# Harmonic conjugates

## 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.

Last edited:

lurflurf
Homework Helper
consider w2

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?

Last edited:
lurflurf
Homework Helper
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

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

lurflurf
Homework Helper
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.

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.