ ♣❤➙♥ ✤➣ ❝❤♦ ❦❤ỉ♥❣ ❝➜♣✳ A0 = ✸✺ ❱➼ ❞ư ✷✳✷✳✶✷ ✭❬✻❪✮ ✳ ❚➻♠ rót ❣å♥ ❍❛r❞② ❝õ❛ +x (x5 + x4 + x3 + x2 + x + 1)ex dx ❚➼❝❤ ♣❤➙♥ ♥➔② ❝â ❝➜♣ ❦❤ỉ♥❣❄ ▲í✐ ❣✐↔✐✳ ❚❛ ❝â f (x) = x5 + x4 + x3 + x2 + x + 1, Ng = s♣❛♥{1, x}, g(x) = x3 + x, Eg = s♣❛♥{βj (x) = 3xj+2 + xj + jxj−1 : j = 0, 1, 2, }, βj (x)ex +x dx = xj ex f (x) = +x + c, 1 1 + x − β0 (x) + β1 (x) + β2 (x) + β3 (x) 9 9 3 ❉♦ ✤â✱ t❛ ✤÷đ❝ f (x)eg(x) dx = ❱➻ u(x) = + x 9 ❱➼ ❞ö ✷✳✷✳✶✸ 1 3 + x ex +x dx + − + x + x2 + x3 ex +x 9 9 3 ❦❤→❝ ✤❛ t❤ù❝ ✭❬✻❪✮ ✳ ♥➯♥ t➼❝❤ ♣❤➙♥ ✤➣ ❝❤♦ ❦❤ỉ♥❣ ❝➜♣✳ ❚➻♠ rót ❣å♥ ❍❛r❞② ❝õ❛ x3 dx e (x2 − 1)2 ❚➼❝❤ ♣❤➙♥ ♥➔② ❝â ❝➜♣ ❦❤ỉ♥❣❄ ▲í✐ ❣✐↔✐✳ ❚❛ ❝â g(x) = f (x) = ✣➦t F = {−1, 1}, x3 , (x2 1 1 = + − + 2 − 1) x + (x + 1) x − (x − 1)2 ❦❤✐ ✤â Q(F) = s♣❛♥{1, xj , 1 : j = 1, 2, }, , (x + 1)j (x − 1)j Eg (F) = s♣❛♥{α±1j (x), x2 , βj (x) : j = 1, 2, } ✸✻ ✈ỵ✐ α±1j (x) = x2 j − , j (x ± 1) (x ± 1)j+1 βj (x) = xj+2 + jxj−1 , 1 , , 1, x , x + x1 f (x) = 2x + − α−11 (x) − α11 (x) x+1 Ng (F) = s♣❛♥ ❚❤❡♦ ✭✷✳✷✳✺✮✱ f (x)eg(x) dx = ❱➻ u(x) = 2x + ❱➼ ❞ö ✷✳✷✳✶✹ x+1 ✭❬✻❪✮ ≡0 ✳ 2x + x+1 x3 e dx − 1 + x−1 x+1 x3 e3 ♥➯♥ t➼❝❤ ♣❤➙♥ ✤➣ ❝❤♦ ❦❤ỉ♥❣ ❝➜♣✳ ❚➻♠ rót ❣å♥ ❍❛r❞② ❝õ❛ 2x6 − 3x5 − 5x4 + x3 + 3x2 + 12x − exp x4 − 6x3 + 13x2 − 12x + x4 + x3 − x2 − x + x2 − dx ❚➼❝❤ ♣❤➙♥ tr➯♥ ❝â ❝➜♣ ❦❤ỉ♥❣❄ ▲í✐ ❣✐↔✐✳ ❚❛ ❝â 1 x4 + x3 − x2 − x + = − + x2 + x x −1 x−1 x+1 2x6 − 3x5 − 5x4 + x3 + 3x2 + 12x − f (x) = x4 − 6x3 + 13x2 − 12x + 38 = + + − + 2x2 + 9x + 23 2 x − (x − 1) x − (x − 2) g(x) = ✣➦t F = {1, 2} 1 , , , λ = 1, 2, j = 1, 2, , j (x − λ) x + (x + 1)2 1 Ng = , , , x − (x − 1) x + 1 1 Ng (F) = s♣❛♥ 1, x, , , , x − x − (x − 1) x + Qg (F) = s♣❛♥ 1, xj , Eg (F) = s♣❛♥ αλj (x), g (x), βj (x); λ = 1, 2, j = 1, 2, ✸✼ ❚❤❡♦ ✭✷✳✷✳✹✮ αλj (x) = − 4x λ)j (x2 1)2 + 2x + j − , j (x − λ) (x − λ)j+1 (x − − 1 g (x) = − + + 2x + 1, (x − 1)2 (x + 1)2 4xj+1 βj (x) = − + 2xj+1 + xj + jxj−1 , (x − 1) 1 + 4g (x) + β1 (x) + 9α21 (x) f (x) = + (x − 1)2 x − ✈ỵ✐ 1 1 − + 2x2 + x + − + (x − 1)2 x − (x + 1)2 x + 1 37 1 1 α21 (x) = − + + + − − 2 (x − 2) 9(x − 2) (x − 1) x − 3(x + 1) 9(x + 1) β1 (x) = − ❚ø ✭✷✳✷✳✺✮✱ s✉② r❛ f (x)eg(x) dx = = ❱➻ u(x) = 1 + + 4g (x) + β1 (x) + 9α21 (x) eg(x) dx (x − 1) x−2 1 + eg(x) dx + + x + eg(x) (x − 1) x−2 x−2 1 + ∈ Ng (F) \ {0} (x − 1) x−2 ♥➯♥ t➼❝❤ ♣❤➙♥ ✤➣ ❝❤♦ ❦❤ỉ♥❣ ❝➜♣✳ ✸✽ ❈❤÷ì♥❣ ✸ ▼❐❚ ❙➮ ⑩P ❉Ư◆● ❚r♦♥❣ ❈❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tỉ✐ sû ❞ư♥❣ ♠ët sè ❦➳t q✉↔ ❝õ❛ ❈❤÷ì♥❣ ✷ ✤➸ ♥❣❤✐➯♥ ❝ù✉ ♠ët sè ✤✐➲✉ ❦✐➺♥ ❝õ❛ ✤❛ t❤ù❝ P k P (x)eax dx ❧➔ ❝➜♣✳ ❈→❝ ✤✐➲✉ s❛♦ ❝❤♦ ❝→❝ t➼❝❤ ♣❤➙♥ ❦✐➺♥ ❝â t❤➸ ✤↕t ✤÷đ❝ ❧➔ ❝→❝ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ①→❝ ✤à♥❤ ❝→❝ ❤➺ sè ❝õ❛ ✤❛ t❤ù❝ tỉ✐ ❜➢t ✤➛✉ ❜➡♥❣ ❝→❝ tr÷í♥❣ ❤đ♣ ✤ì♥ ❣✐↔♥ ❝õ❛ k ♥❤÷ k=2 ❤♦➦❝ P ❈❤ó♥❣ k = ✸✳✶ ▼ët sè ❞↕♥❣ t➼❝❤ ♣❤➙♥ ▲✐♦✉✈✐❧❧❡ ✤➦❝ ❜✐➺t ❱➼ ❞ư ✸✳✶✳✶ ✈ỵ✐ r = 0, ✭❬✻❪✮ ✳ ❈❤♦ P (x) = an xn + · · · + a1 x + a0 P (x)eax dx t❤➻ a ∈ C \ {0}✳ ◆➳✉ j=0 − 2a ❚❛ ❝❤➾ ①➨t tr÷í♥❣ ❤đ♣ j (2j − 1)!!a2j = n = 2l ỵ 2l−1 j Aj+1 xj+1 + aj x = j=0 j=0 j j−1 x 2a 2l − A2l x2l−2 + 2a A2 x2 + A1 + A3 x + 2a = A2l x2l + A2l−1 x2l−1 + A2l−2 + + · · · + A2 + A4 2a ✣✐➲✉ ♥➔② ❝â ♥❣❤➽❛ ❧➔ 2l − A2l = a2l−2 , , 2a A2 A2 + A4 = a2 , A1 + A3 = a1 , = a0 2a 2a A2l = a2l ; A2l−1 = a2l−1 , A2l−2 + P (x)eax dx ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ 2l n = 2l + r✱ ❝➜♣ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ l ❈❤ù♥❣ ♠✐♥❤✳ ✈➔ ❝➜♣ ✸✾ ❍➺ ♣❤÷ì♥❣ tr➻♥❤ t t tữỡ ữỡ ợ t ✳ ❱➼ ❞ö ✸✳✶✳✶ ❧➔ ♠ð rë♥❣ ❝→❝ ❦➳t q✉↔ tr♦♥❣ ❬✽❪ ❈ö t❤➸ tr♦♥❣ ❬✽❪ ✤➣ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ P (x)e−x dx ❧➔ ❝➜♣ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ✤✐➲✉ ❦✐➺♥ trü❝ ❣✐❛♦ +∞ P (x)e−x dx = −∞ ①↔② r❛✳ ❚❤➟t ✈➟②✱ ✈ỵ✐ ♠å✐ j = 0, 1, 2, t❤➻ √ +∞ 2j −x2 x e dx = −∞ π(2j − 1)!! 