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❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ■■ ❑❍❖❆ ❚❖⑩◆ ✖✖✖✖✖✖✖✖✖✖♦✵♦✖✖✖✖✖✖✖✖✖✖ ✣➱ ❚❍➚ ❚❍❆◆❍ ❍⑨▼ ✣■➋❯ ❍➪❆ ❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈ ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ●✐↔✐ ❚➼❝❤ ●✐↔♥❣ ✈✐➯♥ ữợ s ỡ rữợ tr ❦❤â❛ ❧✉➟♥✱ ❡♠ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s tợ s ữớ t t ữợ õ t t õ ❧✉➟♥ ♥➔②✳ ❊♠ ❝ơ♥❣ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ tỵ✐ t♦➔♥ t❤➸ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ tr♦♥❣ ❦❤♦❛ ❚♦→♥✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ P❤↕♠ ❍➔ ◆ë✐ ✷ ✤➣ ❞↕② ❜↔♦ ❡♠ t➟♥ t➻♥❤ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ t↕✐ ❦❤♦❛✳ ◆❤➙♥ ❞à♣ ♥➔② ❡♠ ❝ô♥❣ ①✐♥ ữủ ỷ ỡ t tợ ❜↕♥ ❜➧ ✤➣ ❧✉ỉ♥ ❜➯♥ ❡♠✱ ❝ê ✈ơ✱ ✤ë♥❣ ✈✐➯♥✱ ❣✐ó♣ ✤ï ❡♠ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣✳ ❍➔ ◆ë✐✱ t❤→♥❣ ✵✺ ♥➠♠ ✷✵✶✹ ❙✐♥❤ ✈✐➯♥ ✣é ❚❤à ❚❤❛♥❤ ✐✐ ▲í✐ ❝❛♠ ✤♦❛♥ ❙❛✉ ♠ët t❤í✐ ❣✐❛♥ ♥❣❤✐➯♥ ❝ù✉ ✈ỵ✐ sü ❝è ộ ỹ t ũ sỹ ữợ ♥❤✐➺t t➻♥❤ ❝õ❛ ❚❤✳❙ ❍♦➔♥❣ ◆❣å❝ ❚✉➜♥ ❡♠ ✤➣ ❤♦➔♥ t❤➔♥❤ ❜➔✐ ❦❤â❛ ❧✉➟♥ ❝õ❛ ♠➻♥❤✳ ❊♠ ①✐♥ ❝❛♠ ✤♦❛♥ ❜➔✐ ❦❤â❛ ❧✉➟♥ ❧➔ ❞♦ ❜↔♥ t❤➙♥ ♥❣❤✐➯♥ ❝ù✉ ❝ò♥❣ sỹ ữợ ổ trò♥❣ ❤đ♣ ✈ỵ✐ ❜➜t ❦ý ✤➲ t➔✐ ♥➔♦✳ ❍➔ ◆ë✐✱ t❤→♥❣ ✵✺ ♥➠♠ ✷✵✶✹ ❙✐♥❤ ✈✐➯♥ ✣é ❚❤à ❚❤❛♥❤ ✐✐✐ ▼ư❝ ❧ư❝ ▲í✐ ♥â✐ ✤➛✉ ✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✶ ✶✳✷ ✶✳✸ ✶✳✹ ✶✳✺ ❍➺ t❤è♥❣ sè ♣❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚♦♣♦ tr➯♥ ♠➦t ♣❤➥♥❣ ♣❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❍➔♠ ❣✐↔✐ t➼❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚➼❝❤ ♣❤➙♥ ♣❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ ổ ỹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✈ ✶ ✶ ✸ ✻ ✶✵ ✶✻ ✷ ❍➔♠ ✤✐➲✉ ❤á❛ ✶✽ ❑➳t ❧✉➟♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✹✻ ✹✼ ✷✳✶ ✷✳✷ ✷✳✸ ✷✳✹ ✷✳✺ ❈→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ ✤✐➲✉ ❤á❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❍➔♠ ✤✐➲✉ ❤á❛ tr➯♥ ❤➻♥❤ trá♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ữợ ỏ tr➯♥ ✤✐➲✉ ❤á❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❇➔✐ t♦→♥ ❉✐r✐❝❤❧❡t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❍➔♠ ●r❡❡♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐✈ ✶✽ ✷✷ ✸✵ ✸✻ ✹✸ ▲í✐ ♥â✐ ỵ t ♠ỉ♥ ❦❤♦❛ ❤å❝ ❣➢♥ ❧✐➲♥ ✈ỵ✐ t❤ü❝ t✐➵♥✳ ❚r♦♥❣ t♦→♥ ❤å❝✱ ❣✐↔✐ t➼❝❤ ❝❤✐➳♠ ♠ët ✈à tr➼ q✉❛♥ trå♥❣✳ ❈→❝ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ✤÷đ❝ tr♦♥❣ ❣✐↔✐ t➼❝❤ ❦❤ỉ♥❣ ❝❤➾ →♣ ❞ö♥❣ tr♦♥❣ ❝→❝ ❧➽♥❤ ✈ü❝ ❝õ❛ t♦→♥ ❤å❝ ♠➔ ❝á♥ →♣ ❞ư♥❣ tr♦♥❣ ❝→❝ ♥❣➔♥❤ ❦❤♦❛ ❤å❝ ❦❤→❝ ♥❤÷ t ỵ õ t ❤➔♠ ✤✐➲✉ ❤á❛ ❤❛✐ ❜✐➳♥ ❜➡♥❣ ❝ỉ♥❣ ❝ư ❤➔♠ ❜✐➳♥ ♣❤ù❝✱ ❝❤ó♥❣ t❛ ❦❤ỉ♥❣ ❝❤➾ ♥➢♠ ✤÷đ❝ ❦✐➳♥ t❤ù❝ ❣✐↔✐ t➼❝❤ ♠ët ❝→❝❤ ❤➺ t❤è♥❣ t❤æ♥❣ q✉❛ ✈✐➺❝ ♠ð rë♥❣ ❝→❝ ❦➳t q✉↔ ✈➲ ❤➔♠ ✤✐➲✉ ❤á❛ tr♦♥❣ tr÷í♥❣ ❤đ♣ ❤❛✐ ❜✐➳♥✱ ♠➔ ❝á♥ ❣✐ó♣ ❝❤ó♥❣ t❛ ❣✐↔✐ q✉②➳t ✤÷đ❝ ♠ët sè ❜➔✐ t♦→♥ t❤ü❝ t➳ ❝â ❧✐➯♥ q✉❛♥✳❉♦ ✈➟②✱ ự ỵ tt ỏ ú t t s s ỡ ỵ tt ự ỗ tớ sỷ ỳ ỵ tt õ qt ởt số ỵ t❤✉②➳t t♦→♥ ❤å❝✳ ❱➔ ✤➙② ❧➔ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝ì sð ✤➸ ❣✐↔✐ q✉②➳t ♠ët sè ❜➔✐ t♦→♥ t❤ü❝ t✐➵♥ r q tr t ữủ t ổ ợ t t ữủ sỹ ữợ ủ ỵ ❚❤↕❝ sÿ ❍♦➔♥❣ ◆❣å❝ ❚✉➜♥ ❡♠ r➜t ♠✉è♥ t➻♠ ❤✐➸✉ ỏ ữợ ự ❧➔♠ q✉❡♥ ✈ỵ✐ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝ ❡♠ ✤➣ ❝❤å♥ ✤➲ t➔✐ ❍⑨▼ ✣■➋❯ ❍➪❆ ✷✳ ▼ö❝ ✤➼❝❤✱ ♥❤✐➺♠ ✈ö ♥❣❤✐➯♥ ❝ù✉ ▲➔♠ rã ❝→❝ ❦❤→✐ ♥✐➺♠✱ t➼♥❤ ❝❤➜t✱ ỵ ỏ ự ❝→❝ ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥ ✤➳♥ ❤➔♠ ✤✐➲✉ ❤á❛✳ ✈ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❍➔♠ ✤✐➲✉ ❤á❛ ✸✳ ✣è✐ t÷đ♥❣ ✈➔ ♣❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉ ✲ ✣è✐ t÷đ♥❣ ♥❣❤✐➯♥ ❝ù✉✿ ❍➔♠ ✤✐➲✉ ❤á❛✳ ✲ P❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉✿ ◆❤ú♥❣ ✤à♥❤ t t ỵ q✉❛♥ ❝õ❛ ❤➔♠ ✤✐➲✉ ❤á❛✳ ✹✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ ❑❤â❛ ❧✉➟♥ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ ❞ü❛ tr➯♥ sü ❦➳t ❤đ♣ ữỡ ự ỵ t tê♥❣ ❤ñ♣✳✳✳ ✺✳ ✣â♥❣ ❣â♣ ❝õ❛ ❦❤â❛ ❧✉➟♥ ✲▲➔♠ rã ❝❤✐ t✐➳t ❤ì♥ ❤➺ t❤è♥❣ tr✐ t❤ù❝ ♠ỵ✐✱ ❝❤✉②➯♥ s➙✉ ✈➲ ❜ë ♠æ♥ ❤➔♠ ❜✐➳♥ ♣❤ù❝✳ ✲❑❤â❛ ❧✉➟♥ ❝á♥ ❝✉♥❣ ❝➜♣ t❤➯♠ ❝→❝ t➼♥❤ ❝❤➜t ✈➔ ❝→❝ ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥ ❝õ❛ ❤➔♠ ✤✐➲✉ ❤á❛✳ ✻✳ ❈➜✉ tró❝ ❝õ❛ ❦❤â❛ ❧✉➟♥ ◆❣♦➔✐ ♠ö❝ ❧ö❝✱ ♣❤➛♥ ♠ð ✤➛✉✱ ❦➳t ❧✉➟♥ ✈➔ t t õ ỗ ữỡ ữỡ ✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲✿ ❍➔♠ ❜✐➳♥ ♣❤ù❝✱ ❚♦♣♦ tr➯♥ ♠➦t ♣❤➥♥❣ ♣❤ù❝✱ ❍➔♠ ❣✐↔✐ t➼❝❤✱ ❚➼❝❤ ự ỵ ổ ỹ ữỡ ✤✐➲✉ ❤á❛✿ ❚➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❤➔♠ ✤✐➲✉ ❤á❛✱ ỏ tr trỏ ữợ ỏ ✈➔ tr➯♥ ✤✐➲✉ ❤á❛✱ ❇➔✐ t♦→♥ ❉✐r✐❝❤❧❡t✱ ❍➔♠ ●r❡❡♥✳ ✣é ❚❤à ❚❤❛♥❤ ✈✐ ❑✸✻❈ ❚♦→♥ ✣❍❙P❍◆ ✷ ❈❤÷ì♥❣ ✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✶ ❍➺ t❤è♥❣ sè ♣❤ù❝ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ✭❚r÷í♥❣ sè ♣❤ù❝✮ ❈❤ó♥❣ t❛ ✤à♥❤ ♥❣❤➽❛ t➟♣ sè ♣❤ù❝ C s➢♣ t❤ù tü ❝➦♣ ✭❛✱ ❜✮ tr♦♥❣ ✤â ❛✱ ❜ ❧➔ ❝→❝ sè t❤ü❝ ✈➔ tr♦♥❣ ✤â ♣❤➨♣ ❝ë♥❣✱ ♣❤➨♣ ♥❤➙♥ ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉✿ (a, b) + (c, d) = (a + c, b + d) (a, b)(c, d) = (ac − bd, bc + ad) ❉➵ ❞➔♥❣ ❦✐➸♠ tr❛ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ sè ♣❤ù❝ C t❤ä❛ ♠➣♥ ❝→❝ t✐➯♥ ✤➲ ❝õ❛ tr÷í♥❣✳ ❚ù❝ ❧➔✱ C t❤ä❛ ♠➣♥ q✉② t➢❝ ❦➳t ❤ñ♣✱ ❣✐❛♦ ❤♦→♥✱ ♣❤➙♥ ♣❤è✐ ❝õ❛ ♣❤➨♣ ❝ë♥❣ ✈➔ ♣❤➨♣ ♥❤➙♥✳ ❈❤ó♥❣ t❛ ✈✐➳t a ❝❤♦ sè ♣❤ù❝ (a, 0)✳ ⑩♥❤ ①↕ a → (a, 0) ①→❝ ✤à♥❤ ✤➥♥❣ ❝➜✉ tr÷í♥❣ R → C✳ ❱➟② R ❧➔ t➟♣ ❝♦♥ ❝õ❛ C✳ ◆➳✉ ✤➦t i = (0, 1) t❤➻ (a, b) = a + ib✳ ❑❤✐ ✤â ❝❤ó♥❣ t❛ ❝â t❤➸ ❜ä ❝→❝❤ ✈✐➳t t tổ tữớ số ự ú ỵ r➡♥❣ i = −1✱ ✈➟② ♣❤÷ì♥❣ tr➻♥❤ z + = ❝â ♥❣❤✐➺♠ tr♦♥❣ C✳ ❚❤➟t ✈➟②✱ ✈ỵ✐ ♠é✐ z ∈ C✱ z + = (z + i)(z − i)✳ ❍ì♥ ♥ú❛✱ ♥➳✉ z ✈➔ w ❧➔ ❝→❝ sè ♣❤ù❝✱ t❛ ❝â ❜✐➸✉ ❞✐➵♥ 2 z + w2 = (z + iw)(z − iw) ✣➦t z✱ w ❜ð✐ ❝→❝ sè t❤ü❝ a✱ b ❦❤✐ ✤â t❛ ❝â ✭❛✱ ❜ = ✵✮ a a − ib b = = − i a + ib a + b2 a2 + b a2 + b2 ✶ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❍➔♠ ✤✐➲✉ ❤á❛ ❱➟② t❛ ❝â ❝æ♥❣ t❤ù❝ ❝õ❛ ♣❤➛♥ tû ♥❣❤à❝❤ ✤↔♦ ❝õ❛ sè ♣❤ù❝✳ ❑❤✐ ✈✐➳t z = a + ib (a, b ∈ R) ❣å✐ ❛✱ ❜ ❧➔ ♣❤➛♥ t❤ü❝✱ ✐ ❧➔ ♣❤➛♥ ↔♦ ❝â ♥❣❤➽❛ ❧➔ a = Rez✱ b = Imz✳ ❈❤ó♥❣ t❛ ✤÷❛ r❛ ❤❛✐ ♣❤➨♣ t♦→♥ ✈➔♦ C ♠➔ ❦❤ỉ♥❣ ❧➔ ♣❤➨♣ t♦→♥ tr÷í♥❣✳ ◆➳✉ z = x + iy (x, y ∈ R) t❤➻ t❛ ✤à♥❤ ♥❣❤➽❛ |z| = (x + y ) ❧➔ ♠æ✤✉♥ ❝õ❛ ③ ✈➔ z = x − iy ❧➔ ❧✐➯♥ ❤ñ♣ ❝õ❛ z ú ỵ r 2 |z|2 = zz ◆â✐ r✐➯♥❣✱ ♥➳✉ z = t❤➻ z = z |z| ❈→❝ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ♠ỉ✤✉♥ ✈➔ ❧✐➯♥ ❤đ♣✳ Rez = (z + z), Imz = (z + w) = z + w, (z − z) 2i zw = z w |zw| = |z| |w| z |z| = w |w| |z| = |z| ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ✭▼➦t ♣❤➥♥❣ ♣❤ù❝✮ ❚ø ✤à♥❤ ♥❣❤➽❛ ✈➲ sè ♣❤ù❝✱ ♠é✐ ✤✐➸♠ z ∈ C õ t ỗ t t (Rez, Imz) tr ♠➦t ♣❤➥♥❣ R ✳ P❤➨♣ ❝ë♥❣ ❝→❝ sè ♣❤ù❝ t❤ü❝ ❝❤➜t ❧➔ ♣❤➨♣ ❝ë♥❣ ❝→❝ ✈❡❝tì tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ R✳ ◆➳✉ ③✱ ✇ ∈ C t❤➻ ✤â ❧➔ ✤ë ❞➔✐ ❝õ❛ ✈❡❝tì ③✱ ✇ tỵ✐ ❖ (= (0, 0))✳ ✣➙② ❧➔ ❤❛✐ ❝↕♥❤ ❝õ❛ ❤➻♥❤ ❜➻♥❤ ❤➔♥❤ ✈ỵ✐ ❖✱ ③✱ ✇ ❧➔ ❜❛ ✤➾♥❤ ✈➔ ✤➾♥❤ t❤ù t÷ ❧➔ ③ ú ỵ õ t |z w| ❧➔ ❦❤♦↔♥❣ ❝→❝❤ ❣✐ú❛ ③ ✈➔ ✇✳ ❚➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❦❤♦↔♥❣ ❝→❝❤ ❧➔ t❤ä❛ ♠➣♥ ❜➜t ✤➥♥❣ t❤ù❝ t❛♠ ❣✐→❝✳ |z1 − z2 | ✣é ❚❤à ❚❤❛♥❤ |z1 − z3 | + |z3 − z2 | ✷ ❑✸✻❈ ❚♦→♥ ✣❍❙P❍◆ ✷ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❍➔♠ ✤✐➲✉ ❤á❛ ✈ỵ✐ ❝→❝ sè ♣❤ù❝ z , z , z ✳ ⑩♣ ❞ư♥❣ z ✤÷đ❝ |z + w| ❱ỵ✐ ❜➜t ❦ý z ∈ C ❉♦ ✤â✱ Re(zw) − z2 = (z1 − z3 ) + (z3 − z2 ) t❛ |z| + |w| (z, w ∈ C) − |z| Rez |z| − |z| Imz |z| ✳ ❱➟② |zw| = |z| |w| |z + w|2 = |z|2 + 2Re(zw) + |w|2 |z|2 + |z| |w| + |w|2 = (|z| + |w|)2 ❚❛ ❝â |z1 + z2 + + zn | |z1 | + |z2 | + + |zn | ❚÷ì♥❣ tü |z| − |w| |z − w| ✶✳✷ ❚♦♣♦ tr➯♥ ♠➦t ♣❤➥♥❣ ♣❤ù❝ ❱➻ ♠➦t ♣❤➥♥❣ ự C õ t ỗ t ợ R q ①↕ z → (Rez, Imz) ♥➯♥ t♦♣♦ ❝õ❛ ♠➦t ♣❤➥♥❣ ♣❤ù❝ C ❝❤➼♥❤ ❧➔ t♦♣♦ R✳ ❱➻ ✈➟②✱ t❛ ♥➯✉ ♠ët sè ❦✐➳♥ t❤ù❝ ❧✐➯♥ q✉❛♥ ✤➳♥ t♦♣♦ ❝õ❛ R ♥❤÷ s❛✉✿ ✣à♥❤ ♥❣❤➽❛ ✶✳✸✳ ✭❑❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝✮ ❑❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ❧➔ ♠ët ❝➦♣ ✭X, d✮❀ tr♦♥❣ ✤â ❳ ❧➔ ♠ët t➟♣✱ ❤➔♠ d : X × X → R ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ❦❤♦↔♥❣ ❝→❝❤ ❤❛② ♠➯tr✐❝ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✈ỵ✐ x✱ y ✈➔ z ∈ X✿ ✣é ❚❤à ❚❤❛♥❤ ✸ ❑✸✻❈ ❚♦→♥ ✣❍❙P❍◆ ✷ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❍➔♠ ✤✐➲✉ ❤á❛ (a) d(x, y) ≥ (b) d(x, y) = ⇔ x = y ✭✤è✐ ①ù♥❣✮ (d) d(x, z) ≤ d(x, y) + d(y, z) ✭❜➜t ✤➥♥❣ t❤ù❝ t❛♠ ❣✐→❝✮ ◆➳✉ x ∈ X ✈➔ r > t❤➻ ✤à♥❤ ♥❣❤➽❛ (c) d(x, y) = d(y, x) B(x; r) = {y ∈ X : d(x; y) < r} B(x; r) = {y ∈ X : d(x; y) ≤ r} ✱ t÷ì♥❣ ù♥❣ ✤÷đ❝ ❣å✐ ❧➔ ❤➻♥❤ ❝➛✉ ♠ð ✈➔ ✤â♥❣ ✈ỵ✐ t➙♠ ①✱ ❜→♥ ❦➼♥❤ r✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✹✳ ❈❤♦ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✭❳✱ ❞✮ ❝â t➟♣ G ⊂ X ❧➔ t➟♣ ♠ð ♥➳✉ ✈ỵ✐ ♠å✐ x ∈ G ❝â ♠ët > s❛♦ ❝❤♦ B(x; ) ⊂ G✳ B(x; r) B(x; r) ▼➺♥❤ ✤➲ ✶✳✶✳ ❈❤♦ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝✱ ❦❤✐ ✤â ✭❛✮ ❚➟♣ ❳ ✈➔ ∅ ❧➔ t➟♣ ♠ð❀ ✭❜✮ ◆➳✉ G , , G ❧➔ t➟♣ ♠ð tr♦♥❣ ❳ t❤➻ G ❧➔ t➟♣ ♠ð❀ ✭❝✮ ◆➳✉ {G : j ∈ J} ❧➔ t➟♣ ❝→❝ t➟♣ ♠ð tr♦♥❣ ❳✱ ❏ ❧➔ t➟♣ ❤ñ♣ ❝❤➾ n n k k=1 j sè ❜➜t ❦ý t❤➻ G = ∪ {Gj : j ∈ J} ❧➔ t➟♣ ♠ð✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✺✳ ❚➟♣ F ⊂ X ❧➔ t➟♣ ✤â♥❣ ♥➳✉ X − F ❧➔ t➟♣ ♠ð✳ ▼➺♥❤ ✤➲ ✶✳✷✳ ❈❤♦ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝✱ ❦❤✐ ✤â ✭❛✮ ❚➟♣ ❳ ✈➔ ∅ ❧➔ t➟♣ ✤â♥❣❀ ✭❜✮ ◆➳✉ F , , F ❧➔ t➟♣ ✤â♥❣ tr♦♥❣ ❳ t❤➻ F ❧➔ t➟♣ ✤â♥❣❀ ✭❝✮ ◆➳✉ {F : j ∈ J} ❧➔ t➟♣ ❝→❝ t➟♣ ✤â♥❣ tr♦♥❣ ❳✱ ❏ ❧➔ t➟♣ ❤ñ♣ ❝❤➾ n n k k=1 j sè ❜➜t ❦ý t❤➻ F = ∩ {Fj : j ∈ J} ❧➔ t➟♣ ✤â♥❣✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✻✳ ❈❤♦ t➟♣ A ⊂ X ✳ ❑❤✐ ✤â✱ ♣❤➛♥ tr♦♥❣ ❝õ❛ ❆ ❧➔ ❤ñ♣ ❝õ❛ t➜t ❝↔ ❝→❝ t➟♣ ♠ð tr♦♥❣ ❆✱ ❦➼ ❤✐➺✉ ❧➔ intA✳ ❇❛♦ ✤â♥❣ ❝õ❛ ❆ ❧➔ ❣✐❛♦ ❝õ❛ t➜t ❝↔ ❝→❝ t➟♣ ✤â♥❣ ❝❤ù❛ ❆✱ ❦➼ ❤✐➺✉ A ú ỵ r intA ộ ❚❤❛♥❤ ✹ ❑✸✻❈ ❚♦→♥ ✣❍❙P❍◆ ✷ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❍➔♠ ✤✐➲✉ ❤á❛ ❍➺ q✉↔ ✷✳✻✳ ❈❤♦ ϕ ❧➔ ❤➔♠ ữợ ỏ tr G B(a; r) G✳ ❍➔♠ ϕ ✤÷đ❝ ①→❝ ✤à♥❤ tr➯♥ G ❜ð✐✿ ✭✐✮ ϕ (z) = ϕ(z) ♥➳✉ z ∈ G − B(a; r)❀ ✭✐✐✮ϕ (z) ❧➔ ❤➔♠ ❧✐➯♥ tö❝ tr➯♥ B(a; r)✱ ỏ tr B(a; r) ỗ t ợ (z) ❦❤✐ |z − a| = r✳ ❑❤✐ ✤â✱ ϕ ❧➔ ữợ ỏ ữ tr ❝õ❛ ♠ö❝ ♥➔②✱ ♠ët tr♦♥❣ ♥❤ú♥❣ ♠ö❝ ✤➼❝❤ ❦❤✐ ♥❣❤✐➯♥ ự ữợ ỏ t r t rt ỵ ỹ tự t÷✱ ♥➳✉ ♠✐➲♥ G ✈➔ u : G → R ❧➔ ❤➔♠ ❧✐➯♥ tö❝ ✭G ❧➔ ❜❛♦ ✤â♥❣ tr♦♥❣ C ✮✱ ✤✐➲✉ ❤á❛ tr♦♥❣ G t❤➻ ϕ(z) u(z) ✈ỵ✐ ♠å✐ z G ợ ữợ ỏ ϕ t❤ä❛ ♠➣♥ lim sup ϕ(z) u(a) ✈ỵ✐ ♠å✐ a G u ữợ ỏ u(z) = sup (z) : ữợ ỏ ✈➔ lim sup ϕ(z) u(a) ✭✷✳✶✶✮ ✈ỵ✐ ♠å✐ a ∈ ∂ G ✣➙② ❧➔ ✤✐➸♠ ♠è❝ tr➯♥ ❝♦♥ ✤÷í♥❣ t➻♠ ♥❣❤✐➺♠ ❝õ❛ ❇➔✐ t♦→♥ ❉✐r✐❝❤❧❡t✳ P❤÷ì♥❣ tr➻♥❤ ✭2.10✮ ❝❤♦ t❤➜② ♥➳✉ f : ∂ G → R ❧➔ ❤➔♠ ❧✐➯♥ tư❝ ✈➔ ♥➳✉ f ♠ð rë♥❣ tỵ✐ ❤➔♠ u✱ ✤✐➲✉ ❤á❛ tr➯♥ G t❤➻ u ❝â t❤➸ ♥❤➟♥ ✤÷đ❝ tø t➟♣ ❝→❝ ❤➔♠ ✤✐➲✉ ❤á❛ ✤÷đ❝ ①→❝ ✤à♥❤ r➡♥❣ t❤❡♦ ❝→❝ ❣✐→ trà ❜✐➯♥ f ✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✼✳ ◆➳✉ ♠✐➲♥ G ✈➔ f : ∂ G → R ❧➔ ❤➔♠ ❧✐➯♥ tö❝ t❤➻ ❦➼ ❤✐➺✉ ❤å P❡rr♦♥✱ P(f, G)✱ ỗ tt ữợ ỏ : G → R s❛♦ ❝❤♦ − − ∞ ∞ z→a z→a ∞ ∞ ∞ lim sup ϕ(z) f (a) z→a ✈ỵ✐ ♠å✐ a ∈ ∂ G✳ ❱➻ f ❧✐➯♥ tö❝ ♥➯♥ ❝â ❤➡♥❣ sè M s❛♦ ❝❤♦ |f (a)| M ✈ỵ✐ ♠å✐ a ∈ ∂ G✳ ❱➟② ❤➔♠ ❤➡♥❣ −M ⊂ P(f, G) ✈➔ ❤å P❡rr♦♥ ❦❤→❝ ré♥❣✳ ◆➳✉ u : G → R ❧➔ ❤➔♠ ❧✐➯♥ tö❝✱ ✤✐➲✉ ❤á❛ tr♦♥❣ G ✈➔ f = u | ∂ G ∞ ∞ − ✣é ❚❤à ❚❤❛♥❤ ∞ ✸✸ ❑✸✻❈ ❚♦→♥ ✣❍❙P❍◆ ✷ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❍➔♠ ✤✐➲✉ ❤á❛ t❤➻ ✭2.11✮ trð t❤➔♥❤ ✭✷✳✶✷✮ ✈ỵ✐ ♠é✐ z ∈ G✳ ❚❤➟t ợ f u ữủ ✤à♥❤ ❜ð✐ ✭2.12✮ t❤➻ u ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❉✐r✐❝❤❧❡t ✈ỵ✐ ❣✐→ trà ❜✐➯♥ f ❀ tù❝ ❧➔ ❝â t❤➸ ❣✐↔✐ ❇➔✐ t♦→♥ ❉✐r✐❝❤❧❡t✳ ✣➸ ✭2.12✮ ❧➔ ♥❣❤✐➺♠ ♣❤↔✐ tr↔ ❧í✐ ❤❛✐ ❝➙✉ ❤ä✐ s❛✉ ✭❛✮ u ✤✐➲✉ ❤á❛ tr♦♥❣ G❄ ✭❜✮ lim u(z) = f (a) ✈ỵ✐ ♠é✐ a ∈ ∂ G ❄ ❈➙✉ ❤ä✐ ✤➛✉ t✐➯♥ ❝â t❤➸ tr↔ ❧í✐ ❧➔ ✤ó♥❣ ✈➔ ✤✐➲✉ ♥➔② ✤÷đ❝ ❝❤ù♥❣ tr ỵ t t ọ tự ✤ỉ✐ ❦❤✐ ❝â ❝➙✉ tr↔ ❧í✐ ♣❤õ ✤à♥❤ ✈➔ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ t❤ỉ♥❣ q✉❛ ✈➼ ❞ư✳ ❚✉② ♥❤✐➯♥✱ sû ❞ư♥❣ ❤↕♥ ❝❤➳ ❤➻♥❤ ❤å❝ tr➯♥ G ❜↔♦ ✤↔♠ ❝➙✉ tr↔ ❧í✐ ✭❜✮ ❧➔ ✤ó♥❣ ✈ỵ✐ ❜➜t ❦ý ❤➔♠ ❧✐➯♥ tư❝ f ✳ u(z) = sup {ϕ(z) : ϕ ∈ P(f, G)} za ỵ G f : ∂∞G → R ❧➔ ❤➔♠ ❧✐➯♥ tö❝✳ ❑❤✐ ✤â u(z) = sup {ϕ(z) : ϕ ∈ P(f, G)} ①→❝ ✤à♥❤ ♠ët ❤➔♠ ✤✐➲✉ ❤á❛ u tr➯♥ G✳ ✣➦t |f (a)| t tứ ú ỵ ự M ✈ỵ✐ ♠å✐ a ∈ ∂ G✳ ❱✐➺❝ ❝❤ù♥❣ ♠✐♥❤ ∞ ✭✷✳✶✸✮ ❚ø ✤à♥❤ ♥❣❤➽❛ t❛ ❝â lim sup ϕ(z) M tr♦♥❣ ✤â ϕ ∈ P(f, G) ✈➟② ✭2.13✮ ❧➔ ❤➺ q trỹ t ữủ s r tứ ỵ ỹ ✤↕✐✳ ✣➦t a ∈ G ✈➔ B(a; r) ⊂ G✳ ❑❤✐ ✤â u(a) = sup {ϕ(a) : ϕ ∈ P(f, G)}✳ ❱➟② ❝â ♠ët ❞➣② {ϕ } ⊂ P(f, G) s❛♦ ❝❤♦ u(a) = lim ϕ (z)✳ ✣➦t Φ = max {ϕ , , ϕ }✳ ❚❤❡♦ q ữợ ỏ t ữợ ỏ tr G s (z) = Φ (z) ✈ỵ✐ z ∈ G − B(a; r) ✈➔ Φ ❧➔ ✤✐➲✉ ❤á❛ tr➯♥ B(a; r) ✭t❤❡♦ ❍➺ q✉↔ ✷✳✻✮✳ ϕ(z) M ∀z ∈ G, ϕ ∈ P(f, G) z→z n n n n n n n n n ✣é ❚❤à ❚❤❛♥❤ ✸✹ ❑✸✻❈ ❚♦→♥ ✣❍❙P❍◆ ✷ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❍➔♠ ✤✐➲✉ ❤á❛ ❑❤✐ ✤â ✭✷✳✶✹✮ ϕ Φ Φ ✭✷✳✶✺✮ Φ ∈ P(f, G) ✭✷✳✶✻✮ u(a)✳ ❚ø ✭2.15✮ ✈➔ t❤❡♦ ❝→❝❤ ❝❤å♥ {ϕ }✱ ✤✐➲✉ Φn Φn+1 n n n n ❚❤❡♦ ✭2.16✮✱ Φ (a) ♥➔② ❝❤♦ t❤➜② n n ✭✷✳✶✼✮ ❍ì♥ ♥ú❛✱ t❤❡♦ ✭2.13✮ t❛ ❝â Φ M ✈ỵ✐ ♠å✐ n✳ ❱➟② tø ✭2.14✮ ỵ r s r õ ởt ❤á❛ U tr➯♥ B(a; r) s❛♦ ❝❤♦ U (z) = lim Φ (z) ✤➲✉ t❤❡♦ z tr➯♥ ❜➜t ❦ý ❤➻♥❤ trá♥ ❝♦♥ ♥➔♦ ❝õ❛ B(a; r)✳ ❚ø ✭2.16✮ ✈➔ ✭2.17✮ t❛ ❝â U u ✈➔ U (a) = u(a)✳ ✣➦t z ∈ B(a; r) ✈➔ ❞➣② {ψ } ⊂ P(f, G) s❛♦ ❝❤♦ u(z ) = lim ψ (z )✳ ✣➦t χ = max {ϕ , ψ }✱ X = max {χ , , χ }✱ ✤➦t X ữợ ỏ ợ X õ tr B(a; r)✱ ✤✐➲✉ ❤á❛ tr♦♥❣ B(a; r)✳ ❉♦ ✤â ❝â ❤➔♠ ✤✐➲✉ ❤á❛ U tr➯♥ B(a; r) s❛♦ ❝❤♦ U u ✈➔ U (z ) = u(z )✳ ◆❤÷♥❣ Φ X s❛♦ ❝❤♦ Φ X ✳ ❉♦ ✤â U U u ✈➔ U (a) = U (a) = u(a)✳ ❉♦ ✤â U − U ❧➔ ❤➔♠ ✤✐➲✉ ❤á❛ ➙♠ tr➯♥ B(a; r) ✈➔ (U − U )(a) = 0✳ ❚ø ỵ ỹ t õ U = U ❱➟② U (z ) = u(z )✳ ❱➻ z ❜➜t ❦ý ♥➯♥ u = U tr♦♥❣ B(a; r)✳ ❚ù❝ ❧➔✱ u ✤✐➲✉ ❤á❛ tr➯♥ ♠é✐ ❤➻♥❤ trá♥ ❜➜t ❦ý ❝❤ù❛ tr♦♥❣ G✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✽✳ ❈❤♦ ♠✐➲♥ G ✈➔ f : ∂ G → R ❧➔ ❤➔♠ ❧✐➯♥ tö❝✳ ❍➔♠ ✤✐➲✉ ❤á❛ u(z) = sup {ϕ(z) : ϕ ∈ P(f, G)} ữủ Prr t ợ f ữợ t t tr t rt ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ✈ỵ✐ ♠é✐ ✤✐➸♠ a ∈ ∂ G✱ lim u(z) tỗ t f (a) ❦❤ỉ♥❣ ♣❤↔✐ ❧✉ỉ♥ ✤ó♥❣✳ ❱➼ ❞ư s❛✉ s➩ ♠✐♥❤ ❤å❛ ❝❤♦ tr÷í♥❣ ❤đ♣ ♥➔②✳ ❈❤♦ G = {z : < |z| < 1}✱ T = {z : |z| = 1}✳ ❱➟② ∂G = T ∪ {0}✳ ✣à♥❤ ♥❣❤➽❛ f : ∂G → R ❜ð✐ f (z) = ♥➳✉ z ∈ T ✈➔ f (0) = 1✳ ❱ỵ✐ u(a) = lim Φn (a) n n n n n n n n n n n n n n 0 0 n 0 0 0 ∞ ∞ ✣é ❚❤à ❚❤❛♥❤ z→a ✸✺ ❑✸✻❈ ❚♦→♥ ✣❍❙P❍◆ ✷ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❍➔♠ ✤✐➲✉ ❤á❛ ✤➦t u (z) = (log |z|)(log ) ✳ ❑❤✐ ✤â u ✤✐➲✉ ❤á❛ tr♦♥❣ G✱ u (z) > ✈ỵ✐ z ∈ G✱ u (z) = ✈ỵ✐ z ∈ T ✈➔ u (z) = ♥➳✉ |z| = ✳ ●✐↔ sû v ∈ P(f, G)✳ ❱➻ |f | ♥➯♥ |v(z)| ✈ỵ✐ ♠å✐ z ∈ G✳ ◆➳✉ R = {z : < |z| < 1} t❤➻ lim sup v(z) ≤ u (a) ✈ỵ✐ ♠å✐ a R ỵ ỹ v(z) u (z) ✈ỵ✐ ♠å✐ z ∈ R ✳ ❱➻ ❜➜t ❦ý ♥➯♥ ✈ỵ✐ ♠é✐ z ∈ G✱ v(z) lim u (z) = 0✳ ❉♦ ✤â ❤➔♠ P❡rr♦♥ ❧✐➯♥ ❦➳t ✈ỵ✐ f ỗ t ợ ổ t ❉✐r✐❝❤❧❡t ❦❤æ♥❣ t❤➸ ❣✐↔✐ tr➯♥ ❤➻♥❤ trá♥ ❜à t❤õ♥❣✳ −1 0< 0✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✶✵✳ ❈❤♦ ♠✐➲♥ G ✈➔ a ∈ ∂ G✳ ▼ët ♠↔♥❣ ❝❤➢♥ ❝õ❛ G t↕✐ a ❧➔ ❤å {ψ : r > 0} ❝õ❛ ❝→❝ ❤➔♠ t❤ä❛ ♠➣♥✿ ✭❛✮ ữủ ữợ ỏ tr G(a; r) ✈ỵ✐ ψ (z) 1❀ ✭❜✮ lim ψ (z) = 0❀ ✭❝✮ lim ψ (z) = ✈ỵ✐ w ∈ G ∩ {w : |w − a| = r}✳ ◆➳✉ ψˆ ✤÷đ❝ ①→❝ ✤à♥❤ ❜➡♥❣ ❝→❝❤ ❧➜② ψˆ = ψ tr➯♥ G(a; r) ✈➔ ψˆ (z) = ✈ỵ✐ z ∈ G − B(a; r)✱ t❤➻ ψˆ ❧➔ ❤➔♠ tr➯♥ ✤✐➲✉ ❤á❛✳ ❱➟② ❤➔♠ ψˆ t✐➳♥ tỵ✐ ♠ët ❤➔♠ ♥❤÷♥❣ tr♦♥❣ ✤â z = a = 0✳ ◆➳✉ G ❧➔ ♠✐➲♥ − ∞ ∞ ∞ ∞ r r r z→a z→a r r r r r ✣é ❚❤à ❚❤❛♥❤ r r ✸✻ ❑✸✻❈ ❚♦→♥ ✣❍❙P❍◆ ✷ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❍➔♠ ✤✐➲✉ ❤á❛ ❉✐r✐❝❤❧❡t t❤➻ ❝â ♠ët ♠↔♥❣ ❝❤➢♥ ❝õ❛ G t↕✐ ♠é✐ ✤✐➸♠ t❤✉ë❝ ∂ G✳ ❚❤➟t ✈➟②✱ ♥➳✉ a ∈ ∂ G ✭a = ∞✮ ✈➔ f (z) = |z − a| (1 + |z − a|) ✈ỵ✐ z = ∞, f (∞) = t❤➻ f ❧✐➯♥ tö❝ tr➯♥ ∂ G✳ ❱➟② ❝â ❤➔♠ ❧✐➯♥ tö❝ u : G → R s❛♦ ❝❤♦ u ✤✐➲✉ ❤á❛ tr➯♥ G ✈➔ u(z) = f (z) ✈ỵ✐ z ∈ ∂ G✳ ◆â✐ r✐➯♥❣✱ u(a) = ✈➔ a = ✈ỵ✐ u ∈ G ✳ ✣➦t ∞ −1 ∞ ∞ − ∞ − cr = inf {u(z) : |z − a| = r, z ∈ G} = u(z) : |z − a| = r, z ∈ G− > ✣à♥❤ ♥❣❤➽❛ ψ : G(a; r) → R ❜ð✐ ψ (z) = c1 {u(z), c }✳ ❑❤✐ ✤â✱ {ψ } ❧➔ ♠↔♥❣ ❝❤➢♥✳ r r r r r ỵ G a ∂∞G s❛♦ ❝❤♦ ❝â ♠↔♥❣ ❝❤➢♥ ❝õ❛ G t↕✐ a✳ ◆➳✉ f : ∂∞ G → R ❧✐➯♥ tö❝ ✈➔ u ❧➔ ❤➔♠ P❡rr♦♥ ❧✐➯♥ ❦➳t ✈ỵ✐ f t❤➻ lim u(z) = f (a) z→a ✣➦t {ψ : r > 0} ❧➔ ♠ët ♠↔♥❣ ❝❤➢♥ ❝õ❛ G t↕✐ a ✈➔ ✈ỵ✐ ❣✐↔ t❤✐➳t a = ∞✱ ✈➔ ❣✐↔ sû r➡♥❣ f (a) = ✭❝→❝❤ ❦❤→❝✱ ①➨t ❤➔♠ f − f (a)✮✳ ▲➜② > ✈➔ ❝❤å♥ δ > s❛♦ ❝❤♦ |f (w)| < tr♦♥❣ ✤â w ∈ ∂ G ✈➔ |w − a| < 2δ✱ ✤➦t ψ = ψ ✳ ●✐↔ sû ψˆ : G → R ˆ ữủ (z) = (z) ợ z ∈ G(a; δ) ✈➔ ψ(z) = ✈ỵ✐ z ∈ G − B(a; δ)✳ ❑❤✐ ✤â ψˆ ❧➔ tr➯♥ ✤✐➲✉ ❤á❛✳ ◆➳✉ |f (w)| M ✈ỵ✐ ♠å✐ w ∈ ∂ G t M ữợ ỏ ự ♠✐♥❤✳ r ∞ δ ∞ ❑❤➥♥❣ ✤à♥❤✳ −M ψˆ − ∈ P(f, G) ˆ − = −M − < ◆➳✉ w ∈ ∂ G − B(a; δ) t❤➻ lim sup −M ψ(z) ˆ ˆ − − ✈ỵ✐ ♠å✐ f (w)✳ ❱➻ ψ(z) ♥➯♥ lim sup −M ψ(z) w ∈ ∂ G✳ ◆â✐ r✐➯♥❣✱ ♥➳✉ w ∈ ∂ G ∩ B(a; δ) t❤➻ ˆ − lim sup −M ψ(z) − < f (w) tò② t❤❡♦ ❝→❝❤ ❝❤å♥ δ ✳ ❑❤➥♥❣ ✤à♥❤ ♥➔② ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❉♦ ✤â ˆ − − M ψ(z) u(z) ✭✷✳✶✽✮ ∞ z→w z→w ∞ ∞ z→w ✣é ❚❤à ❚❤❛♥❤ ✸✼ ❑✸✻❈ ❚♦→♥ ✣❍❙P❍◆ ✷ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❍➔♠ ✤✐➲✉ ❤á❛ ✈ỵ✐ ♠å✐ z ∈ G✳ ❚÷ì♥❣ tü t❛ ❝â ˆ + lim inf M ψ(z) lim sup ϕ(z) z→w z→w ✈ỵ✐ ♠å✐ ϕ ∈ P(f, G) w G ỵ ỹ tự tữ + ợ P(f, G) ✈➔ z ∈ G✳ ❉♦ ✤â ϕ(z) M ψ(z) ∞ u(z) ˆ + ; M ψ(z) ❤♦➦❝✱ ❦➳t ❤đ♣ ✈ỵ✐ ✭2.19✮✱ t❛ ❝â ✭✷✳✶✾✮ ˆ ✈ỵ✐ ♠å✐ z ∈ G✳ ◆❤÷♥❣ lim ψ(z) = lim ψ(z) = 0✳ ❱➻ ❜➜t ❦ý ♥➯♥ ✭2.20✮ ✤➣ ❝❤♦ trð t❤➔♥❤ ˆ − − M ψ(z) u(z) z→a ˆ + M ψ(z) z→a lim u(z) = = f (a) z→a ❚❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❍➺ q✉↔ ✷✳✼✳ ▼✐➲♥ G ❧➔ ♠✐➲♥ ❉✐r✐❝❤❧❡t ♥➳✉ ❝â ♠ët ♠↔♥❣ ❝❤➢♥ ❝õ❛ G t↕✐ ♠é✐ ✤✐➸♠ ❝õ❛ ∂∞ G✳ ❇ê ✤➲ ✷✳✶✳ ❈❤♦ ♠✐➲♥ G ⊂ C ✈➔ S ❧➔ t➟♣ ❝♦♥ ❧✐➯♥ t❤æ♥❣ ✤â♥❣ ❝õ❛ C∞ s❛♦ ❝❤♦ ∞ ∈ S ✈➔ S ∩ ∂∞ G = {a}✳ ◆➳✉ G0 ❧➔ ♣❤➛♥ tû ❝õ❛ C∞ − S ✱ ❝❤ù❛ G t❤➻ G0 ❧➔ ♠✐➲♥ ✤ì♥ ❧✐➯♥ tr♦♥❣ ♠➦t ♣❤➥♥❣✳ ✣➦t G , G , , ❧➔ ❝→❝ ♣❤➛♥ tû ❝õ❛ C S ợ G G ú ỵ r ♠é✐ ♠✐➲♥ G ⊂ C✳ ◆➳✉ z ∈ ∂ G t❤➻ G ∪ {z} ❧➔ ❧✐➯♥ t❤æ♥❣✳ ❱➻ G ❧➔ ♣❤➛♥ tû ♥➯♥ ∂ G ⊂ S ✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✶✳✺✱ G ∪S ✭= G ∪ S ✮ ❧➔ ❧✐➯♥ t❤æ♥❣✳ ❱➟② (G ∪ S) = C − G õ t ỵ G ỡ ❧✐➯♥✳ ❈❤ù♥❣ ♠✐♥❤✳ 0 ∞ ∞ n n − n ∞ ∞ n n n n ∞ n=1 ỵ G C ❣✐↔ sû a ∈ ∂∞G s❛♦ ❝❤♦ a ❧➔ ♣❤➛♥ tû ❝õ❛ C∞ − G✱ ❝❤ù❛ a ✈➔ ❦❤æ♥❣ ❤ë✐ tư tỵ✐ ♠ët ✤✐➸♠✳ ❑❤✐ ✤â G ❝â ♠↔♥❣ ❝❤➢♥ t↕✐ a✳ ✣é ❚❤à ❚❤❛♥❤ ✸✽ ❑✸✻❈ ❚♦→♥ ✣❍❙P❍◆ ✷ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❍➔♠ ✤✐➲✉ ❤á❛ ❈❤♦ S ❧➔ ♣❤➛♥ tû ❝õ❛ C − G s❛♦ ❝❤♦ a ∈ S ✳ ❇➡♥❣ ❝→❝❤ ①➨t ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▼♦❜✐✉s t❛ ❝â t❤➸ ❣✐↔ sû r➡♥❣ a = ✈➔ ∞ ∈ S ✳ ●å✐ G ❧➔ ♠ët ♣❤➛♥ tû ❝õ❛ C S ự G trữợ t❛ t❤➜② G ❧➔ ✤ì♥ ❧✐➯♥✳ ❱➻ ∈ G ♥➯♥ ❝â ♥❤→♥❤ ❧ ❝õ❛ log z ✤÷đ❝ ①→❝ ✤à♥❤ tr➯♥ G ✳ ◆â✐ ❝→❝❤ ❦❤→❝ ❧ ①→❝ ✤à♥❤ tr➯♥ G✳ ❱ỵ✐ r > 0✱ ✤➦t ❧ (z) = ❧( ) = ❧(z) − log r ✈ỵ✐ z ∈ G(0; r)✳ ❱➟② −❧ (G(0; r)) ❧➔ t➟♣ ❝♦♥ ❝õ❛ ♥û❛ ♠➦t ♣❤➥♥❣ ❜➯♥ ♣❤↔✐✳ ✣➦t C = G ∩ {z : |z| = r}✳ ❑❤✐ ✤â C ❧➔ ❤ñ♣ ❝õ❛ ♠ët sè ✤➳♠ ✤÷đ❝ ❝→❝ ❝✉♥❣ trá♥ ♠ð γ rí✐ ♥❤❛✉ tø♥❣ ✤æ✐ ♠ët tr♦♥❣ {z : |z| = r}✳ ◆❤÷♥❣ −❧ (γ ) = (iα , iβ ) = {it : α < t < β } ✈ỵ✐ k 1✳ ❉♦ ✤â ❈❤ù♥❣ ♠✐♥❤✳ ∞ ∞ 0 0 z r r r r r k r k k k k k ∞ −❧r (Cr ) = (iαk , iβk ) k=1 ✈➔ ❝→❝ ❦❤♦↔♥❣ ♥➔② rí✐ ♥❤❛✉ tø♥❣ ✤ỉ✐ ♠ët✳ ❍ì♥ ♥ú❛✱ ✤ë ❞➔✐ ❝õ❛ γ ❧➔ r(β − α )✳ ❱➟② (β − α ) 2π ✭✷✳✷✵✮ ◆➳✉ ❧♦❣ ❧➔ ♥❤→♥❤ ❝❤➼♥❤ ❝õ❛ ❧♦❣❛r✐t t❤➻ k k k ∞ k k k=1 hk (z) = Im log z − iαk z − iβk ❧➔ ❤➔♠ ✤✐➲✉ ❤á❛ tr♦♥❣ ♥û❛ ♠➦t ♣❤➥♥❣ ❜➯♥ ♣❤↔✐ ✈➔ < h (z) < π ✈ỵ✐ Rez > 0✳ ◆❣♦➔✐ r❛ k βk ♥➳✉ x > 0✳ ❚ø ✭2.21✮ t❛ ❝â αk ∞ n x dt = π x2 + (y − t)2 hk (x + iy) k=1 ❱➻ ♠é✐ h k ✣é ❚❤à ❚❤❛♥❤ ✭✷✳✷✶✮ x dt x2 + (y − t)2 hk (x + iy) = −∞ ♥➯♥ t ỵ r t õ h = hk k=1 ✤✐➲✉ ❤á❛ ❑✸✻❈ ❚♦→♥ ✣❍❙P❍◆ ✷ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❍➔♠ ✤✐➲✉ ❤á❛ tr♦♥❣ ♥û❛ ♠➦t ♣❤➥♥❣ ❜➯♥ ♣❤↔✐✳ ❉♦ ✤â ψr (z) = h(−❧r (z)) π ✤✐➲✉ ❤á❛ tr♦♥❣ G(0; r)✳ ❚❛ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤ {ψ } ❧➔ ♠↔♥❣ ❝❤➢♥ t↕✐ a✳ ✣➦t r > 0✳ ❑❤✐ ✤â lim Re [−❧ (z)] = +∞✳ ❱➟② ❞➵ ❞➔♥❣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ r➡♥❣ h(z) → ❦❤✐ Rez → +∞✳ ❚ø ✭2.21✮✱ ✭2.22✮ ✈ỵ✐ x > 0✱ r r z→0 ∞ hk (x + iy) h(x + iy) = k=1 βk ∞ = k=1 αk x x + (y − xt )2 dt ∞ (βk − αk ) k=1 2π x ❱➟②✱ lim h(x + iy) = ✤➲✉ t❤❡♦ y s✉② r❛ lim ψ (z) = 0✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ lim ψ (z) = 1✱ w ∈ G ✈ỵ✐ |w| = r t❛ ❞➵ ❞➔♥❣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ r➡♥❣ lim h(z) = π ♥➳✉ α < c < β ✭✷✳✷✷✮ ❱➟② ✤➦t k 1✱ c ∈ (α , β )✳ ❈❤ù♥❣ ♠✐♥❤ ♥❤÷ s❛✉✳ ❑❤➥♥❣ ✤à♥❤✳ ❈❤♦ ❝→❝ sè α, β s❛♦ ❝❤♦ α < α < β < β ✈➔ ♥➳✉ x→+∞ z→0 z→w r k z→ic k k k k z − iα z − iαk z − iβk v(z) = Im log z − iβ u(z) = Im log t❤➻ x > ✈➔ α k < y < βk k , ✳ ❚ø ✤â s✉② r❛ h(x + iy) − hk (x + iy) v(x + iy) = arctan ✭✷✳✷✸✮ ✳ u(x + iy) + v(x + iy) P❤÷ì♥❣ tr➻♥❤ ✭2.22✮ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ♥❤÷ s❛✉✱ ✣é ❚❤à ❚❤❛♥❤ r y − βk x ✹✵ − arctan y−β x ❑✸✻❈ ❚♦→♥ ✣❍❙P❍◆ ✷ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❍➔♠ ✤✐➲✉ ❤á❛ ❱➟② ♥➳✉ x + iy → ic, c < β < β t❤➻ v(x + iy) → 0✳ ❚÷ì♥❣ tü u(x + iy) → ❦❤✐ x + iy → ic ✈ỵ✐ α < α < c✳ ❉♦ ✤â tø ❦❤➥♥❣ ✤à♥❤ t❛ ❝â k k ✭✷✳✷✹✮ lim [h(z) − hk (z)] = ◆❤÷♥❣ hk (x + iy) = arctan y − αk x − arctan y − βk x , ❱➟② lim h (z) = t ủ ợ 2.24 s r ữỡ tr 2.22 ✣➸ ❝â ✤÷đ❝ ✭2.24✮ t❛ sû ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❤➻♥❤ ❤å❝✳ ●å✐ h (z) ❧➔ ❣â❝ tr♦♥❣ ❤➻♥❤✳ ❳➨t t♦➔♥ ❜ë ❝→❝ ❦❤♦↔♥❣ ✭iα , iβ ✮ ♥➡♠ tr➯♥ ✭iα , iβ ✮ ✈➔ tà♥❤ t✐➳♥ ✤✐ ①✉è♥❣ ❞å❝ t❤❡♦ trư❝ ↔♦✱ ❣✐ú ❝❤ó♥❣ ♥➡♠ tr➯♥ ✭iα , iβ ✮ ❝❤♦ tỵ✐ ❦❤✐ trò♥❣ ✈ỵ✐ ✤✐➸♠ ❝✉è✐ ❝õ❛ ❝❤ó♥❣ ✈➔ s❛♦ ❝❤♦ ♠ët tr♦♥❣ ❝→❝ ✤✐➸♠ ❝✉è✐ trò♥❣ ✈ỵ✐ iβ ✳ ❉♦ (β − α ) 2π s➩ ❝â sè β < (β + 2π) s❛♦ ❝❤♦ ♠é✐ ♠ët ❦❤♦↔♥❣ tà♥❤ t✐➳♥ ♥➡♠ tr♦♥❣ ✭iβ , iβ✮✳ ◆➳✉ z = x+iy✱ x > ✈➔ α < y < β ✱ k z→ic j j k j j j k k k k k k k ✣é ❚❤à ❚❤❛♥❤ k ✹✶ ❑✸✻❈ ❚♦→♥ ✣❍❙P❍◆ ✷ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❍➔♠ ✤✐➲✉ ❤á❛ t❤➻ ❣â❝ h (z) t➠♥❣ ❧➯♥ ❦❤✐ tà♥❤ t✐➳♥ ❦❤♦↔♥❣ ✭iα , iβ ✮ ✤✐ ①✉è♥❣✳ ❉♦ ✤â α < Imz < β s✉② r❛ h (z) v(z), ✭✷✳✷✺✮ tr♦♥❣ ✤â v ①→❝ ✤à♥❤ ♥❤÷ tr♦♥❣ ❦❤➥♥❣ ✤à♥❤✱ ✈ỵ✐ ♠å✐ j s❛♦ ❝❤♦ α β ✳ ❚❤ü❝ ❤✐➺♥ t t tữỡ tỹ ợ i , iβ ✮ ✈ỵ✐ β α ✱ ❝â α < (α − 2π) s❛♦ ❝❤♦ sü tà♥❤ t✐➳♥ ♥➡♠ tr♦♥❣ ❦❤♦↔♥❣ ✭iα, iα ✮✳ ❱➟② ♥➳✉ u ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ tr♦♥❣ ❦❤➥♥❣ ✤à♥❤ ✈➔ α < Imz < β ✱ t❤➻ h (z) u(z) ✭✷✳✷✻✮ ✈ỵ✐ ♠å✐ j ✈ỵ✐ β α ✳ ❑➳t ❤ñ♣ ✭2.25✮ ✈➔ ✭2.26✮ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ j k j j k j j j j j k k j k k k k j j j k ❍➺ q✉↔ ✷✳✽✳ ❈❤♦ ♠✐➲♥ G ❦❤ỉ♥❣ ❧➔ ♣❤➛♥ tû ❝õ❛ C∞ − G ❤ë✐ tư tỵ✐ ♠ët ✤✐➸♠✳ ❑❤✐ ✤â G ❧➔ ♠✐➲♥ ❉✐r✐❝❤❧❡t✳ ❍➺ q ỡ rt ỵ ổ õ ỵ ữủ ❤å❛ ❜ð✐ ✈➼ ❞ö s❛✉✳ ❈❤♦ > r > r > ✈ỵ✐ r → 0✳ ❱ỵ✐ ♠é✐ n ✤➦t γ ❧➔ ❝✉♥❣ trá♥ ✤â♥❣ ❝õ❛ ✤÷í♥❣ trá♥ |z| = r ✈ỵ✐ ✤ë ❞➔✐ V (γ )✳ ✣➦t V (γ ) = 2π ✳ ❱➟② G = B(0; 1) − {γ } ∪ {0} ✈➔ ❣✐↔ sû r➡♥❣ lim r C − G = {0} ∪ {γ } {z : |z| 1} ỵ ❝â ♠↔♥❣ ❝❤➢♥ t↕✐ ♠é✐ ✤✐➸♠ ❝õ❛ ∂ G = ∂G✱ trø ✤✐➸♠ ❦❤ỉ♥❣✳ ❈❤ó♥❣ t❛ ❝ơ♥❣ ❝â t❤➸ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ r➡♥❣ ❝â ♠ët ♠↔♥❣ ❝❤➢♥ t↕✐ ❦❤ỉ♥❣ ◆➳✉ r > r > r ✈➔ ♥➳✉ m > n✱ t❤➻ ✤➦t B = B(0; r) − {γ }✳ ●✐↔ sû h ❧➔ ❤➔♠ ❧✐➯♥ tö❝ tr➯♥ B ✱ ✤✐➲✉ ❤á❛ tr➯♥ B ✈ỵ✐ h (z) = ❦❤✐ |z| = r ✈➔ h (z) = ❦❤✐ z ∈ {γ }✳ ❑❤✐ ✤â {h } ❧➔ ❞➣② ❦❤æ♥❣ t➠♥❣ ❝õ❛ ❝→❝ ❤➔♠ ✤✐➲✉ ❤á❛ ❞÷ì♥❣ tr➯♥ G(0; r)✳ ❚❤❡♦ ỵ r {h } tử tợ ởt ✤✐➲✉ ❤á❛ tr➯♥ G(0; r)✳ ❱➻ lim h(z) = ✈ỵ✐ n n n n ∞ n n n n=1 ∞ ∞ n n=1 ∞ m n−1 n m j j=n − m m m m m j m m j=n m ✣é ❚❤à ❚❤❛♥❤ z→w ✹✷ ❑✸✻❈ ❚♦→♥ ✣❍❙P❍◆ ✷ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❍➔♠ ✤✐➲✉ ❤á❛ ♥➯♥ t❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ lim h(z) = 0✳ ✣➦t k ❧➔ ❤➔♠ ✤✐➲✉ ❤á❛ tr➯♥ B(0; r )✱ ❜➡♥❣ ♥➳✉ k ♥➡♠ tr➯♥ {γ } ✈➔ ❜➡♥❣ ♥➳✉ k ♥➡♠ tr➯♥ {z : |z| = r } − {γ }✳ ❑❤✐ ✤â h k tr➯♥ B(0; r ) ✈➔ |w| = r m z→0 m m m m m m m m 2π km (0) = 2π km (rm eiθ )dθ = ❱➻ V (γr ) → 2π, k ♠↔♥❣ ❝❤➢♥ t↕✐ 0✳ m m (0) 2π →0 m 2π − V (γm ) rm ♥➯♥ h(z) → ❦❤✐ z → 0✳ ❱➟② G ❝â ✷✳✺ ❍➔♠ ●r❡❡♥ ❚r♦♥❣ ♣❤➛♥ ♥➔② s➩ ❣✐ỵ✐ t❤✐➺✉ ✈➲ ❤➔♠ ●r❡❡♥ ✈➔ ♥❣❤✐➯♥ ❝ù✉ sỹ tỗ t r õ trá q✉❛♥ trå♥❣ tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✈➔ tr♦♥❣ ❝→❝ ♠æ♥ ❤å❝ ❦❤→❝ ❝õ❛ ❣✐↔✐ t➼❝❤✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✶✶✳ ❈❤♦ ♠✐➲♥ G tr♦♥❣ ♠➦t ♣❤➥♥❣ ✈➔ a ∈ G✳ ❍➔♠ ●r❡❡♥ ❝õ❛ G ❦➻ ❞à t↕✐ a ❧➔ ❤➔♠ g : G → R ❝â ❝→❝ t➼♥❤ ❝❤➜t✿ ✭❛✮ g ✤✐➲✉ ❤á❛ tr♦♥❣ G − {a}❀ ✭❜✮ G(z) = g (z)+log |z − a| ✤✐➲✉ ❤á❛ tr♦♥❣ ❤➻♥❤ trá♥ ①✉♥❣ q✉❛♥❤ a❀ ✭❝✮ lim g (z) =) ✈ỵ✐ ♠é✐ w ∈ ∂ G✳ ❱ỵ✐ ♠✐➲♥ G ✤➣ ❝❤♦ ✈➔ a G g ổ t tt tỗ t tỗ t t ♥❤➜t✳ ❚❤➟t ✈➟②✱ ♥➳✉ h ❝â ❝→❝ t➼♥❤ ❝❤➜t t÷ì♥❣ tü t❤➻ tø ✭❜✮ t❛ ❝â h − g ✤✐➲✉ ❤á❛ tr➯♥ G✳ ◆❤÷♥❣ tø ✭❝✮ s✉② r❛ lim [h (z) − g (z)] = ✈ỵ✐ ♠å✐ w ∈ G h = g t ỵ ỹ ✤↕✐✳ ❍➔♠ ●r❡❡♥ ❧➔ ❞÷ì♥❣✳ ❚❤➟t ✈➟②✱ g ✤✐➲✉ ❤á❛ tr♦♥❣ G − {a} ✈➔ lim g(z) = +∞ ✈➻ g (z) + log |z − a| ✤✐➲✉ ❤á❛ t↕✐ z = a✳ ❚❤❡♦ ♥❣✉②➯♥ a a a z→w ∞ a a a a j→w a a ∞ a a a a z→a ✣é ❚❤à ❚❤❛♥❤ a ✹✸ ❑✸✻❈ ❚♦→♥ P õ tốt ỏ ỵ ❝ü❝ ✤↕✐✱ g (z) > ✈ỵ✐ ♠å✐ z ∈ G − {a}✳ ❉➵ t❤➜② C ❦❤æ♥❣ ❝â ❤➔♠ ●r❡❡♥ ❦➻ ❞à t↕✐ ❦❤æ♥❣ ✭❤♦➦❝ ❦➻ ❞à t↕✐ ♠ët ✤✐➸♠ ❜➜t ❦ý✮✳ ❚❤➟t ✈➟②✱ ❣✐↔ sû g ❧➔ ❤➔♠ ●r❡❡♥ ❦➻ ❞à t↕✐ ❦❤æ♥❣ ✈➔ ✤➦t g = −g t❤➻ g(z) < ✈ỵ✐ ♠å✐ z✳ ❚❛ t❤➜② g ❧➔ ❤➔♠ ❤➡♥❣✱ ♠➙✉ t❤✉➝♥✳ ✣➸ ❝â ✤÷đ❝ ✤✐➲✉ ♥➔② t❛ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤ ♥➳✉ = z = z = t❤➻ g(z ) g(z )✳ ◆➳✉ > t❤➻ ❝â δ > s❛♦ ❝❤♦ |g(z) − g(z )| < t❤ä❛ ♠➣♥ |z − z| < δ✳ ❱➟② g(z) < g(z ) + ♥➳✉ |z − z | < δ✳ ✣➦t r > |z − z | > δ ❦❤✐ ✤â a 0 1 1 1 hr (z) = g(z1 ) + z − z1 log δ r log r ✤✐➲✉ ❤á❛ tr♦♥❣ C − {z }✳ ❑❤✐ ✤â g(z) h (z) ✈ỵ✐ ③ ♥➡♠ tr➯♥ ❜✐➯♥ ❝õ❛ ✈➔♥❤ A = {z : δ < |z − z | < r}✳ ỵ ỹ g(z) h (z) ợ z ∈ A✱ ❤❛② h (z ) g(z )✳ ❈❤♦ r → ∞ t❛ ❝â r r r g(z2 ) 2 lim hr (z2 ) = g(z1 ) + ; r→∞ ✈➻ ❜➜t ❦ý✱ g(z ) g(z ) g r tỗ t ỵ G ❝õ❛ ♠✐➲♥ ❉✐r✐❝❤❧❡t t❤➻ ✈ỵ✐ ♠é✐ a ∈ G ❝â ❤➔♠ ●r❡❡♥ tr➯♥ G ❦ý ❞à t↕✐ a✳ ✣à♥❤ ♥❣❤➽❛ f : ∂G → R ❜ð✐ f (z) = log |z − a| ✈➔ ✤➦t u : G → R ❧➔ ❤➔♠ ❧✐➯♥ tö❝ ❞✉② ♥❤➜t✱ ✤✐➲✉ ❤á❛ tr➯♥ G ✈➔ s❛♦ ❝❤♦ u(z) = f (z) ✈ỵ✐ z ∈ ∂G✳ ❑❤✐ ✤â g (z) = u(z) − log |z − a| ❧➔ ❤➔♠ ●r❡❡♥✳ ❍➔♠ ●r❡❡♥ ❧➔ ❜➜t ❜✐➳♥ ự a ỵ G ✈➔ Ω ❧➔ ❝→❝ ♠✐➲♥ s❛♦ ❝❤♦ ❝â ♠ët ✈➔ ❝❤➾ ♠ët ❤➔♠ ❣✐↔✐ t➼❝❤ f : G → Ω✱ a ∈ G ✈➔ α = f (a)✳ ◆➳✉ ga ✈➔ γα ❧➔ ❝→❝ ❤➔♠ ●r❡❡♥ ❝õ❛ G ✈➔ Ω ❧➛♥ ❧÷đt ❦ý ❞à t↕✐ a ✈➔ α✱ t❤➻ ga (z) = γα (f (z)) ✣é ❚❤à ❚❤❛♥❤ ✹✹ ❑✸✻❈ ❚♦→♥ ✣❍❙P❍◆ ✷ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❍➔♠ ✤✐➲✉ ❤á❛ ❈❤♦ ϕ : G → R ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ϕ = γ ◦ f ✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ ϕ = g t❛ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤ ϕ ❝â ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ ●r❡❡♥ ❦ý ❞à t↕✐ z = a✳ ❘ã r➔♥❣ ϕ ✤✐➲✉ ❤á❛ tr♦♥❣ G − {a}✳ ◆➳✉ w ∈ ∂ G t❤➻ lim ϕ(z) = 0✳ ❑❤✐ ✤â t❛ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤ lim ϕ(z ) = ✈ỵ✐ ❜➜t ❦ý ❞➣② {z } ⊂ G ❦❤✐ z → w✳ ◆❤÷♥❣ ❞➣② {f (z )} ⊂ Ω ♥➯♥ ❝â ❞➣② ❝♦♥ {z } s❛♦ ❝❤♦ f (z ) → w tr♦♥❣ Ω ✭ ❜❛♦ ✤â♥❣ ❝õ❛ C ✮✳ ❱➟② γ (f (z )) → 0✳ ❱➻ ✤✐➲✉ ♥➔② ①↔② r❛ ✈ỵ✐ ❜➜t ❦ý ❞➣② ❝♦♥ ❤ë✐ tư ❝õ❛ {f (z )} ♥➯♥ t❛ ❝â lim ϕ(z ) = lim γ (f (z )) = 0✳ ❉♦ ✤â lim ϕ(z) = ✈ỵ✐ ♠å✐ z ∈ ∂ G✳ ❈✉è✐ ❝ò♥❣✱ ❧➜② ❦❤❛✐ tr✐➸♥ ❝❤✉é✐ ❧ơ② t❤ø❛ ❝õ❛ f t↕✐ z = a✱ ❈❤ù♥❣ ♠✐♥❤✳ α a ∞ n z→w n n nk α n − nk ∞ nk n n α n z→w ∞ f (z) = α + A1 (z − a) + A2 (z − a)2 + ; ❤♦➦❝ f (z) − α = (z − a) [A1 + A2 (z − a) + ] ❉♦ ✤â ✭✷✳✷✼✮ + A (z − a) + | ❧➔ ✤✐➲✉ ❤á❛ ❣➛♥ z = a ✈➻ log |f (z) − α| = log |z − a| + h(z), tr♦♥❣ ✤â h(z) = log |A A = 0✳ ●✐↔ sû γ (w) = ∆(w) − log |w − α| tr♦♥❣ ✤â ∆ ❧➔ ❤➔♠ ✤✐➲✉ ❤á❛ tr➯♥ Ω✳ ❙û ❞ö♥❣ ✭2.27✮✱ t❛ ❝â α ϕ(z) = ∆(f (z)) − log |f (z) − α| = [∆(f (z)) − h(z)] − log |z − a| ❱➻ ∆ ◦ f − h ✤✐➲✉ ❤á❛ ❣➛♥ z = a ♥➯♥ ϕ(z) + log |z − a| ✤✐➲✉ ❤á❛ ❣➛♥ z = a✳ ✣é ❚❤à ❚❤❛♥❤ ✹✺ ❑✸✻❈ ❚♦→♥ ✣❍❙P❍◆ ✷ ❑➳t ❧✉➟♥ ❑❤â❛ ❧✉➟♥ ✤➣ tr➻♥❤ ❜➔② ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ❍➔♠ ✤✐➲✉ ❤á❛✳ ❱✐➺❝ ♥❣❤✐➯♥ ❝ù✉ s➙✉ ✈➲ ❍➔♠ ✤✐➲✉ ❤á❛ ❣â♣ ♣❤➛♥ ❜ê s✉♥❣ t❤➯♠ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ q✉❛♥ trå♥❣ ✈➔♦ ❧➼ t❤✉②➳t ❤➔♠ ❜✐➳♥ ♣❤ù❝✱ ❜ë ♠ỉ♥ ❝â ♣❤➛♥ q✉❛♥ trå♥❣ ✤➦❝ ❜✐➺t ✤è✐ ✈ỵ✐ t♦→♥ ❤å❝ ❝ì ❜↔♥ ✈➔ t♦→♥ ❤å❝ ù♥❣ ❞ư♥❣✳ ❈→❝ ❦✐➳♥ t❤ù❝ ❝➛♥ ❝❤✉➞♥ ❜à ❧➔ ❝→❝ ❦✐➳♥ t❤ù❝ ❝â ❧✐➯♥ q✉❛♥ ✤➳♥ ❍➺ t❤è♥❣ sè ♣❤ù❝✱ ❚♦♣♦ tr➯♥ ♠➦t ♣❤➥♥❣ ự t ự ỵ ổ ❝ü❝ ✤↕✐✳ ✣â ❧➔ ❝ì sð ✤➸ ❤➻♥❤ t❤➔♥❤ ♥➯♥ ❝→❝ ❦✐➳♥ t❤ù❝ ✈➲ ❍➔♠ ✤✐➲✉ ❤á❛✿ ❈→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ ✤✐➲✉ ❤á❛✱ ❍➔♠ ✤✐➲✉ ❤á❛ tr➯♥ ❤➻♥❤ trá♥✱ ữợ ỏ tr ỏ t ❉✐r✐❝❤❧❡t✱ ❍➔♠ ●r❡❡♥✳ ❉♦ t❤í✐ ❣✐❛♥ ♥❣❤✐➯♥ ❝ù✉ ❤↕♥ ❝❤➳ ✈➔ ❦❤↔ ♥➠♥❣ ❜↔♥ t❤➙♥ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ✤➲ t➔✐ ♥➔② ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât ♥❤➜t ✤à♥❤✳ ❱➻ ✈➟②✱ ❡♠ r➜t ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ sü ✤â♥❣ ❣â♣ þ ❦✐➳♥ ❝õ❛ ❝→❝ t❤➛② ❣✐→♦✱ ❝æ ❣✐→♦ ✈➔ ❝→❝ ❜↕♥ s✐♥❤ ✈✐➯♥ tr♦♥❣ ❦❤♦❛ ✤➸ ✤➲ t➔✐ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ✹✻ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬❆❪ ❚➔✐ ❧✐➺✉ t✐➳♥❣ ❱✐➺t ❬✶❪ ◆❣✉②➵♥ P❤ö ❍② ✭✷✵✵✻✮✱ ❇➔✐ t➟♣ ❤➔♠ sè ❜✐➳♥ sè ♣❤ù❝✱ ◆❳❇ ❑❤♦❛ ❤å❝ ✈➔ ❑ÿ t❤✉➟t✳ ❬✷❪ ◆❣✉②➵♥ ❱➠♥ ❑❤✉➯✱ ▲➯ ▼➟✉ ❍↔✐ ✭✷✵✵✶✮✱ ❍➔♠ ❜✐➳♥ ♣❤ù❝✱ ◆❳❇ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✳ ❬❇❪ ❚➔✐ ❧✐➺✉ t✐➳♥❣ ❆♥❤ ❬✸❪ ❏♦❤♥ ❇✳ ❈♦♥✇❛② ✭✶✾✼✸✮✱ ❋✉♥❝t✐♦♥s ♦❢ ❖♥❡ ❈♦♠♣❧❡① ❱❛r✐❛❜❧❡✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ◆❡✇ ❨♦r❦✳ ✹✼ ... t❤ø❛ ✤➣ ❝❤♦ ✈ỵ✐ ❜→♥ ❦➼♥❤ ❤ë✐ tö R✱ t❤➻ R = lim ✣é ❚❤à ❚❤❛♥❤ ✼ an an+1 ❑✸✻❈ ❚♦→♥ ✣❍❙P❍◆ ✷ ❑❤â❛ ❧✉➟♥ tốt ỏ ợ tỗ t↕✐✳ ∞ zn n=0 n! ∞ ✱ t❤❡♦ ▼➺♥❤ ✤➲ ✶✳✻ ❝❤✉é✐ ♥➔② ❝â ❜→♥ ❦➼♥❤ ❤ë✐ tö ❳➨t ❝❤✉é✐ ✳

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