(adsbygoogle = window.adsbygoogle || []).push({}); 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

[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.

2. Relevant 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]

3. 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.

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# Homework Help: Cauchy-Riemann conditions

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