2j ✈➔ +∞ x2j+1 e−x dx = −∞ ❉♦ ✤â✱ ❦❤✐ n P (x) = an x + · · · + a1 x + a0 trü❝ ❣✐❛♦ tr tữỡ ữỡ ợ l j=0 ợ n = 2l + r✱ r ∈ {0, 1} P (x)e−x dx t❤➻ ✤✐➲✉ ❦✐➺♥ ❧➔ ❝➜♣ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ (2j − 1)!! a2j = 2j ✳ ❱➼ ❞ư ✸✳✶✳✸ ✈ỵ✐ r = 1, ✭❬✻❪✮ t❤➻ ✳ ❈❤♦ P (x)e P (x) = an xn + · · · + a1 x + a0 ax3 dx − j=0 l − j=0 ◆➳✉ n = 3l✱t❤➻ P (x)e ax3 dx ◆➳✉ n = 3l + r✱ j 3a 3a (3j − 2)!!!a3j = j (3j − 1)!!!a3j+1 = ❝➜♣ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ l j=0 ✈➔ a ∈ C \ {0} ❝➜♣ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ l ✈➔ ✈➔ l−1 − j=0 j − 3a 3a (3j − 2)!!!a3j = j (3j − 1)!!!a3j+1 = ✭✸✳✶✳✷✮ ✹✵ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝❤➾ ①➨t ❝❤➾ ♥➳✉ n = 3l + 3l+2 3l+2 aj x j = j=0 ỗ ♥❤➜t ❤➺ sè ❝õ❛ x j A j xj + j=2 ❝➜♣ ♥➳✉ ✈➔ j − j−3 x 3a ✱ t❛ ✤÷đ❝ A3l+2 = a3l+2 , A3l+1 = a3l+1 , A3l = a3l , A3l−1 + A2 + P (x)eax dx ỵ 3l A3l+2 = a3l−1 , , 3a A5 A3 = a2 , A4 = a1 , = a0 a 3a 3a ❍➺ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ♥➔② t÷ì♥❣ ✤÷ì♥❣ ✭✸✳✶✳✷✮✳ ▼➺♥❤ ✤➲ ✸✳✶✳✹ ✭❬✻❪✮ ✳ ❈❤♦ k ❧➔ ♠ët sè ♥❣✉②➯♥ ❞÷ì♥❣✱ P (x) = a x n ✈ỵ✐ a ∈ C \ {0}✳ ◆➳✉ n = kl + r✱ ✈ỵ✐ ≤ r ≤ t❤➻ t➼❝❤ ♣❤➙♥ n + · · · + a1 x + a0 k P (x)eax dx ❝➜♣ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ❝→❝ ❤➺ sè ❝õ❛ P t❤ä❛ ♠➣♥ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ❜➟❝ k − s❛✉ l j=0 l−1 j=0 −1 ka j −1 ka j (j − 1)k + i + !k aki+j = 0, i = 0, 1, , {r, k − 2} (j − 1)k + i + !k akj+i = 0, i = {r, k − 2} + 1, , k − a P (x)e xk dx ❇➙② ❣✐í t❛ ①➨t ❝→❝ t➼❝❤ ♣❤➙♥ ❞↕♥❣ ❱➼ ❞ö ✸✳✶✳✺ ✭❬✻❪✮ ✳ ❈❤♦ n j=0 P ỵ ✱ ♥➳✉ s♣❛♥ ✈➔ P ❧➔ ♠ët ✤❛ t❤ù❝✳ aj aj = (j + 1)! P ❧➔ ♠ët ✤❛ t❤ù❝ t❤➻ x − a2 , x2 − a3 x, , xj − ♥➳✉ n j=0 xj ✱ Aj xj − j=1 a P (x)e x dx ✭✸✳✶✳✸✮ a P (x)e x dx ❝➜♣ ♥➳✉ ✈➔ a xj−1 , ✳ ✣✐➲✉ ♥➔② ①↔② r❛ ♥➳✉ ✈➔ ❝❤➾ j+1 n aj x j = ỗ t số k>0 P (x) = an xn +· · ·+a1 