Một số ứng dụng của thặng dư và phép biến đổi laplace

ĐẠI HỌC ĐÀ NẴNG
TRƯỜNG ĐẠI HỌC SƯ PHẠM
KHOA TỐN
——————————–

KHĨA LUẬN TỐT NGHIỆP

Đề tài:

MỘT SỐ ỨNG DỤNG CỦA THẶNG DƯ VÀ
PHÉP BIẾN ĐỔI LAPLACE

Giảng viên hướng dẫn : TS. HOÀNG NHẬT QUY
Sinh viên thực hiện
Lớp

: TRẦN THỊ VY
: 14ST

ĐÀ NẴNG - NĂM 2018


ĐẠI HỌC ĐÀ NẴNG
TRƯỜNG ĐẠI HỌC SƯ PHẠM
KHOA TỐN
——————————–

KHĨA LUẬN TỐT NGHIỆP
Đề tài:

MỘT SỐ ỨNG DỤNG CỦA THẶNG DƯ VÀ

PHÉP BIẾN ĐỔI LAPLACE

Giảng viên hướng dẫn: TS. HOÀNG NHẬT QUY
Sinh viên thực hiện: TRẦN THỊ VY
Chuyên ngành: Sư phạm Toán
Lớp: 14ST

ĐÀ NẴNG - NĂM 2018


õ tốt

t


rữợ tr➻♥❤ ❜➔② ♥ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❦❤â❛ ❧✉➟♥✱ ❡♠ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝
tỵ✐ ❚✐➳♥ s➽ ❍♦➔♥❣ t ữớ t t ữợ ❡♠ ❝â t❤➸ ❤♦➔♥ t❤➔♥❤ ❦❤â❛
❧✉➟♥ ♥➔②✳
❊♠ ❝ô♥❣ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ tỵ✐ t♦➔♥ t❤➸ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ tr♦♥❣ ❦❤♦❛
❚♦→♥✱ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ✲ ✣↕✐ ❤å❝ ✣➔ ◆➤♥❣ ✤➣ ❞↕② ❜↔♦ ❡♠ t➟♥ t➻♥❤ tr♦♥❣ s✉èt q✉→ tr➻♥❤
❤å❝ t➟♣ t↕✐ ❦❤♦❛✳
◆❤➙♥ ❞à♣ ♥➔② ❡♠ ụ ữủ ỷ ớ ỡ t tợ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧ ✤➣
❧✉ỉ♥ ❜➯♥ ❡♠✱ ❝ê ✈ơ✱ ✤ë♥❣ ✈✐➯♥✱ ❣✐ó♣ ✤ï ❡♠ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❦❤â❛
❧✉➟♥ tèt ♥❣❤✐➺♣✳
✣➔ ◆➤♥❣✱ t❤→♥❣ ✵✺ ♥➠♠ ✷✵✶✽
❙✐♥❤ ✈✐➯♥

❚r➛♥ ❚❤à ❱②

❙❱❚❍✿ ❚r➛♥ ❚❤à ❱②

❚r❛♥❣ ✶


▼ö❝ ❧ö❝
✶ ❈❒ ❙Ð ▲➑ ❚❍❯❨➌❚

✹

✶✳✶ ❙è ♣❤ù❝ ✲ ❞➣② sè ♣❤ù❝ ✲ ❝❤✉é✐ sè ♣❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹
✶✳✷ ❍➔♠ sè ❜✐➳♥ ♣❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾
✶✳✸ ❈❤✉é✐ ❤➔♠ ❜✐➳♥ ♣❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷

✷ ❚❍➄◆● ❉× ❱⑨ Ù◆● ❉Ư◆● ❈Õ❆ ❚❍➄◆● ❉×

✶✻

✸ Ù◆● ❉Ư◆● ế PP P



ỵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻
✷✳✷ ❈→❝❤ t➼♥❤ t❤➦♥❣ ❞÷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾
✷✳✸ Ù♥❣ ❞ö♥❣ ❝õ❛ t❤➦♥❣ ❞÷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
ỵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾
✸✳✷ Ù♥❣ ❞ö♥❣ ❝õ❛ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻

✷

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

●❱❍❉✿ ❚❙✳❍♦➔♥❣ ◆❤➟t ◗✉②

▲❮■ ▼Ð ✣❺❯
●✐↔✐ t➼❝❤ ♣❤ù❝ ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ♥❣➔♥❤ ❝ê ✤✐➸♥ ❝õ❛ t♦→♥ ồ ỵ tt ữủ
t ỗ tứ t 19 t õ t sợ ỡ trữợ ✤â✳ ❚ỵ✐ t❤➳ ❦➾ 20 − 21✱
✤➙② ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ❧➽♥❤ ✈ü❝ ✤❛♥❣ ♣❤→t tr✐➸♥ r➜t ♠↕♥❤ tỵ✐ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ ❝→❝
❦❤æ♥❣ ❣✐❛♥ ✈æ ❤↕♥ ❝❤✐➲✉✳ ◆❣♦➔✐ ①✉ ữợ rở tr ữớ t ụ rt q t tỵ✐
❦❤➼❛ ❝↕♥❤ ù♥❣ ❞ư♥❣ ❝õ❛ ♥â tr♦♥❣ t♦→♥ ❤å❝ ❝ơ♥❣ ♥❤÷ ❝→❝ ♥❣➔♥❤ ❦❤♦❛ ❤å❝ ❦❤→❝ ✈➔ ù♥❣
❞ư♥❣ tr♦♥❣ t❤ü❝ t➳✳ ▼ët tr♦♥❣ ♥❤ú♥❣ ù♥❣ ❞ö♥❣ ❝â t➼♥❤ ♥ê✐ ❜➟t tr ữỡ
ỵ tt ừ →♥❤ ①↕ ❜↔♦ ❣✐→❝ tr♦♥❣ ❝ì ❦❤➼✱ Ù♥❣ ❞ư♥❣ tr♦♥❣ ở ỹ ự
rt ỵ tt ỳ t q t ởt ừ ỵ tt t
ự ỹ tr ởt ỵ q trồ õ t ốt õ ỵ tt t
ụ tứ ỵ tt t ồ ỹ ởt ỵ tt
tr t ự ỵ tt t ữ ỵ tt t ữ ♠ët ❝ỉ♥❣ ❝ư q✉❛♥
trå♥❣ ✤➸ ♥❣❤✐➯♥ ❝ù✉ ❜↔♥ ❝❤➜t ❝õ❛ ❝→❝ ✤✐➸♠ ❦➻ ❞à✳ ◆❤ú♥❣ ù♥❣ ❞ö♥❣ ❜❛♥ ✤➛✉ ❝õ❛ ♥â ❧➔ ❞ị♥❣
✤➸ t➼♥❤ ♠ët ❧ỵ♣ ❦❤→ rë♥❣ ❝→❝ t➼❝❤ ♣❤➙♥ ♠➔ ✤ỉ✐ ❦❤✐ ❦❤ỉ♥❣ t❤➸ ❣✐↔✐ q✉②➳t ✤÷đ❝ ❜➡♥❣
ữỡ tổ tữớ t ữợ ❞➜✉ t➼❝❤ ♣❤➙♥ ❝â ❞↕♥❣ ❜➜t t❤÷í♥❣✳
❇✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ❧➔ ♠ët ♣❤➨♣ ❜✐➳♥ ✤ê✐ q✉❛♥ trå♥❣✱ ✤÷đ❝ ✤➦t t❤❡♦ t➯♥ ❝õ❛ ♥❤➔ t♦→♥ ❤å❝
✈➔ t❤✐➯♥ ✈➠♥ ❤å❝ ♥ê✐ t✐➳♥❣ ♥❣÷í✐ P❤→♣ P✐❡rr❡ ❙✐♠♦♥ ▲❛♣❧❛❝❡ ✭✶✼✹✾✲✶✽✷✼✮✳ Ù♥❣ ❞ư♥❣ ❧ỵ♥
♥❤➜t ❝õ❛ ♥â ❧➔ ✤➸ ❣✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✈➔ ❝→❝ t q ỗ ố ừ
ự ử ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ❝❤♦ ♣❤➨♣ ❝❤✉②➸♥ tø ♣❤➨♣ t➼♥❤ ✈✐ t➼❝❤ ♣❤➙♥ tr➯♥ ❤➔♠
s❛♥❣ ❝→❝ ♣❤➨♣ t➼♥❤ ✤↕✐ sè tr➯♥ ↔♥❤ ❝õ❛ ❤➔♠ q✉❛ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡✳ ❈→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❝❤♦
♣❤➨♣ ❝❤✉②➸♥ ♥❤÷ ✈➟② ❣å✐ ❝❤✉♥❣ ❧➔ ♣❤➨♣ t➼♥❤ t♦→♥ tû ✭♦♣❡r❛t✐♦♥❛❧ ❝❛❧❝✉❧✉s ✮✳
❚r♦♥❣ ❦❤â❛ ❧✉➟♥ ♥➔② ❡♠ s➩ tr➻♥❤ ❜➔② sì ❧÷đ❝ ✈➲ ♠ët sè ù♥❣ ❞ư♥❣ ❝õ❛ t❤➦♥❣ ❞÷ ✈➔
♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ tr♦♥❣ ❣✐↔✐ t♦→♥✱ tr♦♥❣ ự ử tỹ t
ố ử ừ õ ỗ ữỡ
ã ữỡ ừ õ tố ❝→❝ ❦❤→✐ ♥✐➺♠✱ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ♥❤➜t ✈➲ sè
♣❤ù❝✱ ❞➣② sè ♣❤ù❝✱ ❝❤✉é✐ sè ♣❤ù❝ ✈➔ ❤➔♠ ✲ ❝❤✉é✐ ❤➔♠ ❜✐➳♥ sè ♣❤ù❝ ♥❤➡♠ t❤✉➟♥ t✐➺♥

