p3383.3theResidue.tk# y✓open

.Det Let felt I G- Isa3) have Laurent expansion abt .z=a

f-(z) =nI⇒Cn CZ -a) h ,valid Ze Brca) .

Thenresidue off at a tented Res Cf

,

a) : E,

.

-

,open

Note Let felt ( Elsa3) have Laurent expansion abt .z=a

f-(z) =nI⇒Cn CZ -am ,valid Ze Brca) .

Let 8pct ) ÷ at reit,o ETE 2T ,

OL re R and 8# c Bp .la)n k

9. see So E ch Cz -al MIE f on V*.

•

a- k=- n

so by uniform convergence (Cor I. . 2.6, pl7), get

n

Eckfqcz -alkdzT.nu?#c2SqfHdz-.k---n(Ex I. 2.7 p 17

11 Spr CZ -asrkdz I { Oz, ,

,if k t - I

,if k=

-

I

C-,C 21T i )

a

← This formula isso

pestf.al-I.fm#Z)dZJ..hndaPTfshfffo@nI.

I'

If,

also f- has a pole of order m at a Lso fcz) =I Cal Z -a)Y

-) .

k= - m , then

Res Cf,a) =

'IFA# ! IIIT Cz -ajmfcz)

II.mdnitgtiemrem ' Etienne# =¥T⇒n*h< producing the cm - D ! > & thentake Linna of the (m- D -trues diff . powerserie

# WTF E,

"

Emt im .§

Open y'Useful Fact Let p, QEIHCG) and Zoe G. (WTF Red Eq , Zo)),XIf q Lzo ) = 0 but pczo) ¥0 and

q'

Ho) to,

then at Zo, Iq has a simple pole Cdet a pole of order 1) and

Res ( Iq , Zo) = P¥ .

q' (Zo)

Why true ?LTGBG

.

L handwriting warning q #g )

(1.) By considering the power series ofQEHCG) at Zo,since qCZot=O&gt⇒to

,

qcz) = ( Z - Zo ) g CZ)for some g EH (G) Stg C. Zo) to . Gog has zero@zoof multiplicity 1),-7② So

pcz, PYgczL and I e Hla) cot Pho)1- =-

- F O .

qcz) CZ -Zo) G gczo)

so Iq has a simplepole at Zo . ( ie pole order m't )

wi m =/

③sores leg,

⇒ E- '¥m⇒o.I [⇐ - zDmTf¥)Iim Cz -EdtPCI by 12) 1 ,⇒Zo q Cz)

= IFZo Pg¥z)

P ,g ECCE) PL Zo) p ( Zo )-1- --

gczo) =p q'Ho)

( r ) ⇒ of'CE ) = g CZ) t ( Z

-

Zo) g'Cz)

⇒ q' ( Zo) = glzo ) to

p 34 Residue Thm ( Thm Ii . 3. I ) TF-

If §→ opens connected ! v (•I

↳ ¥.?.ge#arh:YisiiEF7oin+sP...e. !

te H (G1 Ep , ,. . .

, ph})path

'

L- -

8 be a closed piecewise" in G)Sp , ,. .

>Pnl ,Then

:S,fczgdz = ziti aft

,

[ Res Cf, Pa)) Indy(Pk) .

PI LTGBG .Find O CE CR sit . for each ke { I , nm , n } :

GU l note④ The Laurent Expansion off abt Pk

.

f- Cz) : =a

calm Cz - pwn is valid in Bpl Cpg.)② The balls { Bp

,Lpga) :3hh=

,

are disjoint .

(3) 8"C [ Be Cpk))

'

ien ,fBdPaD'

zeal iz - Peel > e V-E -1 . .,n3••

so for each k =L , in .

, n ,the singular part of the LE off abt ph, ie

Sch :=nI→dnh) Cz - put corn . uniformly on [ Beeped?Define g

: G) Sp . . .. . ,Pn3 → Q by :

g CZ) : I f Cz) - §,

Sch)(z) .

Note g E H ( Gl Ep , ... ,pn3) Lbk f- & Sch) 's are > .

Ate for each k , ge H ( Bpl Cpa) )

since the'balls 9 D

'

n(Ph )3y!

,

are disjoint (by 12dg.

Claim Fix k est, . . . ,n3 . Then g

EH ( B"

Cpa) ) extends tog e HLB Lpp) .4)( f z e Br' l Ph)

f ,⇒ -Z

s Cz)g Cz) : =

j -- I

= I aim Cz - pay - SHH - ¥h s" 'Cz)

n = - so

Iµ÷÷÷÷÷i:¥¥¥i÷::÷:*.s! what.is.

in thepurple box defines a bolam. (thus

continuous) function on Bk (pee) .So limp

,

g ( Z) exists,set igcpz) := lziyypu.gl Z) .

LAnd ph is a remote

. sing . of gon Br Cpa) and age H CBR (Pa)).

Thus g extends from G) Epi , Pn} tog e HCG) .

