(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

If the functions [tex]u(x,y)[/tex] and [tex]v(x,y)[/tex] have continuous second partial derivatives and they satisfy the Cauchy-Riemann equations. Show that [tex]u(x,y)[/tex] and [tex]v(x,y)[/tex] are harmonic functions.

2. Relevant equations

The Cauchy-Riemann equations are given: [tex]\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}[/tex] and [tex]\frac{\partial v}{\partial x} = -\frac{\partial u}{\partial y}[/tex]

And functions are harmonic if [tex]\frac{\partial^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2} = 0[/tex] (Laplace equation)

3. The attempt at a solution

I've been stuck with this one for couple of hours now, and I really can't get much out of it. The only thing that's gone through my head is to differentiate the Cauchy-Riemann equations once more and trying to arrange the terms so that the Laplace equation is satisfied. But to no avail.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Unable to show that functions are harmonic.

**Physics Forums | Science Articles, Homework Help, Discussion**