Finding the Harmonic Conjugate of \phi

[jonmondalson]

Homework Statement

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

Homework Equations

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

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

The Attempt at a Solution

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

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

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

Any help is greatly appreciated.

 

[lurflurf]

consider w²

 

[jonmondalson]

consider w²

Thanks for this lurflurf. So I get
w^2 = u^2 - v^2

I really have no idea what to do with this, could you kindly give me another little hint?

 

[lurflurf]

That is not w² that is Re[w²] what about Im[w²]?
alternatively try to write
(uv)_x=u_xv+uv_x
(uv)_y=u_yv+uv_y
in terms of oposite partials via Cauchy-Riemann equations

 

[jonmondalson]

I'm confused by this. I'm trying to understand, but I don't know how to get any expression for the conjugate.

 

[lurflurf]

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.

 

[jackmell]

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:

g=uv

compute:

\frac{\partial^2 g}{\partial x^2}+\frac{\partial^2 g}{\partial y^2}

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