Cauchy 's Thin for A - like sets(Thin I.2.12 p2o)⇒ , Sy gtz) dz = O

⇒ Sy fczydz = £,Sp s" (⇒ da

.(4)

Fix k. It,chew ez - pan Sth) iz) on8*14

II.jcnh)Iz-panda ] S

,SH) Czsdz

if nt -1 : I z'

- pay = Iz Z'

{ d

'

z -Pat de-0[if n = I °

. Indo tph) 9¥ Eti Sr F-putZ .

eggso SysMlHdz = CY 12 Thi ) Indy ( Ph) .

byso Sp f- IZ) d 2-

Ep ÷ ,

Res ( f,Pa) 121T i) Ind p ( Pa) .

pgI (5)

5

Deaf Let f : 112-5112 be St flan, pig

is Riemann integrable f¥.

The Cauchy principle value integral of tf is :

(P.

V) f: fox) dx %± tiny, fr! fox) dx ;

provided the limit exists .

Note-

( D 18 x dx DNE but CPV ) Igo xdx = O .

⑦ f even function ⇒ fnk faux = 2 for tix) dx ,

Ex ( typical Qual Question) ..Find the principle value of I

•

XZS ⇒ dx .

-a

using the Residue Theorem . .

Consider z6= - I ⇒It= @ilttzth)

f- Cz) =ZI2-6+1

°

LEZM

f- Cz) EHLE l Epo , pi , . ..

, Ps3 ) where pies are the" 6th roots of

-

l"

,N

''

e ' starlikeph

= ei ( Ite t "¥) for k-0,1,-

,5

.

(ie pw smooth

.

-

To apply Res .Then . ..need closed path thatdoesn't passthru thepals

( and also want to somehow pick up [- R

,R ] CIR

.

R•

¥ , H5* Let Rs ,

Pa ...

. Po ✓(KR)#

-

R .- R

Pg" f " Ps

Pg =-

c'

HR : co,IT] -3K us 89 G) = Reit

BY :C- R, R] →a ur 82k Its = Reit-ter is the join of HR and ref =

''semicircle" Suk DX

-11

Ide Snr *Hdz = tzdz t {q ffzldzI

" Irma:3:* . Hofoper-38some"

nice " numberPV If tax

.

Apply Res .Them M HH = gf

2-left

{ p f# tz = ZITI k¥0 [ Res ( f, Pa) ] In¥%pI)iet

'r compute Res(f

, Pa) for k=0, 1,2 .

Res (Ey,

Pk) ="

1¥

f . gicpa,

"

= 6!h÷=tPI3 .Useful Fact Let p, QEIIIE) and Zoe G . (WTF RedEq , Zo))

¥¥¥÷÷÷÷÷÷÷÷:÷÷÷÷÷÷:÷.:¥¥÷÷.2-o=P k .

k> = t ( ⇐ it either)

- 3

=L e- it ⇐- i ;t£

-I2

= I C- i ) L -ok = if - 1)htt ( iz)

so,when R > I ,

fppftzdz = 2mi → C-Dhtt )) -4mi) (E) I - Itt - 1) =Iz .

Next try to show / Syr fcz, dz )R O.

'

p

¥?M often the ML Imax / length) Lemma is helpful.( Prop 2,5 ( 3 ) p 163 • Need te CC8*1 M 8 path .

I SatE)del a- [ mzaxey * Had ] KD

Recall €74 and if It) = Reit ,ostst

. I-

By the ML lemma

Isr.rE÷id4¥ecoiy max

' Hr,-

€,g*lI÷/ )n f :*, I. "' I'⇒ = :# Eso

.12-16 - IIIR> I% . use revere Ding

So

Spr ¥7 ,dz = Spa

,

II dz t Say IET dz

11 when Rsl fasho"

V

'

Iz o If 7¥,

dx.

so PRI: ¥. dx-detlpig.fr#adx=Iz.DEx .

Find Soo XI dx .

f-note fix) : III

,is even

Xbtl.

III. ↳ =

live.sc?IIi.idx-hi:..H.rII.dxwE.'zlEoI--It.

ERevisit .'

Find PV 15 7¥,dx •

&

• Had fcz) = IIe,

E H E e l s poop, ,. .

, pg 3) wipe = e

"

! :#,piecewise smooth curve

• To apply Res.Then . ..

need closed path that doesn't passthru thepals( and also want to somehow pick up [

- R,R ] a IR

.

Let R > 1R•

¥ ..

.45*13.

..

.

.

. Po ✓(KR)#

-.- p,

"

t .

pq =- i

R →0

Then it was not too hard to show fair f-Hdz→ O•

↳ this is often the case when f is a rational functionand 8Th is a Envoirole .

• But sometime,when trying to find a closed path that doesn't

pass thru the pts. . where f- is not different id be,a nice

semicircle circle will not work , Another commonly usedpath to try

'

is :'

ut r 't .÷÷*

) Bither:datedso

•

THEY'

Sqrfczsdz'

o.

-

p ,

"

I 12

. Tthis "

rectangle" is

Pg =- I

met a good choiceforf- CZ) =ZI .

Z6+1