- #1

Kruum

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## Homework Statement

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.

## Homework 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)

## 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.