(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.

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# Unable to show that functions are harmonic.

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