(adsbygoogle = window.adsbygoogle || []).push({}); 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.

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# Harmonic conjugates

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