Homotopy between 2 curveswhat is a curve in S where ra E is a open

r a b R piecewise Ct smooth connected subsetContinuously

i e I a to Lt s th b ri Eti ti r is a differentiable

j ftp.fg.Jfunction

homotopy between 2 curves is an interpolation between

two curves Toca Tsca LNo Vz i ta b r curves Tocb Kcb P

homotopy i Ts a b R SE oilsit VsCt Is o Volt

Blt Is tilt E t E Ia b

forfixeds HsC o I Js a b D a Carne

Tsca X Bcb pwe want the function resit kitJ o I x a BI 52

to be jointly continuous in S and tfamilythe curvesvaries continuously in s

L yBirocb Kcb

aToca

n'is

If Vo and Tz are two curves ins with the sameendpoints and if Zo r are connected by ahomotopy Is then we say Yo and Ji homotopic

The Let 5L be an open connected setf S E holeYo V homotrpic curves in R

Thenfy.fczidz fy.fmDZ

deformation invariance of contour integral

Pfsketch i cut the S interval oil into smallenough segments 0L Soss c CSN I

so that t.si hit are close together

Do a direct computation for comparing integral alongsuch e nearby canes by covering themwith disks

TsiTsa

EI 1 domain with 3 holes

Fo i r are not homotopic

my µ µ

I

IL

claim let r D a diskZo Zi E R be two disinfect points

then any cure from Io to Z ishomotopic to

the straight segments

S

TSH l s t.tt sfixedfor each t VsCt lies on the 5 betweenroot and tilt that segment lies in DC Dis convex

jVsH is indeed a curve in Dending at Zo and Zi

Choose E small enough such thatHSE10,12 ft C a b

Dae BHI CSL

let K theimage of the mapT o D XEa b SL

T is continuous oil xEab is compact

K is compact

dist CK Dc O Cifl of

ftp stk.sE ixhef distCx.ri ixnefiynfzdcx.y7suppose distck.sc then 1 Ex y Kiya

Xi Ek YiESE sit dcxi.yDe.IT by compactness of KI subsequence of Xn called Xm Xm sitfinfXn EEK Then we get

dCXn yn s O as g is

d E Yog 0 as j p

re x'er I Uax Ucrwe have a contradiction i

willif dix yay 20 Ynj eventually be in Ucontradicting of Ynj ER

a 3E Iz distck.SE

By knife EEetinuity of 8 o Dia b 252I 8 such that It IS sales we have

sup I rs.lt r's I c Etea b

it.si 7 s or

For anypair of Si Sz sit Isi Sel S wewill show fg.fi Z Srs.fHdZt

t Zn Zo wo d

o µ D Zha una Bow Ya nWo DDo

cover both curves by disks Do i Dn ofradius2eeSuch that I Zo Zi Ent on s and

Wo wi Wht on TszDnr contains Zin Zhai D Zi Zi

Wi wz

feximple construction we choosezsome to't t'z za

sit Irs ti RsCti I s E oiii iii aI tacti tactin I e Ewit wit

wi wit

L one can then choose Di Dae Zi

fO is frolic on Dm a primitive Fm of f on Dmonthe over lap DmNDmti where both Fm and Fm4are defined they at most differby a constant

def y f yFme 5me C on DmnDm11

Dm Dmti

Fma Fma Fm Ema CFma Wmu Fm Zuma _C

Jubtract themFane Zina Fma Wmu Fml 2mi FmWmu2me

f f d'tMma

Srs f dz SI'fdz t f de t JI t okIof.fi Fotto F Zz 11 1 Fauna En

fat Fo wi Fo Wa t F Wa F wi t tfnlwnDE.w.mnman

I F ZoFocus t Fn Zari Fn writ

0y

i f I e l odlateef r r

i F

by doing local modification within a disks we canturn a canoe 2 Zi Zz Zn B Th

ly2

X wi Zi Zz Zu Bdf wN Wi Wz Zz 23s Zn falsifywew

region marinated open

Det simplyconnected IEgion forany 2 curves withZ

are homotopic the sameendpoints

EI convex region are simply connectedsee the previous discussion on linear interpolation

star shaped domain an origin e c Isit any print 2 Ercan be connected to e by

a straight segment

Zz

interior of t.oycoatoai.FIfailure of simply connected if he hasholes i puncture

The If r is a simply connected done regionf r a hole

Sthen f has a primitive F

PI i pick a Zo C R then defineZ

Fez J four DwZotalong any path fromZoto Z

thingythis is well defined since theintegalonlydepends

ontomotassofpathsfromZotoZlandthere is a uniaes of pathfrom Zo to Z definitionof

SD fimply connect

check F'Cz fezdomain

Conservative Vector fieldZd Ux x y Ugag such that

2yUx 2xVyv Ux Dx t Uydy

U only depends on the homotopyclass

Q F R G holeclosedT came in R

Q f IIIT.dz guffawt

EI f z E J unit circle

dz fz DZ ziti121 1

Jwtdw Thi Efo for

f dw d InCwInCw In.tw ti.argCw

welldefinedismattivalued.fd.tnH O tCf d i argue i 21T i

HW6 2Laurent expansion around a polef Z is meromorphic Zo is a poleOoforder n

Laurent Vexpansion ifczy.cz yn t.cz gn't 20 nAot ZDregular

H at

part

hasZ ZD then Taylorexpand her around a

hot fo t E Z Zo 1 G Z 2oft