Khóa luận tốt nghiệp đại học chuyên ngành giải tích

❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ■■
❑❍❖❆ ❚❖⑩◆

✖✖✖✖✖✖✖✖✖✖♦✵♦✖✖✖✖✖✖✖✖✖✖

✣➱ ❚❍➚ ❚❍❆◆❍

❍⑨▼ ✣■➋❯ ❍➪❆
❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈
❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ●✐↔✐ ❚➼❝❤

●✐↔♥❣ ✈✐➯♥ ❤÷î♥❣ ❞➝♥
❚❤✳s✳ ❍♦➔♥❣ ◆❣å❝ ❚✉➜♥
❍⑨ ◆❐■✱ ✺✴✷✵✶✹


▲í✐ ❝↔♠ ì♥
❚r÷î❝ ❦❤✐ tr➻♥❤ ❜➔② ♥ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❦❤â❛ ❧✉➟♥✱ ❡♠ ①✐♥ ❜➔② tä
❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tî✐ ❚❤↕❝ sÿ ❍♦➔♥❣ ◆❣å❝ ❚✉➜♥ ♥❣÷í✐ ✤➣ t➟♥ t➻♥❤
❤÷î♥❣ ❞➝♥ ✤➸ ❡♠ ❝â t❤➸ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔②✳
❊♠ ❝ô♥❣ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ tî✐ t♦➔♥ t❤➸ ❝→❝ t❤➛②
❝æ ❣✐→♦ tr♦♥❣ ❦❤♦❛ ❚♦→♥✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ P❤↕♠ ❍➔ ◆ë✐ ✷ ✤➣ ❞↕②
❜↔♦ ❡♠ t➟♥ t➻♥❤ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ t↕✐ ❦❤♦❛✳
◆❤➙♥ ❞à♣ ♥➔② ❡♠ ❝ô♥❣ ①✐♥ ✤÷ñ❝ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ tî✐ ❣✐❛
✤➻♥❤✱ ❜↕♥ ❜➧ ✤➣ ❧✉æ♥ ❜➯♥ ❡♠✱ ❝ê ✈ô✱ ✤ë♥❣ ✈✐➯♥✱ ❣✐ó♣ ✤ï ❡♠ tr♦♥❣ s✉èt
q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣✳
❍➔ ◆ë✐✱ t❤→♥❣ ✵✺ ♥➠♠ ✷✵✶✹

❙✐♥❤ ✈✐➯♥

✣é ❚❤à ❚❤❛♥❤

✐✐


▲í✐ ❝❛♠ ✤♦❛♥
❙❛✉ ♠ët t❤í✐ ❣✐❛♥ ♥❣❤✐➯♥ ❝ù✉ ✈î✐ sü ❝è ❣➢♥❣✱ ♥é ❧ü❝ ❝õ❛ ❜↔♥ t❤➙♥
❝ò♥❣ sü ❤÷î♥❣ ❞➝♥ ♥❤✐➺t t➻♥❤ ❝õ❛ ❚❤✳❙ ❍♦➔♥❣ ◆❣å❝ ❚✉➜♥ ❡♠ ✤➣ ❤♦➔♥
t❤➔♥❤ ❜➔✐ ❦❤â❛ ❧✉➟♥ ❝õ❛ ♠➻♥❤✳
❊♠ ①✐♥ ❝❛♠ ✤♦❛♥ ❜➔✐ ❦❤â❛ ❧✉➟♥ ❧➔ ❞♦ ❜↔♥ t❤➙♥ ♥❣❤✐➯♥ ❝ù✉ ❝ò♥❣ sü
❤÷î♥❣ ❞➝♥ ❝õ❛ ❚❤✳❙ ❍♦➔♥❣ ◆❣å❝ ❚✉➜♥ ❦❤æ♥❣ ❤➲ trò♥❣ ❤ñ♣ ✈î✐ ❜➜t
❦ý ✤➲ t➔✐ ♥➔♦✳
❍➔ ◆ë✐✱ t❤→♥❣ ✵✺ ♥➠♠ ✷✵✶✹

❙✐♥❤ ✈✐➯♥

✣é ❚❤à ❚❤❛♥❤

✐✐✐


▼ö❝ ❧ö❝
▲í✐ ♥â✐ ✤➛✉
✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à

✶✳✶
✶✳✷
✶✳✸
✶✳✹
✶✳✺

❍➺ t❤è♥❣ sè ♣❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❚♦♣♦ tr➯♥ ♠➦t ♣❤➥♥❣ ♣❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❍➔♠ ❣✐↔✐ t➼❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❚➼❝❤ ♣❤➙♥ ♣❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
◆❣✉②➯♥ ❧þ ♠æ✤✉♥ ❝ü❝ ✤↕✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✈
✶

✶
✸
✻
✶✵
✶✻

✷ ❍➔♠ ✤✐➲✉ ❤á❛

✶✽

❑➳t ❧✉➟♥
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✹✻
✹✼

✷✳✶
✷✳✷
✷✳✸
✷✳✹
✷✳✺

❈→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ ✤✐➲✉ ❤á❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❍➔♠ ✤✐➲✉ ❤á❛ tr➯♥ ❤➻♥❤ trá♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❍➔♠ ❞÷î✐ ✤✐➲✉ ❤á❛ ✈➔ tr➯♥ ✤✐➲✉ ❤á❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❇➔✐ t♦→♥ ❉✐r✐❝❤❧❡t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❍➔♠ ●r❡❡♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✐✈

✶✽
✷✷
✸✵
✸✻
✹✸


ớ õ
ỵ ồ t

ồ ổ ồ ợ tỹ t r t ồ
t ởt tr q trồ t q ự ữủ
tr t ổ ử tr ỹ ừ t ồ
ỏ ử tr ồ ữ t ỵ õ ồ
t ồ
ỏ ổ ử ự
ú t ổ ữủ tự t ởt tố
tổ q rở t q ỏ tr trữớ
ủ ỏ ú ú t qt ữủ ởt số t
tỹ t õ q ự ỵ tt ỏ
ú t t s s ỡ ỵ tt ự ỗ

tớ sỷ ử ỳ ỵ tt õ qt ởt số
ừ ỵ tt t ồ ỳ tự ỡ s qt
ởt số t tỹ t
r q tr ồ t ữủ t ổ ợ t t ữủ sỹ
ữợ ủ ỵ ừ s ồ rt ố t
ỏ ữợ ự q ợ
ự ồ ồ t


ử ử ự

ró t t ỵ ừ ỏ
ự q ỏ



❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❍➔♠ ✤✐➲✉ ❤á❛

✸✳ ✣è✐ t÷ñ♥❣ ✈➔ ♣❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉

✲ ✣è✐ t÷ñ♥❣ ♥❣❤✐➯♥ ❝ù✉✿ ❍➔♠ ✤✐➲✉ ❤á❛✳
✲ P❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉✿ ◆❤ú♥❣ ✤à♥❤ ♥❣❤➽❛✱ t➼♥❤ ❝❤➜t✱ ✤à♥❤ ❧þ ✈➔ ❝→❝
✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥ ❝õ❛ ❤➔♠ ✤✐➲✉ ❤á❛✳
✹✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉

❑❤â❛ ❧✉➟♥ ✤÷ñ❝ ❤♦➔♥ t❤➔♥❤ ❞ü❛ tr➯♥ sü ❦➳t ❤ñ♣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣
♥❣❤✐➯♥ ❝ù✉✿ ❧þ ❧✉➟♥✱ ♣❤➙♥ t➼❝❤ ✱ tê♥❣ ❤ñ♣✳✳✳
✺✳ ✣â♥❣ ❣â♣ ❝õ❛ ❦❤â❛ ❧✉➟♥

✲▲➔♠ rã ❝❤✐ t✐➳t ❤ì♥ ❤➺ t❤è♥❣ tr✐ t❤ù❝ ♠î✐✱ ❝❤✉②➯♥ s➙✉ ✈➲ ❜ë ♠æ♥
❤➔♠ ❜✐➳♥ ♣❤ù❝✳
✲❑❤â❛ ❧✉➟♥ ❝á♥ ❝✉♥❣ ❝➜♣ t❤➯♠ ❝→❝ t➼♥❤ ❝❤➜t ✈➔ ❝→❝ ✈➜♥ ✤➲ ❧✐➯♥
q✉❛♥ ❝õ❛ ❤➔♠ ✤✐➲✉ ❤á❛✳
✻✳ ❈➜✉ tró❝ ❝õ❛ ❦❤â❛ ❧✉➟♥

