✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✕ ◆●❯❨➍◆ ❚❍➚ ▼❾◆ ❱➋ P❍×❒◆● ❚❘➐◆❍ ❍⑨▼ ❈❆❯❈❍❨ ❱⑨ Ù◆● ❉Ư◆● ❚❍⑩■ ◆●❯❨➊◆✱ ✺✴✷✵✶✼ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✕ ◆●❯❨➍◆ ❚❍➚ ▼❾◆ ❱➋ P❍×❒◆● ❚❘➐◆❍ ❍⑨▼ ❈❆❯❈❍❨ ❱⑨ Ù◆● ❉Ư◆● ❈❤✉②➯♥ ♥❣➔♥❤✿ P❤÷ì♥❣ ♣❤→♣ ❚♦→♥ ❝➜♣ ▼➣ sè✿ ✻✵ ✹✻ ✵✶ ✶✸ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ●■⑩❖ ❱■➊◆ ❍×❰◆● ❉❼◆ ❚❙✳ ❚❘❺◆ ❳❯❹◆ ◗❯Þ ❚❍⑩■ ◆●❯❨➊◆✱ ✺✴✷✵✶✼ ✶ ▼ư❝ ❧ư❝ ▼ð ✤➛✉ ✷ ❈❤÷ì♥❣ ✶✳ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ✹ ✶✳✶✳ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ♠ët ❜✐➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✶✳ ❱➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ❝ë♥❣ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✷✳ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❝ë♥❣ t➼♥❤ tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ♣❤ù❝ ✶✳✶✳✸✳ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ♠ơ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✹✳ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ▲♦❣❛r✐t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✺✳ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ♥❤➙♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳ P❤÷ì♥❣ tr➻♥❤ ❈❛✉❝❤② ♥❤✐➲✉ ❜✐➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✶✳ P❤÷ì♥❣ tr➻♥❤ ❈❛✉❝❤② ❝ë♥❣ t➼♥❤ ♥❤✐➲✉ ❜✐➳♥ ✳ ✳ ✳ ✶✳✷✳✷✳ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ♥❤➙♥ t➼♥❤ ♥❤✐➲✉ ❜✐➳♥ ✳ ✶✳✷✳✸✳ ❍❛✐ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ♥❤✐➲✉ ❜✐➳♥ ❦❤→❝ ✳ ✶✳✸✳ ▼ð rë♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✹✳ ▼ët sè ❜➔✐ t♦→♥ →♣ ❞ö♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✻ ✶✶ ✶✹ ✶✼ ✶✽ ✷✸ ✷✸ ✷✼ ✷✽ ✷✾ ✸✺ ❈❤÷ì♥❣ ✷✳ ▼ët sè ù♥❣ ❞ư♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ✸✼ ✷✳✶✳ ❚ê♥❣ ❝→❝ ❧ô② t❤ø❛ ❝õ❛ sè ♥❣✉②➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✳✶✳ ❚ê♥❣ ❝õ❛ n sè tü ♥❤✐➯♥ ✤➛✉ t✐➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✳✷✳ ❚ê♥❣ ❜➻♥❤ ♣❤÷ì♥❣ ❝õ❛ n sè tü ♥❤✐➯♥ ✤➛✉ t✐➯♥ ✷✳✶✳✸✳ ❚ê♥❣ ❧ô② t❤ø❛ k ❝õ❛ n sè tü ♥❤✐➯♥ ✤➛✉ t✐➯♥ ✳ ✷✳✷✳ ❚ê♥❣ ❧ô② t❤ø❛ ❝õ❛ ❝→❝ sè tr♦♥❣ ❞➣② ❝➜♣ sè ❝ë♥❣ ✳ ✳ ✳ ✷✳✸✳ ❙è ❝➦♣ ❝â t❤➸ rót r❛ tø n ♣❤➛♥ tû ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹✳ ❚ê♥❣ ❝õ❛ ❝❤✉é✐ ❤ú✉ ❤↕♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸✽ ✸✾ ✸✾ ✹✷ ✹✸ ✹✹ ❑➳t ❧✉➟♥ ✹✼ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✹✽ ✷ ▼ð ✤➛✉ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❧➔ ♠ët ♥❤→♥❤ ❝õ❛ t♦→♥ ❤å❝ ❤✐➺♥ ✤↕✐✱ tø ♥➠♠ ✶✼✹✼ ✤➳♥ ✶✼✺✵ ♥❤➔ t♦→♥ ❤å❝ ❏✳ ❉✬❆❧❡♠❜❡rt ✤➣ ❝æ♥❣ ❜è ✸ ❜➔✐ ❜→♦ ❧✐➯♥ q✉❛♥ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠✱ ✤➙② ✤÷đ❝ ①❡♠ ❧➔ ❝→❝ ❦➳t q✉↔ ✤➛✉ t✐➯♥ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠✳ ▼➦❝ ❞ị ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✤➣ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ tr➯♥ ✷✻✵ ♥➠♠✱ ♥❤÷♥❣ ♥â t❤ü❝ sü ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ tr ỹ ỵ tt ự ❝õ❛ t♦→♥ ❤å❝ ❝❤➾ ❦❤♦↔♥❣ ✶✵✵ ♥➠♠ trð ❧↕✐ ✤➙②✳ ✣➛✉ t❤➳ ❦✛ ✷✵✱ ❦➳ t✐➳♣ ♥❤ú♥❣ ✤â♥❣ ❣â♣ q✉❛♥ trồ rt tr ỵ tt ữỡ tr ỵ tt ữỡ tr tr ♥➯♥ r➜t q✉❛♥ trå♥❣ ✈➔ t❤✉ ✤÷đ❝ ♥❤✐➲✉ ❦➳t q✉↔ t❤ó ✈à✱ ❝❤➥♥❣ ❤↕♥ ♥❤÷ ❙✳ P✐♥❝❤❡r❧❡ ✭✶✾✵✻✱ ✶✾✶✷✮❀ ❊✳ P✐❝❛r❞ ✭✶✾✷✽✮❀ ●✳ ❍❛r❞②✱ ❏✳❊✳ ▲✐tt❧❡✇♦♦❞ ❛♥❞ ●✳ P♦❧②❛ ✭✶✾✸✹✮❀ ▼✳ ●❤❡r♠❛♥❡s❝✉ ✭✶✾✻✵✮❀ ❏✳❆❝✁③❡❧ ✭✶✾✻✻✮❀ ❛♥❞ ▼✳ ❑✉❝③♠❛ ✭✶✾✻✽✮✳ ●➛♥ ✤➙②✱ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✤÷đ❝ r➜t ♥❤✐➲✉ ♥❤➔ ❚♦→♥ ❤å❝ ♥ê✐ t✐➳♥❣ ❝õ❛ t❤➳ ❣✐ỵ✐ ♥❣❤✐➯♥ ❝ù✉✱ ✈➔ ❝â ♥❤ú♥❣ õ õ ợ t ỵ tt t♦→♥ ù♥❣ ❞ư♥❣✱ ❝❤➥♥❣ ❤↕♥ ♥❤÷ q✉❛ ❝→❝ ❝✉è♥ s→❝❤ ❝õ❛ ❆✳◆✳ ❙❛r❦♦✈s❦✐✐ ❛♥❞ ●✳P✳❘❡❧❥✉❝❤ ✭✶✾✼✹✮❀ ❏✳ ❆❝✁③❡❧ ❛♥❞ ❩✳ ❉❛✁r♦❝③② ✭✶✾✼✺✮❀ ❏✳ ❉❤♦♠❜r❡s ✭✶✾✼✾✮✳✳✳✳ ❈❤➼♥❤ sü ♣❤→t tr✐➸♥ ♠↕♥❤ ỵ tt ữỡ tr t q✉↔ ❝õ❛ ♥â ✤➣ ✤÷đ❝ ①❡♠ ①➨t ♥❣❤✐➯♥ ❝ù✉ ❝❤♦ ✤è✐ t÷đ♥❣ ❤å❝ s✐♥❤ tr✉♥❣ ❤å❝ ♣❤ê t❤ỉ♥❣✳ ❚❤➸ ❤✐➺♥ q✉❛ ❝→❝ ❦ý t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐ q✉è❝ ❣✐❛✱ ❝→❝ ❜➔✐ t♦➔♥ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❧✉ỉ♥ t❤✉ ❤ót ❇❚❈ q✉❛♥ t➙♠ ✈➔ ❧ü❛ ❝❤å♥✳ ❱➻ ✈➟②✱ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥ t❤↕❝ s➽ ♣❤÷ì♥❣ ♣❤→♣ t♦→♥ ❝➜♣ s➩ t➟♣ tr ợ ữỡ tr ỡ õ ❱➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ✈➔ ù♥❣ ❞ư♥❣✳ ▲✉➟♥ ✈➠♥ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❤❛✐ ❝❤÷ì♥❣✳ ✸ ❈❤÷ì♥❣ ✶✿ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ❝→❝ ✤à♥❤ ♥❣❤➽❛✱ ỵ ự ữỡ tr ❝→❝ ❞↕♥❣ ❝õ❛ ♥â✳ ❚➻♠ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ❝ë♥❣ t➼♥❤✱ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ♥❤➙♥ t➼♥❤✱ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ♠ơ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ▲♦❣❛r✐t✳ ❚r➻♥❤ ❜➔② ♠ð rë♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤②✳ ✣÷❛ r❛ ♠ët sè ❜➔✐ t♦→♥ ✈➟♥ ❞ư♥❣ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ❝ë♥❣ t➼♥❤ ✤➸ ❣✐↔✐ q✉②➳t✳ ▼ët sè ❜➔✐ t♦→♥ t s ọ ữợ ữủ tr tø t➔✐ ❧✐➺✉ ❬✾❪ ❝õ❛ t→❝ ❣✐↔ ❚✐t✉ ❆♥❞r❡❡s❝✉ ✈➔ ■✉r✐❡ ❇♦r❡✐❝♦✳ ❈❤÷ì♥❣ ✷✿ ▼ët sè ù♥❣ ❞ư♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ù♥❣ ❞ư♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② tr♦♥❣ t➼♥❤ tê♥❣ ❧ơ② t❤ø❛ ❝õ❛ sè ♥❣✉②➯♥ ✭tê♥❣ ❝õ❛ n sè tü ♥❤✐➯♥ ✤➛✉ t✐➯♥✱ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣ ❝õ❛ n sè tü ♥❤✐➯♥ ✤➙✉ t✐➯♥✱ tê♥❣ ❧ô② t❤ø❛ k ❝õ❛ n sè tü ♥❤✐➯♥ ✤➛✉ t✐➯♥✮✱ t➼♥❤ tê♥❣ ❧ô② t❤ø❛ ❝õ❛ ❝→❝ sè tr♦♥❣ ❞➣② ❝➜♣ sè ❝ë♥❣✱ t➻♠ sè ❝➦♣ ❝â t❤➸ rót r❛ tø n ♣❤➛♥ tû✱ ❧ü❝ ❧÷đ♥❣ ❝õ❛ ♠ët t➟♣ ❤đ♣ ✈➔ tê♥❣ ❝õ❛ ❝❤✉é✐ ❤ú✉ ❤↕♥✳ ✣➸ ❤♦➔♥ t❤✐➺♥ ❧✉➟♥ trữợ t tổ ỷ ỡ s s tợ r ỵ tớ ữợ t t ú ù tr♦♥❣ q✉→ tr➻♥❤ ①➙② ❞ü♥❣ ✤➲ t➔✐ ✈➔ ❤♦➔♥ t❤✐➺♥ ❧✉➟♥ ✈➠♥✳ ◗✉❛ ✤➙② tỉ✐ ❝ơ♥❣ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ tỵ✐ t➜t ❝↔ ❝→❝ t❤➛② ❝ỉ✱ ❇❛♥ ❣✐→♠ ❤✐➺✉✱ ❑❤♦❛ ❚♦→♥ ✲ ❚✐♥ ✲ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥✱ ❣✐ó♣ ✤ï tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ổ ữủ sỹ õ ỵ t ❝æ ✈➔ ❝→❝ ❜↕♥✳ ❚❤→✐ ◆❣✉②➯♥✱ ♥❣➔② ✵✺ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✼ ❚→❝ ❣✐↔ ❧✉➟♥ ✈➠♥ ❍å❝ ✈✐➯♥ ◆❣✉②➵♥ ❚❤à ▼➟♥ ✹ ❈❤÷ì♥❣ ✶ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ❱✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ❤➔♠ ❝ë♥❣ t➼♥❤ ❝â tø t❤í✐ ❆✳▼✳ ▲❡❣❡♥❞r❡ ❧➔ ♥❣÷í✐ ✤➛✉ t✐➯♥ ❝è ❣➢♥❣ t➻♠ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② f (x + y) = f (x) + f (y) ✈ỵ✐ ♠å✐ x, y ∈ R✳ ❱✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❤➺ t❤è♥❣ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ❝ë♥❣ t➼♥❤ ữủ ữợ tr ố s ❝õ❛ ỉ♥❣ ✧❈♦✉rs❞ ❞✬❆♥❛❧②s❡✧ ♥➠♠ ✶✽✷✶✳ ▼ët ♣❤÷ì♥❣ tr➻♥❤ ❜❛♦ ỗ ởt ữ t ởt ❤➔♠ ❝õ❛ ♥â ✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✳ ❱➼ ❞ư ♥❤÷ f (x) + mx = ✈➔ f (x) + f (x) + sin(x) = ❈→❝ ữỡ tr ỗ t số ữ t ✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥✳ ▼ët ✈➔✐ ✈➼ ❞ư ✈➲ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ x f (x) = ex − ex−t f (t) dt, [1 − xcos(xt)]f (t)dt, f (x) = sin(x) + ✺ ✈➔ x [tf (t) − 1]dt f (x) = P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ tr♦♥❣ ✤â ❝→❝ ➞♥ ❧➔ ❝→❝ ❤➔♠ sè✳ ❱➼ ❞ư ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❧➔ f (x + y) = f (x) + f (y), f (x + y) = f (x)f (y), f (xy) = f (x)f (y), f (xy) = f (x) + f (y), f (x + y) = f (x)g(y) + g(x)f (y), f (x + y) + f (x − y) = 2f (x)f (y), f (x + y) + f (x − y) = 2f (x) + 2f (y), f (x + y) = f (x) + f (y) + f (x)f (y), f (x + y) = g(xy) + h(x − y), f (x) − f (y) = (x − y)h(x + y), f (pr, qs) + f (ps, qr) = 2f (p, q) + 2f (r, s), g(f (x)) = g(x) + β, g(f (x)) = αg(x), α = ✈➔ f (t) = f (2t) + f (2t − 1) P❤↕♠ ✈✐ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ỗ ữỡ tr ữỡ tr s❛✐ ♣❤➙♥✱ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥✳✳✳✳ ❈→❝ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❧➔ ♠ët ❧➽♥❤ ✈ü❝ ❝õ❛ t♦→♥ ❤å❝ tr➯♥ ✷✵✵ ♥➠♠ t✉ê✐✳ ❍ì♥ ✺✵✵✵ ❜➔✐ ❜→♦ ✤➣ ✤÷đ❝ ❝ỉ♥❣ ❜è tr♦♥❣ ❧➽♥❤ ✈ü❝ ♥➔②✳ ❚✉② ♥❤✐➯♥ ✤è✐ ✈ỵ✐ ❧✉➟♥ ✈➠♥ t❤↕❝ s➽ tỉ✐ ❝❤➾ t➟♣ tr✉♥❣ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ✈➔ ♠ët sè ù♥❣ ❞ö♥❣ ❝õ❛ ♥â✳ ◆➠♠ ✶✼✹✼ ✈➔ ✶✼✺✵✱ ❞✬❆❧❛♠❜❡rt ✤➣ ❝æ♥❣ ❜è ✸ ❜➔✐ ❜→♦ tr♦♥❣ ✤â ❜➔✐ t❤ù ♥❤➜t ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✭①❡♠ ❆❝③➨❧ ✭✶✾✻✻✮✮✳ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ❜ð✐ ❞✬❆❧❛♠❜❡rt ✭✶✼✹✼✮✱ ❊✉❧❡r ✭✶✼✻✽✮✱ P♦✐ss♦♥ ✭✶✽✵✹✮✱ ❈❛✉❝❤② ✭✶✽✷✶✮✱ ❉❛r❜♦✉① ✭✶✽✼✺✮ ✈➔ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ ❦❤→❝✳ ❍✐❧❜❡rt ✻ ✭✶✾✵✷✮ ✤➲ ①✉➜t tr♦♥❣ sü ♥è✐ t✐➳♣ ✈ỵ✐ ✈➜♥ ✤➲ ✺ ❝õ❛ ỉ♥❣ ỵ ữỡ ✤➭♣ ✈➔ ♠↕♥❤ ✤➸ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠✱ tr♦♥❣ ✤â ❣✐↔ t❤✐➳t ❦❤↔ ✈✐ ❧➔ ✤✐➲✉ ❦✐➺♥ ❦❤æ♥❣ t❤➸ t❤✐➳✉✳ ◆❤í ✤➲ ①✉➜t ❝õ❛ ❍✐❧❜❡rt ♥❤✐➲✉ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ♣❤÷ì♥❣ tr t ợ ữỡ tr ❦❤→❝ ♥❤❛✉ ❦❤æ♥❣ ❝â ♠ët ✈➔✐ ❤♦➦❝ ➼t ❝→❝ ❣✐↔ t❤✐➳t ✤➲✉✳ ❙ü ♥é ❧ü❝ ♥➔② ✤➣ ❣â♣ ♣❤➛♥ ♣❤→t tr ỵ ữỡ tr ỵ t❤✉②➳t ❝→❝ ❞↕♥❣ q✉② t➢❝ t♦→♥ ❤å❝ ❤✐➺♥ ✤↕✐ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ♥❣➔② ❝➔♥❣ ♣❤→t tr✐➸♥ ♥❤❛♥❤ ❝❤â♥❣ ð ❝✉è✐ t❤➟♣ ❦➾ ✻✳ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ♥❣❤➽❛ ❧➔ t➻♠ t➜t ❝↔ ❝→❝ ❤➔♠ sè t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠✳ ✣➸ t❤✉ ✤÷đ❝ ♠ët ♥❣❤✐➺♠✱ ❝→❝ ❤➔♠ sè ♣❤↔✐ ợ ởt t trữ r ữ t tử ỗ ✤÷đ❝ ❤❛② ✤ì♥ ✤✐➺✉✮✳ ✶✳✶✳ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ♠ët ❜✐➳♥ ✶✳✶✳✶✳ ❱➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ❝ë♥❣ t➼♥❤ P❤➛♥ ợ t ữỡ tr t ✈➔ ①→❝ ✤à♥❤ ♥❣❤✐➺♠ ❝õ❛ ♥â ✭✤÷đ❝ tr➼❝❤ tø t➔✐ ❧✐➺✉❬✼❪✮✳ ❈❤♦ f : R → R tr♦♥❣ ✤â R ❧➔ t➟♣ sè t❤ü❝✱ f ❧➔ ❤➔♠ sè t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ f (x + y) = f (x) + f (y) ✭✶✳✶✮ ✈ỵ✐ ♠å✐ x, y ∈ R✳ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ♥➔② ✤➣ ✤÷đ❝ ❜✐➳t ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤②✳ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✭✶✳✶✮ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ✤➛✉ t✐➯♥ ❜ð✐ ❆✳▼✳ ▲❡❣❡♥❞r❡ ✭✶✼✾✶✮ ✈➔ ❈✳❋✳ ●❛✉ss ✭✶✽✵✾✮ ♥❤÷♥❣ ❆✳▲✳ ❈❛✉❝❤② ✭✶✽✷✶✮ ❧➔ ♥❣÷í✐ ✤➛✉ t✐➯♥ t➻♠ r❛ ♥❣❤✐➺♠ tr ợ tử Pữỡ tr õ tr➼ q✉❛♥ trå♥❣ tr♦♥❣ t♦→♥ ❤å❝ ♥â ✤÷đ❝ ✤➲ ❝➟♣ tỵ✐ tr♦♥❣ ❤➛✉ ❤➳t ❝→❝ ❦❤➼❛ ❝↕♥❤ ❝õ❛ t♦→♥ ❤å❝✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶ ❍➔♠ sè f : R → R ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ❝ë♥❣ t➼♥❤ ♥➳✉ ♥â t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② f (x + y) = f (x) + f (y) ✈ỵ✐ ♠å✐ x, y ∈ R ✣à♥❤ ♥❣❤➽❛ ✶✳✷ ❍➔♠ sè f : R → R ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ t✉②➳♥ t➼♥❤ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ♥â ❝â ❞↕♥❣ f (x) = cx (∀x ∈ R), tr õ ởt số tũ ỵ ỗ t t t f (x) = cx ❧➔ ♠ët ✤÷í♥❣ ❦❤ỉ♥❣ t❤➥♥❣✱ ✤✐ q✉❛ ❣è❝ ❞♦ ✤â ♥â ✤÷đ❝ ❣å✐ ❧➔ t✉②➳♥ t➼♥❤✳ ❍➔♠ sè t✉②➳♥ t➼♥❤ t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤②✳ ❈→❝ ❝➙✉ ❤ä✐ ✤÷đ❝ ✤÷❛ r❛ ❧➔ ❝â ❤➔♠ ♥➔♦ ❦❤→❝ t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ❤❛② ❦❤æ♥❣❄ ❚❛ t❤➜② r➡♥❣ ❝❤➾ ❝â ♥❣❤✐➺♠ ❧✐➯♥ tư❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ❧➔ t✉②➳♥ t➼♥❤✳ ✣➙② ❧➔ ❦➳t q✉↔ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ❜ð✐ ❈❛✉❝❤② ỵ f : R → R ❧➔ ❧✐➯♥ tư❝ ✈➔ t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ❝ë♥❣ t➼♥❤ ✭✶✳✶✮✳ ❑❤✐ ✤â f t✉②➳♥ t➼♥❤✱ ♥❣❤➽❛ ❧➔ f (x) = cx tr♦♥❣ ✤â c ❧➔ ởt số tũ ỵ ự rữợ t t ố x rỗ t ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✮ t❤❡♦ ❜✐➳♥ y t❛ ✤÷đ❝ f (x) = f (x)dy [f (x + y) − f (y)]dy = 1+x f (u)du − = x f (y)dy, u = x + y ❱➻ ❤➔♠ sè f ❧✐➯♥ tö❝ ♥➯♥ s✉② r❛ f (x) = f (1 + x) − f (x) ✭✶✳✷✮ ❚ø t➼♥❤ ❝ë♥❣ t➼♥❤ ❝õ❛ f t❛ ❝â f (1 + x) = f (1) + f (x) ✭✶✳✸✮ ❚❤❛② ✭✶✳✸✮ ✈➔♦ ✭✶✳✷✮ t❛ ❝â f (x) = f (1) = c ❙✉② r❛ f (x) = cx + d t❤❛② ✈➔♦ ✭✶✳✶✮ s✉② r❛ d = 0✳ ❚r♦♥❣ ỵ t sỷ t tử f ✤➸ ❦➳t ❧✉➟♥ r➡♥❣ f ❦❤↔ t➼❝❤✳ ❚➼♥❤ t➼❝❤ ♣❤➙♥ ❝õ❛ f ❜➢t ❜✉ë❝ ♥❣❤✐➺♠ f ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✽ ❈❛✉❝❤② ❝ë♥❣ t➼♥❤ ❧➔ t✉②➳♥ t➼♥❤✳ ❉♦ ✤â ♠é✐ ♥❣❤✐➺♠ ❦❤↔ t➼❝❤ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❈❛✉❝❤② ❝ë♥❣ t➼♥❤ ❝ơ♥❣ t✉②➳♥ t➼♥❤✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✸ ▼ët ❤➔♠ f : R → R ✤÷đ❝ ❣å✐ ❧➔ ❦❤↔ t➼❝❤ ✤à❛ ♣❤÷ì♥❣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ♥â ❧➔ t➼❝❤ ♣❤➙♥ tr➯♥ ♠å✐ ❦❤♦↔♥❣ ❤ú✉ ❤↕♥✳ ❚❤❡♦ tr➯♥ ♠é✐ ♥❣❤✐➺♠ ❦❤↔ t➼❝❤ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❈❛✉❝❤② ❝ë♥❣ t➼♥❤ ❝ơ♥❣ ❧➔ t✉②➳♥ t➼♥❤✳ ❚❛ ✤÷❛ r❛ ♠ët ❝→❝❤ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ✤÷❛ r❛ ❜ð✐ ❙❤❛♣✐r♦ ✶✾✼✸✳ ●✐↔ sû f ❧➔ ♠ët ♥❣❤✐➺♠ ❦❤↔ t➼❝❤ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❈❛✉❝❤② ❝ë♥❣ t➼♥❤✳ ❉♦ ✤â f (x + y) = f (x) + f (y) ✤ó♥❣ ✈ỵ✐ ♠å✐ x, y ∈ R✳ ❚ø ✤â sû ❞ư♥❣ t➼♥❤ ❦❤↔ t➼❝❤ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ f t❛ ✤÷đ❝ y f (x)dz yf (x) = y [f (x + z) − f (z)]dz = x+y y f (u)du − = x x+y f (z)dz f (u)du − = y x f (u)du − f (u)du ❱➳ ♣❤↔✐ ❝õ❛ ✤➥♥❣ t❤ù❝ tr➯♥ ❜➜t ❜✐➳♥ ❦❤✐ t❛ t❤❛② ✤ê✐ ✈❛✐ trá ❝õ❛ x ✈➔ y tø ✤â s✉② r❛ yf (x) = xf (y) ✈ỵ✐ ♠å✐ x, y ∈ R✳ ❉♦ ✤â ✈ỵ✐ x = t ữủ f (x) = c, x ợ c ❧➔ ♠ët ❤➡♥❣ ❜➜t ❦ý✳ ✣✐➲✉ ♥➔② s✉② r❛ f (x) = cx ✈ỵ✐ ♠å✐ x ∈ R \ {0}✳ ❈❤♦ x = ✈➔ y = ð ✭✶✳✶✮ t❛ ✤÷đ❝ f (0) = 0✳ ◆❤÷ ✈➟② f ❧➔ ♠ët ❤➔♠ t✉②➳♥ t➼♥❤ tr➯♥ R✳ ▼➦❝ ❞ò ❝❤ù♥❣ ỵ ỗ ❝→❝ ♣❤➨♣ t➼♥❤ ✈✐ ♣❤➙♥✱ t➼❝❤ ♣❤➙♥ ♥❤÷♥❣ ♥â ❧↕✐ ❦❤ỉ♥❣ ❤✐➺✉ q✉↔ ❝❛♦ ✈➔ ❝â ♥❤✐➲✉ ❦✐➳♥ t❤ù❝✳ ●✐í t❛ s➩ tr➻♥❤ ❜➔② ♠ët ❝→❝❤ ❝❤ù♥❣ ♠✐♥❤ ❦❤→❝ s➩ ❣✐ó♣ t❛ ❤✐➸✉ ❤ì♥ ✈➲ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❈❛✉❝❤② ❝ë♥❣ t➼♥❤✳ ❚❛ ①➨t ✤à♥❤ ♥❣❤➽❛ s❛✉✳ ✸✹ ❚r÷í♥❣ ❤đ♣ ✶✳ ●✐↔ sû s + t t❤✉ë❝ [0, ) ①➨t m n +s + +t 2 = A (m + n) + s + t = (m + n)f + f (s + t ) 1 + nf + f (s ) + f (t ) = mf 2 1 = mf + f (s ) + nf + f (t ) 2 = A(s) + A(t) A(s + t) = A ❚r÷í♥❣ ❤đ♣ ✷✳ ●✐↔ sû s + t t❤✉ë❝ [ , 1) t❤➻ s +t = t↕✐ ✤â z ∈ 0, ✳ ❉♦ ✈➟② m n + s + + t = A (m + n) + s + t 2 1 A (m + n) + + z = A (m + n + 1) + z 2 1 (m + n + 1)f + f (z ) = (m + n)f +f + f (z ) 2 1 +f + z = (m + n)f + f (s + t ) (m + n)f 2 (m + n)f + f (s ) + f (t ) 1 + f (s ) + nf + f (t ) mf 2 m n A +s +A +t 2 A(s) + A(t) A(s + t) = A = = = = = = = +z ❱➟② A ❧➔ ❝ë♥❣ t➼♥❤ tr➯♥ R✳ ✸✺ ✶✳✹✳ ▼ët sè ❜➔✐ t♦→♥ →♣ ❞ö♥❣ ❚r♦♥❣ ♠ö❝ ♥➔② t❛ ✤÷❛ r❛ ♠ët sè ❜➔✐ t♦→♥ ✈➟♥ ❞ư♥❣ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ❝ë♥❣ t➼♥❤ ✤➸ ❣✐↔✐ q✉②➳t✳ ▼ët sè ❜➔✐ t♦→♥ ❧➔ ✤➲ t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐ ❝→❝ ữợ ữủ tr tứ t t ❚✐t✉ ❆♥❞r❡❡s❝✉ ✈➔ ■✉r✐❡ ❇♦r❡✐❝♦✳ ❇➔✐ t♦→♥ ✶✳✶ ✭❆▼▼ ✷✵✵✶✮ ❚➻♠ t➜t ❝↔ ❝→❝ ❤➔♠ sè f : R → R t❤ä❛ ♠➣♥ f (x2 + y + f (y)) = 2y + f (x) ✈ỵ✐ ♠å✐ sè t❤ü❝ x, y ∈ R✳ ❇➔✐ t♦→♥ ✶✳✷ ❚➻♠ t➜t ❝↔ ❝→❝ ❤➔♠ sè f, g, h : R → R s❛♦ ❝❤♦ f (x + y) = f (x)g(y) + h(y) ✈ỵ✐ ♠å✐ sè t❤ü❝ x, y ∈ R✳ ❇➔✐ t♦→♥ ✶✳✸ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ♠å✐ ❤➔♠ ❝ë♥❣ t➼♥❤ f tr R+ ữợ tr tr ởt R+ ❝â ❞↕♥❣ f (x) = f (1)x ✈ỵ✐ ♠å✐ x ∈ R+ ❇➔✐ t♦→♥ ✶✳✹ ✭❚✉②♠❛❛❞❛ ✷✵✵✸✮ ❚➻♠ t➜t ❝↔ ❝→❝ ❤➔♠ sè f : R+ → R t❤ä❛ ♠➣♥ 1 1 f (x + ) + f (y + ) = f (x + ) + f (y + ) x y y x ✈ỵ✐ ♠å✐ x, y ∈ R+ ✳ ❇➔✐ t♦→♥ ✶✳✺ ✭❙❛♥❦t✲P❡t❡rs❜✉r❣✮ ❚➻♠ ♠å✐ ❤➔♠ sè f : R → R t❤ä❛ ♠➣♥ f (f (x + y)) = f (x) + f (y) ✈ỵ✐ ♠å✐ sè t❤ü❝ x, y ∈ R✳ ❇➔✐ t♦→♥ ✶✳✻ ❚➻♠ t➜t ❝↔ ❝→❝ ❝➦♣ ❝õ❛ ❤➔♠ sè f, g : R → R t❤ä❛ ♠➣♥ f (x) + f (y) = g(x + y) ❇➔✐ t♦→♥ ✶✳✼ ❚➻♠ t➜t ❝↔ ❝→❝ ❤➔♠ sè f : N → N t❤ä❛ ♠➣♥ f (m2 + f (n)) = f (m)2 + n ✈ỵ✐ ♠å✐ sè t❤ü❝ m, n ∈ N✳ ✸✻ ❇➔✐ t♦→♥ ✶✳✽ ❚➻♠ t➜t ❝↔ ❝→❝ ❤➔♠ sèf : R → R t❤ä❛ ♠➣♥ f (f (x) + yz) = x + f (y)f (z) ✈ỵ✐ ♠å✐ sè t❤ü❝ x, y, z ∈ R✳ ❇➔✐ t♦→♥ ✶✳✾ ❚➻♠ t➜t ❝↔ ❝→❝ ❤➔♠ sè f : R → R s❛♦ ❝❤♦ f (f (x)2 + y) = x2 + f (y) ❇➔✐ t♦→♥ ✶✳✶✵ ✭❇✉❧❣❛r✐❛ ✷✵✵✹✮ ❚➻♠ t➜t ❝↔ ❝→❝ ❤➔♠ sè f : R → R t❤ä❛ ♠➣♥ (f (x) − f (y))f x+y x−y = f (x) + f (y) ✈ỵ✐ ♠å✐ sè t❤ü❝ x, y ∈ R ✈➔ x = y ✳ ❇➔✐ t♦→♥ ✶✳✶✶ ✭■♥❞✐❛ ✷✵✵✸✮ ❚➻♠ t➜t ❝↔ ❝→❝ ❤➔♠ sè f : R → R t❤ä❛ ♠➣♥ f (x + y) + f (x)f (y) = f (x) + f (y) + f (xy) ✈ỵ✐ ♠å✐ sè t❤ü❝ x, y ∈ R✳ ✸✼ ❈❤÷ì♥❣ ✷ ▼ët sè ù♥❣ ❞ư♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ❚r♦♥❣ ♣❤➛♥ ♥➔②✱ t❛ tr➻♥❤ ❜➔② ♠ët ✈➔✐ ù♥❣ ❞ư♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ✤÷đ❝ tr➼❝❤ tø t➔✐ ❧✐➺✉ ❬✼❪✳ ❙û ❞ư♥❣ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ❝ë♥❣ t➼♥❤ ①→❝ ✤à♥❤ tê♥❣ ❧ô② t❤ø❛ k ❝õ❛ n sè tü ♥❤✐➯♥ ✤➛✉ t✐➯♥ ✈ỵ✐ k = 1, 2, 3✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ sè ❝➦♣ ❝â t❤➸ tr♦♥❣ sè n ♣❤➛♥ tû ❝â t❤➸ ✤÷đ❝ ①→❝ ✤à♥❤ sû ❞ư♥❣ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ❝ë♥❣ t➼♥❤✳ ❍ì♥ ♥ú❛✱ t❛ sû ❞ư♥❣ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ❝ë♥❣ t➼♥❤ ✤➸ t➻♠ tê♥❣ ❝õ❛ ❝❤✉é✐ ❤ú✉ ❤↕♥✳ ✷✳✶✳ ❚ê♥❣ ❝→❝ ❧ô② t❤ø❛ ❝õ❛ sè ♥❣✉②➯♥ ✣➦t fk (n) = 1k + 2k + + nk ợ n số ữỡ ✈➔ k ❧➔ sè ♥❣✉②➯♥ ❦❤æ♥❣ ➙♠✳ fk (n) ❧➔ ỵ tờ ụ tứ tự k n sè tü ♥❤✐➯♥ ✤➛✉ t✐➯♥✳ ❚➻♠ ❝æ♥❣ t❤ù❝ ❝õ❛ fk (n) ✤➣ t❤✉ ❤ót sü q✉❛♥ t➙♠ ❝õ❛ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ ❦❤♦↔♥❣ t❤í✐ ❣✐❛♥ ❤ì♥ ✸✵✵ ♥➠♠✱ ❜➢t ✤➛✉ tø t❤í✐ ❝õ❛ ❏❛♠❡s ❇❡r♥♦✉❧❧✐ ✭✶✻✺✺✲✶✼✵✺✮✳ ❈â ♥❤✐➲✉ ♣❤÷ì♥❣ ♣❤→♣ ❦❤→❝ ♥❤❛✉ ✤➣ ✤÷đ❝ sû ❞ư♥❣ ✤➸ t➻♠ tê♥❣ fk (n) ✭❝❤➥♥❣ ❤↕♥ ❱❛❦✐❧ ✭✶✾✾✻✮✮✳ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✱ t❛ s➩ ✈➟♥ ❞ư♥❣ ♣❤÷ì♥❣ ✸✽ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ✤➸ t➼♥❤ tê♥❣ fk (n) ✈ỵ✐ k = 1, 2, ✈➔ ợ k tũ ỵ ú ỵ r fk : N → N ❧➔ ❤➔♠ sè tr♦♥❣ ✤â k = 0, 1, 2, ✷✳✶✳✶✳ ❚ê♥❣ ❝õ❛ n sè tü ♥❤✐➯♥ ✤➛✉ t✐➯♥ ❈❤♦ ❤➔♠ f1 t❤ä❛ ♠➣♥ f1 (m + n) = + + + + m + (m + 1) + + (m + n) = f1 (m) + (m + 1) + (m + 2) + + (m + n) ✭✷✳✷✮ = f1 (m) + f1 (n) + mn ✈ỵ✐ ♠å✐ m, n ∈ N ❳➨t ❤➔♠ sè g1 : N → R ①→❝ ✤à♥❤ ❜ð✐ g1 (x) = f1 (x) − x2 ✈ỵ✐ x ∈ N ✭✷✳✸✮ ❑❤✐ ✤â✱ tø ✭✷✳✷✮ t❛ ❝â g1 (m + n) = g1 (m) + g1 (n), ✈ỵ✐ m, n ∈ N ✭✷✳✹✮ ◆❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ❝ë♥❣ t➼♥❤ ✭✷✳✹✮ tr➯♥ N ✤÷đ❝ ❝❤♦ ❜ð✐ g1 (n) = cn, ✭✷✳✺✮ tr♦♥❣ ✤â c ❧➔ ♠ët ❤➡♥❣ sè✳ ❚ø ✭✷✳✺✮ ✈➔ ✭✷✳✸✮ t❛ ❝â f1 (n) = cn + n2 ❉♦ f1 (1) = t❛ ❝â 1=c+ tr♦♥❣ ✤â c=1− ✤â ❧➔ ✭✷✳✻✮ 1 = 2 n n2 f1 (n) = + 2 n(n + 1) = ❱➟② f1 (n) = + + + + n = n(n + 1) ✸✾ ✷✳✶✳✷✳ ❚ê♥❣ ❜➻♥❤ ♣❤÷ì♥❣ ❝õ❛ n sè tü ♥❤✐➯♥ ✤➛✉ t✐➯♥ ❳➨t ❤➔♠ f2 t❤ä❛ ♠➣♥ f2 (m + n) = 12 + 22 + + m2 + (m + 1)2 + + (m + n)2 = f2 (m) + [12 + 22 + + n2 ] + 2m[1 + + + n] + m2 n = f2 (m) + f2 (n) + 2mf1 (n) + m2 n = f2 (m) + f2 (n) + mn2 + m2 n + mn ✭✷✳✼✮ ✈ỵ✐ ♠å✐ m, n ∈ N✳ ✣à♥❤ ♥❣❤➽❛ g2 : N → R ❜ð✐ n2 n3 g2 (n) = f2 (n) − − , ✈ỵ✐ n ∈ N ❚ø ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✼✮ t❛ ❝â g2 (m + n) = g2 (m) + g2 (n) ✈ỵ✐ ♠å✐ m, n ∈ N✳ ❙✉② r❛ g2 (n) = cn ❤❛② n2 n3 + f2 (n) = cn + ✭✷✳✽✮ ❑➳t ❤đ♣ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ f2 (1) = t❛ ❝â 1=c+ 1 − =⇒ c = ❙✉② r❛ t❛ ❝â tê♥❣ ❝➛♥ t➻♠ ❧➔ n n2 n3 n + 3n2 + 2n3 n(n + 1)(2n + 1) f2 (n) = + + = = 6 ✷✳✶✳✸✳ ❚ê♥❣ ❧ô② t❤ø❛ k ❝õ❛ n số tỹ t ợ k tũ ỵ sû ❞ö♥❣ ❦❤❛✐ tr✐➸♥ ♥❤à t❤ù❝ ◆❡✇t♦♥ ①→❝ ✤à♥❤ ❤➔♠✱ tt q tr ỗ t fk ♥❤÷ s❛✉ fk (n + m) = 1k + 2k + + nk + (n + 1)k + + (n + m)k k k Cki ni 1k−i = fk (n) + Cki ni mk−i + + i=0 k i=0 Cki ni [1k−i + + mk−i ] = fk (n) + i=0 k Cki ni fk−i (m) = fk (n) + i=0 k Cki ni fk−i (m) ✈ỵ✐ m, n, k ∈ N = fk (n) + fk (m) + i=1 ❉♦ ✤â✱ t❛ ❝â k Cki ni fk−i (m) ✈ỵ✐ m, n, k ∈ N ✭✷✳✾✮ fk (m + n) − fk (m) − fk (n) = i=1 t ữủ ổ tự tr ỗ ❚❛ s➩ ①➨t ♠ët ✈➔✐ tr÷í♥❣ ❤đ♣ ❝ư t❤➸ ✤è✐ ợ k ú ỵ r fk (1) = ✈ỵ✐ ♠å✐ k ∈ N ✈➔ f0 (m) = m✳ ✭❆✮ ❚ø ❝æ♥❣ t❤ù❝ ✭✷✳✾✮✱ ①➨t n = 1✳ ❚❛ ❝â k Cki fk−i (m), fk (m + 1) − fk (m) − fk (1) = i=1 ♥❣❤➽❛ ❧➔ k Cki fk−i (m) ✈ỵ✐ m ∈ N, k (m + 1) − = i=1 ✕ ❱ỵ✐ k = t❛ ✤÷đ❝ m2 + 2m = 2f1 (m) + f0 (m) = 2f1 (m) + m ❤❛② f1 (m) = m(m + 1) ✹✶ ✕ ❱ỵ✐ k = t❛ ✤÷đ❝ m3 + 3m2 + 3m = 3f2 (m) + 3f1 (m) + f0 (m) 3m(m + 1) = 3f2 (m) + +m ❤❛② f2 (m) = m(m + 1)(2m + 1) ✭❇✮ ❚r÷í♥❣ ❤đ♣ tê♥❣ q✉→t✳ ❱➳ ♣❤↔✐ ❝õ❛ ✭✷✳✾✮ ❧➔ ✤è✐ ①ù♥❣ ✈ỵ✐ t÷ì♥❣ ù♥❣ ❝õ❛ m ✈➔ n✳ ❉♦ ✤â t❛ t❤✉ ✤÷đ❝ k k Cki ni fk−i (m) Cki mi fk−i (n) ✈ỵ✐ m, n ∈ N = i=1 i=1 ❚❤❛② t❤➳ ✈ỵ✐ m = ✈➔ sû ❞ư♥❣ fk (1) = t❛ ❝â k k Cki ni fk−i (1) Cki ni fk−i (n), = i=1 i=1 ♥❣❤➽❛ ❧➔ k Cki fk−i (n) = (1 + n)k − i=1 ❚ø ✤â t❛ ❝â k Cki fk−i (n) ✈ỵ✐ n ∈ N k kfk−1 (n) = (1 + n) − − i=2 ❤❛② fk−1 (n) = k i i i=2 Ck n fk−i (n) (1 + n)k − − k ✈ỵ✐ k, n ∈ N ❙û ❞ư♥❣ f0 (n) = n t❛ ❝â t❤➸ ①→❝ ✤à♥❤ ✤÷đ❝ fk n✳ ❈❤➥♥❣ ❤↕♥ ✈ỵ✐ k = tr♦♥❣ ✭✷✳✶✵✮ t❛ ❝â n2 + 2n − f0 (n) n(n + 1) f1 (n) = = 2 ✭✷✳✶✵✮ ✹✷ ❚÷ì♥❣ tü k = tr♦♥❣ ✭✷✳✶✵✮ ❝❤♦ n3 + 3n2 + 3n − 3f1 (n) − f0 (n) f2 (n) = 3 n n + n + 2 = n(n + 1)(2n + 1) = ✷✳✷✳ ❚ê♥❣ ❧ô② t❤ø❛ ❝õ❛ ❝→❝ sè tr số ợ số ữỡ n, k ∈ N ✈➔ h ∈ R ✤à♥❤ ♥❣❤➽❛ sk (n, h) = 1k + (1 + h)k + + (1 + (n − 1)h)k ), ✭✷✳✶✶✮ tê♥❣ ❝õ❛ ❝→❝ sè tü ♥❤✐➯♥ ❜➟❝ k tr♦♥❣ ❝➜♣ sè ❝ë♥❣✳ ố ữ trữợ õ t ởt q tr t♦→♥ sk (m + n; h) = 1k + (1 + h)k + · · · + (1 + (n − 1)h)k +(1 + nh)k + · · · + (1 + (m + n − 1)h)k = sk (n; h) + (1 + nh)k + (1 + h + nh)k + · · · +(1 + (m − 1)h + nh)k k Cki sk−i (m; h)(nh)i ; = sk (n; h) + sk (m; h) + i=1 ♥❣❤➽❛ ❧➔ sk (n, h) t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ k Cki sk−i (m; h)(nh)i ; sk (m + 1; h) − sk (n; h) + sk (m; h) + ✭✷✳✶✷✮ i=1 ✈ỵ✐ k ∈ N, h ∈ R, m, n = 1, 2, ú ỵ r s0 (n, h) = n, sk (1, h) = t❛ ①→❝ ✤à♥❤ s1 (n, h) ✈➔ s2 (n, h)✳ ✣➛✉ t✐➯♥ ❝❤♦ n = tr♦♥❣ ✭✷✳✶✷✮ t❛ t❤✉ ✤÷đ❝ k Cki sk−i (m; h)(h)i sk (m + 1; h) − sk (m; h) = sk (1; h) + i=1 ❞♦ ✤â k k Cki sk−i (m; h)(h)i (1 + mh) = + i=1 ✭✷✳✶✸✮ ✹✸ ✈ỵ✐ m = 1, 2, , h ∈ R, k ∈ N✳ ❚÷ì♥❣ tü k = tr♦♥❣ ✭✷✳✶✸✮ ❝❤♦ m2 h2 + 2mh = 2s1 (m; h)h + s0 (m; h)h2 ; ❞♦ ✤â s1 (m; h) = 1− h h m + m2 2 ❈❤♦ k = tr♦♥❣ ✭✷✳✶✸✮ t❛ ❝â m3 h3 + 3m2 h2 + 3mh = 3s2 (m; h)h + 3s1 (m; h)h2 + s0 (m; h)h3 ; ❞♦ ✤â s2 (m; h) = h2 1−h+ h h2 m+h 1− m + m ✷✳✸✳ ❙è ❝➦♣ ❝â t❤➸ rót r❛ tø n ♣❤➛♥ tû ❈❤♦ f2 (n) ❧➔ ❦➼ ❤✐➺✉ sè ❝➦♣ ❝â t❤➸ rót r❛ tø n ♣❤➛♥ tû✳ ❳➨t ❤❛✐ t ợ n m tữỡ ự õ số ❝➦♣ ❝â t❤➸ rót r❛ m + n ♣❤➛♥ tû ❜➡♥❣ sè ❝➦♣ tr♦♥❣ t➟♣ A ❝ë♥❣ ✈ỵ✐ sè ❝➦♣ tr♦♥❣ t➟♣ B ❝ë♥❣ ✈ỵ✐ ♠ët ✤✐➸♠ tø ♠é✐ t➟♣✳ ❉♦ ✤â t❛ ❝â f2 (m + n) = f2 (m) + f2 (n) + mn ❚↕✐ ✤â ❣✐↔♠ ①✉è♥❣ g2 (m + n) = g2 (m) + g2 (n), tr♦♥❣ ✤â ❉♦ ✤â n2 g2 (n) = f2 (n) − n2 f2 (n) = cn + ❱➻ f2 (2) = 1, t❛ ❝â = 2c + ✹✹ ❤♦➦❝ c=− ❉♦ ✤â n(n − 1) = Cn2 ◆➳✉ f3 (n) ❧➔ ❦➼ ❤✐➺✉ sè ❜ë ✸ ❝â t❤➸ rót r❛ tø n ♣❤➛♥ tû ❦❤✐ ✤â t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ f3 (n) = Cn3 ✳ ❇➙② ❣✐í✱ t❛ t t ợ n m tữỡ ự f3 (m + n) s➩ ❧➔ sè ❜ë ❜❛ ❝õ❛ t➟♣ A✱ ❝ë♥❣ ✈ỵ✐ sè ❜ë ❜❛ ❝õ❛ t➟♣ B ❝ë♥❣ ✈ỵ✐ ♠ët sè ❤↕♥❣ ❦➳t ❤đ♣ ❝õ❛ sè ❜ë ❜❛ ✈ỵ✐ ✈➔✐ ♣❤➛♥ tû ❝õ❛ ♠é✐ t➟♣ ✈➟② f2 (n) = f3 (m + n) = f3 (m) + f3 (n) + mf2 (n) + nf2 (m) = f3 (m) + f3 (n) + (mn2 + nm2 ) − mn ✣à♥❤ ♥❣❤➽❛ g3 : N → R ❜ð✐ n3 n g3 (n) = f3 (n) − + ✈ỵ✐ n ∈ N, t❛ ❝â g3 (m + n) = g3 (m) + g3 (n) ❉♦ ✤â f3 (n) = cn − n2 n3 + ❱➻ f3 (3) = 1, t❛ ❝â c= ✈➔ f3 (n) = n(n − 1)(n − 2) = Cn3 ✷✳✹✳ ❚ê♥❣ ❝õ❛ ❝❤✉é✐ ❤ú✉ ❤↕♥ ✭✐✮ ❈❤♦ S(n) = 1.2 + 2.3 + + n(n + 1) ✈ỵ✐ n ∈ N, ✭✷✳✶✹✮ ✹✺ tr♦♥❣ ✤â S : N → N ❞♦ ✤â S(m + n) = S(n) + S(m) + mn2 + nm2 + 2mn ❉♦ ✤â f (m + n) = f (m) + f (n), tr♦♥❣ ✤â n3 f (n) = S(n) − n − ✈ỵ✐ n ∈ N ✈➔ f : N → R✳ ❉♦ ✤â f ❧➔ ❝ë♥❣ t➼♥❤ ✈➔ f (n) = cn✳ ❱➟② S(n) = cn + n2 + n3 ❉♦ ✤â S(1) = 2✱ t❛ ❝â S(n) = n(n + 1)(n + 2) ✭✷✳✶✺✮ ✭✐✐✮ ❈❤♦ t(n) = 1.3 + 2.5 + + n(n + 2) ✈ỵ✐ n ∈ N, tr õ t : N N ú ỵ r➡♥❣ t(1) = 3✳ ❇➙② ❣✐í t(m + n) = t(n) + t(m) + mn2 + nm2 + 3nm ✈ỵ✐ m, n ∈ N ✣à♥❤ ♥❣❤➽❛ f : N → R ❜ð✐ f (n) = t(n) − n3 − n2 ✈ỵ✐ n ∈ N ◗✉❛♥ ❤➺ tr✉② t♦→♥ tr➯♥ trð t❤➔♥❤ f (m + n) = f (m) + f (n) ✈ỵ✐ m, n ∈ N ✣â ❧➔ f ❧➔ ❝ë♥❣ t➼♥❤ ✈➔ f (n) = cn ✈ỵ✐ n ∈ N✳ ❉♦ t(1) = t❛ ❝â t(n) = n(n + 1)(2n + 1) ✈ỵ✐ n ∈ N ✭✷✳✶✼✮ ✭✐✐✐✮ ❚ê♥❣ ❝õ❛ t➼❝❤ ❤é♥ t↕♣✳ s(n) = 1.2.3 + 2.3.4 + + n(n + 1)(n + 2), ✈ỵ✐ n ∈ N, ✭✷✳✶✽✮ ✹✻ ❱ỵ✐ s : N N ú ỵ r s(1) = 6✳ ❳➨t m, n ≥ t❛ ❝â s(n + m − 1) = s(n − 1) + n(n + 1)(n + 2) + {(n + 1)(n + 2)(n + 3) + · · · + (n + m − 1)(n + m)(n + m + 1)} = s(n − 1) + (n3 + 3n2 + 2n) + (m − 1)n3 +n2 (6 + · · · + 3m) + · · · + n{[1 · + · · · + (m − 1)m] +[1 · + · · · + (m − 1)(m + 1)] +[2 · + m(m + 1)]} + s(m − 1) = s(n − 1) + s(m − 1) + mn3 + 3n2 (1 + + · · · + m) +n{[1 · + · · · + (m − 1)m] +[1 · + · · · + (m − 1)(m + 1)][1 · + · · · + m(m + 1)]} m(m + 1) = s(n − 1) + s(m − 1) + mn3 + 3n2 (m − 1)m(m + 1) m(m − 1)(2m + 5) +n +n + m(m − 1)(2m + 5) (m − 1)m(m + 1) +n n m(m + 1)(m + 2) +n ✭sû ❞ö♥❣ ✤➥♥❣ t❤ù❝ ✭✷✳✶✺✮✱ ✭✷✳✶✻✮✮ 3 = s(n − 1) + s(m − 1) + mn3 + m2 n2 + m3 n 3 + n m + nm2 − mn 2 ữ trữợ f : N → R ①→❝ ✤à♥❤ ♥❤÷ s❛✉ f (n) = s(n − 1) − n4 − n + n ✈ỵ✐ n ∈ N 4 ❈❤♦ f (m+n) = f (m)+f (n) ✭❝ë♥❣ t➼♥❤✮ ✈➔ f (n) = cn✳ ❙û ❞ö♥❣ s(1) = 6✱ t❛ ❝â s(n − 1) = [n4 + 2n3 − n2 − 2n]; ♥❣❤➽❛ ❧➔ s(n) = n(n + 1)(n + 2)(n + 3) ✈ỵ✐ n ∈ N ✭✷✳✶✾✮ ✹✼ ❑➳t ❧✉➟♥ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❧➔ ♠ët ❝❤õ ✤➲ ❧✐➯♥ q✉❛♥ tỵ✐ ❤➜✉ ❤➳t ❝→❝ ❦❤➼❛ ❝↕♥❤ ❝õ❛ ❚♦→♥ ❤å❝✳ ❚✉② ♥❤✐➯♥ ✈ỵ✐ ♣❤↕♠ ✈✐ ❝õ❛ ❧✉➟♥ ✈➠♥ ❚❤↕❝ s➽ ❝❤✉②➯♥ ♥❣➔♥❤ ♣❤÷ì♥❣ ♣❤→♣ ❚♦→♥ ❝➜♣ tỉ✐ t➟♣ tr✉♥❣ tr➻♥❤ ❜➔② ✈➲✿ ✧P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ✈➔ ù♥❣ ❞ö♥❣✧✳ ▲✉➟♥ ✈➠♥ ❝â ♥❤ú♥❣ ♥ë✐ ❞✉♥❣ s❛✉✿ ✲ ❚r➻♥❤ ❜➔② ❝→❝ ❦✐➳♥ t❤ù❝ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ❝ë♥❣ t➼♥❤✱ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ♠ơ✱ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ❧♦❣❛r✐t✱ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ♥❤➙♥ t➼♥❤✱ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ♥❤✐➲✉ ❜✐➳♥ ✈➔ ♠ð rë♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤②✳ ✲ ❚r➻♥❤ ❜➔② ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤②✳ ❚➻♠ ♥❣❤✐➺♠ ❝õ❛ ♥â tr➯♥ t➟♣ sè t❤ü❝ ✈➔ sè ♣❤ù❝✱ ❝❤➾ r❛ ♥❣❤✐➺♠ ❧✐➯♥ tö❝ ❝õ❛ ♥â ❧➔ t✉②➳♥ t➼♥❤✱ ①→❝ ✤à♥❤ ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ❤➔♠ sè ♠ơ ❈❛✉❝❤② ♠➔ ❦❤ỉ♥❣ ❝➛♥ ✤✐➲✉ ❦✐➺♥ ❝❤➼♥❤ q✉② ♥❤÷ ❧✐➯♥ tö❝✱ ❜à ❝❤➦♥ ❤❛② ❦❤↔ ✈✐✳ ◆❣❤✐➯♥ ❝ù✉ ♥❣❤✐➺♠ ❝õ❛ ❝→❝ ❞↕♥❣ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ♥❤✐➲✉ ❜✐➳♥✳ ✲ ❚r➻♥❤ ❜➔② ù♥❣ ❞ư♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② tr♦♥❣ t➼♥❤ tê♥❣ ❧ô② t❤ø❛ ❝õ❛ sè ♥❣✉②➯♥ ✭tê♥❣ ❝õ❛ n sè tü ♥❤✐➯♥ ✤➛✉ t✐➯♥✱ tê♥❣ ❜➻♥❤ ♣❤÷ì♥❣ ❝õ❛ n sè tü ♥❤✐➯♥ ✤➙✉ t✐➯♥✱ tê♥❣ ❧ô② t❤ø❛ k ❝õ❛ n sè tü ♥❤✐➯♥ ✤➛✉ t✐➯♥✮✱ t➼♥❤ tê♥❣ ❧ô② t❤ø❛ ❝õ❛ ❝→❝ sè tr♦♥❣ ❞➣② ❝➜♣ sè ❝ë♥❣✱ t➻♠ sè ❝➦♣ ❝â t❤➸ rót r❛ tø n ♣❤➛♥ tû✱ ❧ü❝ ❧÷đ♥❣ ❝õ❛ ♠ët t➟♣ ❤đ♣ ✈➔ tê♥❣ ❝õ❛ ❝❤✉é✐ ❤ú✉ ❤↕♥✳ ✹✽ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❆✳ ❚✐➳♥❣ ❱✐➺t ❬✶❪ ❚r➛♥ ✣ù❝ ▲♦♥❣✱ ❍♦➔♥❣ ◗✉è❝ ❚♦➔♥✱ ◆❣✉②➵♥ ✣➻♥❤ ❙❛♥❣ ✭✷✵✵✺✮✱ ●✐→♦ tr➻♥❤ ●✐↔✐ t➼❝❤✱ t➟♣ ✶✱ ◆❳❇ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍◆✳ ❬✷❪ ◆❣✉②➵♥ ❱➠♥ ▼➟✉ ✭✶✾✾✼✮✱ P❤÷ì♥❣ tr➻♥❤ ❤➔♠✱ ◆❳❇ ●✐→♦ ❞ö❝✳ ❬✸❪ ◆❣✉②➵♥ ❱➠♥ ◆❤♦✱ ▲➯ ❍♦➔♥❣ P❤á t t t ữợ ⑩✲ ❚❤→✐ ❇➻♥❤ ❉÷ì♥❣✱ ◆❳❇ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍◆✳ ❇✳ ❚✐➳♥❣ ❆♥❤ ❬✹❪ ❏✳ ❆❝➨❧ ✭✶✾✻✻✮✱ ▲❡❝t✉r❡s ♦♥ ❋✉♥❝t✐♦♥❛❧ ❊q✉❛t✐♦♥s ❛♥❞ ❚❤❡✐r ❛♣♣❧✐✲ ❝❛t✐♦♥s✳ ❬✺❪ ❈❤r✐st♦♣❤❡r ●✳ ❙♠❛❧❧ ✭✷✵✵✼✮✱ ❋✉♥❝t✐♦♥❛❧ ❊q✉❛t✐♦♥s ❛♥❞ ❍♦✇ t♦ ❙♦❧✈❡ ❚❤❡♠✱ ❙♣r✐♥❣❡r✳ ❬✻❪ P✳ ❑❛♥♥❛♣♣❛♥ ✭✷✵✵✶✮✱ ✧❆♣♣❧✐❝❛t✐♦♥ ♦❢ ❈❛✉❝❤②✬s ❊q✉❛t✐♦♥ ✐♥ ❈♦♠✲ ❜✐♥❛t♦r✐❝s ❛♥❞ ●❡♥❡t✐❝s✧✱ ▼❛t❤✇❛r❡ & ❙♦❢t ❈♦♠♣✉t✐♥❣✱ ✭✽✮✱ PP✳ ✻✶✲✻✹✳ ❬✼❪ P✳ ❑✳ ❙❛❤♦♦✱ P✳ ❑❛♥♥❛♣♣❛♥ ✭✷✵✶✶✮✱ ■♥tr♦❞✉❝t✐♦♥ t♦ ❋✉♥❝t✐♦♥❛❧ ❊q✉❛t✐♦♥s✱ ❈❤❛♣♠❛♥ & ❍❛❧❧✴❈❘❈✳ ❬✽❪ ❙♦♦♥✲▼♦ ❏✉♥❣ ✭✷✵✶✵✮✱ ❍②❡rs✕❯❧❛♠✕❘❛ss✐❛s ❙t❛❜✐❧✐t② ♦❢ ❋✉♥❝t✐♦♥❛❧ ❊q✉❛t✐♦♥s ✐♥ ◆♦♥❧✐♥❡❛r ❆♥❛❧②s✐s✱ ❙♣r✐♥❣❡r✳ ❬✾❪ ❚✐t✉ ❆♥❞r❡❡s❝✉✱ ■✉r✐❡ ❇♦r❡✐❝♦ ✭✷✵✵✼✮✱ ❋✉♥❝t✐♦♥❛❧ ❊q✉❛t✐♦♥s✱ ❊❧❡❝✲ tr♦♥✐❝ ❊❞✐t✐♦♥✳