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

#### Mmmm

1. Homework Statement
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. Homework 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...

Related Calculus and Beyond Homework Help News on Phys.org

"Finding the conjugate harmonic of a function u(x,y)"

### Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving