✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ✖✖✖✖✖✖✖✖✖✖✖✖✖✖ P❍❸▼ ❚❍➚ ◆●➴❈ ❚❍⑩❈ ❚❘■➎◆ ❈❍➓◆❍ ❍➐◆❍ ❑■➎❯ ❍❆❘❚❖●❙✲❈❍■❘❑❆ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✶✾ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ✖✖✖✖✖✖✖✖✖✖✖✖✖✖✲ P❍❸▼ ❚❍➚ ◆●➴❈ ❚❍⑩❈ ❚❘■➎◆ ❈❍➓◆❍ ❍➐◆❍ ❑■➎❯ ❍❆❘❚❖●❙✲❈❍■❘❑❆ ❈❍❯❨➊◆ ◆●⑨◆❍✿ ❚❖⑩◆ ●■❷■ ❚➑❈❍ ▼❶ ❙➮✿ ✽ ✹✻ ✵✶ ✵✷ ▲❯❾◆ ữớ ữợ ●❙✳ ❚❙❑❍✳ ◆●❯❨➍◆ ◗❯❆◆● ❉■➏❯ ❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✶✾ ▲í✐ ❝❛♠ ✤♦❛♥ ❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ ❝æ♥❣ tr➻♥❤ tr➯♥ ❧➔ tổ ự ữợ sỹ ữợ ◆❣✉②➵♥ ◗✉❛♥❣ ❉✐➺✉✳ ❈→❝ ❦➳t q✉↔ ♥➯✉ tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ tr✉♥❣ t❤ü❝ ✈➔ ❝❤÷❛ tø♥❣ ✤÷đ❝ ❝ỉ♥❣ ❜è tr♦♥❣ ❜➜t ❦ý ❝æ♥❣ tr➻♥❤ ❦❤♦❛ ❤å❝ ♥➔♦ ❦❤→❝✳ ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✻ ♥➠♠ ✷✵✶✾ ❚→❝ ❣✐↔ P❤↕♠ ❚❤à ◆❣å❝ ❳⑩❈ ◆❍❾◆ ❈Õ❆ ❑❍❖❆ ❈❍❯❨➊◆ ▼➷◆ ❳⑩❈ ◆❍❾◆ ❈Õ❆ ◆●×❮■ ❍×❰◆● ❉❼◆ ●❙✳❚❙❑❍ ◆❣✉②➵♥ ◗✉❛♥❣ ❉✐➺✉ ✐ ▲í✐ ❝↔♠ ì♥ ❚r♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ✤➸ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ tỉ✐ ✤➣ ♥❤➟♥ ✤÷đ❝ sü ❣✐ó♣ ✤ï t t ữớ ữợ ❚ỉ✐ ❝ơ♥❣ ♠✉è♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❜ë ♠ỉ♥ ●✐↔✐ t➼❝❤✱ ❑❤♦❛ ❚♦→♥✱ ✤➣ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐✱ ữợ tổ õ t t tèt ❧✉➟♥ ✈➠♥ ♥➔②✳ ❉♦ t❤í✐ ❣✐❛♥ ❝â ❤↕♥✱ ❜↔♥ t❤➙♥ t→❝ ❣✐↔ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❧✉➟♥ ✈➠♥ ❝â t❤➸ ❝â ♥❤ú♥❣ t❤✐➳✉ sât✳ ❚→❝ ❣✐↔ ♠♦♥❣ ♠✉è♥ ♥❤➟♥ ữủ ỵ ỗ õ õ ỹ ❝õ❛ ❝→❝ t❤➛② ❝æ✱ ✈➔ ❝→❝ ❜↕♥✳ ❚æ✐ ①✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥✳ ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✻ ♥➠♠ ✷✵✶✾ ❚→❝ ❣✐↔ P❤↕♠ ❚❤à ◆❣å❝ ✐✐ ▼ư❝ ❧ư❝ ▲í✐ ❝❛♠ ✤♦❛♥ ▲í✐ ❝↔♠ ì♥ ▼ư❝ ❧ư❝ ▲➼ ❞♦ ❝❤å♥ ✤➲ t➔✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▼ö❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉ ✐ ✐✐ ✐✐✐ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ◆❤✐➺♠ ✈ö ♥❣❤✐➯♥ ❝ù✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ❈➜✉ tró❝ ❧✉➟♥ ✈➠♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✷ ✶✳✶ ❍➔♠ ❝❤➾♥❤ ❤➻♥❤ ♠ët ❜✐➳♥ ✈➔ ♥❤✐➲✉ ❜✐➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỏ ữợ ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✸ ❈❤✉é✐ ❋♦✉r✐❡r ❝õ❛ ❤➔♠ sè ❧✐➯♥ tö❝ ✈➔ sü ❤ë✐ tö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ t tr rts rở ỵ t tr ❍❛rt♦❣s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ rtsr rở tr ỗ t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✽ ✽ ✾ ❑➳t ❧✉➟♥ ✷✸ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✷✹ ✐✐✐ ▼ð ✤➛✉ ✶✳ ▲➼ ❞♦ ❝❤å♥ ✤➲ t➔✐✳ ❚❤→❝ tr✐➸♥ ❝❤➾♥❤ ❤➻♥❤ ❧➔ ♠ët ❜➔✐ t♦→♥ q✉❛♥ trå♥❣ ❝õ❛ ❣✐↔✐ t➼❝❤ ♣❤ù❝ ♠ët ❜✐➳♥✳❚r♦♥❣ C ♠å✐ ♠✐➲♥ ♣❤➥♥❣ ✤➲✉ ❧➔ ♠✐➲♥ ❝❤➾♥❤ ❤➻♥❤✳ ✣✐➲✉ ♥➔② ❝â ♥❣❤➽❛ tỗ t ởt ổ t rë♥❣ ❧➯♥ ♠ët ♠✐➲♥ rë♥❣ ❤ì♥ t❤➟t sü✳ ❚✉② ♥❤✐➯♥ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥❤✐➲✉ ❝❤✐➲✉ ✭C n ✱ ♥ ≥ 2✮ t❤➻ ❝→❝ ❦➳t q✉↔ tr➯♥ ❦❤ỉ♥❣ ❝á♥ ✤ó♥❣ ♥ú❛ ỵ rts õ r ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ tr➯♥ ❧➙♥ ❝➟♥ ❝õ❛ ❜✐➯♥ ♠ët s♦♥❣ ✤➽❛ ✤➲✉ ♠ð rë♥❣ ❝❤➾♥❤ ❤➻♥❤ ❧➯♥ s♦♥❣ ✤➽❛✳ ✣à♥❤ ỵ ữủ r t tr tr ỗ t ởt sè ❧✐➯♥ tư❝ tr➯♥ ✤➽❛ ✤ì♥ ✈à✳ ✣➙② ❧➔ ♠ët ♠ð rë♥❣ r➜t s→♥❣ t↕♦ ✈➔ ❧➔ ❝↔♠ ❤ù♥❣ ✤➸ ❝→❝ ♥❤➔ t♦→♥ ❤å❝ ✤✐ s❛✉ ♥❣❤✐➯♥ ❝ù✉✳ ✷✳ ▼ö❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉✳ ▲✉➟♥ ✈➠♥ ♥❣❤✐➯♥ ❝ù✉✿ ❚❤→❝ tr✐➸♥ ❝❤➾♥❤ ❤➻♥❤ ❦✐➸✉ ❍❛t♦❣s ✲ ❈❤✐r❦❛✳ ❈❤ó♥❣ tỉ✐ ❝ì ❜↔♥ tr➻♥❤ ❜➔② t❤❡♦ ♠ët ❜➔✐ ❜→♦ ❝❤✉②➯♥ ❦❤↔♦ ❝õ❛ ❇❛rr❡t ✈➔ ❇❤❛r❛❧✐✳ ✸✳ ◆❤✐➺♠ ✈ö ♥❣❤✐➯♥ ❝ù✉ ◆❣❤✐➯♥ ❝ù✉ ❤❛✐ ✤à♥❤ ỵ ỡ ỵ t tr ts ỵ r rở tr ởt ỗ t Pữỡ ự ũ ữỡ tt ỵ tt t❤➳ ✈à ✈➔ ❣✐↔✐ t➼❝❤ ♣❤ù❝✳ ✺✳ ❈➜✉ tró❝ ❧✉➟♥ ỗ ữỡ ữỡ ✶✿ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✳ ❈❤÷ì♥❣ ♥➔② tỉ✐ s➩ ♥❤➢❝ ❧↕✐ ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ❝õ❛ ●✐↔✐ t➼❝❤ ự ử ữỡ ữỡ ỵ t tr rt rở r ữỡ s tr ỵ rts ỵ rtsr t tr tr ỗ t ữỡ tự ❝❤✉➞♥ ❜à ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ t❛ s➩ ♥❤➢❝ ❧↕✐ ♠ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✈➲ ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ ♠ët ❜✐➳♥ ✈➔ ♥❤✐➲✉ ❜✐➳♥ s➩ ✤÷đ❝ ❞ò♥❣ ✈➲ s❛✉✳ ❑❤→✐ ♥✐➺♠ q✉❛♥ trå♥❣ ❧➔ ♠✐➲♥ ❝❤➾♥❤ ❤➻♥❤✱ ♠✐➲♥ ❣✐↔ ỗ ũ ợ ỵ tử t ỗ ởt ✈➔ ♥❤✐➲✉ ❜✐➳♥ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ❍➔♠ f ①→❝ ✤à♥❤ tr♦♥❣ ♠✐➲♥ D ⊂ C ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ✤÷đ❝ ❣å✐ ❧➔ ❝❤➾♥❤ ❤➻♥❤ t↕✐ z0 ∈ D ♥➳✉ tỗ t r > f C t↕✐ ♠å✐ z ∈ ∆(z0 , r) ⊂ D ◆➳✉ f ❝❤➾♥❤ ❤➻♥❤ t↕✐ ♠å✐ z ∈ D t❤➻ t❛ ♥â✐ f ❝❤➾♥❤ ❤➻♥❤ tr➯♥ D✳ ❱➼ ❞ö ✶✳✶✳✷✳ ❈→❝ ❤➔♠ ✤❛ t❤ù❝ ❝❤➾♥❤ ❤➻♥❤ tr➯♥ t♦➔♥ ♠➦t ♣❤➥♥❣ ♣❤ù❝ C✳ ❈→❝ ❤➔♠ ❤ú✉ t✛ ❝❤➾♥❤ ❤➻♥❤ tr➯♥ C trø r❛ t↕✐ ❝→❝ ✤✐➸♠ ♠➔ ♥â ❦❤æ♥❣ ①→❝ ✤à♥❤✳ ❈æ♥❣ tự t s ỵ t t t ự ởt ỵ ✶✳✶✳✸✳ ❈❤♦ ❤➔♠ f (z) ❝❤➾♥❤ ❤➻♥❤ tr➯♥ ♠✐➲♥ D ✈➔ γ ❧➔ ♠ët ❝❤✉ t✉②➳♥ tr♦♥❣ D s❛♦ ❝❤♦ ♠✐➲♥ γ ❣✐ỵ✐ ❤↕♥ ❜ð✐ γ ♥➡♠ tr♦♥❣ D✳ ❑❤✐ ✤â ✈ỵ✐ ♠å✐ z0 ∈ γ ✱ t❛ ❝â ❛✮ f (z0 ) = 2πi ✷ γ f (z) dz z − z0 ✭✶✳✶✮ ❱ỵ✐ n ≥ t❛ ❝â ❜✮ f (n) (z0 ) = ❈❤ù♥❣ ợ >0 ỵ ❤✐➺✉ n! 2πi γ f (z) dz (z − z0 )n+1 ✤õ ❜➨ ✤➸ ❤➻♥❤ trá♥ Cδ ❧➔ ❜✐➯♥ ❝õ❛ ∆(z0 , δ) ∆(z0 , δ) ⊂ γ , ✭✶✳✷✮ ♣❤➛♥ ♠➦t ♣❤➥♥❣ ✈➔ ✤➦t Dγ,δ = γ \∆(z0 , δ) ❉♦ Dγ,δ ❧➔ ♠✐➲♥ ✷✲❧✐➯♥✱ ♥➯♥ t❛ ❝â γ∪Cδ− f (ν) dν = ν − z0 ❚ø ✤â ❝â ✤➥♥❣ t❤ù❝ γ ❇➔♥❣ ❝→❝❤ t❤❛♠ sè ❤â❛ Cδ f (ν) dν = ν − z0 Cδ f (η) dη η − z0 η = a + δeiφ , dη = iδeiφ dφ 2π f (η) dη = η − z0 ✭✶✳✸✮ t❛ ❝â f (z0 + ρeiϕ ) iϕ ρe dϕ ρeiϕ 2π f (z0 + ρeiϕ )dϕ =i 2π [f (z0 + ρeiϕ ) − f (z0 )]dϕ + 2πif (z0 ) =i ❈❤♦ δ→0 t❛ ❝â 2π [f (z0 + ρeiϕ ) − f (z0 )]dϕ = lim δ→0 ❱➟② t❛ ❝â lim δ→0 γ f (η) dη = 2πif (z0 ) η − z0 ✭✶✳✹✮ ❑➳t ❤ñ♣ ❧↕✐ t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❜✳ ❇➡♥❣ ❝→❝❤ ✤↕♦ ữợ t t õ ổ tự ❝❤ù♥❣ ♠✐♥❤✳ ◆❤í ❝ỉ♥❣ t❤ù❝ t➼❝❤ ♣❤➙♥ ❈❛✉❝❤② t❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❦➳t q✉↔ s❛✉ ✈➲ ❜✐➸✉ ❞✐➵♥ ✤à❛ ♣❤÷ì♥❣ ởt t ởt ộ tứ ỵ ✶✳✶✳✹✳ ❈❤♦ f ❧➔ ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ tr➯♥ ♠✐➲♥ ♠ð D✳ ❑❤✐ ✤â ✈ỵ✐ ♠å✐ a ∈ D✱ ❤➔♠ f ❝â t❤➸ ❦❤❛✐ tr✐➸♥ t❤➔♥❤ ❝❤✉é✐ ❧ô② t❤ø❛ tr♦♥❣ ♠å✐ ❧➙♥ ❝➟♥ ✤õ ♥❤ä ❝õ❛ a ∞ cn (z − a)n f (z) = n=0 ✸ ✭✶✳✺✮ ❍ì♥ ♥ú❛ ❝→❝ ❤➺ sè ❝õ❛ ❝❤✉é✐ ❧➔ ✤÷đ❝ t➼♥❤ t❤❡♦ ❝ỉ♥❣ t❤ù❝ f (n) (a) n! cn := ❚ø ✤à♥❤ ỵ tr ú t õ t ❝❤➾♥❤ ❤➻♥❤ ♥❤✐➲✉ ❜✐➳♥ ♥❤÷ s❛✉✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✺✳ ❍➔♠ f ✤÷đ❝ ❣å✐ ❧➔ ❝❤➾♥❤ ❤➻♥❤ t↕✐ z ∈ Cn ♥➳✉ f ❝â t❤➸ ❦❤❛✐ tr✐➸♥ ✤÷đ❝ t❤➔♥❤ ❝❤✉é✐ ❧✉ÿ t❤ø❛ tr♦♥❣ ❧➙♥ ❝➙♥ ❝õ❛ z ✳ ❍➔♠ f ❝❤➾♥❤ ❤➻♥❤ tr➯♥ ♠✐➲♥ D ♥➳✉ ♥â ❝❤➾♥❤ ❤➻♥❤ t↕✐ ♠å✐ z D ữỡ tỹ ữ ỵ ❝❤♦ ❤➔♠ ♠ët ❜✐➳♥ ♣❤ù❝✱ ❝❤ó♥❣ t❛ ❝â ❦➳t q✉↔ s ỵ sỷ U = U (a, r) = {z ∈ Cn : |zj − aj | < rj ∀j = 1, , n} ❧➔ ✤❛ ✤➽❛ t➙♠ a ✤❛ ❜→♥ ❦➼♥❤ r = (r1 , , rn ) ✈➔ Γ = {z ∈ Cn : |zj − aj | = rj ∀j = 1, , n} ◆➳✉ f ❧➔ ❤➔♠ ❧✐➯♥ tö❝ tr➯♥ U ✈➔ ❝❤➾♥❤ ❤➻♥❤ tr♦♥❣ U t❤➻ f (z) = 2πi n Γ f (η)dη1 · · · dηn (η1 − z1 ) · · · (ηn − zn ) ∀z ∈ U ỵ s ữủ ự tữỡ tỹ ữ ởt ỵ sỷ {fn} ❤ë✐ tư ✤➲✉ tr➯♥ ♠å✐ t➟♣ ❝♦♠♣❛❝t tr♦♥❣ D tỵ✐ ❤➔♠ f ✱ t❤➻ ❤➔♠ f ❝❤➾♥❤ ❤➻♥❤ tr➯♥ D✳ ❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ z0 ∈ D✳ t➼❝❤ ♣❤➙♥ ❈❛✉❝❤② ✈ỵ✐ ♠å✐ ❈❤å♥ z ∈ U (z0 , r) fn (z) = ❉♦ (fn ) ❤ë✐ tư ✤➲✉ tỵ✐ f r >0 tr➯♥ 2πi ∂D(z0 , r) ✤õ ❜➨ ✤➸ U (z0 , r) ⊂ D✳ ❚❤❡♦ ❝æ♥❣ t❤ù❝ t❛ ❝â ∂D(z0 ,r) fn (η) dη η−z ❜➡♥❣ ❝→❝❤ t✐➳♥ ợ ữợ t t ữủ fn (z) = ✈ỵ✐ ♠å✐ z ∈ D(z0 , r)✳ ❱➻ t❤➳ f 2πi ∂D(z0 ,r) ❝❤➾♥❤ ❤➻♥❤ tr➯♥ fn (η) dη η−z D(z0 , r)✳ ❙û ❞ö♥❣ ✤à♥❤ ỵ tr ú t õ ỵ s t➼♥❤ ❝♦♠♣❛❝t ❝õ❛ ❤å ❝→❝ ❤➔♠ ❝❤➾♥❤ ❤➻♥❤✿ ✹ ✣à♥❤ ỵ sỷ D ởt tr C F H(D) ỵ t õ F ❜à ❝❤➦♥ ✤➲✉ tr➯♥ ❝→❝ t➟♣ ❝♦♠♣❛❝t ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ♠å✐ ❞➣② {fn } ⊂ F ❝❤ù❛ ♠ët ❞➣② ❝♦♥ fnk ❤ë✐ tö ✤➲✉ tr➯♥ ❝→❝ t➟♣ ❝♦♠♣❛❝t✳ ỏ ữợ ỗ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✳ ❈❤♦ D ❧➔ t➟♣ ♠ð tr♦♥❣ C✳ ❍➔♠ u : D → [−∞, +∞) ✤÷đ❝ ❣å✐ ❧➔ ỏ ữợ tr D u ỷ tö❝ tr➯♥ tr➯♥ D, u = −∞ tr➯♥ ❜➜t ❦➻ t❤➔♥❤ ♣❤➛♥ ❧✐➯♥ t❤æ♥❣ ❝õ❛ D ✈➔ t❤ä❛ ♠➣♥ ❜➜t tự ữợ tr ữợ tr D ợ x D tỗ t r > s❛♦ ❝❤♦ ∆(x, ρ) ⊂ D ✈➔ ✈ỵ✐ ♠å✐ ≤ r < r t❛ ❝â u(x) ≤ 2π u(x + reit )dt ỵ u ỏ ữợ tr t D1 v ỏ ữợ tr t D2 ⊂ D1 ✳ ●✐↔ sû ✈ỵ✐ ♠å✐ x ∈ D1 ∩ ∂D2 t❛ ❝â lim sup v(z) ≤ u(x) z→x ❑❤✐ ✤â ❤➔♠ u˜ = max{u, v} tr➯♥D2 u tr D1 \ D2 ỏ ữợ tr D1 ❑➳t q✉↔ s❛✉ ✤➙② ❣✐ó♣ ❝❤ó♥❣ t❛ trì♥ õ ỏ ữợ ỵ u ỏ ữợ tr t D ⊂ C ✈ỵ✐ u = −∞✳ ❍➔♠ θ ❧➔ ❤➔♠ ①→❝ ✤à♥❤ ❜ð✐ − 1− 1x θ(x) = λe ♥➳✉ x < ♥➳✉ x ≥ ❱ỵ✐ r > ❞÷ì♥❣ t❛ ✤➦t θr (z) = z θ 2 r r z ∈ C ❑❤✐ õ u r ỏ ữợ trỡ tr➯♥ Dr ✈➔ ❤ì♥ ♥ú❛ u ∗ χ ↓ u tr➯♥ D ▼è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ ✈➔ ữợ ữủ t ữ s ✺ ❇ê ✤➲ ✷✳✷✳✷✳ ●å✐ G(reiθ ) = ✤✐➲✉ ❦✐➺♥✿ N n=−N N n=−N gn (r)einθ ✈➔ ❣✐↔ sû G ∈ C ∞ (∆; C) t❤ä❛ ♠➣♥ |gn (r)| 0 ✤õ ♥❤ä s❛♦ ❝❤♦ n0 ỗ t số tỹ |F (reiθ ) − σN (θ, r)| < δ/2 s❛♦ ❝❤♦ ∀(θ, r) ∈ [0, 2π) × [0, 1] ❑➳t q tr q ỵ r ợ r [0, 1] t ỵ r →♣ ❞ö♥❣ ❝❤♦ ❤➔♠ t✉➛♥ ❤♦➔♥ ✭✷✳✽✮ ❝è ✤à♥❤✱ ✭✷✳✽✮ ❝❤➼♥❤ ❧➔ F (rei )✳ ❚✉② ♥❤✐➯♥✱ ❦✐➸♠ tr❛ ❝❤ù♥❣ ỵ r t t r t tử ỗ {F (rei )}r[0,1] C(T)✱ ❝→❝❤ ❝❤å♥ N ◆➳✉ t❛ ✈✐➳t tr♦♥❣ ✭✷✳✽✮ ❧➔ t❤è♥❣ ♥❤➜t ✈ỵ✐ r ∈ [0, 1]✳ N aj (r)eijθ , σN (θ, r) = j=−N t❤➻ t❛ t❤➜② ♥❣❛② C−j (r) ✭✐✮ ✭✐✐✮ ✭✐✐✐✮ |aj (r)| ≤ |aj (r)| ∀r ∈ [0, 1]✳ ❱ỵ✐ ♠é✐ j = 1, 2, , N, t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿ C−j ∈ C ∞ ([0, 1]; C), C−j tr✐➺t t✐➯✉ tỵ✐ ❝➜♣ ✈ỉ ❤↕♥ t↕✐ r0 , ✈➔ |a−j (r) − C−j (r)| ≤ δ/2(2N + 1) ∀r ∈ [0, 1]✱ ❚❤❡♦ ✤✐➲✉ ❦✐➺♥ ✈➲ ❤➺ sè ❋♦✉r✐❡r t❛ ❝â ✤→♥❤ ❣✐→ {An (r)}n∈Z , |aj (r)| ≤ |aj (r)| ≤ rj ∀j = 1, 2, , N, r ∈ [0, 1) ❈❤♦ < R0 < ❧➔ ♠ët sè ✤õ ♥❤ä s❛♦ ❝❤♦ R0 ≤ δ 4(2N + 1) 1/j ∀j = 1, 2, , N ✶✸ t❛ ❝❤å♥ ❤➔♠ ❱ỵ✐ ♠é✐ j = 1, 2, , N t❛ ✤✐♥❤ ♥❣❤➽❛ ❤➔♠ Cj (r) αj (r)rj , ♥➳✉ r ≤ R0 , βj (r), ♥➳✉ r ≥ R0 , Cj (r) := ♥❤÷ s❛✉✿ t❤ä❛ ♠➣♥ ∗ Cj ∈ C ∞ ([0, 1]; C), ∗ αj ❜à tr✐➺t t✐➯✉ tỵ✐ ❝➜♣ ✈ỉ ❤↕♥ t↕✐ ∗ αj t❤ä❛ ♠➣♥ ✭✐ ✮ ✭✐✐ ✮ ✭✐✐✐ ✮ r = 0, |aj (s)| sj |αj (r)| ≤ sup s≤1 ∗ ✭✐✈ ✮ βj ∀s ∈ [0, R0 ], t❤ä❛ ♠➣♥ R0j δ |βj (r) − aj (r)| ≤ 2(2N + 1) ❈✉è✐ ❝ò♥❣✱ ✤à♥❤ ♥❣❤➽❛ C0 (r) ❧➔ ❤➔♠ C∞ |C0 (r) − a0 (r)| < ✈➔ t❤ä❛ ♠➣♥ C0 − C0 (0) ∀r ∈ [R0 , 1] ❜➜t ❦ý s❛♦ ❝❤♦ δ ∀r ∈ [0, 1] 2(2N + 1) tr✐➺t t✐➯✉ tỵ✐ ❝➜♣ ✈ỉ ❤↕♥ t↕✐ r = ❇➙② ❣✐í ✤➦t N iθ Cj (r)eijθ G(re ) = j=−N ❈❤ó♥❣ t❛ ❝â ♠ët sè ✤→♥❤ ❣✐→ ✈➲ ❝→❝ ❤➺ sè Cj ✳ ✣➛✉ t✐➯♥ ①➨t C−j (r), j = 1, 2, , N ú ỵ r ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ s❛✉ ✈➝♥ ✤ó♥❣ ♥➳✉ C−j N N |C−j (r) − a−j (r)| ≤ j=1 j=1 δN δ ≤ 2(2N + 1) 2(2N + 1) N N rj j |C−j (r)|r ≤ |a−j (r)| + j=1 j=1 N |a−j (r)|rj + ≤ j=1 ❚✐➳♣ t❤❡♦✱ t❛ ①➨t Cj (r), j = 1, 2, , N ∗ ❝❤➜t ✭✐✐✐ ✮ tr♦♥❣ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ Cj (r) N δ 2(2N + 1) ∀r ∈ [0, 1] ≤ r ≤ R0 ✳ t❛ t❤✉ ✤÷đ❝ N rj αj (r) − j=1 ✶✹ ✭✷✳✾✮ δ 2(2N + 1) rữợ t |Cj (r) aj (r)| j=1 ≡ ✈ỵ✐ ❜➜t ❦ý j = 1, 2, , N |aj (r)| rj ✭✷✳✶✵✮ ❙û ❞ö♥❣ t➼♥❤ N 2R0j sup ≤ s≤1 j=1 N δ 4(2N + 1) ≤ j=1 ≤ N j=1 ❱➔ ❦❤✐ ①➨t R0 ≤ r ≤ 1, |Cj (r)| ≤ rj |aj (s)| sj sup s≤1 |aj (s)| sj δN , 2(2N + 1) N sup j=1 s≤1 ✭✷✳✶✶✮ |aj (s)| sj ∀r ∈ [0, R0 ] ∗ t❛ sû ❞ö♥❣ t➼♥❤ ❝❤➜t ✭✐✈ ✮ tr♦♥❣ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ✭✷✳✶✷✮ Cj (r) ✤➸ t❤✉ ✤÷đ❝ N N |Cj (r) − aj (r)| ≤ j=1 j=1 N j=1 |Cj (r)| ≤ rj R0j δ δN ≤ , 2(2N + 1) 2(2N + 1) N R0j δ |aj (r)| + rj 2(2N + 1)R0j j=1 N ≤ j=1 ✭✷✳✶✸✮ |aj (r)| δN + j r 2(2N + 1) ∀r ∈ [R0 , 1] ✭✷✳✶✹✮ ❚ø ✭✷✳✶✵✮ ✈➔ ✭✷✳✶✷✮✱ t❛ ❝â N j=−N |Cj (r)| ≤ rj N |a−j (r)|rj + j=1 δN + |a0 (r)| 2(2N + 1) δ + + 2(2N + 1) N ≤ sup j=−N s≤1 N ≤ sup j=−N s≤1 N sup j=1 s≤1 |aj (s)| sj |aj (s)| δ(N + 1) + j s 2(2N + 1) |aj (s)| k +