Complex analysis lec notes, Lecture notes of Complex analysis

Complex analysis lec notes covering imp theorems and its proof

Typology: Lecture notes

2022/2023

Uploaded on 03/16/2026

pallavi-aunoor
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Laucent
P
eKpaicn
(6)-Z=
is.
Comerslt
Seon
Developneat
S
be
Buopose.
2 anlz-a
be
it
Lauret
is
KemGxeble
ingulanly
if
fos
an=0
PAGE
No.
bne
a
DATE
an
essenal sisgulany
iR
inhojab
riany
oegaive
inheqersn
isclaled
singuari
ty
fc2)
=
an
le-a)"
n0
ialated
eingwany
at
hee
is.
Rith
annlai0,R
a0n
(a,0,R).
4is
analyic
enovable
sinsulanhy
atz=a
On
Ba
0<
12-al<k
removable
analytic
fun
Lat
pf3
pf4
pf5
pf8
pf9
pfa
pfd

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Laucent

P

eKpaicn

(6)-Z=

is.

Comerslt

Seon Developneat

S

be

Buopose.

(^2) anlz-a (^) be (^) it (^) Lauret is (^) KemGxeble (^) ingulanly if

fos

an=

PAGE No.

bne a

DATE

an (^) essenal (^) sisgulany iR inhojab riany (^) oegaive (^) inheqersn

isclaled singuari ty

fc2) =^ an^ le-a)"

n

ialated eingwany at

hee is.

Rith annlai0,R

a0n (a,0,R).

4isanalyic enovable sinsulanhy atz=a

On Ba 0< 12-al But hen

Bud tben^ Pcz=^ an^ (2-a)

analahe

Consersh

(z-fczy of (za)

singulanly

(z-ao

Wan at z=a

PAGE No.

B(a,e)

A-m-nttt

DATE

(z-a

on 02]z-ake

(z-a)

negatie pasax

Vemovahle s

fCx) han ale al order mat 2c

is a Dole ot ordel

Suppose

uppase

f(ann(a,os))=C -f( ann(a,o,& eooplex nymber

aralyt

EsoSuaAhat

PAGE No.

fes-c haapole

DATE

|eAsiune thene is a pojd cin &&a isc B schthatb BIe;t)n(ann (a,o,s) = )

Z’a z-a)

m is (^) ghe (^) order o (^) tbis pole (^) then Aintz-es-)

4hen

Tesidue

Keaide

Nolaim ohee

Lauren)

4hat

\im

het daose

Contogdici

-fco(z-as

1z-al

snce 8)

hae

Containedben

to

he isolated singalan hen a a

cloed yechhable

han

PAGEDATE No.

isolated aingdanly

Resla) a-A=aut

does not pass throna any of the =0(K;arKes(,k

1So laed

an -a

Do tooisk gcais) intersects G.This is Singa

hen tae

4hen

possible

anh(ako

Senes 4his

apanon Hence

pnmbre (^) hence

PAGE No. DATE

beLauret

(z -a ho

4hen

=Res (gar)n(&kaK)T

poder senen

(2-a)

aa

is analyh

pole nai Za

f osder

s mordlerat

DATE

Rosca)=bod

PAGENo.

abon 41z)=beth(z-a)t--

yemavaklengulan

z) polk (z-a"z)then (rm (a)

(z-ay

Suppsehasa

(m-

TO=bo

put 6Muppose

&

Sns ide

analyhc

As

f and are

le hane

PAGE No.

(Pe.P are 4he no. af zenos paleo)

Counhed (^) ceeording 4o (^) thet

DATE

Xenos a poles

real number then

mie

Hente (^) the (^) nmernnaphic hun

fo

e

inside an

thesc

PAGEDATE Na.

han

maps hsanch

Ancther statemehet^ C s^ be^ asinple closed Contaur,^ uopase^ 4lat analyh

2)= z+z-)

Acz)= 2z1-2zt222-2Z

nea paly ide

hauehae^ fcz)+qcz)h

-3 -2Z han

plane

6

DATEPAGENo.

he mhlicileshepolhncmialpayncnatinside^ zl=

Deexnine Ze0s^ Counting

insde

Fc2s42) hao Same

54

4

Counhag i the mulipliciteA,

Hencep pcz hoo^0 psecisely s,

nanelyn

Lame

Rouhe's

,

By