❝❤♦ ✈✐➺❝ tr➻♥❤ ❜➔② ữỡ ữỡ [3]
ã ữỡ ừ õ tr sỡ ữủ ỵ tt t ữ
ỵ ỡ t t ữ ứ õ ữ r ỵ tữ ự ử ừ t
ữ tr ởt số t
ã ữỡ ✸ ❝õ❛ ❦❤â❛ ❧✉➟♥ tr➻♥❤ ❜➔② tâ♠ t➢t ❝→❝ ✈➜♥ ✤➲ ❝ì ❜↔♥ ♥❤÷ ✤à♥❤ ♥❣❤➽❛✱ t➼♥❤
❝❤➜t ✈➔ ✤✐➲✉ ❦✐➺♥ tỗ t ừ ờ õ ữ r❛ ù♥❣ ❞ö♥❣ ❝õ❛
❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ tr♦♥❣ ❦➽ t❤✉➟t ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥
t➼♥❤ ❤➺ sè ❤➡♥❣ ✈➔ t➼♥❤ t♦→♥ tr♦♥❣ ❝→❝ ♠↕❝❤ ✤✐➺♥✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝á♥ t➻♠ ❤✐➸✉
❦❤→✐ ♥✐➺♠ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ♥❣÷đ❝ ✈➔ ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ t➻♠ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡
♥❣÷đ❝ ❝õ❛ ởt trữợ t ứ ♠ù❝ ✤ë ♥❤➡♠ ♣❤ö❝ ✈ö ❝❤♦
✈✐➺❝ tr➻♥❤ ❜➔② ù♥❣ ❞ö♥❣ ❝õ❛ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ♥❤÷ ✤➣ ♥➯✉ ð tr➯♥✳
❉♦ t❤í✐ ❣✐❛♥ t❤ü❝ ❤✐➺♥ ❦❤æ♥❣ ♥❤✐➲✉✱ ❦✐➳♥ t❤ù❝ ✈➝♥ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❦❤✐ ❧➔♠ ❦❤â❛ ❧✉➟♥
❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ ❤↕♥ ❝❤➳ ✈➔ s❛✐ sât✳ ❚→❝ ❣✐↔ ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ sü õ ỵ ỳ
ỵ ừ qỵ t ❝ỉ ✈➔ ❜↕♥ ✤å❝✳ ❳✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ✦
❙❱❚❍✿ ❚r➛♥ ❚❤à ❱②

❚r❛♥❣ ✸


❈❤÷ì♥❣ ✶

❈❒ ❙Ð ▲➑ ❚❍❯❨➌❚
❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ỵ tt ỡ số ự ộ sè ♣❤ù❝✱
❤➔♠ ✲ ❝❤✉é✐ ❤➔♠ ❜✐➳♥ ♣❤ù❝ ❧➔♠ ❝ì sð ự ỵ t t
ữỡ ✷✱ ✸ ♣❤ö❝ ✈ö ✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ❧➔ ♠ët sè ù♥❣ ❞ư♥❣ ❝õ❛ t❤➦♥❣ ❞÷ ✈➔ ♣❤➨♣
❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡✳ ❚✉② ♥❤✐➯♥✱ ð ✤➙② tæ✐ ①✐♥ ♣❤➨♣ t❤æ♥❣ q✉❛ ♣❤➛♥ ự
ỵ õ ữủ tr ❝❤✐ t✐➳t ð ❝→❝ ❣✐→♦ tr➻♥❤ ❤✐➺♥ ❤➔♥❤✳

✶✳✶

❙è ♣❤ù❝ ✲ ❞➣② sè ♣❤ù❝ ✲ ❝❤✉é✐ sè ♣❤ù❝

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶ ●✐↔ sû t❛ ✤➦t C = {(a, b) : a, b∈R}✳ ❚r➯♥ C ①→❝ ✤à♥❤ ❤❛✐ ♣❤➨♣
t♦→♥ s❛✉ ✿

(a, b) + (c, d) = (a + c, b + d)
(a, b).(c, d) = (ac − bd, ad − bc)

❑❤✐ ✤â✱ t➟♣ C ❧➔ ♠ët tr÷í♥❣ ✈➔ t❛ ❣å✐ ❧➔ tr÷í♥❣ sè ♣❤ù❝✳
▼é✐ ♣❤➛♥ tû z = (a, b) ❣å✐ ❧➔ ♠ët sè ♣❤ù❝✳
❑➼ ❤✐➺✉ a = Rez ❣å✐ ❧➔ ♣❤➛♥ t❤ü❝ ❝õ❛ z✱
b = Imz ❣å✐ ❧➔ ♣❤➛♥ ↔♦ ❝õ❛ z
ộ số ự z = (a, 0) ữủ ỗ ♥❤➜t ✈ỵ✐ sè t❤ü❝ a∈R✳
✣ì♥ ✈à ↔♦✿ ✣➦t i = (0, 1)✱ t❛ ❝â ✿
i2 = (0, 1).(0, 1) = (−1, 0) = −1.

✹


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

●❱❍❉✿ ❚❙✳❍♦➔♥❣ ◆❤➟t ◗✉②

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷ ✭❉↕♥❣ ✤↕✐ sè ❝õ❛ sè ♣❤ù❝✮ ●✐↔ sû z = (a, b)∈C✳ ❑❤✐ ✤â✱ t❛ ❣å✐

❜✐➸✉ t❤ù❝ z = a + bi ❧➔ ❞↕♥❣ ✤↕✐ sè ❝õ❛ z✳ ❱➔ t❛ ❣å✐ z = a − bi ❧➔ sè ♣❤ù❝ ❧✐➯♥ ❤ñ♣
❝õ❛ z✳
∗

✶✳

✷✳
✸✳
✹✳

▼ët sè t➼♥❤ ❝❤➜t✿

z.z = a2 + b2 ≥0

✳

z1 + z2 + ... + zn = z1 + z2 + ... + zn
z1 .z2 ....zn = z1 .z2 ....zn

✳

✳

z + z = 2.Rez ✱ z − z = 2.Imz ✱ z = z

✳
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✸ ✭▼æ✤✉♥ ❝õ❛ sè ♣❤ù❝✮ ▼é✐ sè ♣❤ù❝ z = a + bi✱ ♠ỉ✤✉♥ ❝õ❛ z
✤÷đ❝ ①→❝ ✤à♥❤ ❧➔
|z| =
∗

✶✳
✷✳
✸✳
✹✳
✺✳

√
√
z.z = a2 + b2 ≥ 0.

▼ët sè t➼♥❤ ❝❤➜t✿

|z|≥0✱ |z| = 0⇔z = 0
|z1 + z2 |≤|z1 | + |z2 |
|z1 .z2 |≤|z1 |.|z2 |

✳

✳

✳

✳
|Rez|≤|z|✱ Imz≤|z| ✳
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✹ ✭❆r❣✉♠❡♥t ❝õ❛ sè ♣❤ù❝✮ ●✐↔ sû z = a + bi=0✳ ●â❝ ϕ t↕♦ ❜ð✐
→
✈❡❝tì −
Oz ✈➔ ❝❤✐➲✉ ❞÷ì♥❣ ❝õ❛ trư❝ Ox ❣å✐ ❧➔ ❛r❣✉♠❡♥t ❝õ❛ z
ã rt argz = ợ −π≤ϕ≤π ✳
• ❆r❣✉♠❡♥t ♣❤ư ✿ ❑➼ ❤✐➺✉ Argz = argz + k2 ợ kZ
ữủ ❝õ❛ sè ♣❤ù❝✮ ❱ỵ✐ ♠é✐ z=0✱ ✤➦t r = |z|✱
ϕ = argz ✳ ❑❤✐ ✤â✱ sè ♣❤ù❝ z ❝â t❤➸ ữủ t ữợ s
||z1 | |z2 |||z1 − z2 |

z = r(cosϕ+isinϕ)

✤÷đ❝ ❣å✐ ❧➔ ❞↕♥❣ ❧÷đ♥❣ ❣✐→❝ ❝õ❛ sè ♣❤ù❝ ❝õ❛ z✳
❙❱❚❍✿ ❚r➛♥ ❚❤à ❱②

❚r❛♥❣ ✺


õ tốt

t

ú ỵ õ t ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ t➼♥❤ ❝❤➜t s❛✉ ✤➙② ✿
n

z = rn (cosnϕ+isinnϕ)
(cosϕ+isinϕ)n = cosnϕ+isinnϕ.

