# Homework Help: Finding the conjugate harmonic of a function u(x,y)

1. Mar 15, 2010

### Mmmm

1. The problem statement, all variables and given/known data
Let u(x,y) be harmonic in a simply connected domain $\Omega$. Use the Cauchy-Riemann equations to obtain the formula for the conjugate harmonic

$$v(x,y)=\int^{(x,y)}_{x_0,y_0} (u_xdy-u_ydx)$$

where $(x_0,y_0)$ is any fixed point of $\Omega$ and the integration is along any path in $\Omega$ joining $(x_0,y_0)$ and (x,y).

2. Relevant equations

Cauchy Riemann eqns

$$u_x=v_y, u_y=v_x$$

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?

$$u_x=v_y$$
$$\Rightarrow v(x,y) = \int^y_{y_0} u_xdy$$
$$v(x,y) = \int^x_{x_0} u_ydx$$

Differentiating each wrt the other variable in an attempt to link the two eqns doesnt seem to get me anywhere...