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

  • Thread starter Mmmm
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1. Homework Statement
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. Homework 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...
 

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