◆❣♦➔✐ ♠ö❝ ❧ö❝✱ ♣❤➛♥ ♠ð ✤➛✉✱ ❦➳t ❧✉➟♥ ✈➔ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✱ ❦❤â❛
❧✉➟♥ ❣ç♠ ❤❛✐ ❝❤÷ì♥❣✿
❈❤÷ì♥❣ ✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲✿ ❍➔♠ ❜✐➳♥ ♣❤ù❝✱ ❚♦♣♦ tr➯♥
♠➦t ♣❤➥♥❣ ♣❤ù❝✱ ❍➔♠ ❣✐↔✐ t➼❝❤✱ ❚➼❝❤ ♣❤➙♥ ♣❤ù❝✱ ◆❣✉②➯♥ ❧þ ♠æ✤✉♥
❝ü❝ ✤↕✐✳
❈❤÷ì♥❣ ✷✳ ❍➔♠ ✤✐➲✉ ❤á❛✿ ❚➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❤➔♠ ✤✐➲✉ ❤á❛✱ ❍➔♠
✤✐➲✉ ❤á❛ tr➯♥ ❤➻♥❤ trá♥✱ ❍➔♠ ❞÷î✐ ✤✐➲✉ ❤á❛ ✈➔ 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♥❣✱ ♣❤➨♣ ♥❤➙♥ ✤÷ñ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉✿
(a, b) + (c, d) = (a + c, b + d)
(a, b)(c, d) = (ac − bd, bc + ad)

❉➵ ❞➔♥❣ ❦✐➸♠ tr❛ ✤÷ñ❝ ✤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❤❡♦ ❝➦♣ t❤æ♥❣
t❤÷í♥❣ ❝õ❛ ❝→❝ sè ♣❤ù❝✳
❈❤ó þ r➡♥❣ i = −1✱ ✈➟② ♣❤÷ì♥❣ tr➻♥❤ z + 1 = 0 ❝â ♥❣❤✐➺♠ tr♦♥❣
C✳ ❚❤➟t ✈➟②✱ ✈î✐ ♠é✐ z ∈ C✱ z + 1 = (z + i)(z − i)✳ ❍ì♥ ♥ú❛✱ ♥➳✉ z
✈➔ w ❧➔ ❝→❝ sè ♣❤ù❝✱ t❛ ❝â ❜✐➸✉ ❞✐➵♥
2

2

2

z 2 + w2 = (z + iw)(z − iw)

✣➦t z✱ w ❜ð✐ ❝→❝ sè t❤ü❝ a✱ b ❦❤✐ ✤â t❛ ❝â ✭❛✱ ❜ = ✵✮
1
a
a − ib
b
= 2
=
−

i
a + ib a + b2
a2 + b 2
a2 + b2
✶


õ tốt

ỏ

t õ ổ tự ừ tỷ ừ số ự
t z = a + ib (a, b R) ồ tỹ
õ a = Rez b = Imz
ú t ữ r t C ổ t
trữớ z = x + iy (x, y R) t t |z| = (x + y )
ổ ừ z = x iy ủ ừ z ú ỵ r
2

2

1
2

|z|2 = zz

õ r z = 0 t
1
z
= 2

z
|z|

t t ỡ ừ ổ ủ
1
Rez = (z + z),
2

Imz =

(z + w) = z + w,

1
(z z).
2i

zw = z w.

|zw| = |z| |w| .
z
|z|
.
=
w
|w|
|z| = |z| .

t ự ứ số ự

ộ z C õ t ỗ t t (Rez, Imz) tr t

R P ở số ự tỹ t ở tỡ
tr ổ R C t õ ở ừ tỡ tợ
(= (0, 0)) ừ ợ
tự tữ
ú ỵ õ t |z w| ỳ
t ỡ ừ tọ t tự t

2

|z1 z2 |

ộ

|z1 z3 | + |z3 z2 |


P


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❍➔♠ ✤✐➲✉ ❤á❛

✈î✐ ❝→❝ sè ♣❤ù❝ z , z , z ✳ ⑩♣ ❞ö♥❣ z
✤÷ñ❝
1

2

3

1

|z + w|

❱î✐ ❜➜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 + 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 ♣❤➥♥❣ ♣❤ù❝ C ❝â t❤➸ ✤ç♥❣ ♥❤➜t ✈î✐ 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♥❤ ♥❣❤➽❛ ✶✳✸✳ ✭❑❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝✮ ❑❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ❧➔ ♠ët
❝➦♣ ✭X, d✮❀ tr♦♥❣ ✤â ❳ ❧➔ ♠ët t➟♣✱ ❤➔♠ d : X × X → R ✤÷ñ❝ ❣å✐ ❧➔
❤➔♠ ❦❤♦↔♥❣ ❝→❝❤ ❤❛② ♠➯tr✐❝ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✈î✐ x✱ y ✈➔
z ∈ X✿

✣é ❚❤à ❚❤❛♥❤

✸

❑✸✻❈ ❚♦→♥ ✣❍❙P❍◆ ✷

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❍➔♠ ✤✐➲✉ ❤á❛

(a) d(x, y) ≥ 0
(b) d(x, y) = 0 ⇔ x = y

✭✤è✐ ①ù♥❣✮
(d) d(x, z) ≤ d(x, y) + d(y, z) ✭❜➜t ✤➥♥❣ t❤ù❝ t❛♠ ❣✐→❝✮
◆➳✉ x ∈ X ✈➔ r > 0 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♥❣ ✤÷ñ❝ ❣å✐ ❧➔ ❤➻♥❤ ❝➛✉ ♠ð ✈➔ ✤â♥❣ ✈î✐ t➙♠
①✱ ❜→♥ ❦➼♥❤ r✳
✣à♥❤ ♥❣❤➽❛ ✶✳✹✳ ❈❤♦ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✭❳✱ ❞✮ ❝â t➟♣ G ⊂ X ❧➔ t➟♣
♠ð ♥➳✉ ✈î✐ ♠å✐ x ∈ G ❝â ♠ët > 0 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

1

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

1

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 ✳ ❈❤ó þ r➡♥❣ intA
−

✣é ❚❤à ❚❤❛♥❤

✹

❑✸✻❈ ❚♦→♥ ✣❍❙P❍◆ ✷


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❍➔♠ ✤✐➲✉ ❤á❛

❝â t❤➸ ré♥❣✱ A ❝â t❤➸ ❧➔ ❳✳ ◆➳✉ A = {a + bi : a, b ∈ R} t❤➻ A = C
✈➔ intA = ∅✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✶✳✶ ✈➔ ▼➺♥❤ ✤➲ ✶✳✷ t❛ ❝â A ❧➔ t➟♣
✤â♥❣✱ intA ❧➔ t➟♣ ♠ð✳ ❇✐➯♥ ❝õ❛ ❆ ❦➼ ❤✐➺✉ ❜ð✐ ∂A ✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐
∂A = A ∩ (X − A) ✳
−

−

−

−

−

▼➺♥❤ ✤➲ ✶✳✸✳ ❈❤♦ A, B

❧➔ t➟♣ ❝♦♥ ✤â♥❣ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝

(X, d)✳ ❑❤✐ ✤â✿

✭❛✮ A ♠ð ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ A = intA❀
✭❜✮ A ✤â♥❣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ A = A ❀
✭❝✮ intA = X − (X − A) ; A = X − int(X − A); ∂A = A − intA❀
✭❞✮ ✭❆ ∪ ❇✮ ❂ A ∪ B ❀
✭❡✮ x ∈ intA ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❝â > 0 s❛♦ ❝❤♦ B(x ; ) ⊂ A❀
✭❢✮ x ∈ A ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✈î✐ ♠å✐ > 0✱ B(x ; ) ∩ A = ∅✳
✣à♥❤ ♥❣❤➽❛ ✶✳✼✳ ❑❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ (X, d) ❧➔ ❧✐➯♥ t❤æ♥❣ ♥➳✉ ❝❤➾ ❝â
❝→❝ t➟♣ ❝♦♥ ∅ ✈➔ X ❝õ❛ X ❧➔ ✈ø❛ ♠ð ✈ø❛ ✤â♥❣✳ ◆➳✉ A ⊂ X t❤➻ A ❧➔
t➟♣ ❝♦♥ ❧✐➯♥ t❤æ♥❣ ♥➳✉ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ (A, d) ❧➔ ❧✐➯♥ t❤æ♥❣✳
−