x+a0 ✈➔ a ∈ C\{0} ❑❤✐ ✤â✱ ❝➜♣ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ❈❤ù♥❣ ♠✐♥❤✳ ✈ỵ✐ a j−1 x j+1 t❛ ✤÷đ❝ a An = an−1 , , n+1 a a aA1 A2 − A3 = a2 , A1 − A2 = a1 , − = a0 An = an , An−1 − ❍➺ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ tữỡ ữỡ ợ ✳ ❈❤♦ P (x)an xn + · · · + a1 x + a0 ✈➔ a ∈ C \ {0} ◆➳✉ n = 2l + t❤➻ a P (x)e x2 dx ❝➜♣ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ l j=0 ◆➳✉ n = 2l✱ l (2a)j a2j = (2j + 1)!! ✈➔ j=0 (2a)j a2j+1 = (2j + 2)!! ✭✸✳✶✳✹✮ a P (x)e x2 dx t❤➻ l j=0 ❈❤ù♥❣ ♠✐♥❤✳ ❝➜♣ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ l−1 (2a)j a2j = (2j + 1)!! ✈➔ j=0 ❚❤❡♦ ✣à♥❤ ỵ (2a)j a2j+1 = (2j + 2)!! a P (x) ❧➔ ♠ët ✤❛ t❤ù❝ t❤➻ P (x)e x2 dx ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ P ∈ s♣❛♥ x2 − 2l+1 ●✐↔ sû n = 2l + 2a 2a j−2 , , xj − x , j+1 2l+1 j Aj xj − aj x = ✈➔ j=0 j=2 2a j−2 x j+1 ỗ t số ợ xj t ữủ 2a A2l+1 = a2l−1 , , 2l + 2a 2a 2a A2 − A4 = a2 , − A3 = a1 , − A3 = a0 A2l+1 = a2l+1 , A2l = a2l , A2l−1 − ❍➺ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ♥➔② tữỡ ữỡ ợ ự tữỡ tỹ ợ n = 2l✳ ▼➺♥❤ ✤➲ ✸✳✶✳✼ ✭❬✻❪✮ ✳ ❈❤♦ k ❧➔ ♠ët sè ♥❣✉②➯♥ ❞÷ì♥❣✱ P (x) = a x n ✈ỵ✐ a ∈ C \ {0}✳ ◆➳✉ n = kl + r✱ ✈ỵ✐ ≤ r ≤ t❤➻ t➼❝❤ ♣❤➙♥ n + · · · + a1 x + a0 k P (x)ea/x dx ❝➜♣ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ❝→❝ ❤➺ sè ❝õ❛ P t❤ä❛ ♠➣♥ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ❜➟❝ k s❛✉ l j=0 l−1 j=0 ❱➼ ❞ö ✸✳✶✳✽ t❤ù❝ ❜➟❝ ✭❬✻❪✮ ✳ l ≤n−1 (ka)j akj+i = 0, (kj + i + 1)!(k) i = 0, 1, , r; (ka)j akj+i = 0, (kj + i + 1)!(k) i = r + 1, , k − ❈❤♦ a=0 t❤➻ P (x) ax e dx ❝➜♣ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ (x−c)n ✈➔ c n j j=n−l ❧➔ ❤➡♥❣ sè✳ ◆➳✉ n ❧➔ sè ♥❣✉②➯♥ ❞÷ì♥❣ ✈➔ n j (n−j) aP (c) = j P ❧➔ ✤❛ ✭✸✳✶✳✺✮ ✹✷ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â a0 a1 ak + n−1 + · · · + n−l n x x x ❝➜♣ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ eax dx ✭✸✳✶✳✻✮ n aj an−j =0 (j − 1)! j=n−l ✭✸✳✶✳✼✮ ❚❤❡♦ ✣à♥❤ ỵ t t sỡ ♥➳✉ a0 a1 al + n−1 + · · · + n−l n x x x n−2 n−l n−1 a a a + A + · · · + A =A1 − − − l xn xn−1 xn−1 xn−2 xn−l+1 xn−l (n − 1)A1 (n − 2)A2 − aA1 (n − l)A1 − aAl−1 aA1 = + + ··· + − n−l n n−1 n−l+1 x x x x ✈ỵ✐ A1 , A2 , , An−1 số ỗ t số t ữủ (n − 1)A1 = a0 , (n − 2)A2 − aA1 = a1 , , (n − l)Al − aAl−1 = al−1 , −aAl = al ữỡ tr t t tữỡ ữỡ ợ ❑❤❛✐ tr✐➸♥ ❚❛②❧♦r ❝õ❛ P ①✉♥❣ q✉❛♥❤ P (x) ax e dx = eax n (x − c) ❉♦ ✤â✱ t❤❛② aj = x=c ✈➔ ✤➦t u=x−c t❛ ✤÷đ❝ P (c) P (c) P ( l)(c) + + ··· + un 1!un−1 l!un−1 eau du P (j) (c) tr♦♥❣ ✭✸✳✶✳✼✮✱ t❛ ✤÷đ❝ ✭✸✳✶✳✺✮✳ j! ✸✳✷ ❈→❝ t➼❝❤ ♣❤➙♥ ❑✐➸✉ ▲✐♦✉✈✐❧❧❡ ❚r♦♥❣ ♠ư❝ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ✈➜♥ ✤➲ ✈➲ t➼❝❤ ♣❤➙♥ ❑✐➸✉ ▲✐♦✉✈✐❧❧❡✳ ❈→❝ t➼❝❤ ♣❤➙♥ ♥➔② ❝â ❞↕♥❣ F (x) log G(x)dx, tr♦♥❣ ✤â ❝→❝ ❤➔♠ F, G F (x) arctan G(x)dx ❧➔ ♣❤➙♥ t❤ù❝ ✈ỵ✐ G ✈➔ F (x) tanh−1 G(x)dx ❦❤→❝ ❤➡♥❣✳ ❙û ❞ö♥❣ ❝→❝ ✤➥♥❣ t❤ù❝ + ix log 2i − ix 1+x tanh−1 x = log 1−x arctan x = ✈➔ ✹✸ t❤➻ t❛ ❝❤➾ ❝➛♥ ♥❣❤✐➯♥ ❝ù✉ t➼❝❤ ♣❤➙♥ ✤➛✉ t✐➯♥✳ ❑❤æ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t✱ t❛ ❣✐↔ sû G(x) = a(x − α1 ) (x − αn ) (x − β1 ) (x − βm ) ❑❤✐ ✤â✱ ❜➡♥❣ ❝→❝❤ sû ❞ö♥❣ t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ sè ❧♦❣❛r✐t t❛ ❝â t❤➸ ①→❝ ✤à♥❤ ✤÷đ❝ ❤➔♠ f s❛♦ ❝❤♦ F (x) log G(x)dx = f (x) log xdx ❦❤✐ ✤â t❛ ❝â t❤➸ sû ❞ö♥❣ t✐➯✉ ❝❤✉➞♥ t➼❝❤ ♣❤➙♥ ▲✐♦✉✈✐❧❧❡ ✲ ❍❛r❞② ✤➸ ♥❣❤✐➯♥ ❝ù✉ ✈➲ t➼♥❤ ❝❤➜t ❝➜♣ ❝õ❛ t➼❝❤ ♣❤➙♥ ♥➔②✳ ❱➻ P (x) log xdx ❧➔ ❝➜♣ ✈ỵ✐ ♠å✐ ✤❛ t❤ù❝ ❤➔♠ ♣❤➙♥ t❤ù❝ t❤ü❝ sỹ tự tỹ sỹ ỵ f ✭❬✻❪✮ Qp ✈ỵ✐ ❤➺ sè tr➯♥ ❈❤ù♥❣ ♠✐♥❤✳ ❱➻ ✳ t = f Qp : ú ỵ r ✈➔ ♠å✐ ❉♦ ✤â✱ t❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ n l=1 f ✤÷đ❝ ❣å✐ ❧➔ ♣❤➙♥ f (x) log xdx ❧➔ ❤➔♠ ❝➜♣ βl ∈ N \ {0} x − λl ✈➔ : λ ∈ C \ {0} x−λ Q = N ⊕ s♣❛♥ log x dx = (log x)2 + c x u(x) = Ð ✤➙②✱ ♠ët ❤➔♠ sè 