Discussed

2 Complex Number G t R2

complex'Emingai.ie z II eyc2 y Imczs Jgjiudx2zE y Zziuseful for additionand subtraction

n z r e O r O E IR mod 2112

r e O p ei0 pp pilotoON On 24 0 p.e.io

cr.eio 71crieioj.fr ri ei0 0 fhgrsino

r modulus of Z absolute rake of Z0 argument phase r coso

X T cosDg risino

Ex fromSteini IZ Z I IZ Zal Z Za EE

EZ

in

bIz il 12 11 3

atb 3

I 1answer ellipse

z2 x iyI2 11 I xtiy il CXD iyl jcx.itJfxD2tyTtlxtiTty 3 is 27 1 2 1

Ic 12 11 I 2til 3

y

CD Yz I 2 Xtiy r 0

I z E 1zpI x ig r e ie

ZZ x2 iy5 x i2ayx2ty2 1212

y.eio.gr e it rz IR

Re AZlb 0 a b EE

a A tick D bitibz2 xtiy

aztb article Craig bitibz CElReca b soAX Azytb t eC

Relazeb Aix aid bAix day bi 70

Recz 70Ex 03

E

2 Holomorphic function exist complex derivativeonholes

fCz Iz 1212choiceif2 to arg Zez Sinti I

fogi 124 is not holomorphic at any point Zoe Ee.g ifZo O we need to check

fc2tn fczo 64k 10thI lollim line b himh o h h o

cpk h InteionInigo e Toh doesnot exist Because if h

approach 0 from different directions the result e 0h arenot the same hence the limit doesn't exist

Remarts Having complex derivative at 2omeans.fmh1gqfCZothfCZo t f Zo h t

smallconstant linearterm term openconnectedsubset

e will see if fcoimspa.hoIcouadomIthen.fhas all the derives I f't is bolt f G isHE

flZoth f f Zo as h 20hremainderterm

f Zoth f Zo and R Zoh 70f Zo RCZo h ash20h hZoth fczosth.fcZD h.RCZo.HN

Small

Power Series

a series in number it is an infinitesum

AotA 1 Azt Info An In Ana series if the poutine

Sn Aot Ait 1 Anhas a limit i.e Lim Sa exist

n b

lanl oA necessary condition is 0 as n A

M1aSz is

2C1 Lnot sufficient i e Eth is not convergent ma h

Absolute Convergence In An absolutelyconverge iff Infant convergePowerseries Info an.z Tato'HanTazi i i

wesay a powerseries age at a point 2 Zo if2 An2on converge

Fact If Info An Z is absolutely convergent at Z Zo

then it is abs conv for all Z sit 12111201n En tant III convergent

if 12151701 then 121 El2dmi lanl 121 E lad 121En lan e Enniolaal 12inch

Ex ee 1tz E Z01 1

series a powerseriesthisS m is always convergent for all 2E E

by ratio test zmanthp lim hi's 17 so

n s

i p Ll i by ratio test In IT is absolutely come

Xtiye eiht polar form of a aplx numbed

Radius of convergence of a power series an't

A 31230 such if 121212 then Ian abs conuif 121 R then is divergent

Th the radius of convergence existsand is given by

f limsupn nolaulthis

Ex I 12 124231 I l Z an _In o

mmsgp 111th L Th R I

r if 17121 then I Zn diverge 121 doesnotgoto zeroif Hal E Z FE

Dax diverges at2 1z makes sense as long as 2 I

IZ only make sense for 121 l

Pf Assume for simplicity thatL lingspop fault is not 0 D

R 11LCa Need to show if 121212 then Eau Z absscour

III Lcl E of room7 E o small enough such that gang's

A a121 Lte I Lee eThen Lliningp tant't L ir

n7 N large enough salt n N

Ianthe Lte

then 2nFµ lard 121 En Lte 121

In k44

converges

b if 121 R then 121 L 1 Is o smallenoughtanh

im i i ii If i7 an infinite occurrence or subsequence

Am.INT Lao e

So for such he Ni Nz Nz 3IanilZl L e

n 121 lzl.CL o A

as n P in this Seg hi Nz 3 A fix hawker l

T i IfRecife

Retain

alfcxu.gsImCfczD

iey.i.ex.eim.iexi.ie ex

IT x

Source targetTanotherway

fCZ holk Xtiy i utiV u ReffczD Im CfczD

ucx.gs vcx.gs

i g gU V differentiable

Ex Ey It Ex

is conformal

dfoJ JodfP Trotategosendforward

thetangentvector