−

−

−

−

−

−

0

0

0

−

0

▼➺♥❤ ✤➲ ✶✳✹✳ ❈❤♦ x0 ∈ X ✈➔ {Dj : j ∈ J} ❧➔ t➟♣ ❤ñ♣ ❝→❝ t➟♣ ❝♦♥
❧✐➯♥ t❤æ♥❣ ❝õ❛ X s❛♦ ❝❤♦ x0 ∈ Dj ✈î✐ j ∈ J ✳ ❑❤✐ ✤â

D=

{Dj : j ∈ J} ❧➔ ❧✐➯♥ t❤æ♥❣✳

✣à♥❤ ♥❣❤➽❛ ✶✳✽✳ ❚➟♣ ❝♦♥ K ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ X ❧➔ ❝♦♠♣❛❝t

♥➳✉ ✈î✐ ♠å✐ t➟♣ G ❝õ❛ ❝→❝ t➟♣ ♠ð tr♦♥❣ X t❤ä❛ ♠➣♥ t➼♥❤ ❝❤➜t
{G : G ∈ G }

K⊂

✭✶✳✶✮

❝â ♠ët sè ❤ú✉ ❤↕♥ ❝→❝ t➟♣ G , . . . , G tr♦♥❣ G s❛♦ ❝❤♦
K ⊂ G ∪ G ∪ . . . ∪ G ✳ ❚➟♣ ❤ñ♣ ❝→❝ t➟♣ tr♦♥❣ G t❤ä❛ ♠➣♥ (1.1)

✤÷ñ❝ ❣å✐ ❧➔ ♣❤õ ❝õ❛ K ✳ ◆➳✉ ♠é✐ t➟♣ tr♦♥❣ G ❧➔ t➟♣ ♠ð t❤➻ G ✤÷ñ❝
❣å✐ ❧➔ ♣❤õ ♠ð ❝õ❛ K ✳
✣à♥❤ ♥❣❤➽❛ ✶✳✾✳ ❑❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♠♣❛❝t
♥➳✉ ♠å✐ ❞➣② tr♦♥❣ X ✤➲✉ ❝â ❞➣② ❝♦♥ ❤ë✐ tö✳
1

1

✣é ❚❤à ❚❤❛♥❤

2

n

n

✺

❑✸✻❈ ❚♦→♥ ✣❍❙P❍◆ ✷


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❍➔♠ ✤✐➲✉ ❤á❛

❇ê ✤➲ ✶✳✶✳ ✭❇ê ✤➲ ♣❤õ ▲❡❜❡s❣✉❡✮ ◆➳✉ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥
❝♦♠♣❛❝t ✈➔ G ❧➔ ♣❤õ ♠ð ❝õ❛ X t❤➻ ❝â

> 0 s❛♦ ❝❤♦✱ ♥➳✉ x ∈ X t❤➻

❝â t➟♣ G ∈ G ✈î✐ B(x; ) ⊂ G✳

+b
✣à♥❤ ♥❣❤➽❛ ✶✳✶✵✳ ▼ët →♥❤ ①↕ ❝â ❞↕♥❣ S(z) = az
✤÷ñ❝ ❣å✐ ❧➔
cz + d

→♥❤ ①↕ ♣❤➙♥ t✉②➳♥ t➼♥❤✳ ◆➳✉ a✱ b✱ c ✈➔ d t❤ä❛ ♠➣♥ ad − bc = 0 t❤➻
S(z) ✤÷ñ❝ ❣å✐ ❧➔ →♥❤ ①↕ ▼♦❜✐✉s✳
dz − b
t❤ä❛ ♠➣♥
◆➳✉ S ❧➔ →♥❤ ①↕ ▼♦❜✐✉s t❤➻ S (z) = −cz
+a
S(S (z)) = S S(z) = z ✳ ❚ù❝ ❧➔ S ❧➔ →♥❤ ①↕ ♥❣÷ñ❝ ❝õ❛ S ✳ ◆➳✉ S
✈➔ T ❧➔ →♥❤ ①↕ ♣❤➙♥ t✉②➳♥ t➼♥❤ t❤➻ S ◦ T ❝ô♥❣ ❧➔ →♥❤ ①↕ ♣❤➙♥ t✉②➳♥
t➼♥❤✳ ❚rø ❦❤✐ ❝â ❝→❝ ✤✐➲✉ ❦✐➺♥ ❦❤→❝✱ ❝❤ó♥❣ t❛ ❝❤➾ ①➨t →♥❤ ①↕ ♣❤➙♥
t✉②➳♥ t➼♥❤ ❧➔ →♥❤ ①↕ ▼♦❜✐✉s✳
❈❤ó♥❣ t❛ ❝ô♥❣ ♣❤↔✐ ❝❤ó þ r➡♥❣ S ✤÷ñ❝ ①→❝ ✤à♥❤ tr➯♥ C ✈î✐
a
−d
S(∞) =
✈➔
S( ) = ∞✳ ❱➻ S ❝â →♥❤ ①↕ ♥❣÷ñ❝ ♥➯♥ t❛ ❝â →♥❤
①↕ C → cC ✳ c
✣à♥❤ ♥❣❤➽❛ ✶✳✶✶✳ ◆➳✉ G ❧➔ t➟♣ ♠ð tr♦♥❣ C ✈➔ (R, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥
♠➯tr✐❝ ✤➛② t❤➻ ❦➼ ❤✐➺✉ C(G, R) ❧➔ t➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ ❤➔♠ ❧✐➯♥ tö❝
G → R✳
−1

−1

−1

−1

∞

∞

∞

▼➺♥❤ ✤➲ ✶✳✺✳ C(G, R) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤➛②✳
✣à♥❤ ❧þ ✶✳✶✳ ✭✣à♥❤ ❧þ ❉✐♥✐✮ ❈❤♦ C(G, R) ✈➔ ❣✐↔ sû ❞➣②

{fn } ⊂ C(G, R) ❧➔ ❞➣② ✤ì♥ ✤✐➺✉ t➠♥❣ ✈➔ lim fn (z) = f (z) ✈î✐ ♠å✐
z ∈ G tr♦♥❣ ✤â f ∈ C(G, R)✳ ❑❤✐ ✤â fn → f ✳

✶✳✸ ❍➔♠ ❣✐↔✐ t➼❝❤

◆➳✉ a ⊂ C ✈î✐ n 0 t❤➻ ❝❤✉é✐ a ❤ë✐ tö tî✐ ③ ♥➳✉ ✈î✐ > 0 ❝â
sè ♥❣✉②➯♥ ◆ s❛♦ ❝❤♦ a − z < tr♦♥❣ ✤â n N ✳ ❈❤✉é✐ a
❤ë✐ tö t✉②➺t ✤è✐ ♥➳✉ |a | ❤ë✐ tö✳
∞

n

n

n=0

m

n

n

n=0

n

▼➺♥❤ ✤➲ ✶✳✻✳ ◆➳✉

✣é ❚❤à ❚❤❛♥❤

an ❤ë✐ tö t✉②➺t ✤è✐ t❤➻
✻

an ❤ë✐ tö✳

❑✸✻❈ ❚♦→♥ ✣❍❙P❍◆ ✷


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❍➔♠ ✤✐➲✉ ❤á❛

◆➳✉ ❞➣② {a } ⊂ R t❤➻ ✤à♥❤ ♥❣❤➽❛
n

lim inf an = lim [inf {an , an+1 , . . .}]

n→∞

lim sup an = lim [sup {an , an+1 , . . .}]
n→∞

❈❤✉é✐ ❧ô② t❤ø❛ ❝õ❛ ❛ ❧➔ ❝❤✉é✐ ✈æ ❤↕♥ a (z − a) ✳ ▼ët ✈➼ ❞ö ✤ì♥
❣✐↔♥ ❝õ❛ ❝❤✉é✐ ❧ô② t❤ø❛ ❧➔ ❝❤✉é✐ z ✳ ❉➵ t❤➜②
∞

n

∞

n

n=0

n

n=0

❱➟②

1 − z n+1 = (1 − z)(1 + z + . . . + z n ),
1 − z n+1
.
1 + z + ... + z =
1−z
lim z n = 0
n