1 , , λ ∈ C, j = 2, 3, x x−λ log x j dx = j−1 (x − λ) ✣➦t ♥➯♥ t❛ ❝➛♥ ①❡♠ ①➨t ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤♦➦❝ ❜➟❝ tû t❤ù❝ ♥❤ä ❤ì♥ ❜➟❝ ♠➝✉ t❤ù❝✳ N = s♣❛♥ ❑❤✐ ✤â ε = s♣❛♥ C✳ P ✈➔ Q = N ⊕ ε 1 , ,λ ∈ C x x−λ λ ∈ C, j ≥ 2, j ∈ N ♥➯♥ dx log x j−1 − x (1 − λ) (j − 1) (x − λ)j−1 u(x) log xdx ✈➔ ❦❤ỉ♥❣ ❝➜♣ ✈ỵ✐ u(x) log xdx ∀u ∈ N \ {0} ❧➔ ❝➜♣✳ ❑❤✐ ✤â✱ t❤❡♦ t✐➯✉ g ✈➔ ❤➡♥❣ sè c s❛♦ ❝❤♦ c u(x) = g (x) + ✳ ✣✐➲✉ ổ ỵ t k = t u ❝â ❝ü❝ ✤ì♥ t↕✐ x = λk x ❱➻ λk = ♥➯♥ ∃m ≥ ✈➔ ❤➔♠ ♣❤➙♥ t❤ù❝ r ❧✐➯♥ tö❝ t↕✐ x = λk s❛♦ ❝❤♦ r(λk ) = ✈➔ r(x) g(x) = ✳ ❙✉② r❛ (x − λk )m ❝❤✉➞♥ t➼❝❤ ♣❤➙♥ ▲✐♦✉✈✐❧❧❡ r tỗ t tự g (x) = r (x)(x − λk )m − m(x − λk )m−1 r(x) (x − λk )2m ✹✹ ❉♦ ✤â (x − λk )m g (x) = r (x) − mr(x) x − λk ❍❛② c mr(x) (x − λk )m (u(x) − ) = r (x) − x x k ổ ỵ ♣❤↔✐ ❦❤ỉ♥❣ ❧✐➯♥ tư❝ t↕✐ ❍➺ q✉↔ ✸✳✷✳✷ x = λk ✳ ✳ ❈→❝ t➼❝❤ ♣❤➙♥ ❑✐➸✉ ▲✐♦✉✈✐❧❧❡ tr➯♥ ❝â rót ❣å♥ ❍❛r❞② ❧➔ tê♥❣ ❝õ❛ ✭❬✻❪✮ ❝→❝ ❤➔♠ sè ❝➜♣ ✈ỵ✐ ❤ú✉ ❤↕♥ ❝→❝ t➼❝❤ ♣❤➙♥ ❦❤ỉ♥❣ ❝➜♣ ❝â ❞↕♥❣ log x dx✱ ✈ỵ✐ x−λ λ ∈ C \ {0} ❱➼ ❞ư ✸✳✷✳✸ ▲í✐ ❣✐↔✐✳ ✭❬✻❪✮ ✳ ❚➻♠ rót ❣å♥ ❍❛r❞② ❝õ❛ 3x2 − 5x + log xdx✳ x(x − 2)2 ❚❛ ❝â ♣❤➙♥ t➼❝❤ s❛✉ 2 3x2 − 5x + = + + x(x − 2) x x − (x − 2)2 ✈ỵ✐ 1 ∈ N, , ∈ ε x−2 x (x − 2)2 ❉♦ ✤â 3x2 − 5x + log xdx = x(x − 2)2 =2 ❱➻ ∈ N \ {0} x−2 ♥➯♥ log x log x log x dx + dx + dx x−2 (x − 2) x log x log x dx + log − + (log x)2 −3 x−2 x x−2 log x dx x−2 ❧➔ ❦❤ỉ♥❣ ❝➜♣✳ ✹✺ ❑➌❚ ▲❯❾◆ ▲✉➟♥ ✈➠♥ ✤➣ ✤↕t ✤÷đ❝ ❝→❝ ♠ư❝ t✐➯✉ ✤➲ r❛ ❧➔ ♥❣❤✐➯♥ ❝ù✉ ❝➙✉ tr↔ ❧í✐ ❝❤♦ ❝➙✉ ❤ä✐ ♥➯✉ tr➯♥ ✤è✐ ✈ỵ✐ ❧ỵ♣ t➼❝❤ ♣❤➙♥ ▲✐♦✉✈✐❧❧❡✳ ✣â ❧➔ ❧ỵ♣ ❝→❝ t➼❝❤ ♣❤➙♥ ❝õ❛ ❤➔♠ sè ❝â ❞↕♥❣ f (x)eg(x) dx tr♦♥❣ ✤â f, g ❧➔ ❝→❝ ❤➔♠ sè ❤ú✉ t✛✱ g ❦❤ỉ♥❣ ❧➔ ❤➔♠ ❤➡♥❣✳ ❈ư t❤➸ ▲✉➟♥ ✈➠♥ ✤➣ tr➻♥❤ ❜➔② ❝❤✐ t✐➳t ♠ët sè ❦➳t q✉↔ s❛✉ ✶✳ ❍➺ t❤è♥❣ ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ trữớ ỵ ▲✐♦✉✈✐❧❧❡ tê♥❣ q✉→t tr➯♥ ♠ët tr÷í♥❣ ✈✐ ♣❤➙♥ ✈➔ ♠ët sè ❤➺ q✉↔ ✤➦❝ ❜✐➺t ❝õ❛ ♥â ✤è✐ ✈ỵ✐ ❝→❝ ❤➔♠ sè ❝➜♣✳ ✷✳ ◆❣❤✐➯♥ ❝ù✉ ✤÷❛ r❛ ✤÷đ❝ ♠ët t❤✉➟t t♦→♥ rót ❣å♥ ❍❛r❞② ✤➸ ①→❝ ✤à♥❤ ❧✐➺✉ ❝→❝ t➼❝❤ ♣❤➙♥ ✤➣ ❝❤♦ ❝â ♣❤↔✐ ❧➔ ❤➔♠ ❝➜♣ ❤❛② ❦❤æ♥❣✱ ✈➔ ❦❤✐ ✤➣ ❦❤➥♥❣ ✤à♥❤ t❤➻ ❧✐➺✉ ❝â t❤➸ t➼♥❤ ✤÷đ❝ ❣✐→ trà ❝❤➼♥❤ ①→❝ ❤❛② ❦❤ỉ♥❣✳ ✸✳ ◆❣❤✐➯♥ ❝ù✉ ♠ët sè ✤✐➲✉ ❦✐➺♥ ❝õ❛ ✤❛ t❤ù❝ P s❛♦ ❝❤♦ ❝→❝ t➼❝❤ ♣❤➙♥ k P (x)eax dx ❧➔ ❝➜♣✳ ❈→❝ ✤✐➲✉ ❦✐➺♥ ✤↕t ✤÷đ❝ ❧➔ ❝→❝ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ①→❝ ✤à♥❤ ❝→❝ ❤➺ sè ❝õ❛ ✤❛ t❤ù❝ P ✹✻ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ❬✶❪ ❏✳ ❇❛❞❞♦✉r❛✱ ❛r✐t❤♠s✱ ■♥t❡❣r❛t✐♦♥ ✐♥ ❢✐♥✐t❡ t❡r♠s ✇✐t❤ ❡❧❡♠❡♥t❛r② ❢✉♥❝t✐♦♥s ❛♥❞ ❞✐❧♦❣✲ ❏✳ ❙②♠❜♦❧✐❝ ❈♦♠♣✉t✳✱ ✹✶ ✭✷✵✵✻✮✱ ✾✵✾✲✾✹✷✳ ❬✷❪ ▼✳ ❇r♦♥st❡✐♥✱ ❚❤❡ tr❛♥s❝❡♥❞❡♥t❛❧ ❘✐s❝❤ ❞✐❢❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥✱ ❏✳ ❙②♠❜♦❧✐❝ ❈♦♠♣✉t✳✱ ✾ ✭✶✾✾✵✮✱ ✹✾✲✻✵✳ ❬✸❪ ▼✳ ❇r♦♥st❡✐♥✱ ■♥t❡❣r❛t✐♦♥ ♦❢ ❡❧❡♠❡♥t❛r② ❢✉♥❝t✐♦♥s✱ ❏✳ ❙②♠❜♦❧✐❝ ❈♦♠♣✉t✳✱ ✾ ✭✶✾✾✵✮✱ ✶✶✼✲✶✼✸✳ ❬✹❪ ▼✳ ❇r♦♥st❡✐♥✱ ❙②♠❜♦❧✐❝ ■♥t❡❣r❛t✐♦♥ ■✿ ❚r❛♥s❝❡♥❞❡♥t❛❧ ❋✉♥❝t✐♦♥s✳ ❙❡❝♦♥❞ ❡❞✐✲ t✐♦♥✳ ❲✐t❤ ❛ ❢♦r❡✇♦r❞ ❜② ❇✳ ❋✳ ❈❛✈✐♥❡ss✳ ❱♦❧✳ ✶ ❆❧❣♦r✐t❤♠s ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✐♥ ▼❛t❤❡♠❛t✐❝s✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ❇❡r❧✐♥ ❍❡✐❞❡❧❜❡r❣✱ ✷✵✵✺✳ ❬✺❪ ❚✳ ❈r❡s♣♦✱ ❩✳ ❍❛❥t♦✱ ❆❧❣❡❜r❛✐❝ ❣r♦✉♣s ❛♥❞ ❞✐❢❢r❡♥t✐❛❧ ●❛❧♦✐s ❚❤❡♦r❡②✱ ●r❛❞✲ ✉❛t❡ ❙t✉❞✐❡s ✐♥ ▼❛t❤❡♠❛t✐❝s✱ ❱♦❧✳ ✶✷✷✱ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ ✭✷✵✶✶✮✳ ❬✻❪ ❏✳ ❈r✉③✲❙❛♠♣❡❞r♦✱ ▼✳ ❚❡t❧❛❧♠❛t③✐✲▼♦♥t✐❡❧✱ ❍❛r❞②✬s ❘❡❞✉❝t✐♦♥ ❢♦r ❛ ❈❧❛ss ♦❢ ▲✐♦✉✈✐❧❧❡ ■♥t❡❣r❛❧s ♦❢ ❊❧❡♠❡♥t❛r② ❋✉♥❝t✐♦♥s✱ ❚❤❡ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ▼♦♥t❤❧②✱ ❱♦❧✳ ✶✷✸✱ ◆♦✳ ✺ ✭▼❛② ✷✵✶✻✮✱ ♣♣✳ ✹✹✽✲✹✼✵✳ ❬✼❪ ❏✳ ❍✳ ❉❛✈❡♥♣♦rt✱❚❤❡ ❘✐s❝❤ ❞✐❢❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ♣r♦❜❧❡♠✱ ❙■❆▼ ❏✳ ❈♦♠♣✉t✳✱ ✶✺ ✭✶✾✽✻✮✱ ✾✵✸✲✾✶✽✳ ❬✽❪ P✳ ❉✐❛❝♦♥✐s✱ ❙✳ ❩❛❜❡❧❧✱ ❈❧♦s❡❞ ❢♦r♠ s✉♠♠❛t✐♦♥ ❢♦r ❝❧❛ss✐❝❛❧ ❞✐str✐❜✉t✐♦♥s✿ ✈❛r✐❛t✐♦♥s ♦♥ ❛ t❤❡♠❡ ♦❢ ❞❡ ▼♦✐✈r❡✱ ❙t❛t✐st✳ ❙❝✐✳✱ ✻ ✭✶✾✾✶✮✱ ✷✽❝♦♥t❡♥t ✹✲✸✵✷✳ ✹✼ ❬✾❪ ●✳ ❍✳ ❍❛r❞②✱ ❚❤❡ ■♥t❡❣r❛t✐♦♥ ♦❢ ❋✉♥❝t✐♦♥s ♦❢ ❛ ❙✐♥❣❧❡ ❱❛r✐❛❜❧❡ ✳ ❈❛♠❜r✐❞❣❡ ❚r❛❝ts ✐♥ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ▼❛t❤❡♠❛t✐❝❛❧ P❤②s✐❝s✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈✳ Pr❡ss✱ ❲❛r❡❤❛✉s❡✱ ❊♥❣❧❛♥❞✱ ✭✶✾✵✺✮✳ ❬✶✵❪ ❏✳ ▲✐♦✉✈✐❧❧❡✱ ❞❛♥t❡s✱ ▼➨♠♦✐r❡ s✉r ❧➼♥t❡❣r❛t✐♦♥ ❞ó♥❡ ❝❧❛ss❡ ❞❡ ❢♦♥❝t✐♦♥s tr❛♥s❝❡♥✲ ❏✳ ❘❡✐♥❡ ❆♥❣❡✇✳ ▼❛t❤✳✱ ✶✸ ✭✶✽✸✺✮✱ ✾✸✲✶✶✽✳ ❬✶✶❪ ❊✳ ❆✳ ▼❛r❝❤✐s♦tt♦✱ ●✳✲❆ ❩❛❦❡r✐✱ ❆♥ ✐♥✈✐t❛t✐♦♥ t♦ ✐♥t❡❣r❛t✐♦♥ ✐♥ ❢✐♥✐t❡ t❡r♠s✱ ❈♦❧❧✳ ▼❛t❤✳ ❏✳✱ ✷✺ ✭✶✾✾✹✮✱ ✷✾✺✲✸✵✽✳ ❬✶✷❪ ❉✳ ●✳ ▼❡❛❞✱ ❈❧❛ssr♦♦♠ ♥♦t❡s✿ ■♥t❡❣r❛t✐♦♥✱ ❆♠❡r✳ ▼❛t❤✳ ▼♦♥t❤❧②✱ ✻✽ ✭✶✾✻✶✮✱ ✶✺✷✲✶✺✻✳ ❬✶✸❪ ❘✳ ❍✳ ❘✐s❝❤✱ ❚❤❡ ♣r♦❜❧❡♠ ♦❢ ✐♥t❡❣r❛t✐♦♥ ✐♥ ❢✐♥✐t❡ t❡r♠s✱ ❚r❛♥s✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✶✸✾ ✭✶✾✻✾✮✱ ✶✻✼✲✶✽✾✳ ❬✶✹❪ ▼✳ ❘♦s❡♥❧✐❝❤t✱ ■♥t❡❣r❛t✐♦♥ ✐♥ ❢✐♥✐t❡ t❡r♠s✱ ❆♠❡r✳ ▼❛t❤✳ ▼♦♥t❤❧②✱ ✼✾ ✭✶✾✼✷✮✱ ✾✻✸✲✾✼✷✳ ❬✶✺❪ ●✳ ❋✳ ❙✐♠♠♦♥s✱ ❈❛❧❝✉❧✉s ✇✐t❤ ❆♥❛❧②t✐❝ ●❡♦♠❡tr②✱ ▼❝●r❛✇ ❍✐❧❧✱ ◆❡✇ ❨♦r❦✱ ✭✶✾✽✺✮✳ ❬✶✻❪ ▼✳ ❋✳ ❙✐♥❣❡r✱ ❇✳ ❉✳ ❙❛✉♥❞❡rs✱ ❇✳ ❋✳ ❈❛✈✐♥❡ss✱ ❆♥ ❡①t❡♥s✐♦♥ ♦❢ ▲✐♦✉✈✐❧❧❡✬s t❤❡♦r❡♠ ♦♥ ✐♥t❡❣r❛t✐♦♥ ✐♥ ❢✐♥✐t❡ t❡r♠s✱ ❙■❆▼ ❏✳ ❈♦♠♣✉t✳✱ ✶✹ ✭✶✾✽✺✮✱ ✾✻✻✲✾✾✵✳ ❬✶✼❪ P✳ ▲✳ ❚❝❤❡❜②❝❤❡❢✱ ■♥t➨❣r❛t✐♦♥ ❞❡s ❉✐❢❢➨r❡♥t✐❡❧❧❡s ■rr❛t✐♦♥♥❡❧❧❡s✱ ❖❡✉✈r❡s ❞❡ P✳ ▲✳ ❚❝❤❡❜②❝❤❡❢✱ ■♠♣r✐♠❡r✐❡❞❡ ❧⑩❝❛❞➨♠✐❡ ■♠♣➨r✐❛❧❡ ❞❡s ❙❝✐❡♥❝❡s✱ ❙t✳ P➨t❡rs✲ ❜♦✉r❣ ✶ ✭✶✽✾✾✮ ✶✹✼✲✶✻✽✳ ... ❇➻♥❤ ✣à♥❤ ✲ ✷✵✷✶ ❉❆◆❍ ▼Ư❈ ❈⑩❈ ❑Þ ❍■➏❯ R ✿ ❱➔♥❤ ✈✐ ♣❤➙♥ Q(x) rữớ tự ợ số ỳ t R(x) rữớ tự ợ số tỹ C(x) rữớ tự ợ số ự (2j − 1)!! = 1.3.5 (2j − 1) (2j)!! = 2.4 (2j) (3j − 2)!!! =... arctan x ❝ơ♥❣ ❧➔ ❤➔♠ sè ❝➜♣✳ ✶✳✷✳✷ ỵ ỵ F trữớ ợ số ∈ F✳ ◆➳✉ y = α ❝â ♥❣❤✐➺♠ y tr♦♥❣ ♠ð rë♥❣ ❝➜♣ ❝õ❛ F ✭✈ỵ✐ ❝→❝ ❤➡♥❣ sè ố t tỗ t số c1 , c2 , , cn ✈➔ ❝→❝ ♣❤➛♥ tû u1 , u2 , , un... b ✭tù❝ ❧➔ t❤ù❝✮✳ ❇➙② ❣✐í✱ ♥➳✉ k = 1✱ t❛ ❝â a b ab ✈➔ ❞♦ ✤â sè ❤↕♥❣ t❤ù ❤❛✐ ❦❤ỉ♥❣ t❤➸ rót ❣å♥ ✭s❛✉ b2 − ❦❤✐ rót ❣å♥ t❛ ❝â ❧ơ② t❤ø❛ ✤➛✉ t✐➯♥ ❝õ❛ F[t]✮✳ b ▼➔ b ❜➜t ❦❤↔ q✉② ❝❤♦ ♥➯♥ ♣❤↔✐ ❝❤✐❛ ❤➳t b✳