- #1

- 125

- 1

## Homework Statement

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

[tex]\bigtriangledown^2 u=\bigtriangledown^2 v=0[/tex]

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

## Homework Equations

Cauchy-Riemann conditions:

[tex]\frac{\delta u}{\delta x} = \frac{\delta v}{\delta y}[/tex]

[tex]\frac{\delta u}{\delta y} = -\frac{\delta v}{\delta x}[/tex]

## The Attempt at a Solution

I expanded [tex]\bigtriangledown^2 u = \frac{\delta u}{\delta x}\frac{\delta u}{\delta x} + \frac{\delta u}{\delta y}\frac{\delta u}{\delta y}[/tex] and using the Cauchy-Riemann conditions I found

[tex]\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[/tex]

What I can't figure out how to do is prove that this is equal to zero.