◆➳✉ |z| < 1 t❤➻

✳ ❉♦ ✤â ❝❤✉é✐ ♥➔② ❤ë✐ tö ✈➔ ❝â tê♥❣ ❜➡♥❣

∞

zn =

◆➳✉ |z| > 1 t❤➻ lim |z|

n

n=0

1
.
1−z

✳ ❉♦ ✤â ❝❤✉é✐ ♥➔② ♣❤➙♥ ❦ý✳

=∞

✣à♥❤ ❧þ ✶✳✷✳ ❱î✐ ❝❤✉é✐ ❧ô② t❤ø❛ ✤➣ ❝❤♦

∞

an (z − a)n ✤à♥❤ ♥❣❤➽❛ sè

n=0

❘ t❤ü❝ ♠ð rë♥❣ ❦❤æ♥❣ ➙♠ ❜ð✐
1
1
= lim sup |an | n ,
R
❑❤✐ ✤â✱ t❛ ❝â

✭❛✮◆➳✉ |z − a| < R t❤➻ ❝❤✉é✐ ❤ë✐ tö t✉②➺t ✤è✐❀
✭❜✮ ◆➳✉ |z − a| > R t❤➻ ❝❤✉é✐ ♥➔② ❦❤æ♥❣ ❜à ❝❤➦♥✳ ❱➟② ❝❤✉é✐ ♥➔②

♣❤➙♥ ❦ý❀

✭❝✮ ◆➳✉ 0 < r < R t❤➻ ❝❤✉é✐ ❤ë✐ tö ✤➲✉ tr➯♥ {z : |z| < r}✳
◆❣♦➔✐ r❛✱ ❘ ❧➔ sè ❞✉② ♥❤➜t ❝â t➼♥❤ ❝❤➜t (a), (b)✳

❙è R ✤÷ñ❝ ❣å✐ ❧➔ ❜→♥ ❦➼♥❤ ❤ë✐ tö ❝õ❛ ❝❤✉é✐ ❧ô② t❤ø❛✳

▼➺♥❤ ✤➲ ✶✳✼✳ ◆➳✉

an (z − a)n ❧➔ ❝❤✉é✐ ❧ô② t❤ø❛ ✤➣ ❝❤♦ ✈î✐ ❜→♥

❦➼♥❤ ❤ë✐ tö R✱ t❤➻

R = lim

✣é ❚❤à ❚❤❛♥❤

✼

an
an+1

❑✸✻❈ ❚♦→♥ ✣❍❙P❍◆ ✷


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❍➔♠ ✤✐➲✉ ❤á❛

❦❤✐ ❣✐î✐ ❤↕♥ ♥➔② tç♥ t↕✐✳
∞ zn
n=0 n!
∞

✱ t❤❡♦ ▼➺♥❤ ✤➲ ✶✳✻ ❝❤✉é✐ ♥➔② ❝â ❜→♥ ❦➼♥❤ ❤ë✐ tö
❳➨t ❝❤✉é✐
✳ ❉♦ ✤â✱ ❝❤✉é✐ ♥➔② ❤ë✐ tö tî✐ ♠ët sè ♣❤ù❝ ✈➔ ❤ë✐ tö ✤➲✉ tr➯♥ ♠é✐
t➟♣ ❝♦♠♣❛❝t ❝õ❛ C✳ ❈❤ó♥❣ t❛ ✤➦t ❝❤✉é✐ ♥➔② ❜ð✐
∞
z

e = exp z =
n=0

❧➔ ❤➔♠ ♠ô✳

zn
,
n!

✣à♥❤ ♥❣❤➽❛ ✶✳✶✷✳ ◆➳✉ t➟♣ G ♠ð tr♦♥❣ C ✈➔ f : G → C t❤➻ f ❦❤↔

✈✐ t↕✐ ✤✐➸♠ a ∈ G ♥➳✉

f (a + h) − f (a)
h→0
h
lim

tç♥ t↕✐✱ ❣✐î✐ ❤↕♥ ♥➔② ❦þ ❤✐➺✉ f (a) ✈➔ ✤÷ñ❝ ❣å✐ ❧➔ ✤↕♦ ❤➔♠ ❝õ❛ f t↕✐
❛✳ ◆➳✉ f ❦❤↔ ✈✐ t↕✐ ♠é✐ ✤✐➸♠ ❝õ❛ G t❤➻ f ❦❤↔ ✈✐ tr➯♥ G✳ ❈❤ó þ r➡♥❣
♥➳✉ f ❦❤↔ ✈✐ tr➯♥ G t❤➻ f (a) ✤à♥❤ ♥❣❤➽❛ ❤➔♠ f : G → C✳ ◆➳✉ f
❧✐➯♥ tö❝ t❤➻ f ❦❤↔ ✈✐ ❧✐➯♥ tö❝✳ ◆➳✉ f ❦❤↔ ✈✐ t❤➻ f ❦❤↔ ✈✐ ❝➜♣ ❤❛✐✱ t✐➳♣
tö❝✱ ♠ët ❤➔♠ ❦❤↔ ✈✐ s❛♦ ❝❤♦ ✤↕♦ ❤➔♠ ❝→❝ ❝➜♣ ❦❤↔ ✈✐ ✤÷ñ❝ ❣å✐ ❧➔ ❦❤↔
✈✐ ✈æ ❤↕♥✳
▼➺♥❤ ✤➲ ✶✳✽✳ ◆➳✉ f : G → C ❦❤↔ ✈✐ t↕✐ ✤✐➸♠ a ∈ G t❤➻ f ❧✐➯♥ tö❝
t↕✐ a✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✸✳ ❍➔♠ f : G → C ❣✐↔✐ t➼❝❤ ♥➳✉ f ❦❤↔ ✈✐ ❧✐➯♥ tö❝

tr➯♥ G✳

✣à♥❤ ❧þ ✶✳✸✳ ❈❤♦ f, g ❣✐↔✐ t➼❝❤ tr➯♥ G ✈➔ Ω✱ ❣✐↔ sû f (G) ⊂ Ω ❦❤✐ ✤â
g ◦ f ❣✐↔✐ t➼❝❤ tr➯♥ G ✈➔
(g ◦ f ) (z) = g (f (z))f (z)
✈î✐ ♠å✐ z ∈ G✳

▼➺♥❤ ✤➲ ✶✳✾✳ ❈❤♦ f (z) =
❑❤✐ ✤â

✣é ❚❤à ❚❤❛♥❤

∞

an (z − a)n ❝â ❜→♥ ❦➼♥❤ ❤ë✐ tö R > 0✳

n=0

✽

❑✸✻❈ ❚♦→♥ ✣❍❙P❍◆ ✷


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❍➔♠ ✤✐➲✉ ❤á❛

✭❛✮ ❱î✐ ♠é✐ k

1 ❝❤✉é✐

✭✶✳✷✮

∞

n(n − 1) . . . (n − k + 1)an (z − a)n−k
n=k

❝â ❜→♥ ❦➼♥❤ ❤ë✐ tö ❘❀

✭❜✮ ❍➔♠ f ❦❤↔ ✈✐ ✈æ ❤↕♥ tr➯♥ B(a; R) ✈➔ ❤ì♥ ♥ú❛✱ f

❜ð✐ (1.1) ✈î✐ ♠å✐ k

✭❝✮ ❱î✐ n

0✱

❍➺ q✉↔ ✶✳✶✳ ◆➳✉ ❝❤✉é✐
f (z) =

∞

1 (n)
f (a).
n!

an (z − a)n ❝â ❜→♥ ❦➼♥❤ ❤ë✐ tö R > 0 t❤➻

n=0

an (z − a)n ❣✐↔✐ t➼❝❤ tr♦♥❣ B(a; R)✳

n=0

❉♦ ✤â✱

(z) ✤➣ ❝❤♦

1 ✈➔ |z − a| < R❀
an =

∞

(k)

zn
exp z =
n=0 n!
∞

❣✐↔✐ t➼❝❤✳

▼➺♥❤ ✤➲ ✶✳✶✵✳ ◆➳✉ G ♠ð✱ ❧✐➯♥ t❤æ♥❣ ✈➔ f

: G → C ❦❤↔ ✈✐ ✈î✐

f (z) = 0✱ z ∈ G t❤➻ f ❧➔ ❤➔♠ ❤➡♥❣✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✹✳ ◆➳✉ G ❧➔ t➟♣ ♠ð ❧✐➯♥ t❤æ♥❣ tr♦♥❣ C ✈➔ ❢ ❧➔ ❤➔♠

