Harmonic conjugates

  • #1

Homework Statement


If [itex]w = u(x,y)+iy(x,y)[/itex] is an analytic function then
[itex]\phi(x,y) = u(x,y)v(x,y)[/itex]
is harmonic, where u and v are the real and imaginary parts of w.
What is the harmonic conjugate of [itex]\phi[/itex]?

Homework Equations


So I know for analytic functions the Cauchy-Riemann equations:
[itex]\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}[/itex] and
[itex]\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}[/itex]

And for a harmonic function:
[itex]\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}=0[/itex] and
[itex]\frac{\partial^2v}{\partial x^2} + \frac{\partial^2v}{\partial y^2}=0[/itex]

The Attempt at a Solution


I tried to find a function [itex]\Phi[/itex] that would satisfy:
[itex]\frac{\partial \Phi}{\partial dy}=\frac{\partial \phi}{\partial dx}[/itex] and
[itex]\frac{\partial \Phi}{\partial dx}=-\frac{\partial \phi}{\partial dy}[/itex]

for which I obtained:
[itex]\Phi = \int \frac{\partial u}{\partial x} v + \frac{\partial v}{\partial x} u \partial y[/itex] and
[itex]\Phi = \int \frac{\partial u}{\partial y} v + \frac{\partial v}{\partial y} u \partial x[/itex][/quote]

But I have no idea if this is correct, or relevant to finding the harmonic conjugate.

Any help is greatly appreciated.
 
Last edited:

Answers and Replies

  • #2
lurflurf
Homework Helper
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consider w2
 
  • #3
consider w2
Thanks for this lurflurf. So I get
[itex]w^2 = u^2 - v^2[/itex]

I really have no idea what to do with this, could you kindly give me another little hint?
 
Last edited:
  • #4
lurflurf
Homework Helper
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132
That is not w2 that is Re[w2] what about Im[w2]?
alternatively try to write
(uv)x=uxv+uvx
(uv)y=uyv+uvy
in terms of oposite partials via Cauchy-Riemann equations
 
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  • #5
I'm confused by this. I'm trying to understand, but I don't know how to get any expression for the conjugate.
 
  • #6
lurflurf
Homework Helper
2,432
132
If cannot guess the conjugate what you started doing with integrals would also work, the next step is to use the Cauchy-Riemann equations to that your integrals and partials are with respect to the same variables thus giving an answer without partials.
 
  • #7
1,796
53
May I make a suggestion: If f=u+v is analytic, then u and v are harmonic and so satisfy Laplace's equation each of them. That is, u_xx+u_yy=0, and v_xx+v_yy=0 and also if f is analytic then u and v satisfy the Cauchy Riemann equations. Well there you go: For:

[tex]g=uv[/tex]

compute:

[tex]\frac{\partial^2 g}{\partial x^2}+\frac{\partial^2 g}{\partial y^2}[/tex]

using the chain rule, and use what I've just said to show it's zero.
 

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