if Ddomain given ain harmonic be tosaid is y)u(x,function valuedrealA
D,in continuous are they &exist u & u,u ,u (i) yyyxxx
0u uueqution Laplace satisfiesu (ii)
yyxx2
:Function Harmonic25Section
Din harmonic are v&uthen D,domain ain analytic
is y) v(x,i y)u(x, f(z) If :1 Theorem
)2.(.........., )1(.......,
,Ddomain ain analytic is
y) v(x,i y)u(x, f(z) :Proof
xyyyyyxy
xxyxyxxx
xyyx
vuvuvuvu
Dinthroughoutvuvu
Use the results:
(i) f(z) is analytic at a point,
then Re f(z) & Im f(z) have continuous partial
derivatives of all orders at that point.
(ii) Continuity of partial derivatives of u & v
uxy = uyx, vxy = vyx
uxx+ uyy= 0 & vxx +vyy= 0 ( from 1 and 2)
Hence proved.
u. of Conjugate Harmonic be said is Then v
)1(...........,u
equations. CRsatisfy sderivative
partialfirst thereand Ddomain ain functions
harmonic twobe vandu Let :Conjugate Harmonic
:
Dinoutthroughvuv
Definition
xyyx
(1) as samenot iswhich &
then, of conjugate harmonic a is if For,
. of conjugate harmonic a is imply not oes of conjugate harmonic a is v
:1markRe
xyyx uvuvvu
vudu
(1) as same iswhich &..,- as
- of conjugate harmonic a is of conjugate harmonic a is
:2Remark
xyyx
xyyx
vuvueiuvuv
vuuv
u. of conjugate harmonic a is viff Ddomain ain analytic is
y) v(x,i y)u(x, f(z)function a If
:2 Theorem
D.in constant is f(z) (c)D.in analytic is f(z) (b)
D.in z valuedreal is f(z) (a)if
Din constant bemust f(z) that Prove D.domain ain analytic be f(z)Let
Q.7 74, Page
D.y)(x, 0y)v(x,
y),(x, i y)(x,u f(z)
D z function valuedreal a is f(z)Given (a)
)2()(&
)1(,
D.domain ain analytic is f(z)Since
:Solution
v
viuzf
vuvu
xx
xyyx
D. zconstant )(
0)()2(
),(),(0),(
,
0,0
0 y)v(x,
zf
Dzzf
Dyxyxuyxu
vuvu
vv
yx
xyyx
yx
)3(.......)(,
.,RCsatisfy v- &u
Din analytic isf(z)
),(),(f(z)
),( i),( f(z)
)(
xxyyx vvuvu
vizequations
yxviyxu
yxvyxu
b
(1) and (3) ux = vy, ux = -vy
ux = 0
uy = -vx, uy = vx
vx = 0
f ’(z) = ux + i vx = 0 z D
f(z) constant z D
.tan)(.0)(,0
f(z)Let Din constant f(z)
)(
DztconszfDzzfthencIf
c
c
Din constant is f(z) :(b) caseby Din analytic is)(
Din analytic is)()(
)(
)().(f(z)
0.cf(z),.0 c Assume
2
222
zfzf
zf
czf
czfzfc
Then
harmonic. is00,2
0),1(2)-(1 2 ),( (a) when conjugate harmonic
a find & harmonic is that Show Q.10
uuu
uxuuyuyxyxuv
u
yyxx
yyy
xxx
xuxvxyyv
yuvNowvuvu
uv
yx
xy
xyyx
2)()(2
)1(2, i.e.
satisfied are Equations CR of conjugate harmonic a is
2
cxyyv
cxx
xx
22
2
2
)(
2)(
0
sinsinh,cos.sinhsinsinh,sincosh
ysin sinh),()(
yyxx
yyy
xxx
uu
yxuyxuyxuyxu
xyxub
cyxvcx
xyxu
xyxvxyv
yxvvuvu
uv
y
x
y
xyyx
cos.cosh)(
0)(cos.sinh
)(cossinh)(cos.cosh
sincosh,
of conjugate harmonic a be Let
Problem:
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.
)2(, conjugate harmonic
)1(, u of conjugate harmonic a is v
:Solution
xyyx
xyyx
vuvuuofaisv
vuvu
constantVv
cψ(x),cφ(y)
0(x)ψ0,(y)φ
(x)ψVv(y),φVv
ψ(x)Vvφ(y),Vv
Vv,Vv(2)&(1)
21
xxyy
yyxx
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