✭❈ỉ♥❣ t❤ù❝ M oivre✮
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✻ ✭❉↕♥❣ ♠ơ ❝õ❛ sè ♣❤ù❝✮ ✣➦t eiϕ = cosϕ + isinϕ✳ ❑❤✐ õ số
ự z õ t t ữủ ữợ s ✿
z = reiϕ , r = |z|, ϕ = arg z

✤÷đ❝ ❣å✐ ❧➔ ❞↕♥❣ ♠ơ ❝õ❛ sè ♣❤ù❝✳
∗

▼ët sè t➼♥❤ ❝❤➜t✿

❈❤♦ z1 = r1eiϕ ✱ z2 = r2eiϕ ✱ t❛ ❝â ✿
✶✳ z1.z2 = r1.r2ei(ϕ +ϕ ) ✳
✷✳ zz1 = rr1 ei(ϕ −ϕ ) ✳

2
2
✸✳ ❈æ♥❣ t❤ù❝ Euler ✿
1

2

1

1

2

2

cosϕ=

eiϕ + e−iϕ
eiϕ − e−iϕ
, sin ϕ =
.
2
2i

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✼ ✭ ❈➠♥ ❜➟❝ n ❝õ❛ sè ♣❤ù❝✮ ❱ỵ✐ n≥1 ✈➔ z∈C✱ ❝➠♥ ❜➟❝ n ❝õ❛ z ❧➔
sè ♣❤ù❝ w∈C t❤ä❛ ♠➣♥ ✿ wn = z✳
▼é✐ sè ♣❤ù❝ z=0 s➩ ❝â n ❝➠♥ ❜➟❝ n ❦❤→❝ ♥❤❛✉ ✈➔ ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ √z ✈➔ ✤÷đ❝ ①→❝
✤à♥❤ ♥❤÷ s❛✉ ✿
n

√
n
z=

√
ϕ + 2kπ
ϕ + 2kπ
n
r(cos
+ sin
), k = 0, 1, 2, .., n − 1 .
n
n

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✽ ✭❉➣② sè ♣❤ù❝ ❤ë✐ tö✮ ❈❤♦ (z )⊂C ✈➔ z∈C✳ ❚❛ ♥â✐ ❞➣② (z ) ❤ë✐
n

n

tư tỵ✐ z ✤÷đ❝ ❦➼ ❤✐➺✉ ✈➔ ①→❝ ✤à♥❤ ♥❤÷ s❛✉ ✿

lim zn = z ⇔ lim |z − zn | = 0.

n

n

ự ợ ồ > 0 tỗ t n0 s❛♦ ❝❤♦ ∀n > n0 t❛ ❝â ✿ |zn − z| < ε✳
❙❱❚❍✿ ❚r➛♥ ❚❤à ❱②

❚r❛♥❣ ✻


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

●❱❍❉✿ ❚❙✳❍♦➔♥❣ ◆❤➟t ◗✉②

◆➳✉ zn = xn + iyn✱ z = x + yi t❤➻ t❛ ❝â ✿
lim zn = z ⇔

n→∞


 lim xn = x
n→∞

 lim y = y
n
n

số ự t tợ ữủ ❤✐➺✉ ✈➔ ①→❝ ✤à♥❤ ♥❤÷ s❛✉ ✿
lim zn = ∞ ⇔ lim |zn | = ∞

n→∞

n→∞

tù❝ ❧➔✱ ✈ỵ✐ ♠å✐ M > 0 tỗ t n0 s n > n0 t õ |zn| > M
ỵ ❝❤✉➞♥ Cauchy✮ ❉➣② (zn) ❤ë✐ tö tr♦♥❣ C ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ♥â ❧➔
❞➣② Cauchy✱ tù❝ ❧➔ ❞➣② t❤ä❛ ♠➣♥ ✿ ∀ε > 0✱ ∃n0 ✿ ∀m, n > n0 t❛ ❝â ✿ |zm − zn| < ε✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✾ ✭❈❤✉é✐ sè ♣❤ù❝✮ ❈❤♦ ❞➣② sè ♣❤ù❝ (un)⊂C✳ ❑❤✐ ✤â✱ ❜✐➸✉ t❤ù❝
s❛✉ ✤➙② ✤÷đ❝ ❣å✐ ❧➔ ❝❤✉é✐ sè ♣❤ù❝ ✿
∞

u1 + u2 + ... + un + ... =

un .
n=1

❚ê♥❣ r✐➯♥❣ ❝õ❛ ❝❤✉é✐ ❧➔ ✿ Sn = u1 + u2 + ... + un✳
❑❤✐ ✤â✱ t❛ ❝â ❞➣② tê♥❣ r✐➯♥❣ (Sn)⊂C✳❚❛ ✤à♥❤ ♥❣❤➽❛ sü ❤ë✐ tö ✈➔ ♣❤➙♥ ❦➻ ❝õ❛ ộ
số ự ữ s
tỗ t ợ n→∞
lim Sn = S=∞ t❤➻ t❛ ♥â✐ ❝❤✉é✐ ❤ë✐ tö ✈➔ ❝â tê♥❣ ❜➡♥❣ S ✈➔
t❛ ✈✐➳t ✿
∞
un = S.
n=1

✲ ❈❤✉é✐ ❦❤ỉ♥❣ ❤ë✐ tư t❤➻ t❛ ♥â✐ ❝❤✉é✐ ✤â ♣❤➙♥
ỵ Cauchy ộ ở tử ợ ồ > 0
tỗ t n0 s❛♦ ❝❤♦ ✿ ∀n > n0✱ ∀p≥1 t❛ ❝â ✿
|un+1 + un+2 + ... + un+p | < ε.

▼ët sè ❦❤→✐ ♥✐➺♠ ❚ỉ ♣ỉ tr➯♥ ♠➦t ♣❤➥♥❣ ♣❤ù❝
• r✲❧➙♥ ❝➟♥✿ r✲❧➙♥ ❝➟♥ ❝õ❛ ✤✐➸♠ a∈C ❧➔ ❤➻♥❤ trá♥ t➙♠ a ❜→♥ ❦➼♥❤ r✱ ✤÷đ❝ ❦➼ ❤✐➺✉
✈➔ ①→❝ ✤à♥❤ ♥❤÷ s❛✉ ✿

D(a, r) = {z ∈ C : |z − a| < r} .

❙❱❚❍✿ ❚r➛♥ ❚❤à ❱②

❚r❛♥❣ ✼


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

●❱❍❉✿ ❚❙✳❍♦➔♥❣ ◆❤➟t ◗✉②

▲➙♥ ❝➟♥✿ ❚➟♣ U C ồ ừ a tỗ t r > 0 s❛♦ ❝❤♦ ✿ D(a, r)⊂U ✳
• ❚➟♣ ♠ð✱ t➟♣ ✤â♥❣✿ ❚➟♣ G⊂C ✤÷đ❝ ❣å✐ ❧➔ t➟♣ ♠ð G ừ ồ
ã

aG

F C ữủ ❣å✐ ❧➔ t➟♣ ✤â♥❣ ♥➳✉ t➟♣ C\F ❧➔ t➟♣ ♠ð✳
• ✣✐➸♠ tr♦♥❣✿ ❈❤♦ t➟♣ X⊂C✳ ✣✐➸♠ a∈X ❣å✐ ❧➔ ✤✐➸♠ tr♦♥❣ ❝õ❛ X ♥➳✉ X ❧➔ ♠ët
❧➙♥ ❝➟♥ ❝õ❛ a✳ ❚➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ ✤✐➸♠ tr♦♥❣ ❝õ❛ X ❣å✐ ❧➔ ♣❤➛♥ tr♦♥❣ ❝õ❛ X ✱ ❦➼
❤✐➺✉ ❧➔ ✿ IntX
IntX t ợ t ự tr X
ã ✣✐➸♠ tư✿ ❈❤♦ X⊂C✱ ✤✐➸♠ a∈C ✤÷đ❝ ❣å✐ ❧➔ ✤✐➸♠ tư ❝õ❛ X ♥➳✉ ✈ỵ✐ ♠å✐ ❧➙♥ ❝➟♥
Ua ❝õ❛ a t❛ ❝â Ua ∩X\{a}=0✳ ❚➟♣ t➜t ❝↔ ❝→❝ ✤✐➸♠ tö ❝õ❛ X ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔✿ X ✈➔
❣å✐ ❧➔ t➟♣ ❞➝♥ ①✉➜t t❤ù ♥❤➜t ❝õ❛ X ✳
• ✣✐➸♠ ❝ỉ ❧➟♣✿ ✣✐➸♠ a∈X ✤÷đ❝ ❣å✐ ❧➔ ✤✐➸♠ ❝ỉ ❧➟♣ ❝õ❛ X tỗ t Ua
ừ a s Ua∩X = a✳
❚➟♣ t➜t ❝↔ ❝→❝ ✤✐➸♠ tö ✈➔ ✤✐➸♠ ❝ỉ ❧➟♣ ❝õ❛ X ✤÷đ❝ ❣å✐ ❧➔ ❜❛♦ ✤â♥❣ ❝õ❛ X ✱ ❦➼ ❤✐➺✉
X ✳ ▼é✐ a∈X ✤÷đ❝ ❣å✐ ❧➔ ✤✐➸♠ ❞➼♥❤ ❝õ❛ X ✳
❚➟♣ X⊂C ❧➔ ✤â♥❣ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ X = X ✳
• ✣✐➸♠ ❜✐➯♥✿ ✣✐➸♠ a⊂C ữủ ồ ừ X ợ ồ ❧➙♥ ❝➟♥ Ua ❝õ❛
a t❤➻ Ua ∩X = φ✳ ❚➟♣ t➜t ❝↔ ❝→❝ ✤✐➸♠ ❜✐➯♥ ❝õ❛ X ❦➼ ❤✐➺✉ ❧➔✿ ∂X ✳ ❱➔ t❛ ❝â ✿

X = IntX ∪ ∂X.
•

❚➟♣ ❈♦♠♣❛❝t✿ ❚➟♣ X⊂C ✤÷đ❝ ❣å✐ ❧➔ t➟♣ ❝♦♠♣❛❝t ♥➳✉ ♠å✐ ❞➣② tr♦♥❣ X ❝❤ù❛

♠ët ❞➣② ❝♦♥ ❤ë✐ tư tỵ✐ ♠ët tở X
ã DC ữủ ồ ♠ët ♠✐➲♥ ♥➳✉ ♥â t❤ä❛ ♠➣♥ 2 ✤✐➲✉ ❦✐➺♥ s❛✉ ✿
i. ❉ ❧➔ t➟♣ ♠ð
ii. ❉ ❧✐➯♥ t❤æ♥❣✱ tù❝ ❧➔ ợ ồ a, bD tỗ t ởt ữớ LD ố a ợ b
ã ữợ tr ởt t ●✐↔ sû γ ❧➔ ♠ët ❝❤✉ t✉②➳♥ tr♦♥❣ C✳ ❑❤✐
✤â✱ γ ❝❤✐❛ ♠➦t ♣❤➥♥❣ ♣❤ù❝ t❤➔♥❤ ❤❛✐ ♠✐➲♥ ❝â ❜✐➯♥ ❝❤✉♥❣ ❧➔ γ ✿ ▼✐➲♥ ❜à ❝❤➦♥ ❦➼
❤✐➺✉ ❧➔ Dγ + ✭❤❛② Dγ ✮✱ ♠✐➲♥ ❝á♥ ❧↕✐ ❦➼ ❤✐➺✉ ❧➔ Dγ −✳
❑❤✐ ✤â✱ ❝❤✐➲✉ ❞÷ì♥❣ tr➯♥ γ ❧➔ ❝❤✐➲✉ ♠➔ ❦❤✐ ♥❣÷í✐ q✉❛♥ s→t ✤✐ t❤❡♦ ❝❤✐➲✉ ✤â t❤➻ s➩
♥❤➻♥ t❤➜② ♠✐➲♥ Dγ + ð ❜➯♥ tr→✐✳
❙❱❚❍✿ ❚r➛♥ ❚❤à ❱②

