ftp.fg · 2020. 10. 30. · Fane Zina Fma Wmu Fml2mi FmWmu 2me f f d't Mma Srs f dz SI'fdz t fde t...
Transcript of ftp.fg · 2020. 10. 30. · Fane Zina Fma Wmu Fml2mi FmWmu 2me f f d't Mma Srs f dz SI'fdz t fde t...
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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
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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
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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
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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
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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
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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
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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
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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
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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
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part
hasZ ZD then Taylorexpand her around a
hot fo t E Z Zo 1 G Z 2oft