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MATH F112 (Mathematics-II)

Complex Analysis

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Lecture 26

Harmonic Functions

Dr Trilok Mathur,

Assistant Professor,

Department of Mathematics

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ifdomaingivenainharmonic

betosaidisfunctionvaluedrealA

D

x,yu )(

,incontinuousarethey&exist D& uu,, uui yyyxxx)(

uuu

uii

yyxx 0

)(

2

equtionLaplacesatisfies

plane.complextheinharmonicis

:Example 23),( 32 yyxyxu

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inharmonicarethen,

domainainanalyticisIf

1:Theorem

Dv&uD

x,yi vx,yuzf )()()(

?trueconverseIs:Remark

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equationsCRsatisfiesanddomain

ainfunctionsharmonictwobeandLet

D

vu

...)1.....(

Dvu,vu xyyx

inoutthrough

.ofConjugateHarmonicbesaidisThen uv

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.ofconjugateharmonicais

ofconjugateharmonicais

1:Remark

vu

uv

(1)assamenotiswhich,&then

,ofconjugateharmonicaisifFor,

xyyx uvuv

vu

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vu

uv

-ofconjugateharmonicais

ofconjugateharmonicais

:2Remark

)1(

&..

,

assameiswhich

-as

xyyx

xyyx

vuvuei

uvuv

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.ofconjugateharmonicaisiffdomainainanalyticis

functionA

:2Theorem

uvD

x,yi vx,yuzf )()()(

.)(. 2

zzfEx

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Ex. Find all the points where the function

analytic.is)(2)( 22 yxixyxf

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.inconstantis

.inanalyticis

.invaluedrealis

ifinconstantbemustthatProve

.domainainanalyticbeLet

Q.7Page

Dzfc

Dzfb

Dzzfa

Dzf

Dzf

,

)()(

)()(

)()(

)(

)(

78

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)2()(

)1(,

)(

xx

xyyx

viuzf

vuvu

Dzf

and

.domainainanalyticisSince

:Solution

Dzzfa functionvaluedrealaisGiven )()(

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.

where

Dx,yx,yv

,x,yi vx,yuzf

)(0)(

)()()(

Dyxyxuyxu

vuvu

vv

yx

xyyx

yx

),(),(0),(

,

0,0

.constant Dzzf

Dzzf

)(

,0)()2(

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Ex. Consider the function f(z)= u(x, y)+ i v(x, y)in a domain D,

where

vis a harmonic conjugate of uand

uis also a harmonic conjugate of v.

Then show thatf (z)is constant

throughout inD.

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)-(),(

whenconjugateharmonic

afind&harmonicisthatShowQ.10

yxyxua

v

u

12)(

harmonic.isu

uu

uxu

uyu

yyxx

yyy

xxx

0

0,2

0),1(2

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uv ofconjugateharmonicais

xyyx vuvu

,i.e.

satisfiedareEquationsCR

)1(2 yuv xy

Then

)(2 2

xyyv

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xuxv yx 2)(

cxyyv

cxx

xx

22

2

2

2)(

)(

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yxyxub sinsinh),()( ,sincosh yxux

,sinsinh yxuxx

,cossinh yxuy

yxuyy sinsinh

0 yyxx uu

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xyyx vuvuuv

,ofconjugateharmonicabeLet

yxvy sincosh

)(coscosh xyxv

)(cossinh xyxvx

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cyxv

cx

x

yxuv

xyxv

yx

x

coscosh

)(

0)(

cossinh

)(cossinh

But

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Show that if v and V are harmonic

conjugates of u in a domain D, then

v(x, y) and V(x, y) can differ at most

by an additive constant.

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.ofconjugateharmonic

afindIfQ.

uv

yx

xyxu ,),(

22

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