❚r❛♥❣ ✽


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣
•

●❱❍❉✿ ❚❙✳❍♦➔♥❣ ◆❤➟t ◗✉②

▼✐➲♥ ✤ì♥ ❧✐➯♥✿ ▼✐➲♥ D ữủ ồ ỡ ợ ồ t D t

D D
tỗ t t γ1, γ2, ... s❛♦ ❝❤♦ ❝→❝ ♠✐➲♥ Dγ , Dγ , ... ❦❤æ♥❣ ❜❛♦ ❤➔♠
tr♦♥❣ D t❤➻ t❛ ♥â✐ D ❧➔ ♠✐➲♥ ✤❛ ❧✐➯♥✳

Dγ + ⊂D

1

✶✳✷

2

❍➔♠ sè ❜✐➳♥ ♣❤ù❝

✣à♥❤ ♥❣❤➽❛ sỷ DC ởt t tũ ỵ r D ❝❤♦ ❤➔♠ sè ❜✐➳♥ ♣❤ù❝
w = f (z)✱

♥➳✉ ù♥❣ ợ ộ tr zD tữỡ ự ợ ởt ♠ët sè ❣✐→ trà w
✭❦➸ ❝↔ w = ∞✮✳ ❑❤✐ ✤â✱ t❛ ❣å✐ ❜✐➳♥ z ❧➔ ❜✐➳♥ ✤ë❝ ❧➟♣ ❤❛② ❧➔ ✤è✐ sè✱ ❝á♥ w ❣å✐ ❧➔
❜✐➳♥ ♣❤ö t❤✉ë❝ ❤❛② ❧➔ ❤➔♠ sè✳
◆➳✉ ù♥❣ ✈ỵ✐ ♠ët ❣✐→ trà ❝õ❛ z t÷ì♥❣ ù♥❣ ❝❤➾ ♠ët ❣✐→ trà ❝õ❛ w t❤➻ ❤➔♠ wz = f (z)
✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ✤ì♥ trà✱ ♥❣÷đ❝ ❧↕✐ ❤➔♠ f (z) ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ✤❛ trà✳
❚➟♣ ❤đ♣ t➜t ❝↔ ♥❤ú♥❣ ✤✐➸♠ z∈D ✤÷đ❝ ❣å✐ ❧➔ ♠✐➲♥ ①→❝ ✤à♥❤ ❝õ❛ f (z)✱ ❝á♥ t➟♣ E
t➜t ❝↔ ❝→❝ ❣✐→ trà t÷ì♥❣ ù♥❣ ❝õ❛ w = f (z) ✤÷đ❝ ❣å✐ ❧➔ t➟♣ ❜✐➳♥ t❤✐➯♥ ❝õ❛ ❤➔♠ sè✳
◆➳✉ ✈✐➳t ✿ w = f (z) = u(z) + iv(z) = u(x, y) + iv(x, y)✱ z = x + yi∈D t❤➻ ✈✐➺❝ ❝❤♦
♠ët ❤➔♠ ❜✐➳♥ ♣❤ù❝ w = f (z)✱ z∈D tữỡ ữỡ ợ u, v ừ ❤❛✐
❜✐➳♥ t❤ü❝ x, y ✈ỵ✐ (x, y)∈D✳
✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✷ ✭●✐ỵ✐ ❤↕♥ ❝õ❛ ❤➔♠ ♣❤ù❝✮ ●✐↔ sû f : D−→C ✈➔ z0∈D ❧➔ ♠ët
✤✐➸♠ tư ❝õ❛ D✳ ❙è ♣❤ù❝ a ✤÷đ❝ ❣å✐ ❧➔ ❣✐ỵ✐ ❤↕♥ ❝õ❛ f ❦❤✐ z∈D ✈➔ z−→z0 ✈➔ ✈✐➳t ❧➔ ✿
lim

z→z0 ,z∈D

f (z) = a

♥➳✉ ✈➔ ❝❤➾ ợ ồ > 0 tỗ t > 0 ✿ ∀z∈D✱0 < |z − z0| < δ t❛ ❝â ✿
|f (z) − a| < ε✳
●✐ỵ✐ ❤↕♥ ✈ỉ ❝ị♥❣ ✿
lim

z→∞,z∈D

f (z) = a ⇔ ∀ε > 0, ∃M > 0 : ∀z ∈ D, |z| > M

t❛ ❝â ✿ |f (z) − a| < ε✳
●✐ỵ✐ ❤↕♥ ❜➡♥❣ ✈ỉ ❝ị♥❣ ✿
lim

z→z0 ,z∈D

❙❱❚❍✿ ❚r➛♥ ❚❤à ❱②

f (z) = ∞ ⇔ ∀M > 0, ∃δ > 0 : ∀z ∈ D, |z − z0 | < δ

❚r❛♥❣ ✾


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

●❱❍❉✿ ❚❙✳❍♦➔♥❣ ◆❤➟t ◗✉②

t❛ ❝â ✿ |f (z)| > M ✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✸ ✭❍➔♠ sè ♣❤ù❝ ❧✐➯♥ tö❝✮ ●✐↔ sû f : D−→C ❧➔ ❤➔♠ ♣❤ù❝ ①→❝

✤à♥❤ tr➯♥ D ✈➔ z0∈D✳ ❚❛ ♥â✐ ❤➔♠ sè f ❧✐➯♥ tö❝ t↕✐ z0 ♥➳✉ ♠ët tr♦♥❣ ❤❛✐ ✤✐➲✉ ❦✐➺♥
s❛✉ t❤ä❛ ♠➣♥ ✿
i) z0 ❧➔ ✤✐➸♠ ❝æ ❧➟♣ ❝õ❛ D
ii) ◆➳✉ z0 ❦❤æ♥❣ ❧➔ ✤✐➸♠ ❝æ ❧➟♣ ❝õ❛ D t❤➻ lim f (z) = f (z0 )✳
z→z
◆➳✉ ✈✐➳t f (z) = u(z) + iv(z) t❤➻ f ❧✐➯♥ tö❝ t↕✐ z0∈D ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❝→❝ ❤➔♠ u, v
❧✐➯♥ tö❝ t↕✐ z0✳
❍➔♠ f ✤÷đ❝ ❣å✐ ❧➔ ❧✐➯♥ tư❝ tr➯♥ D ♥➳✉ ♥â ❧✐➯♥ tö❝ t↕✐ ♠å✐ ✤✐➸♠ t❤✉ë❝ D✳
✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✹ ✭❍➔♠ sè ♣❤ù❝ ❧✐➯♥ tư❝ ✤➲✉✮ ❍➔♠ f ✤÷đ❝ ❣å✐ ❧➔ ❧✐➯♥ tö❝ ✤➲✉
tr➯♥ D ♥➳✉ ∀ε > 0✱ ∃δ > 0 s❛♦ ❝❤♦✿ ∀z1, z2∈D ✿ |z1 − z2| < t |f (z1) f (z2)| <
ỵ ✶✳✷✳✶ ◆➳✉ f ❧✐➯♥ tö❝ tr➯♥ t➟♣ ❝♦♠♣❛❝t K⊂C t❤➻ f tử tr K
ỵ f ❧✐➯♥ tö❝ tr➯♥ t➟♣ ❝♦♠♣❛❝t K⊂C t❤➻ ❤➔♠ t❤ü❝ |f (z)| ✤↕t ❣✐→
trà ❧ỵ♥ ♥❤➜t ✈➔ ♥❤ä ♥❤➜t tr➯♥ K
ỵ f tử tr t ❝♦♠♣❛❝t K⊂C t❤➻ f (K) ❝ô♥❣ ❧➔ t➟♣
❝♦♠♣❛❝t✳
✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✺ ✭❈❤✉é✐ ❤➔♠✮ ●✐↔ sû ❞➣② ❤➔♠ (fn) ①→❝ ✤à♥❤ tr➯♥ t➟♣ D⊂C✳ ❑❤✐
✤â✱ ❜✐➸✉ t❤ù❝ s❛✉ ✤➙② ✤÷đ❝ ❣å✐ ❧➔ ❝❤✉é✐ ❤➔♠ ①→❝ ✤à♥❤ tr➯♥ D ✿
0

∞

f1 (z) + f2 (z) + ... + fn (z) + ... =

❚ê♥❣ r✐➯♥❣ ❝õ❛ ❝❤✉é✐ ❤➔♠ ❧➔ ❤➔♠ s❛✉ ✿

fk (z).
k=1

Sn (z) = f1 (z) + f2 (z) + ... + fn (z).

