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Harmonic conjugates

  1. Oct 30, 2011 #1
    1. The problem statement, all variables and given/known data
    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]?

    2. Relevant 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]

    3. 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: Oct 30, 2011
  2. jcsd
  3. Oct 30, 2011 #2

    lurflurf

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    consider w2
     
  4. Oct 30, 2011 #3
    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: Oct 30, 2011
  5. Oct 30, 2011 #4

    lurflurf

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    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
     
    Last edited: Oct 30, 2011
  6. Oct 30, 2011 #5
    I'm confused by this. I'm trying to understand, but I don't know how to get any expression for the conjugate.
     
  7. Nov 1, 2011 #6

    lurflurf

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    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.
     
  8. Nov 1, 2011 #7
    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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