❧✐➯♥ tö❝ s❛♦ ❝❤♦ z = exp f (z) ∀z ∈ G t❤➻ f ❧➔ ♥❤→♥❤ ❝õ❛ ❧♦❣❛r✐t✳
✣✐➲✉ ❦✐➺♥ ❈❛✉❝❤② ✲ ❘❡✐♠❛♥♥✳ ❈❤♦ f : G → C ❣✐↔✐ t➼❝❤ ✈➔ ✤➦t
u(x, y) = Ref (x + iy)✱ v(x, y) = Imf (x + iy) ✈î✐ x + iy ∈ G✳ ❳➨t
❣✐î✐ ❤↕♥
f (z + h) − f (z)
h→0
h
h→0

f (z) = lim

t❤❡♦ ❤❛✐ ❝→❝❤ ❦❤→❝ ♥❤❛✉✳ ✣➛✉ t✐➯♥✱ ❝❤♦
❱î✐ sè t❤ü❝ h ❦❤→❝ ❦❤æ♥❣✱ t❛ ❝â

①➨t ❣✐→ trà t❤ü❝ ❝õ❛ h✳

f (z + h) − f (z) f (x + h + iy) − f (x + iy)
=
h
h
u(x + h, y) − u(x, y)
v(x + h, y) − v(x, y)
=
n+i
h
h
h→0
∂u
∂v
f (z) =
(x, y) + i (x, y)
∂x
∂x

❈❤♦

✣é ❚❤à ❚❤❛♥❤

✱ t❛ ✤÷ñ❝

✭✶✳✸✮

✾

❑✸✻❈ ❚♦→♥ ✣❍❙P❍◆ ✷


õ tốt

ỏ

h 0 t tr ừ h ự ợ số tỹ h ổ
u(x, y + h) u(x, y) v(x, y + h) v(x, y)
f (z + ih) f (z)
= i
+
ih
h
h
v
u
f (z) = i (x, y) + (x, y)
y
y
(1.2) (1.3)

tữỡ ự tỹ ừ

u v
=
,
x y


t õ



u
v
= .
y
x

sỷ u v õ tử tứ ừ
ữỡ tr tr t ữủ
2u
2v
=
x2
xy

õ

v`
a

2u

2y
=

.
y 2
yx

2u 2u
+
= 0.
x2 y 2

ởt õ tử tọ ữỡ tr tr
ữủ ồ ỏ
ỵ sỷ u v tr tỹ ữủ tr
G õ tử õ f : G C
f (z) = u(z) + iv(z) t u v tọ


ỵ G tr C tr trỏ
f : G C ỏ t u õ ủ ỏ

ự
f : [a, b] C ợ [a, b] R õ

õ số M > 0 s ợ t ý
P = {a = t < t < . . . < t = b} ừ [a, b]
0

1

m
m

|(tk ) (tk1 )|

v(; P ) =

M.

k=1

ộ



P


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❍➔♠ ✤✐➲✉ ❤á❛

❇✐➳♥ ♣❤➙♥ t♦➔♥ ♣❤➛♥ ❝õ❛ γ ❧➔ V (γ) ✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐
V (γ) = sup v(γ; P ) : P ❧➔ ♣❤➙♥ ❤♦↕❝❤ ❝õ❛ [a, b]
❘ã r➔♥❣✱ V (γ) M < ∞✳
❉➵ ❞➔♥❣ t❤➜② r➡♥❣ γ ❝â ❜✐➳♥ ♣❤➙♥ ❜à ❝❤➦♥ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ Reγ ✈➔
Imγ ❝â ❜✐➳♥ ♣❤➙♥ ❜à ❝❤➦♥✳ ◆➳✉ γ ❧➔ ❣✐→ trà t❤ü❝✱ ❦❤æ♥❣ ❣✐↔♠ t❤➻ γ ❧➔
❜✐➳♥ ♣❤➙♥ ❜à ❝❤➦♥ ✈➔ V (γ) = γ(b) − γ(a)✳
▼➺♥❤ ✤➲ ✶✳✶✶✳ ❈❤♦ γ : [a, b] → C ❝â ❜✐➳♥ ♣❤➙♥ ❜à ❝❤➦♥✳ ❑❤✐ ✤â

✭❛✮ ◆➳✉ P ✱ Q ❧➔ ♣❤➨♣ ♣❤➙♥ ❤♦↕❝❤ ❝õ❛ [a, b] ✈➔ P ⊂ Q t❤➻

v(γ; P )

v(γ; Q)❀

✭❜✮ ◆➳✉ σ : [a, b] → C ❝â ❜✐➳♥ ♣❤➙♥ ❜à ❝❤➦♥ ✈➔ α✱ β ∈ C t❤➻ αγ+βσ

❝â ❜✐➳♥ ♣❤➙♥ ❜à ❝❤➦♥ ✈➔ V (αγ + βσ)

|α| V (γ) + |β| V (σ)✳

▼➺♥❤ ✤➲ ✶✳✶✷✳ ◆➳✉ γ : [a, b] → C ❧➔ trì♥ tø♥❣ ❦❤ó❝ t❤➻ γ ❝â ❜✐➳♥
♣❤➙♥ ❜à ❝❤➦♥ ✈➔
b

V (γ) =

✣à♥❤ ❧þ ✶✳✻✳ ❈❤♦ γ

γ (t) dt
a

: [a, b] → C ❝â ❜✐➳♥ ♣❤➙♥ ❜à ❝❤➦♥ ✈➔ ❣✐↔ sû

f : [a, b] → C ❧✐➯♥ tö❝✳ ❑❤✐ ✤â ❝â sè ♣❤ù❝ I s❛♦ ❝❤♦ ✈î✐

> 0 ❝â

δ > 0 ✈î✐ P = {t0 < t1 < . . . < tm } ❧➔ ♣❤➨♣ ♣❤➙♥ ❤♦↕❝❤ ❝õ❛ [a, b] ✤➦t
P = max {(tk − tk−1 ) : 1

m} < δ t❤➻

k

m

I−

f (τk ) [γ(tk ) − γ(tk−1 )] <
k=1

❝❤♦♥ ✤✐➸♠ τk ✱ tk−1

tk ✳

τk

❙è I ✤÷ñ❝ ❣å✐ ❧➔ t➼❝❤ ♣❤➙♥ ❝õ❛ f t÷ì♥❣ ù♥❣ ✈î✐ γ ∈ [a, b] ✈➔ ✤÷ñ❝
❦þ ❤✐➺✉ ❜ð✐
b

I=

b

f dγ =
a

✣é ❚❤à ❚❤❛♥❤

f (t)dγ(t).
a

✶✶

❑✸✻❈ ❚♦→♥ ✣❍❙P❍◆ ✷


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❍➔♠ ✤✐➲✉ ❤á❛

▼➺♥❤ ✤➲ ✶✳✶✸✳ ❈❤♦ f ✱ g ❧➔ ❝→❝ ❤➔♠ ❧✐➯♥ tö❝ tr➯♥ [a, b] ✈➔ γ ✱ σ✳ ❑❤✐
✤â✱ ✈î✐ ❜➜t ❦ý ✈æ ❤÷î♥❣ α✱ β

✭❛✮
✭❜✮

b

b

b

gdγ ❀

(αf + βg)dγ = α f dγ + β

a
b

a

a
b

b

f d(αγ + βσ) = α f dγ + β
a

a

f dσ ✳

a

▼➺♥❤ ✤➲ ✶✳✶✹✳ ❈❤♦ γ : [a, b] → C ❝â ❜✐➳♥ ♣❤➙♥ ❜à ❝❤➦♥ ✈➔
f : [a, b] → C ❧✐➯♥ tö❝✳ ◆➳✉ a = t0 < t1 < . . . < tn = b t❤➻
b

n

tk

f dγ =

f dγ.

k=1t

a

k−1

✣à♥❤ ❧þ ✶✳✼✳ ◆➳✉ f trì♥ tø♥❣ ❦❤ó❝ ✈➔ f : [a, b] → C ❧✐➯♥ tö❝ t❤➻
b

b

f dγ =
a

f (t)γ (t)dt.
a

✣à♥❤ ♥❣❤➽❛ ✶✳✶✻✳ ◆➳✉ f : [a, b] → C ❧➔ ✤÷í♥❣ ❝♦♥❣ ❦❤↔ tr÷í♥❣ ✈➔

❤➔♠ f ✤÷ñ❝ ①→❝ ✤à♥❤✱ ❧✐➯♥ tö❝ tr➯♥ γ t❤➻ t➼❝❤ ♣❤➙♥ ✤÷í♥❣ ❝õ❛ f t❤❡♦
γ ❧➔
b

b

f dγ =
a

f (γ(t))dγ(t)
a

❚➼❝❤ ♣❤➙♥ ✤÷í♥❣ ♥➔② ✤÷ñ❝ ❦þ ❤✐➺✉ ❧➔

f=
γ

f (z)dz
γ

✳

▼➺♥❤ ✤➲ ✶✳✶✺✳ ❈❤♦ γ ❧➔ ✤÷í♥❣ ❝♦♥❣ ❦❤↔ tr÷í♥❣ ✈➔ ❣✐↔ sû f ❧➔ ❤➔♠
❧✐➯♥ tö❝ tr➯♥ {γ}✳ ❑❤✐ ✤â✿

