Post on 28-Jan-2021
transcript
Today Residue Theorem
index of a curve around a point windingnumberofa curve around a pt
Example of applications for residue thm staying.rs hc3ufs2
except at poles Zi Zk
ResidueThmi Let f I Q be a hole functionon an openset 1 and let C be a simple closed
curve in SL such that the interiorof C is also contain inrand Zi Zk areinside C Then
fca da ziti 2 Reszi f R
mode'sL.ir SCi
say Cetti Then use
f DZ Thi Req f
Indeed holds by expanding fcz near 2i into formA t GafLZ CzZi ni z z ni l Z Zi
fat dz J 9 DZ Ziti A l Kii ReszifZi Celzi
Rink The limitation on the curve C to be simplecanbe remove ice we can consider crossings i
Ea no
dZ zitiE p
windingnumber
C Cit G a c of C around0
integration along C int along C then int along CzPics
CAhtfors
Lemme If a piecewise smooth closed curve 8 does notpass through a point then the value of theintegral
zeta da n
is a multiple of zitit
L
Firatimi z adZ a dot a d logcz alog 2 a Re togaas i ImKogaAD
log Iz al t i argot awell defined is only welldefinedfor IZ al 0 upto a multiple
of ziti
informally fzztadZ i.ch mentof E as
check i J Fa DZ Ziti zr
o J adZ 41T ir
Formalproof i If r is parametrized by zits instepthen we can consider the fruition
tet Z'es a t phits f dsixzcss a hcpl fzfa.dzIt is defined and is continuous on the closedinternal Exits i
hits a t
Z'Ct is notwhenever 2kt is continuous o continuum atHct E t
httThen the combination e 2Ct a has derivativevanishes everywhere in t Ctap except at possibly finitemany points This function Hits is continuous henceHtt const along Art CEx.BZ TXie
het heeThus e 2Ct a e Zen a
2 L a
hot ZC2Cd a
eHCP zCf a I i hip ziti th
Det mcr a i z a DZ winding numberof T around a
Rinka MC r a Ncr a
Slightymoregeneralversionofresiduthin
f I E has finitely many poles2 i Zk
T is a cane in 52 avoiding these poles
fat dz ziti Ei Resz f aa
A fc If 1O LIt
her owe can defer aslong as 8 avoid E y
ri
f minus because 2 windsclockwise
ner e IT t1 Nct 1 1
fez dz Thi Resof ncr oRes f no 1
ziti d ti t BC.tlI Zhi B d
ni
u caus E Usb
Mi b of f
f da f fo dwfoisw fez argument principle
next time
Application of Residue ThmI evaluate definite real integral
MIRIf dx trim ftp.txfdx
F X
fit zzfat has poles at
R O R roots of ZHI i eiCzei CZ i opole at 2 i 2 i
Cet CR
fez dz Thi Res fatorder 1 pole at fctI
Reszife lim fiz Cz Dilimz i Z i AtiIZeitz i zit
f Ct dZ ZITI IT TL
claim
ffg.zt.dze kid
E TAR Oas R p
70
Thus pliff If'gdZfit dz fzi.hr
fatdefcfcz7dZTL
Rink for fad QCD such thatQLD has no roots along the real line
and degPox E deg Q 12 proofas above
then fj fan DX can be evaluated in the sameway
Rink quiz can we close up the contour frombelow like Lyell
DieEX I f EI dx o c allI exFirstchecke is this well defined a Ico
Near the ftp.eeaf dx f ecaDxdx CPAnear o
If e d J e a du enRU againexpdecay
2What are the zeros of e 11 0e Ti l t e'T 11 0
but z Ti t Thi n nC 2 all have same valueifor et X3Tri
Gor Cmi
fit
R C RThi
f 3miwant Jc f z dz
know fo f't ziti Resz i fi C eat
IIi
or applyL'hopital
t.EEiqfiH tziI.iG i Tear ae AZIng e'I i I'z
z Zo
Iim CHED'tz i e4De a ay
fez ok S coitus fi eafIiieeuduUtpI I I
thCAHd J e
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asR pone can also show Iz fit DZ oalong vertical
edges 14 Ja f de OFE
ni I Iit Iz Is 114time It Is elimp I e g I
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Tinta 70 i ca li Sin Tia 70