◆➳✉ ✈ỵ✐ z∈D ❞➣② ❤➔♠ (Sn(z)) ❤ë✐ tư t❤➻ t❛ ♥â✐ ❝❤✉é✐ t÷ì♥❣ ù♥❣ ❤ë✐ tư ❤❛② ❦❤↔
tê♥❣ t↕✐ z✳
❚➟♣ D1 = {z|z∈D} ❣å✐ ❧➔ ♠✐➲♥ ❤ë✐ tö ❝õ❛ ❝❤✉é✐✳
❚➟♣ D2 = D\D1 ❣å✐ ❧➔ ♠✐➲♥ ♣❤➙♥ ❦➻ ❝õ❛ ❝❤✉é✐✳
❚r➯♥ ♠✐➲♥ ❤ë✐ tö D1 ❝õ❛ ❝❤✉é✐✱ t❛ ①→❝ ✤à♥❤ ❤➔♠ S : D1−→C ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ✿
S(z) = lim Sn (z), z ∈ D1 .
n→∞

❙❱❚❍✿ ❚r➛♥ ❚❤à ❱②

❚r❛♥❣ ✶✵


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

●❱❍❉✿ ❚❙✳❍♦➔♥❣ ◆❤➟t ◗✉②

❑❤✐ ✤â✱ ❤➔♠ S(z) ✤÷đ❝ ❣å✐ ❧➔ tê♥❣ ❝õ❛ ❝❤✉é✐ tr➯♥ D1 ✈➔ ✈✐➳t ✿
∞

fk (z), z ∈ D1 .

S(z) =
k=1

❈❤✉é✐ ❤➔♠ ✤÷đ❝ ❣å✐ ❧➔ ❤ë✐ tư ✤➲✉ tỵ✐ ❤➔♠ S(z) tr➯♥ D1 ♥➳✉ ❞➣② ❤➔♠ (Sn(z)) ❤ë✐
tö ✤➲✉ ✈➲ S(z) tr➯♥ D1✱ tù❝ ❧➔ ✿
∀ε > 0, ∃n0 : ∀n > n0 , ∀z ∈ D1

t❛ ❝â ✿ |Sn(z) − S(z)| < ε✳
✣à♥❤ ỵ Cauchy ộ ở tử tr➯♥ D⊂C ❦❤✐ ✈➔ ❝❤➾
❦❤✐ ✿
∀ε > 0, ∃n0 : ∀n > n0 , ∀p ≥ 1, ∀z ∈ D

t❛ ❝â ✿ |fn+1(z) + fn+2(z) + ... + fn+p(z)| <
ỵ W eierstrass sỷ ❝❤✉é✐ sè ❞÷ì♥❣

∞

an
n=1

❤ë✐ tư ✈➔

|fn (z)| ≤ an , ∀n > n0 , ∀z ∈ D

❦❤✐ ✤â ❝❤✉é✐ ❤➔♠ ❤ë✐ tử tr D
ỵ sỷ ộ ❤ë✐ tö ✤➲✉ tr➯♥ D ✈➔ ❝→❝ ❤➔♠ fk ❧✐➯♥ tư❝ tr➯♥ D
t❤➻ ❤➔♠ tê♥❣
∞
S(z) =

fk (z)
k=1

❝ơ♥❣ ❧✐➯♥ tư❝ tr➯♥ D✳
✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✻ ✭❈❤✉é✐ ❧ô② t❤ø❛✮ ❈❤✉é✐ ❤➔♠ ❝â ❞↕♥❣ s❛✉ ✿
∞

cn (z − z0 )n = c0 + c1 (z − z0 ) + c2 (z − z0 )2 + ... + cn (z − z0 )n + ...
n=0

❣å✐ ❧➔ ❝❤✉é✐ ❧ơ② t❤ø❛ t↕✐ z0 ✳
❇→♥ ❦➼♥❤ ❤ë✐ tư ❝õ❛ ❝❤✉é✐ ❧ơ② t❤ø❛ ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉ ✿
R=

❙❱❚❍✿ ❚r➛♥ ❚❤à ❱②

1
limsupn→∞

n

|cn |

.

❚r❛♥❣ ✶✶


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

●❱❍❉✿ ❚❙✳❍♦➔♥❣ ◆❤➟t ◗✉②

❑❤✐ ✤â✱ ❝❤✉é✐ ❧ô② t❤ø❛ ❤ë✐ tö ✭t✉②➺t ✤è✐✮ tr➯♥ |z − z0| < R ✈➔ ♣❤➙♥ ❦➻ tr➯♥
|z − z0 | > R✳
✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✼ ✭❍➔♠ ♠ô✮ ❍➔♠ ♠ô ❧➔ ❤➔♠ ❝â ❞↕♥❣ w = ez ✱ z∈C✳
✯ ❚➼♥❤ ❝❤➜t✿
ez1 .ez2 = ez1 +z2 ,

e z1
= ez1 −z2 , ez2 = 0, ∀z ∈ C.
e z2

❱ỵ✐ z = x + yi t❛ ❝â ✿
|ez | = ex , Argez = y + 2kπ, k Z.

ỵ
eiz eiz
,
2iz
z
e e
shz =
= i. sin iz,
2

eiz + e−iz
cosz =
2
ez + e−z
chz =
= cosiz.
2

sin z =

✶✳✸

❈❤✉é✐ ❤➔♠ ❜✐➳♥ ♣❤ù❝

✶✳✸✳✶ ❈❤✉é✐ ❚❛②❧♦r ❝õ❛ ❤➔♠ ❣✐↔✐ t➼❝❤
✣à♥❤ ♥❣❤➽❛✿ ◆➳✉ ❤➔♠ f (z) ❣✐↔✐ t➼❝❤ tr♦♥❣ ❤➻♥❤ trá♥ |z − a| < R✳ ❑❤✐ ✤â✱ ✈ỵ✐

♠å✐ z∈G ✤➲✉ ❝â t❤➸ ❦❤❛✐ tr✐➸♥ ♠ët ❝→❝❤ ❞✉② ♥❤➜t t❤➔♥❤ ❝❤✉é✐ ❧ơ② t❤ø❛ ❝õ❛ z − a
♥❤÷ s❛✉ ✿
+∞
cn (z − a)n .

f (z) =
n=0

❚r♦♥❣ ✤â✱ ❤➺ sè cn ✤÷đ❝ ①→❝ ữ s cn = f
ú ỵ
f (n) (a) =

n!
.
2πi

(n)

(a)
✳
n!

f (ζ)dζ
(ζ − a)n+1

Γ

❧➔ ❝❤✉ t✉②➳♥ ❜❛♦ q✉❛♥❤ ✤✐➸♠ a ✈➔ ♥➡♠ ❤♦➔♥ t♦➔♥ tr♦♥❣ G✳
❉♦ ✤â✱
Γ

cn =

1
2πi

f (ζ)
dζ, n = 0, ±1, ±2, ...
(ζ − a)n+1

Γ

❙❱❚❍✿ ❚r➛♥ ❚❤à ❱②

❚r❛♥❣ ✶✷


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

●❱❍❉✿ ❚❙✳❍♦➔♥❣ ◆❤➟t ◗✉②

❚â♠ ❧↕✐✱ t❛ ❝â ✿
+∞

cn (z − a)n =

f (z) =





 1 .
2πi
n=0

f (ζ)dζ 
n
n+1 (z − a) .
(ζ − a)

+∞

n=0

Γ

✶✳✸✳✷ ❑❤æ♥❣ ✤✐➸♠ ❝õ❛ ❤➔♠ ❣✐↔✐ t➼❝❤
✣à♥❤ ♥❣❤➽❛✿ ●✐↔ sû f (z) ❧➔ ❤➔♠ ❣✐↔✐ t➼❝❤ tr♦♥❣ G✳ ✣✐➸♠ a ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣
✤✐➸♠ ❝õ❛ ❤➔♠ f (z) ♥➳✉ f (a) = 0 ✳
◆➳✉ ❦❤❛✐ tr✐➸♥ ❚❛②❧♦r ❝õ❛ ❤➔♠ f (z) t↕✐ ❧➙♥ ❝➟♥ ✤✐➸♠ a ❝â ❞↕♥❣ ✿
f (z) = cm (z − a)m + cm+1 (z − a)m+1 + ... ✈ỵ✐ cm = 0 ✳
❚❛ ❝â ✿
+∞

+∞
n

cm+k (z − a)k = (z − a)m ϕ(z).

m

cn (z − a) = (z − a)

f (z) =
n=m

k=0

❚r♦♥❣ ✤â✱ m ✤÷đ❝ ❣å✐ ❧➔ ❝➜♣ ❝õ❛ ❦❤ỉ♥❣ ✤✐➸♠ ❤❛② t❛ ♥â✐ a ❧➔ ❦❤æ♥❣ ✤✐➸♠ ❝➜♣ m
❝õ❛ ❤➔♠ f (z) ✳
❑❤✐ m = 1 t❤➻ a ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ✤✐➸♠ ❝➜♣ ✶ ✭❦❤ỉ♥❣ ✤✐➸♠ ✤ì♥✮✳
∗ ✣✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸ a ❧➔ ❦❤æ♥❣ ✤✐➸♠ ❝➜♣ m ❝õ❛ ❤➔♠ f (z) ❧➔ ✿
f (a) = f (a) = f (a) = ... = f (m−1) (a) = 0

f (m)(a) = 0
ỵ ❝➛♥ ✈➔ ✤õ ✤➸ ✤✐➸♠ a ❧➔ ❦❤æ♥❣ ✤✐➸♠ ❝➜♣ m ❝õ❛ ❤➔♠ f (z) ❧➔
f (z) ❝â t❤➸ ❜✐➵✉ ữợ f (z) = (z a)m ϕ(z) ✳
Ð ✤➙② ϕ(z) ❧➔ ❤➔♠ ❣✐↔✐ t➼❝❤ t↕✐ a ✈➔ ϕ(a) = 0✳