✭❛✮

f❀

f =−
−γ

γ

✭❜✮ f |f | |dz| V (γ) sup [|f (z)| : z ∈ {γ}]❀
✭❝✮ ◆➳✉ c ∈ C t❤➻ f (z)dz = f (z − c)dz✳
γ

γ

γ

✣é ❚❤à ❚❤❛♥❤

c+γ

✶✷

❑✸✻❈ ❚♦→♥ ✣❍❙P❍◆ ✷


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❍➔♠ ✤✐➲✉ ❤á❛

▼➺♥❤ ✤➲ ✶✳✶✻✳ ❈❤♦ [a, b] × [c, d] → C ❧➔ ❤➔♠ ❧✐➯♥ tö❝ ✈➔ ✤à♥❤ ♥❣❤➽❛
g : [c, d] → C ❜ð✐
b

g(t) =

ϕ(s, t)ds.
a

❦❤✐ ✤â g ❧✐➯♥ tö❝✳ ❍ì♥ ♥ú❛✱ ♥➳✉ tç♥ t↕✐

[a, b] × [c, d] t❤➻ g ❦❤↔ ✈✐ ❧✐➯♥ tö❝ ✈➔

∂ϕ
✈➔ ❤➔♠ ♥➔② ❧✐➯♥ tö❝ tr➯♥
∂t

b

∂ϕ
(s, t)ds.
∂t

g (t) =

▼➺♥❤ ✤➲ ✶✳✶✼✳ ❈❤♦ f

a

: G → C ❣✐↔✐ t➼❝❤ ✈➔ ❣✐↔ sû B(a; r) ⊂ G

(r > 0)✳ ◆➳✉ γ(t) = a + reit ✱ 0
f (z) =

2π t❤➻

t

f (w)
dw
w−z

1
2πi
γ

✈î✐ |z − a| < r✳

✣à♥❤ ❧þ ∞✶✳✽✳ ❈❤♦ f ❣✐↔✐ t➼❝❤ tr♦♥❣ B(a; R)✳ ❑❤✐ ✤â

an (z − a)n ✈î✐ |z − a| < R tr♦♥❣ ✤â an =

f (z) =
n=0

1 (n)
f (a) ✈➔
n!

❝❤✉é✐ ♥➔② ❝â ❜→♥ ❦➼♥❤ ❤ë✐ tö ❧➔ ❧î♥ ❤ì♥ ❤♦➦❝ ❜➡♥❣ R✳

▼➺♥❤ ✤➲ ✶✳✶✽✳ ❈❤♦ γ ❧➔ ✤÷í♥❣ ❝♦♥❣ ❦❤↔ tr÷í♥❣ ✤â♥❣ tr♦♥❣ C ✈➔ ❣✐↔
sû Fn ✈➔ F ❧➔ ❝→❝ ❤➔♠ ❧✐➯♥ tö❝ tr➯♥ {γ}✳ ◆➳✉ F = u − lim Fn t❤➻

F = lim
γ

Fn .
γ

❍➺ q✉↔∞✶✳✷✳ ◆➳✉ f : G → C ❣✐↔✐ t➼❝❤ ✈➔ a ∈ G t❤➻

an (z − a)n ✈î✐ |z − a| < R tr♦♥❣ ✤â R = d(a; ∂G)✳

f (z) =
n=0

❍➺ q✉↔ ✶✳✸✳ ◆➳✉ f : G → C ❣✐↔✐ t➼❝❤ t❤➻ f ❦❤↔ ✈✐ ✈æ ❤↕♥✳
❍➺ q✉↔ ✶✳✹✳ ◆➳✉ f : G → C ❣✐↔✐ t➼❝❤ ✈➔ B(a; r) ⊂ G t❤➻
f (n) (a) =

n!
2πi

f (w)
dw
(w − a)n+1
γ

✣é ❚❤à ❚❤❛♥❤

✶✸

❑✸✻❈ ❚♦→♥ ✣❍❙P❍◆ ✷


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❍➔♠ ✤✐➲✉ ❤á❛

tr♦♥❣ ✤â γ(t) = a + reit ✱ 0

2π ✳

t

✣à♥❤ ❧þ ✶✳✾✳ ✭✣→♥❤ ❣✐→ ❈❛✉❝❤②✮ ❈❤♦ f ❣✐↔✐ t➼❝❤ tr➯♥ B(a; R) ✈➔
❣✐↔ sû |f (z)|

M ✈î✐ ♠å✐ z ∈ B(a; R)✳ ❑❤✐ ✤â
n!M
.
Rn

f (n) (a)

▼➺♥❤ ✤➲ ✶✳✶✾✳ ◆➳✉ f ❣✐↔✐ t➼❝❤ tr➯♥ ❤➻♥❤ trá♥ B(a; R) ✈➔ ❣✐↔ sû γ
❧➔ ✤÷í♥❣ ❝♦♥❣ ❦❤↔ tr÷í♥❣ ✤â♥❣ tr♦♥❣ B(a; R) t❤➻

f = 0.
γ

▼➺♥❤ ✤➲ ✶✳✷✵✳ ◆➳✉ γ : [0, 1] → C ❧➔ ✤÷í♥❣ ❝♦♥❣ ❦❤↔ tr÷í♥❣ ✤â♥❣
✈➔ a ∈ {γ} t❤➻

1
2πi

dz
z−a
γ

❧➔ sè ♥❣✉②➯♥✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✼✳ ◆➳✉ γ ❧➔ ✤÷í♥❣ ❝♦♥❣ ❦❤↔ tr÷í♥❣ ✤â♥❣ tr♦♥❣ C t❤➻

✈î✐ a ∈ {γ}

(z − a)−1 dz

n(γ; a) =
γ

✤÷ñ❝ ❣å✐ ❧➔ ❝❤➾ sè ❝õ❛ γ t↕✐ a ❤❛② ✤÷ñ❝ ❣å✐ ❧➔ sè ❧➛♥ q✉❛② ❝õ❛ γ q✉❛
a✳
▼➺♥❤ ✤➲ ✶✳✷✶✳ ◆➳✉ γ ✈➔ σ ❧➔ ✤÷í♥❣ ❝♦♥❣ ❦❤↔ tr÷í♥❣ ✤â♥❣ ❝â ❝ò♥❣
✤✐➸♠ ✤➛✉ t❤➻

✭❛✮ n(γ; a) = −n(−γ; a) ✈î✐ a ∈ {γ}❀
✭❜✮ n(γ + σ; a) = n(γ; a) + n(σ; a) ✈î✐ a ∈ {γ} ∪ {σ}✳

✣à♥❤ ❧þ ✶✳✶✵✳ ❈❤♦ γ ❧➔ ✤÷í♥❣ ❝♦♥❣ ❦❤↔ tr÷í♥❣ ✤â♥❣ tr♦♥❣ C✳ ❑❤✐ ✤â
n(γ; a) ❧➔ ❤➡♥❣ sè✱ ❧➔ ♣❤➛♥ tû ❝õ❛ G = C − {γ}✳ ❱➟② n(γ; a) = 0 ✈î✐
♠é✐ ♣❤➛♥ tû ❦❤æ♥❣ ❜à ❝❤➦♥ ❝õ❛ G✳

●å✐ H(G) ❧➔ t➟♣ ❤ñ♣ ❝→❝ ❤➔♠ ❣✐↔✐ t➼❝❤ tr➯♥ G✳

✣é ❚❤à ❚❤❛♥❤

✶✹

❑✸✻❈ ❚♦→♥ ✣❍❙P❍◆ ✷


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❍➔♠ ✤✐➲✉ ❤á❛

✣à♥❤ ❧þ ✶✳✶✶✳ ❈❤♦ G ❧➔ t➟♣ ❝♦♥ ❧✐➯♥ t❤æ♥❣ ♠ð ❝õ❛ C✳ ❑❤✐ ✤â✱ ❝→❝
✤✐➲✉ ❦✐➺♥ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿

✭❛✮ G ❧➔ ✤ì♥ ❧✐➯♥❀
✭❜✮ n(γ; a) = 0 ✈î✐ ♠é✐ ✤÷í♥❣ ❝♦♥❣ ❦❤↔ tr÷í♥❣ ✤â♥❣ γ tr♦♥❣ G ✈➔

✤✐➸♠ a ∈ C − G❀

✭❝✮ C − G ❧➔ ❧✐➯♥ t❤æ♥❣❀
✭❞✮ ✈î✐ ❜➜t ❦ý f tr♦♥❣ H(G) ❝â ❞➣② ✤❛ t❤ù❝ ❤ë✐ tö tî✐ f tr♦♥❣ H(G)❀
✭❡✮ ✈î✐ ❜➜t ❦ý f tr♦♥❣ H(G) ✈➔ ❜➜t ❦ý ✤÷í♥❣ ❝♦♥❣ ❦❤↔ tr÷í♥❣ ✤â♥❣
∞

γ tr♦♥❣ G✱

f = 0❀

✭❢✮ ❍➔♠ f tr♦♥❣ H(G) ❝â ♥❣✉②➯♥ ❤➔♠❀
✭❣✮ ✈î✐ ❜➜t ❦ý f tr♦♥❣ H(G) s❛♦ ❝❤♦ f (z) = 0 ✈î✐ ♠å✐ z ∈ G ❝â
γ

❤➔♠ g tr♦♥❣ H(G) s❛♦ ❝❤♦ f (z) = [g(z)]2 ❀

✭✐✮ G ❧➔ ✤ç♥❣ ♣❤æ✐ tr➯♥ ❤➻♥❤ trá♥ ✤ì♥ ✈à❀
✭❥✮ ◆➳✉ u : G → R ❧➔ ✤✐➲✉ ❤á❛ t❤➻ ❝â ❤➔♠ ✤✐➲✉ ❤á❛ v : G → R s❛♦

❝❤♦ f = u + iv ❣✐↔✐ t➼❝❤ tr➯♥ G✳

✣à♥❤ ❧þ ✶✳✶✷✳ ✭❈æ♥❣ t❤ù❝ t➼❝❤ ♣❤➙♥ ❈❛✉❝❤② t❤ù ♥❤➜t✮ ❈❤♦ G
❧➔ t➟♣ ❝♦♥ ♠ð ✈➔ f : G → C ❧➔ ❤➔♠ ❣✐↔✐ t➼❝❤✳ ◆➳✉ γ ❧➔ ✤÷í♥❣ ❝♦♥❣
❦❤↔ tr÷í♥❣ ✤â♥❣ tr♦♥❣ G s❛♦ ❝❤♦ n(γ; w) = 0 ∀z ∈ C − G✱ t❤➻ ✈î✐

a ∈ G − {γ}
f (z)
dz.
z−a

1
2πi

n(γ; a)f (a) =

γ

✣à♥❤ ❧þ ✶✳✶✸✳ ✭❈æ♥❣ t❤ù❝ t➼❝❤ ♣❤➙♥ ❈❛✉❝❤② t❤ù ❤❛✐✮ ❈❤♦ G ❧➔
t➟♣ ❝♦♥ ♠ð ✈➔ f : G → C ❧➔ ❤➔♠ ❣✐↔✐ t➼❝❤✳ ◆➳✉ γ1 , . . . , γm ❧➔ ✤÷í♥❣
❝♦♥❣ ❦❤↔ tr÷í♥❣ ✤â♥❣ tr♦♥❣ G s❛♦ ❝❤♦ n(γ1 ; w) + . . . + n(γm ; w) = 0

∀z ∈ C − G t❤➻ ✈î✐ a ∈ G − {γ}
m

f (a)

m

n(γk ; a) =
k=1

k=1

1
2πi

f (z)
dz.
z−a
γ

✣à♥❤ ❧þ ✶✳✶✹✳ ✭✣à♥❤ ❧þ ❈❛✉❝❤② t❤ù ♥❤➜t✮ ❈❤♦ G ❧➔ t➟♣ ❝♦♥ ♠ð
tr♦♥❣ ♠➦t ♣❤➥♥❣ ✈➔ f : G → C ❧➔ ❤➔♠ ❣✐↔✐ t➼❝❤✳ ◆➳✉ γ1 , . . . , γm ❧➔

✣é ❚❤à ❚❤❛♥❤

✶✺

❑✸✻❈ ❚♦→♥ ✣❍❙P❍◆ ✷


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❍➔♠ ✤✐➲✉ ❤á❛

✤÷í♥❣ ❝♦♥❣ ❦❤↔ tr÷í♥❣ ✤â♥❣ tr♦♥❣ G s❛♦ ❝❤♦

n(γ1 ; w) + . . . + n(γm ; w) = 0 ✈î✐ ♠å✐ w ∈ C − G t❤➻
m

f = 0.

k=1 γk

✣à♥❤ ❧þ ✶✳✶✺✳ ❈❤♦ G ❧➔ t➟♣ ♠ð ✈➔ f : G → C ❧➔ ❤➔♠ ❣✐↔✐ t➼❝❤✳ ◆➳✉
γ ❧➔ ✤÷í♥❣ ❝♦♥❣ ❦❤↔ tr÷í♥❣ ✤â♥❣ tr♦♥❣ G s❛♦ ❝❤♦ n(γ; w) = 0 ✈î✐
♠å✐ w ∈ C − G t❤➻ ✈î✐ a ∈ G − {γ}

f (k) (a)n(γ; a) =

k!
2πi

f (z)
dz.
(z − a)k+1
γ

✶✳✺ ◆❣✉②➯♥ ❧þ ♠æ✤✉♥ ❝ü❝ ✤↕✐
✣à♥❤ ❧þ ✶✳✶✻✳ ✭◆❣✉②➯♥ ❧þ ♠æ✤✉♥ ❝ü❝ ✤↕✐ t❤ù ♥❤➜t✮ ◆➳✉ f ❣✐↔✐
t➼❝❤ tr➯♥ ♠✐➲♥ G ✈➔ ✤✐➸♠ a ∈ G ✈î✐ |f (a)|

|f (z)| ∀z ∈ G t❤➻ f ❧➔

❤➔♠ ❤➡♥❣✳

✣à♥❤ ❧þ ✶✳✶✼✳ ✭◆❣✉②➯♥ ❧þ ♠æ✤✉♥ ❝ü❝ ✤↕✐ t❤ù ❤❛✐✮ ❈❤♦ G ❧➔
t➟♣ ♠ð ❜à ❝❤➦♥ tr♦♥❣ C ✈➔ ❣✐↔ sû ❤➔♠ f ❧✐➯♥ tö❝ tr➯♥ G− ✱ ❣✐↔✐ t➼❝❤
tr♦♥❣ G✳ ❑❤✐ ✤â

max |f (z)| : z ∈ G− = max {|f (z)| : z ∈ ∂G} .

✣à♥❤ ♥❣❤➽❛ ✶✳✶✽✳ ◆➳✉ f : G → R ✈➔ a ∈ G− ❤♦➦❝ a = ∞ t❤➻ ❣✐î✐

❤↕♥ tr➯♥ ❝õ❛ f (z) ❦❤✐ z t✐➳♥ tî✐ a✱ ❦þ ❤✐➺✉ ❜ð✐ lim sup f (z)✱ ✤÷ñ❝ ①→❝
✤à♥❤ ❜ð✐
z→a

lim sup f (z) = lim+ sup {f (z) : z ∈ G ∩ B(a; r)} .
z→a

r→0

❚÷ì♥❣ tü✱ ❣✐î✐ ❤↕♥ ❞÷î✐ ❝õ❛ f (z) ❦❤✐ z t✐➳♥ tî✐ a ✱ ❦þ ❤✐➺✉ ❜ð✐
lim inf f (z)✱ ✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐
z→a

lim inf f (z) = lim+ inf {f (z) : z ∈ G ∩ B(a; r)} .
z→a

✣é ❚❤à ❚❤❛♥❤

r→0

✶✻

❑✸✻❈ ❚♦→♥ ✣❍❙P❍◆ ✷


õ tốt

ỏ

t lim f (z) tỗ t = lim sup f (z) =
lim inf f (z) G C t ỵ G ừ G C
ồ rở ừ G õ G = G G
G = G {} G ổ
za

