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

au

bda

ea 2

I

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

l ea 2

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Tinta 70 i ca li Sin Tia 70