✶✳✸✳✸ ❈❤✉é✐ ▲❛✉r❡♥t ❝õ❛ ❤➔♠ ❣✐↔✐ t➼❝❤
✣à♥❤ ♥❣❤➽❛✿ ◆➳✉ f (z) ❧➔ ❤➔♠ ❣✐↔✐ t➼❝❤✱ ✤ì♥ trà tr♦♥❣ ❤➻♥❤ ✈➔♥❤ ❦❤➠♥

r < |z − a| < R✳

❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ z∈G t❛ ❝â ✿
+∞

cn (z − a)n .

f (z) =
n=−∞

❚r♦♥❣ ✤â✱ ❤➺ sè cn ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉ ✿
❙❱❚❍✿ ❚r➛♥ ❚❤à ❱②

❚r❛♥❣ ✶✸


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

●❱❍❉✿ ❚❙✳❍♦➔♥❣ ◆❤➟t ◗✉②

1
f (ζ)
dζ, n = 0, ±1, ±2, ...
2πi Γ (ζ − a)n+1
❜❛♦ q✉❛♥❤ ✤✐➸♠ a ✈➔ ♥➡♠ ❤♦➔♥ t♦➔♥ tr♦♥❣ G✳

cn =

❧➔ ❝❤✉ t✉②➳♥
❈❤✉é✐ tr➯♥ ✤÷đ❝ ❣å✐ ❧➔ ❝❤✉é✐ ▲❛✉r❡♥t ❝õ❛ ❤➔♠ ❣✐↔✐ t➼❝❤
ữ ỵ ộ rt ợ ộ r ❝â ❧ơ② t❤ø❛ ♥❣✉②➯♥ ➙♠✳
◆❣÷í✐ t❛ t❤÷í♥❣ ✈✐➳t ❝❤✉é✐ ▲❛✉r❡♥t ữợ

+

+
n

cn (z a) =
n=

n=1

+

cn
cn (z a)n .
n +
(z − a)
n=0

❚r♦♥❣ ✤â✱ ♣❤➛♥ ❧ơ② t❤ø❛ ❞÷ì♥❣ ❣å✐ ❧➔ ♣❤➛♥ ✤➲✉✱ ♣❤➛♥ ❧ô② t❤ø❛ ➙♠ ❣å✐ ❧➔ ♣❤➛♥
❝❤➼♥❤✳

✶✳✸✳✹ ❈❤✉é✐ ▼❛❝✲▲❛✉r✐♥ ❝õ❛ ♠ët sè ❤➔♠ t❤ỉ♥❣ ❞ư♥❣
z2 z3
zn
+
+ ... +
+ ...
2!

3!
n!

✶✳

ez = 1 + z +

✷✳

sinz = z −

z3 z5 z7
z 2n−1
+
−
+ ... + (−1)n−1
+ ...
3!
5!
7!
(2n − 1)!

✸✳

cosz = 1 −

z2 z4 z6
z 2n
+
−

+ ... + (−1)n−1
+ ...
2!
4!
6!
(2n)!

✹✳

(1 + z)α = 1 + αz +

✺✳

ln(1 + z) = z −

✻✳

1
= 1 + z + z 2 + ... + z n + ...
1−z

α(α − 1)...(α − n + 1) n
α(α − 1) 2
z + ... +
z + ...
2!
n!

z2 z3 z4
zn

+
−
+ ... + (−1)n−1 + ...
2
3
4
n

✶✳✸✳✺ P❤➙♥ ❧♦↕✐ ✤✐➸♠ ❜➜t t❤÷í♥❣ ❝õ❛ ❤➔♠ ❣✐↔✐ t➼❝❤
✣à♥❤ ♥❣❤➽❛✿ ✣✐➸♠ z = a ✤÷đ❝ ❣å✐ ❧➔ ✤✐➸♠ ❜➜t t❤÷í♥❣ ♥➳✉ f (z) ❦❤ỉ♥❣ ❣✐↔✐ t➼❝❤
t↕✐ a✳

P❤➙♥ ❧♦↕✐✿

◆➳✉ z→a
lim f (z) = L✳ ❑❤✐ ✤â z = a ✤÷đ❝ ❣å✐ ❧➔ ✤✐➸♠ ❜➜t t❤÷í♥❣ ❜ä ✤÷đ❝✳
+ ◆➳✉ ❦❤❛✐ tr✐➸♥ ▲❛✉r❡♥t ❝õ❛ f (z) t↕✐ a ♠➔ ♣❤➛♥ ❝❤➼♥❤ ❝â ❤ú✉ ❤↕♥ sè ❤↕♥❣ t❤➻ a
❧➔ ❝ü❝ ✤✐➸♠ ❝õ❛ ❤➔♠ f (z) ✭❝ü❝ ✤✐➸♠ ❝➜♣ m✮✳
+

❙❱❚❍✿ ❚r➛♥ ❚❤à ❱②

❚r❛♥❣ ✶✹


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

●❱❍❉✿ ❚❙✳❍♦➔♥❣ ◆❤➟t ◗✉②

◆➳✉ ♣❤➛♥ ❝❤➼♥❤ ❝õ❛ ❦❤❛✐ tr✐➸♥ ▲❛✉r❡♥t ❝õ❛ ❤➔♠ f (z) t↕✐ a ❝â ✈ỉ sè sè ❤↕♥❣ t❤➻

✤✐➸♠ a ✤÷đ❝ ❣å✐ ❧➔ ✤✐➸♠ ❜➜t t❤÷í♥❣ ❝èt ②➳✉✳
(z)
tr♦♥❣ ✤â f2(z) ♥❤➟♥ z = a ổ m
ỵ f (z) = ff21(z)
f1 (a)=0 t❤➻ f (z) ♥❤➟♥ z = a ❧➔ ❝ü❝ ✤✐➸♠ ❝➜♣ m✳
+

❙❱❚❍✿ ❚r➛♥ ❚❤à ❱②

❚r❛♥❣ ✶✺


ữỡ

ì ệ
ế ì
ỵ tt t ❞÷ ❧➔ ♠ët ❝ỉ♥❣ ❝ư q✉❛♥ trå♥❣ ✤➸ ♥❣❤✐➯♥ ❝ù✉ ❜↔♥ ❝❤➜t ❝õ❛ ❝→❝
✤✐➸♠ ❦➻ ❞à✳ ◆❤ú♥❣ ù♥❣ ❞ö♥❣ ❜❛♥ ừ ỵ tt t ữ ũ t ởt ❧ỵ♣
❦❤→ rë♥❣ ❝→❝ t➼❝❤ ♣❤➙♥ ♠➔ ✤ỉ✐ ❦❤✐ t❛ ❦❤ỉ♥❣ t❤➸ ❣✐↔✐ q✉②➳t ✤÷đ❝ ❦❤✐ sû ❞ư♥❣ ❝→❝
♣❤÷ì♥❣ ♣❤→♣ t❤ỉ♥❣ tữớ t ữợ t õ t
tữớ



ỵ



f ❣✐↔✐ t➼❝❤ tr♦♥❣ B(a, r)\{a} ✱ ❧✐➯♥ tö❝ tr➯♥ Γ = ∂B(a, R)✳ ❚➼❝❤ ♣❤➙♥
1

Res[f, a] =
f (z)dz ❣å✐ ❧➔ t❤➦♥❣ ❞÷ ❝õ❛ ❤➔♠ f t↕✐ ✤✐➸♠ a✳
2πi Γ
❈❤♦ ❤➔♠ f ❣✐↔✐ t➼❝❤ tr♦♥❣ ♠✐➲♥ R < |z| < ∞✱ ❧✐➯♥ tö❝ tr➯♥ Γ = ∂B(0, R)✳ ❚➼❝❤
1
♣❤➙♥ Res[f, ∞)] = 2πi
f (z)dz ❣å✐ ❧➔ t❤➦♥❣ ❞÷ ❝õ❛ ❤➔♠ f t

ỵ ữ ừ f t↕✐ ✤✐➸♠ a ❧➔ ❤➺ sè c−1 ❝õ❛ ❦❤❛✐ tr✐➸♥
▲❛✉r❡♥t t↕✐ ✤✐➸♠ ✤â✳
Res[f, a] = c−1 ✳
❈❤ù♥❣ ♠✐♥❤✿ ❑❤❛✐ tr✐➸♥ ▲❛✉r❡♥t ❤➔♠ f t↕✐ ✤✐➸♠ a ✿
✶✻


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

●❱❍❉✿ ❚❙✳❍♦➔♥❣ ◆❤➟t ◗✉②
∞

f (z) =
n=1

1
❱ỵ✐ cn = 2πi

Γ

∞

c−n
+
cn (z − a)n
(z − a)n n=0

✳

f (ζ)
dζ, n = 0, ±1, ±2, ...
(ζ − a)n+1

1
▼➦t ❦❤→❝✱ t❤❡♦ ✤à♥❤ ♥❣❤➽❛ t❤➦♥❣ ❞÷ t↕✐ ♠ët ✤✐➸♠ t❤➻ Res[f, a] = 2i
f (z)dz.