za



za







ỵ ỵ ổ ỹ tự
G C f t tr G sỷ õ số M s
lim sup |f (z)|
za

ộ

M a G õ |f (z)|

M z G

P


❈❤÷ì♥❣ ✷

❍➔♠ ✤✐➲✉ ❤á❛
❚r♦♥❣ ♣❤➛♥ ♥➔② t❛ s➩ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ❤➔♠ ✤✐➲✉ ❤á❛ ✈➔ ❣✐↔✐ q✉②➳t ❜➔✐
t♦→♥ ❉✐r✐❝❤❧❡t✳ ❇➔✐ t♦→♥ ❉✐r✐❝❤❧❡t ❜❛♦ ❣ç♠ ✈✐➺❝ ①→❝ ✤à♥❤ ♠✐➲♥ G s❛♦
❝❤♦ ✈î✐ ❜➜t ❦ý ❤➔♠ ❧✐➯♥ tö❝ f : ∂G → R✱ ❝â ❤➔♠ ❧✐➯♥ tö❝ u : G → R
s❛♦ ❝❤♦ u(z) = f (z) ✈î✐ z ∈ ∂G ✈➔ u ✤✐➲✉ ❤á❛ tr♦♥❣ G✳ ◆â✐ ❝→❝❤ ❦❤→❝
t❛ s➩ ①→❝ ✤à♥❤ ♠✐➲♥ G s❛♦ ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ▲❛♣❧❛❝❡ ❣✐↔✐ ✤÷ñ❝ ✈î✐ ❝→❝
❣✐→ trà ❜✐➯♥ tò② þ✳
−

✷✳✶ ❈→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ ✤✐➲✉ ❤á❛
✣à♥❤ ♥❣❤➽❛ ✷✳✶✳ ◆➳✉ G ❧➔ ♠ët t➟♣ ♠ð ❝õ❛ C t❤➻ u : G → R ❧➔ ❤➔♠

✤✐➲✉ ❤á❛ ♥➳✉ ✉ ❝â ❝→❝ ✤↕♦ ❤➔♠ r✐➯♥❣ ❝➜♣ ❤❛✐ ❧✐➯♥ tö❝ ✈➔
∂ 2u ∂ 2u
+
= 0.
∂x2 ∂y 2

P❤÷ì♥❣ tr➻♥❤ ♥➔② ✤÷ñ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ▲❛♣❧❛❝❡✳
❚❛ ①➨t ❝→❝ t➼♥❤ ❝❤➜t s❛✉ ❝õ❛ ❤➔♠ ✤✐➲✉ ❤á❛✳
✭✶✮ ✭t❤❡♦ ✣à♥❤ ❧þ ✶✳✹✿ ❍➔♠ f ❣✐↔✐ t➼❝❤ tr➯♥ ♠✐➲♥ G ♥➳✉ Ref = u ✈➔
Imf = v ❧➔ ❝→❝ ❤➔♠ ✤✐➲✉ ❤á❛ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❈❛✉❝❤② ✲ ❘✐❡♠❛♥♥✳✮
✭✷✮ ✭t❤❡♦ ✣à♥❤ ❧þ ✶✳✶✶✭❥✮✿ ▼✐➲♥ G ❧➔ ✤ì♥ ❧✐➯♥ ♥➳✉ ✈î✐ ♠é✐ ❤➔♠ ✤✐➲✉

❤á❛ u tr➯♥ G t❤➻ s➩ ❝â ❤➔♠ ✤✐➲✉ ❤á❛ v tr➯♥ G s❛♦ ❝❤♦ f = u + iv ❣✐↔✐
t➼❝❤ tr➯♥ G✳✮
✣à♥❤ ♥❣❤➽❛ ✷✳✷✳ ◆➳✉ f : G → C ❧➔ ❤➔♠ ❣✐↔✐ t➼❝❤ t❤➻ Ref = u ✈➔
Imf = v ✤÷ñ❝ ❣å✐ ❧➔ ❧✐➯♥ ❤ñ♣ ✤✐➲✉ ❤á❛✳
✶✽


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❍➔♠ ✤✐➲✉ ❤á❛

❚❤❡♦ ✣à♥❤ ❧þ ✶✳✶✶✭❥✮ ♠é✐ ❤➔♠ ✤✐➲✉ ❤á❛ tr➯♥ ♠✐➲♥ ✤ì♥ ❧✐➯♥ ❧✉æ♥ ❝â
❧✐➯♥ ❤ñ♣ ✤✐➲✉ ❤á❛✳ ◆➳✉ ✉ ❧➔ ❤➔♠ ✤✐➲✉ ❤á❛ tr➯♥ G ✈➔ ❤➻♥❤ trá♥ D ⊂ G
t❤➻ ❝â ❤➔♠ ✤✐➲✉ ❤á❛ v tr➯♥ D s❛♦ ❝❤♦ u + iv ❣✐↔✐ t➼❝❤ tr➯♥ D✳ ▼➦t
❦❤→❝✱ ♠é✐ ❤➔♠ ✤✐➲✉ ❤á❛ ❧✉æ♥ ❝â ❧✐➯♥ ❤ñ♣ ✤✐➲✉ ❤á❛✳ ◆➳✉ v ✈➔ v ❧➔
❧✐➯♥ ❤ñ♣ ✤✐➲✉ ❤á❛ ❝õ❛ u t❤➻ i(v − v ) = (u + iv ) − (u + iv ) ❧➔ ❤➔♠
❣✐↔✐ t➼❝❤✳ ❉♦ ✤â v = v + c✱ c ❧➔ ❤➡♥❣ sè✳
1

1

1

2

1

2

2

2

▼➺♥❤ ✤➲ ✷✳✶✳ ◆➳✉ u : G → C ❧➔ ❤➔♠ ✤✐➲✉ ❤á❛ t❤➻ u ❧➔ ❦❤↔ ✈✐ ✈æ
❤↕♥✳

✣➦t z = x +iy ⊂ G ✈➔ ❝❤å♥ δ s❛♦ ❝❤♦ B(z ; δ) ⊂ G✳
❑❤✐ ✤â u ❝â ❧✐➯♥ ❤ñ♣ ✤✐➲✉ ❤á❛ v tr➯♥ B(z ; δ)✳ ❚ù❝ ❧➔✱ f = u + iv ❣✐↔✐
t➼❝❤✳ ❉♦ ✤â✱ ❤➔♠ ♥➔② ❦❤↔ ✈✐ ✈æ ❤↕♥ tr➯♥ B(z ; δ)✳ ❱➟② u ❦❤↔ ✈✐ ✈æ
❤↕♥✳
▼➺♥❤ ✤➲ ♥➔② ❝❤♦ t❤➜② ❤➔♠ ✤✐➲✉ ❤á❛ ❝â t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ ❣✐↔✐ t➼❝❤✳
✣à♥❤ ❧þ s❛✉ t÷ì♥❣ tü ❝æ♥❣ t❤ù❝ t➼❝❤ ♣❤➙♥ ❈❛✉❝❤②✳
❈❤ù♥❣ ♠✐♥❤✳

0

0

0

0

0

0

✣à♥❤ ❧þ ✷✳✶✳ ✭✣à♥❤ ❧þ ❣✐→ trà tr✉♥❣ ❜➻♥❤✮ ❈❤♦ u : G → R ❧➔

❤➔♠ ✤✐➲✉ ❤á❛ ✈➔ ❤➻♥❤ trá♥ ✤â♥❣ B(a; r) ⊂ G✳ ◆➳✉ γ ❧➔ ✤÷í♥❣ trá♥

|z − a| = r t❤➻
2π

1
u(a) =
2π

u(a + reiθ )dθ.
0

❈❤♦ ❤➻♥❤ trá♥ D s❛♦ ❝❤♦ B(a; r) ⊂ D ⊂ G ✈➔ f
❧➔ ❤➔♠ ❣✐↔✐ t➼❝❤ tr➯♥ D s❛♦ ❝❤♦ u = Ref ✳ ❚ø ❝æ♥❣ t❤ù❝ t➼❝❤ ♣❤➙♥
❈❛✉❝❤②✱ t❛ ❝â

❈❤ù♥❣ ♠✐♥❤✳

2π

f (a) =

1
2π

f (a + reiθ )dθ.
0

s♦ s→♥❤ ♣❤➛♥ t❤ü❝ ❤❛✐ ✈➳ t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳
✣à♥❤ ♥❣❤➽❛ ✷✳✸✳ ❍➔♠ ❧✐➯♥ tö❝ u : G → R ❝â t➼♥❤ ❝❤➜t ❣✐→ trà tr✉♥❣
✣é ❚❤à ❚❤❛♥❤

✶✾

❑✸✻❈ ❚♦→♥ ✣❍❙P❍◆ ✷