complex6_ harmonic function & conjugate

23
if D domain given a in harmonic be to said is y) u(x, function valued real A D, in continuous are they & exist u & u , u , u (i) yy y xx x 0 u u u equtio Laplace satisfies u (ii) yy xx 2 : Functio Harmonic 25 Section

Transcript of complex6_ harmonic function & conjugate

Page 1: complex6_ harmonic function & conjugate

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

Page 2: complex6_ harmonic function & conjugate

Din harmonic are v&uthen D,domain ain analytic

is y) v(x,i y)u(x, f(z) If :1 Theorem

Page 3: complex6_ harmonic function & conjugate

)2.(.........., )1(.......,

,Ddomain ain analytic is

y) v(x,i y)u(x, f(z) :Proof

xyyyyyxy

xxyxyxxx

xyyx

vuvuvuvu

Dinthroughoutvuvu

Page 4: complex6_ harmonic function & conjugate

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.

Page 5: complex6_ harmonic function & conjugate

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

Page 6: complex6_ harmonic function & conjugate

(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

Page 7: complex6_ harmonic function & conjugate

(1) as same iswhich &..,- as

- of conjugate harmonic a is of conjugate harmonic a is

:2Remark

xyyx

xyyx

vuvueiuvuv

vuuv

Page 8: complex6_ harmonic function & conjugate

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

Page 9: complex6_ harmonic function & conjugate

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

Page 10: complex6_ harmonic function & conjugate

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

Page 11: complex6_ harmonic function & conjugate

D. zconstant )(

0)()2(

),(),(0),(

,

0,0

0 y)v(x,

zf

Dzzf

Dyxyxuyxu

vuvu

vv

yx

xyyx

yx

Page 12: complex6_ harmonic function & conjugate

)3(.......)(,

.,RCsatisfy v- &u

Din analytic isf(z)

),(),(f(z)

),( i),( f(z)

)(

xxyyx vvuvu

vizequations

yxviyxu

yxvyxu

b

Page 13: complex6_ harmonic function & conjugate

(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

Page 14: complex6_ harmonic function & conjugate

.tan)(.0)(,0

f(z)Let Din constant f(z)

)(

DztconszfDzzfthencIf

c

c

Page 15: complex6_ harmonic function & conjugate

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

Page 16: complex6_ harmonic function & conjugate

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

Page 17: complex6_ harmonic function & conjugate

xuxvxyyv

yuvNowvuvu

uv

yx

xy

xyyx

2)()(2

)1(2, i.e.

satisfied are Equations CR of conjugate harmonic a is

2

Page 18: complex6_ harmonic function & conjugate

cxyyv

cxx

xx

22

2

2

)(

2)(

Page 19: complex6_ harmonic function & conjugate

0

sinsinh,cos.sinhsinsinh,sincosh

ysin sinh),()(

yyxx

yyy

xxx

uu

yxuyxuyxuyxu

xyxub

Page 20: complex6_ harmonic function & conjugate

cyxvcx

xyxu

xyxvxyv

yxvvuvu

uv

y

x

y

xyyx

cos.cosh)(

0)(cos.sinh

)(cossinh)(cos.cosh

sincosh,

of conjugate harmonic a be Let

Page 21: complex6_ harmonic function & conjugate

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.

Page 22: complex6_ harmonic function & conjugate

)2(, conjugate harmonic

)1(, u of conjugate harmonic a is v

:Solution

xyyx

xyyx

vuvuuofaisv

vuvu

Page 23: complex6_ harmonic function & conjugate

constantVv

cψ(x),cφ(y)

0(x)ψ0,(y)φ

(x)ψVv(y),φVv

ψ(x)Vvφ(y),Vv

Vv,Vv(2)&(1)

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xxyy

yyxx