# Harmonic conjugates

1. Oct 30, 2011

### jonmondalson

1. The problem statement, all variables and given/known data
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 $\phi$?

2. Relevant 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
$\frac{\partial^2v}{\partial x^2} + \frac{\partial^2v}{\partial y^2}=0$

3. 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.

Last edited: Oct 30, 2011
2. Oct 30, 2011

### lurflurf

consider w2

3. Oct 30, 2011

### jonmondalson

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?

Last edited: Oct 30, 2011
4. Oct 30, 2011

### lurflurf

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
5. Oct 30, 2011

### jonmondalson

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

6. Nov 1, 2011

### 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.

7. Nov 1, 2011

### 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.