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Homework Help: Finding the conjugate harmonic of a function u(x,y)

  1. Mar 15, 2010 #1
    1. The problem statement, all variables and given/known data
    Let u(x,y) be harmonic in a simply connected domain [itex]\Omega[/itex]. Use the Cauchy-Riemann equations to obtain the formula for the conjugate harmonic

    [tex]v(x,y)=\int^{(x,y)}_{x_0,y_0} (u_xdy-u_ydx)[/tex]

    where [itex](x_0,y_0)[/itex] is any fixed point of [itex]\Omega[/itex] and the integration is along any path in [itex]\Omega[/itex] joining [itex](x_0,y_0)[/itex] and (x,y).

    2. Relevant equations

    Cauchy Riemann eqns

    [tex]u_x=v_y, u_y=v_x[/tex]

    3. The attempt at a solution

    At first this just looks like a simple bit of integration but for some reason I just cannot get the result. How do I get rid of the dependance on x and y of the constants of integration?

    [tex]u_x=v_y [/tex]
    [tex]\Rightarrow v(x,y) = \int^y_{y_0} u_xdy[/tex]
    [tex] v(x,y) = \int^x_{x_0} u_ydx[/tex]

    Differentiating each wrt the other variable in an attempt to link the two eqns doesnt seem to get me anywhere...
  2. jcsd
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