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7/21/2019 Lecture 26 (1)
1/22
BITSPilaniPilani Campus
MATH F112 (Mathematics-II)
Complex Analysis
7/21/2019 Lecture 26 (1)
2/22
BITSPilaniPilani Campus
Lecture 26
Harmonic Functions
Dr Trilok Mathur,
Assistant Professor,
Department of Mathematics
7/21/2019 Lecture 26 (1)
3/22
BITS Pilani, Pilani Campus
ifdomaingivenainharmonic
betosaidisfunctionvaluedrealA
D
x,yu )(
,incontinuousarethey&exist D& uu,, uui yyyxxx)(
uuu
uii
yyxx 0
)(
2
equtionLaplacesatisfies
plane.complextheinharmonicis
:Example 23),( 32 yyxyxu
7/21/2019 Lecture 26 (1)
4/22
BITS Pilani, Pilani Campus
inharmonicarethen,
domainainanalyticisIf
1:Theorem
Dv&uD
x,yi vx,yuzf )()()(
?trueconverseIs:Remark
7/21/2019 Lecture 26 (1)
5/22
BITS Pilani, Pilani Campus
equationsCRsatisfiesanddomain
ainfunctionsharmonictwobeandLet
D
vu
...)1.....(
Dvu,vu xyyx
inoutthrough
.ofConjugateHarmonicbesaidisThen uv
7/21/2019 Lecture 26 (1)
6/22
BITS Pilani, Pilani Campus
.ofconjugateharmonicais
ofconjugateharmonicais
1:Remark
vu
uv
(1)assamenotiswhich,&then
,ofconjugateharmonicaisifFor,
xyyx uvuv
vu
7/21/2019 Lecture 26 (1)
7/22BITS Pilani, Pilani Campus
vu
uv
-ofconjugateharmonicais
ofconjugateharmonicais
:2Remark
)1(
&..
,
assameiswhich
-as
xyyx
xyyx
vuvuei
uvuv
7/21/2019 Lecture 26 (1)
8/22BITS Pilani, Pilani Campus
.ofconjugateharmonicaisiffdomainainanalyticis
functionA
:2Theorem
uvD
x,yi vx,yuzf )()()(
.)(. 2
zzfEx
7/21/2019 Lecture 26 (1)
9/22BITS Pilani, Pilani Campus
Ex. Find all the points where the function
analytic.is)(2)( 22 yxixyxf
7/21/2019 Lecture 26 (1)
10/22BITS Pilani, Pilani Campus
.inconstantis
.inanalyticis
.invaluedrealis
ifinconstantbemustthatProve
.domainainanalyticbeLet
Q.7Page
Dzfc
Dzfb
Dzzfa
Dzf
Dzf
,
)()(
)()(
)()(
)(
)(
78
7/21/2019 Lecture 26 (1)
11/22BITS Pilani, Pilani Campus
)2()(
)1(,
)(
xx
xyyx
viuzf
vuvu
Dzf
and
.domainainanalyticisSince
:Solution
Dzzfa functionvaluedrealaisGiven )()(
7/21/2019 Lecture 26 (1)
12/22BITS Pilani, Pilani Campus
.
where
Dx,yx,yv
,x,yi vx,yuzf
)(0)(
)()()(
Dyxyxuyxu
vuvu
vv
yx
xyyx
yx
),(),(0),(
,
0,0
.constant Dzzf
Dzzf
)(
,0)()2(
7/21/2019 Lecture 26 (1)
13/22BITS Pilani, Pilani Campus
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.
7/21/2019 Lecture 26 (1)
14/22BITS Pilani, Pilani Campus
)-(),(
whenconjugateharmonic
afind&harmonicisthatShowQ.10
yxyxua
v
u
12)(
harmonic.isu
uu
uxu
uyu
yyxx
yyy
xxx
0
0,2
0),1(2
7/21/2019 Lecture 26 (1)
15/22BITS Pilani, Pilani Campus
uv ofconjugateharmonicais
xyyx vuvu
,i.e.
satisfiedareEquationsCR
)1(2 yuv xy
Then
)(2 2
xyyv
7/21/2019 Lecture 26 (1)
16/22BITS Pilani, Pilani Campus
xuxv yx 2)(
cxyyv
cxx
xx
22
2
2
2)(
)(
7/21/2019 Lecture 26 (1)
17/22BITS Pilani, Pilani Campus
yxyxub sinsinh),()( ,sincosh yxux
,sinsinh yxuxx
,cossinh yxuy
yxuyy sinsinh
0 yyxx uu
7/21/2019 Lecture 26 (1)
18/22BITS Pilani, Pilani Campus
xyyx vuvuuv
,ofconjugateharmonicabeLet
yxvy sincosh
)(coscosh xyxv
)(cossinh xyxvx
7/21/2019 Lecture 26 (1)
19/22BITS Pilani, Pilani Campus
cyxv
cx
x
yxuv
xyxv
yx
x
coscosh
)(
0)(
cossinh
)(cossinh
But
7/21/2019 Lecture 26 (1)
20/22BITS Pilani, Pilani Campus
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.
7/21/2019 Lecture 26 (1)
21/22BITS Pilani, Pilani Campus
.ofconjugateharmonic
afindIfQ.
uv
yx
xyxu ,),(
22
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22