õ Res[f, a] = c1

ỵ

z = a ❧➔ ❝ü❝ ✤✐➸♠ ❝➜♣ ✶ ❝õ❛ ❢ t❤➻ ✿
Res[f, a] = lim (z − a)f (z) ✳
z→a
ϕ(z)
✱ tr♦♥❣ ✤â ϕ(z) ✈➔ ψ(z) ❧➔ ❝→❝ ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ t↕✐ z0 ✈ỵ✐
✐✐✳ ◆➳✉ f (z) = ψ(z)

t❤➻ Res[f, a] = ψϕ(a)
✳
(a)
✐✐✐✳ ◆➳✉ a ❧➔ ❝ü❝ ✤✐➸♠ ❝➜♣ m ❝õ❛ f t❤➻ ✿

1
dm−1
Res[f, a] = lim
[(z − a)m f (z)] ✳
z→a (m − 1)! dz m−1
ϕ(a) = 0, ψ(a) = 0, ψ (a) = 0

❈❤ù♥❣ ♠✐♥❤✿

❚❛ t❤ø❛ ♥❤➟♥ t➼♥❤ ❝❤➜t ✐✳ ✈➔ ✐✐✳
✐✐✐✳ ●✐↔ t❤✐➳t a ❧➔ ❝ü❝ ✤✐➸♠ ❜ë✐ ♠ ❝õ❛ f (z)✳ ▲ó❝ ✤â✱ t❛ ❝â ❦❤❛✐ tr✐➸♥ ▲❛✉r❡♥t ❝õ❛
f (z) tr♦♥❣ ❧➙♥ ❝➟♥ a ✿
∞

c−m
c−1
f (z) =
+
cn (z − a)n .
m + ... +
(z − a)
z − a n=0

❍❛② ❧➔ ✿
(z − a)m f (z) = c−m + ... + c−1 (z − a)m−1 + c0 (z − a)m + ...
⇒

dm−1
[(z − a)m f (z)] = (m − 1)!c−1 + m(m − 1)...2c0 (z − a) + ...
dz m1

q ợ t ữủ
dm1
[(z − a)m f (z)] = (m − 1)!c−1
z→a dz m−1
1
dm−1
⇒
lim m−1 [(z − a)m f (z)] = c−1 = Res[f, a].
(m − 1)! z→a dz
lim

❙❱❚❍✿ ❚r➛♥ ❚❤à ❱②

❚r❛♥❣ ✶✼


õ tốt

t

ỵ ❤➔♠ f (z) ❝â ♠ët sè ❤ú✉ ❤↕♥ ✤✐➸♠ ❦➻ ❞à ❝æ ❧➟♣ tr♦♥❣ ♠✐➲♥

✤â♥❣ ❝õ❛ ♠➦t ♣❤➥♥❣ ♣❤ù❝ t❤➻ tê♥❣ t➜t ❝↔ ❝→❝ t❤➦♥❣ ❞÷ ✭❦➸ ❝↔ t❤➦♥❣ ❞÷ t↕✐ ✤✐➸♠
z = 0✮ ❜➡♥❣ ❦❤æ♥❣ ✿
n
Res[f, ak ] + Res[f, ∞] = 0 ✳

❈❤ù♥❣ ♠✐♥❤✿ ●å✐ Γ ✈ỵ✐ k = 1, .., n ❧➔ ❝→❝ ✤÷í♥❣ trá♥ |z − a | = R ✤õ ❜➨ ✤➸
k=1

k

k

k

❝❤➾ ❜❛♦ r✐➯♥❣ tø♥❣ ✤✐➸♠ ak ✈➔ Γ ❧➔ ✤÷í♥❣ trá♥ |z| = R ừ ợ t tt
ữớ trỏ k ✳ ❚❤❡♦ ❝æ♥❣ t❤ù❝ t➼❝❤ ♣❤➙♥ ❈❛✉❝❤②✱ t❛ ❝â ✿
n

f (z)dz = −

f (z)dz =
k=1 Γ

Γ

f (z)dz.

Γ−

k

❈❤✉②➸♥ ✈➳✱ s❛✉ ✤â ❝❤✐❛ ❤❛✐ ✈➳ ❝❤♦ 2πi✱ t❛ ✤÷đ❝ ✿
n

1
1
f (z)dz = 0.
f (z)dz +

k=1 2πi Γk
Γ− 2πi

❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ✷✳✶✳✶ ✈➲ t❤➦♥❣ ❞÷✱ s✉② r❛ ✿
Res [f, a1 ] + Res [f, a2 ] + ... + Res [f, ak ] + Res [f, ∞] = 0
n

⇒

Res[f, ak ] + Res[f, ] = 0

k=1

ỵ f (z) t tr õ D ợ ữớ


❦➼♥ Γ trø ♠ët sè ❤ú✉ ❤↕♥ ✤✐➸♠ a1, a2, ..., an ❜➜t t❤÷í♥❣ ❝ỉ ❧➟♣ ♥➡♠
tr♦♥❣ DΓ t❤➻ ✿
n

f (z)dz = 2i

Res[f,ak ].
k=1



ự ỵ t ♣❤➙♥ ❈❛✉❝❤②✱ t❛ ❝â ✿
n

f (z)dz =
Γ

f (z)dz.
k=1 Γ

k

◆❤÷♥❣ t❤❡♦ ✤à♥❤ ♥❣❤➽❛ t❤➦♥❣ ❞÷ t❤➻ ✿
❙❱❚❍✿ ❚r➛♥ ❚❤à ❱②

❚r❛♥❣ ✶✽


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

●❱❍❉✿ ❚❙✳❍♦➔♥❣ ◆❤➟t ◗✉②

f (z)dz = 2πiRes[f,ak ].
Γk

❉♦ ✤â t❛ ✤÷đ❝ ✿
n

f (z)dz =

2πiRes[f,ak ]
k=1

Γ

n

⇒

f (z)dz = 2πi

Res[f,ak ].
k=1

Γ

❇ê ✤➲ ❏♦r❞❛♥ ❈❤♦ ✤÷í♥❣ ❝♦♥❣ Γ

✈➔ ❤➔♠ f (z) ❣✐↔✐ t➼❝❤
β} ♥❣♦↕✐ trø ❤ú✉ ❤↕♥ ✤✐➸♠ ❜➜t t❤÷í♥❣✳ ❑❤✐ ✤â✱
R

tr♦♥❣ ♥û❛ ♠➦t ♣❤➥♥❣ D = {Imz
t❛ ❝â ✿

= {|z| = R, Imz β}

✶✳ ◆➳✉ z→∞
lim zf (z) = 0 t❤➻
lim

R→+∞
ΓR

f (z)dz = 0.

✷✳ ◆➳✉ z→∞
lim f (z) = 0✱ ✈ỵ✐ α > 0 t❤➻
lim

R→+∞
ΓR

✷✳✷

f (z)eiαz dz = 0.

❈→❝❤ t➼♥❤ t❤➦♥❣ ❞÷

✷✳✷✳✶ ❈→❝❤ t➻♠ ❝ü❝ ✤✐➸♠ ❝➜♣ ♠

❈❤♦ z = a = ∞ ❧➔ ✤✐➸♠ ❜➜t t❤÷í♥❣ ❝ỉ ❧➟♣ ❝õ❛ f (z) ✿
• ◆➳✉ lim f (z) = L=∞ t❤➻ z = a ❧➔ ❝ü❝ ✤✐➸♠ ❝➜♣ ✵ ✳
z→a
• ◆➳✉

 lim f (z) = ∞
z→a

 lim [(z − a)m f (z)] = L ∈ C\{0}
z→a

t❤➻ z = a ❧➔ ❝ü❝ ✤✐➸♠ ❝➜♣ ♠✳
❙❱❚❍✿ ❚r➛♥ ❚❤à ❱②

❚r❛♥❣ ✶✾


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣
•

●❱❍❉✿ ❚❙✳❍♦➔♥❣ ◆❤➟t ◗✉②

◆➳✉ z→a
lim f (z) ổ tỗ t t z = a t t❤÷í♥❣ ❝èt ②➳✉✳
❱➼ ❞ư ✶✿ ❚➻♠ ✈➔ ♣❤➙♥ ❧♦↕✐ ✤✐➸♠ ❜➜t t❤÷í♥❣ ❝ỉ ❧➟♣ ❝õ❛ ❤➔♠
f (z) =

●✐↔✐✿

sin2 z
z 2 (1 − z)3

❍➔♠ f (z) ❝â ❤❛✐ ✤✐➸♠ ❜➜t t❤÷í♥❣ ❧➔ z = 0 ✈➔ z = 1✳
• ❉♦
2
lim f (z) = lim

z→0

z→0

sin z
1

3
z
(z − 1)

= −1 = ∞

♥➯♥ z = 0 ❧➔ ❝ü❝ ✤✐➸♠ ❝➜♣ ✵ ❝õ❛ f (z)✳
• ❚❛ ❝â ✿



 lim f (z) = ∞
z→1


 0 = lim[(z − 1)3 f (z)] = sin2 1.
z→1

❙✉② r❛ ✿ z = 1 ❧➔ ❝ü❝ ✤✐➸♠ ❝➜♣ ✸✳
❱➼ ❞ö ✷✿ ❳→❝ ✤à♥❤ ✤✐➸♠ ❜➜t t❤÷í♥❣ ❝ỉ ❧➟♣ ❝õ❛ ❤➔♠
f(z) = cos

1
z−i

●✐↔✐✿

❍➔♠ f (z) ❝â ✤✐➸♠ ❜➜t t❤÷í♥❣ ❝ỉ ❧➟♣ ❧➔ z = i
1
lim

cos
ổ tỗ t z = i ❧➔ ✤✐➸♠ ❜➜t t❤÷í♥❣ ❝èt ②➳✉✳
z→i
z−i

✷✳✷✳✷ P❤÷ì♥❣ ♣❤→♣ t➼♥❤ t❤➦♥❣ ❞÷
❈→❝❤ ✶✿ ❉ị♥❣ ✤à♥❤ ♥❣❤➽❛

✭❍➺ sè ❝õ❛ z −1 a tr♦♥❣ ❦❤❛✐ tr✐➸♥ f (z) q✉❛♥❤ ✤✐➸♠ z = a✮✳
1
•Res[f (z), ∞] = −c−1 ✭❍➺ sè ❝õ❛ tr♦♥❣ ❦❤❛✐ tr✐➸♥ f (z) q✉❛♥❤ ✤✐➸♠ z = ∞✮✳
z
❈→❝❤ ✷✿ ❉ò♥❣ ❝ü❝ ✤✐➸♠
•◆➳✉ a = ∞ ❧➔ ❝ü❝ ✤✐➸♠ ✤ì♥ t❤➻ ✿
Res[f (z), a] ❂ lim [(z − a)f (z)] ✳
z→a
•◆➳✉ a = ∞ ❧➔ ❝ü❝ ✤✐➸♠ ❝➜♣ ♠ (m ≥ 2) t❤➻ ✿
•Res[f (z), a] = c−1

