## Cauchy-Riemann conditions

1. The problem statement, all variables and given/known data
The functions u(x,y) and v(x,y) are the real and imaginary parts, respectively, of an analytic function w(z).
Assuming that the required derivatives exist, show that

$$\bigtriangledown^2 u=\bigtriangledown^2 v=0$$

Solutions of Laplace's equation such as u(x,y) and v(x,y) are called harmonic functions.

2. Relevant equations
Cauchy-Riemann conditions:

$$\frac{\delta u}{\delta x} = \frac{\delta v}{\delta y}$$
$$\frac{\delta u}{\delta y} = -\frac{\delta v}{\delta x}$$

3. The attempt at a solution
I expanded $$\bigtriangledown^2 u = \frac{\delta u}{\delta x}\frac{\delta u}{\delta x} + \frac{\delta u}{\delta y}\frac{\delta u}{\delta y}$$ and using the Cauchy-Riemann conditions I found

$$\bigtriangledown^2 u = \frac{\delta v}{\delta y}\frac{\delta v}{\delta y} + \frac{\delta v}{\delta x}\frac{\delta v}{\delta x}=\bigtriangledown^2 v$$

What I can't figure out how to do is prove that this is equal to zero.
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 Recognitions: Homework Help Science Advisor Your eqs. for del^2 are wrong. $$\nabla^2 u=\partial_x\partial_x u+\partial_y\partial_y u.$$
 Dang, you're right. Can I dot it into an element of length like this? $$\bigtriangledown^2 u \bullet d\vec{r}^2 = \frac{\delta}{\delta x}\frac{\delta u}{\delta x} dx^2 + \frac{\delta}{\delta y}\frac{\delta u}{\delta y} dy^2$$

## Cauchy-Riemann conditions

There is a hint in the problem that says I need to construct vectors normal to the curves $$u(x,y)=c_i$$ and $$v(x,y)=c_j$$. Wow, I'm pretty lost.
 Recognitions: Gold Member Science Advisor Staff Emeritus The Cauchy-Riemann equations are $$\frac{\partial u}{\partial x}= \frac{\partial v}{\partial y}$$ $$\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$$ which is what you have, allowing for your peculiar use of $\delta$ rather than $\partial$! Now just do the obvious: differentiate both sides of the first equation with respect to x and differentiate both sides of the second equation with respect to y and compare them. Are you sure that the hint is for this particular problem? A normal vector to u(x,y)= c is $$\frac{\partial u}{\partial x}\vec{i}+ \frac{\partial u}{\partial y}\vec{j}$$ and a normal vector to v(x,y)= c is $$\frac{\partial v}{\partial x}\vec{i}+ \frac{\partial v}{\partial y}\vec{j}$$. Using the Cauchy-Riemann equations, that second equation is $$-\frac{\partial u}{\partial y}\vec{i}+ \frac{\partial u}{\partial x}\vec{j}$$ which tells us the the two families of curves are orthogonal but that does not directly tell us about $\nabla^2 u$ and $\nabla^2 v$.
 Thanks for your reply. There is a part b) to the problem, and it is this: b) Show that $$\frac{\partial u}{\partial x}\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}\frac{\partial v}{\partial y} = 0$$ I solved it easily using the Cauchy-Riemann equations, so I figured that the hint was for the first part.