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11 Argument Principle - MIT OpenCourseWare

argument is something with more structure more akin to the logician's notion of derivation : a series of statements with intermediate steps providing the transition from premises to conclusion



11 Argument Principle - MIT OpenCourseWare

1 The argument principle says Ind( ý 1 0) = 1 ? = 2 Likewise has no poles and one zero inside 2 so Ind( ý 2 0) = 1?0 = 1 For 3 a zero of is on the curve i e (?1) = 0 so the argument principle doesn’t apply The image of 3 is shown in the ?gure below – it goes through 0 Re(z ) Im(z ) 1 2 3 1 2 2 1 Re(w ) Im(w ) w f



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California State University Long Beach

What is the argument principle?

11.1 Introduction The argument principle (or principle of the argument) is a consequence of the residue theorem. It connects the winding number of a curve with the number of zeros and poles inside the curve. This is useful for applications (mathematical and otherwise) where we want to know the location of zeros and poles.

What is the analytic proof of the argument principle?

Here is the analytic proof. The argument principle requires the function to have no zeros or poles on So we ?rst show that this is true of ? ? The argument is goes as follows. Zeros: The fact that 0 ? ð

When is an argument valid?

An argument is valid if, whenever the premises are true, then the conclusion is true. 2 Logical Implication A formulaAlogically implies B if A )B isatautology. Theorem: An argument is valid i the conjunction of its premises logically implies the conclusion.

What is the structure of an argument?

argument is something with more structure, more akin to the logician's notion of derivation : a series of statements with intermediate steps providing the transition from premises to conclusion.

Topic11Notes

Jeremy

Orloff

11

ArgumentPrinciple

11.1

Introduction

The connects is and poles.11.2Principleoftheargument

Setup.

andsomezeros inside (noton). Let m be thepolesofinside. Let n be the zeros ofinside. Write mult k =themultiplicityofthezeroatk

Likewisewritemult

k =theorderofthepole at We

Theorem

11.1.Withtheabovesetup

mult k mult k .Withthis in mind, suppose hasazerooforderat

TheTaylor

seriesfornear is where

Thisimplies

Since isnever0, isanalyticnear

Thisimpliesthat

is asimplepoleof and0 1 mult 1

11ARGUMENTPRINCIPLE2

Likewise,

if is apoleoforderthentheLaurentseriesfornear is where Thus, Again wehavethat is a simple poleof and 0 1 mult The sumoftheresiduesÊ mult k mult k

Wewrite

f for f So theTheorem11.1says, f f (1)Ê

Cauchy"s

is definedas Ind 1Ê (In One and isafunction,thený isanothercurve.Wesaymapstoý. We here sure youareverycomfortablewithit. Let with ff(theunitcircle).Let

Describethecurve

Solution:

Clearlyý

2 traverses theunitcircletwiceasgoesfrom0to.

Let withØØ(the-axis).Let.Describethe

curve

11ARGUMENTPRINCIPLE3

Solution:

the origin centered at.Bycheckingatafewpoints: We

seethatthecircleistraversedinaclockwisemannerasgoesfromØtoØ.Re()Im()0=(1)�=(�1)() =Re()Im()1 =(0)(�)()()12

() =1(1 )=() =1+1 The curve ismappedtoý.

11.2.2

Argumentprinciple

You

Theorem

Indý

f f (2)

Proof.

Theorem11.1showedthat

So change

ýand

so

Indý

The Note in theintegralisnotaproblem. Here

Corollary.

Indý

f (3)

411ARGUMENTPRINCIPLE

Proof.

ApplyingtheargumentprincipleinEquation2tothefunction í µí µ,weget í µí µí µí µ í µí µIndí µÃ½í µ,í µí µí µ Now, (becauseí µ Ind í µÃ½í µ,Indí µÃ½í µ,( í µwindsaround0í µwindsaround-1) (same inbothequations) f (poles ofí µ=polesof í µ)

Example

11.5.Letí µí µ í µ

curves. 1. circle ofradius2. 2. circle ofradius1/2. 3. circle ofradius1. answers. í µí µhaszerosat,.Ithasnopoles. So, í µhasnopolesandtwozerosinsideí µ f

Likewise

hasnopolesandonezeroinsideí µ so Ind For a image ofí µ is 1 2 3 2 2

1Re()Im()wfz z2z

The image of3differentcirclesunderí µí µ í µ

11.2.3

Rouché'stheorem.

Theorem

isasimpleclosedcurve f f

2í µí µ

1 =..í µâ„Ž/Ã½í µ,0/=í µ f f

2í µí µ

0 1 â„Ž â„Ží µâ„Ž<1Ã½í µ1â„Ž=Ã³í µÃ³í µí µ í µ 1 00

1 1í µâ„ŽÃ½í µ,0= 0.í µ

í µ=.í µÃ½í µ,0/=0 = 0í µí µí µ .í µÃ½í µ,0/= = 0⇒..í µâ„Ž/Ã½í µ,0/=.í µÃ½í µ,0/.í µí µâ„Ží µ f f f f

Corollary.

11ARGUMENTPRINCIPLE6

Proof.

Sincethefunctionsareanalytic

f and f are both0.SoEquation4shows f f QED. We thinkofasasmallperturbationof.

Example

11.7.Showall5zerosof

areinsidethecurve

Solution:

Let and 5) areinside

Alsoclearly,ððððon

ThecorollarytoRouchéstheoremsaysall

5 roots of mustalsobeinsidethecurve.

Example

11.8.Show

has one root inthelefthalf-plane.

Solution:

Let ,

Considerthecontourfromtoalongthe-axisand

then the left semicircle ofradiusbackto.Thatis,thecontour R shown below.Re()Im()iR iRC C 1 To R On ,so So

ððon

On R withandðð.So,

ð forlargeðð ð

(since). So

ððon

R The

Therefore,

the half-plane.

Theorem.

Fundamentaltheoremofalgebra.

Rouchés

Proof.

Let n n n be anthorderpolynomial.Let n and .Chooseansuchthat n

Thenonððwehave

n n n n n

11ARGUMENTPRINCIPLE7

On í µ=í µwehaveí µ.í µ/=í µ n

Rouchés

infinity, Note. or equal tomax.1,í µí µ n 1 0 11.3

Nyquistcriterionforstability

The system systems ourselves Note. you

11.3.1

Systemfunctions

A

Typically,

Let with quite We

Denition.

half-plane. The For the would analysis. Example11.9.Isthesystemwithsystemfunctioní µ.í µ/ =stable?.í µ2/.í µ 2 4 5/

Solution:

Example11.10.Isthesystemwithsystemfunctioní µ.í µ/ =stable?.í µ 2

4/.í µ

2 4 5/

Solution:

Example11.11.Isthesystemwithsystemfunctioní µ.í µ/ =stable?.í µ2/.í µ 2 4/

Solution:

the imaginary axis, the system ismarginallystable.

Terminology.

now

11ARGUMENTPRINCIPLE8

clear androllsoffthetonguealittleeasier! We small cross ateachpoleandasmallcircleateachzero.

Givezero-polediagramsforeachofthesystems

1 2 3 .í µ2/.í µ 2 4

5/.í µ

2

4/.í µ

2 4

5/.í µ2/.í µ

2 4/

Solution:

a single 1()1 Re( )Im() 2()1 Re( )Im() 3()1

Pole-zero

diagramsforthethreesystems. This in The 1 corresponds to amodeí µ 1 / =e 1 left half-plane.

Physically

initial called 'noinput")unstable. To to thehomogeneoussolutionsí µ.í µ/ =e called always settleddowntoequilibrium. If negative

11ARGUMENTPRINCIPLE9

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