❙❱❚❍✿ ❚r➛♥ ❚❤à ❱②

❚r❛♥❣ ✷✵


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣
Res[f (z), a]

●❱❍❉✿ ❚❙✳❍♦➔♥❣ ◆❤➟t ◗✉②

lim [(z − a)m f (z)](m−1) ✳

❂ (m −1 1)! z→a

❱➼ ❞ö ✸✿ ❚➼♥❤ Res[f (z), 2] ❝õ❛ f (z) = z
●✐↔✐✿

2

− 2z + 3
z−2

✳

✯ ❈→❝❤ ✶✿ ❚❛ ❝â ✿

❂

z 2 − 2z + 3
3
2 + (z − 2) +
(0 < |z − 2| < ∞)
z−2
z−2
❱➟② Res[f (z), 2] ❂c−1 = 3✳

✳

✯ ❈→❝❤ ✷✿ ❚❛ ❝â ✿

z 2 − 2z + 3
= ∞ ✈➔ lim [(z − 2)f (z)] = 3✳

z→2
z→2
z→2
z−2
⇒ z = 2 ❧➔ ❝ü❝ ✤✐➸♠ ✤ì♥ ❝õ❛ f (z)✳ ◆➯♥ ✿
lim f (z) = lim

❂ z→2
lim [(z − 2)f (z)] ❂ lim (z 2 − 2z + 3) = 3✳
z→2
❱➼ ❞ö ✹✿ ❚➼♥❤ Res[f (z), 1] ❝õ❛ f (z) = z(z −1 1)2 ✳

Res[f (z), 2]

●✐↔✐✿

✯ ❈→❝❤ ✶✿ ❚❛ ❝â ✿ z(z −1 1)2 = (z −1 1)2 . 1 − (11 − z)
1
[1 + (1 − z) + (1 − z)2 + ...]
2
(z − 1)
1
1
=
−
+ 1 − ...(0 < |z − 1| < 1)✳
(z − 1)2 z − 1

=

❱➟② Res[f (z), 1] ❂ c−1 = −1✳
1
lim f (z) = lim
= ∞ ✈➔ lim [(z − 1)2 f (z)] = 1✳
✯ ❈→❝❤ ✷✿ ❚❛ ❝â ✿ z→1
z→1 z(z − 1)2
z→1
⇒ z = 1 ❧➔ ❝ü❝ ✤✐➸♠ ❝➜♣ ✷ ❝õ❛ f (z)✳
1
1
❱➟② Res[f (z), 1] ❂ 1!1 z→1
lim [(z − 1)2 f (z)] ❂ lim ( ) ❂ − lim 2 = −1✳
z→1 z
z→1 z

❱➼ ❞ö ✺✿ ❚➼♥❤ Res[f (z), ∞] ❝õ❛ ❝→❝ ❤➔♠ ✿
✶✮
✷✮

2
f (z) = e z
g(z) =

●✐↔✐✿
✶✮ ❚❛ ❝â ✿

z4

1
+1

2
ez =

∞
n=0

❙❱❚❍✿ ❚r➛♥ ❚❤à ❱②

1 2 n
2
4
( ) =1+ +
+ ..✳
n! z
z 2!z 2

❚r❛♥❣ ✷✶


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

●❱❍❉✿ ❚❙✳❍♦➔♥❣ ◆❤➟t ◗✉②

⇒ c−1 = 2✳

❱➟② Res[f (z), ∞] ❂ −c−1 = −2✳
✷✮ ❚❛ ❝â ✿ g(z) = z14 . 1−1
1−

z4
(−1)n
1
= 4 ∞
n=0
z
(z 4 )n
⇒ c−1 = 0✳

❂ z14 − z18 + z112 − ...

❱➟② Res[f (z), ] c1 = 0

ú ỵ

ợ g(a) = 0 h(a) = 0 ✈➔ g (a) = 0
✶✮ ◆➳✉ a = ∞ ❧➔ ❝ü❝ ✤✐➸♠ ✤ì♥ ✈➔ f (z) = h(z)
g(z)
t❤➻ ✿
Res [f (z), a] =

✷✮ ❑❤✐ t➼♥❤

❱➼ ❞ö ✻✿

❤↕♥✳

h(z)
g (z)

=
z=a

h(a)
.
g (a)

❣✐ỵ✐ ❤↕♥ ❝â ❞↕♥❣ ❤♦➦❝ ∞
✱ t❛ ❝â t❤➸ ❞ị♥❣ q✉② t➢❝ ▲✬❍♦s♣✐t❛❧✳
∞ z
❚➻♠ t❤➦♥❣ ❞÷ ❝õ❛ f (z) = z2e+ 1 t↕✐ ❝→❝ ✤✐➸♠ ❜➜t t❤÷í♥❣ ❝ỉ ❧➟♣
0
0

❤ú✉

●✐↔✐✿

❍➔♠ f (z) ❝â ❤❛✐ ❝ü❝ ✤✐➸♠ ✤ì♥ ❧➔ z = ±i✳
❚❛ ❝â ✿
Res [f (z), i] =

ez
(z 2 + 1)

=
z=i

ez
2z

z

Res [f (z), −i] =

✷✳✸

e
(z 2 + 1)

=
z=−i

=
z=i
z

e
2z

ei
2i
=−

z=−i

1
.
2iei

Ù♥❣ ❞ö♥❣ ❝õ❛ t❤➦♥❣ ❞÷

✷✳✸✳✶ ❚➼♥❤ t➼❝❤ ♣❤➙♥ ❞å❝ t❤❡♦ ✤÷í♥❣ ❝♦♥❣ ❦➼♥
❱➼ ❞ư ✼✿ ❚➼♥❤ t➼❝❤ ♣❤➙♥ I = z e+ 1 dz✳
●✐↔✐✿
z

|z|=2

❙❱❚❍✿ ❚r➛♥ ❚❤à ❱②

2

❚r❛♥❣ ✷✷


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

●❱❍❉✿ ❚❙✳❍♦➔♥❣ ◆❤➟t ◗✉②

z

❍➔♠ f (z) = z2e+ 1 ❝â ❤❛✐ ✤✐➸♠ ❜➜t t❤÷í♥❣ ❝ỉ ❧➟♣ ❧➔ z = ±i ♥➡♠ tr♦♥❣ ❤➻♥❤ trá♥
D : |z| 2✳
⑩♣ ử ỵ ử t õ ✿
I = 2πi(Res[f (z), i] + Res[f (z), −i])
1
ei
−
= 2πi(

)
2i
2iei
ei − e−i
) ❂ 2πi.sin1 ✳
= 2πi(
2i
❚➼♥❤ t➼❝❤ ♣❤➙♥ I =

❱➼ ❞ö ✽✿
●✐↔✐✿

|z−1|=1

z+2
dz ✳
(z − 1)2 (z + 1)

❍➔♠ f (z) = (z − z1)+2(z2 + 1) ❝â ♠ët ❝ü❝ ✤✐➸♠ ❝➜♣ ✷ ❧➔ z = 1 ♥➡♠ tr♦♥❣ ❤➻♥❤ trá♥
D : |z 1| 1
ử ỵ t ❝â ✿
z+2
π
I = 2πiRes[f (z), 1] = 2πi. lim (
) = −i ✳
z→1 z + 1
2

❱➼ ❞ö ✾✿ ❚➼♥❤ t➼❝❤ ♣❤➙♥ I =
●✐↔✐✿

|z|=1

z4
dz
2z 5 − 1

✳

4

❍➔♠ f (z) = 2z5z − 1 ❝â ✺ ✤✐➸♠ ❜➜t t❤÷í♥❣ ❝ỉ ❧➟♣ ak ✱ k = 1, 5 ❧➔ ❝➠♥ ❜➟❝ ✺ ❝õ❛ 21 ✳
❚❛ ❝â ✿
1
z4
= .
5
2z − 1
2z

1

1
= .
1
2z
1− 5
2z
1
1

1
+ 6 + 11 + ...
=
2z 4z
8z
1
1
⇒ c−1 = ⇒ Res[f (z), ∞] = − ✳
2
2
f (z) =

∞
n=0

1
(2z 5 )n

⑩♣ ❞ö♥❣ ✤à♥❤ ❧➼ ✷✳✶✳✸✱ t❛ ✤÷đ❝ ✿
I = −2πi.Res[f (z), ∞] = πi.

❱➼ ❞ö ✶✵✿ ❚➼♥❤ t➼❝❤ ♣❤➙♥ I =
●✐↔✐✿

|z|=2

dz
✳
+1

z4

❍➔♠ f (z) = z4 1+ 1 ❝â ✹ ✤✐➸♠ ❜➜t t❤÷í♥❣ ❝ỉ ❧➟♣ ak ✱ k = 1, 4 ❧➔ ❝➠♥ ❜➟❝ ✹ ❝õ❛ −1 ✳
⑩♣ ❞ö♥❣ ✤à♥❤ ❧➼ ✷✳✶✳✸ ✈➔ ❱➼ ❞ư ✺✱ t❛ ✤÷đ❝ ✿
❙❱❚❍✿ ❚r➛♥ ❚❤à ❱②

❚r❛♥❣ ✷✸