Unable to show that functions are harmonic.

[Kruum]

Homework Statement

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

Homework Equations

The Cauchy-Riemann equations are given: $$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$$ and $$\frac{\partial v}{\partial x} = -\frac{\partial u}{\partial y}$$

And functions are harmonic if $$\frac{\partial^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2} = 0$$ (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.

 

[Dick]

Are you forgetting that u,x,y=u,y,x (the commas indicate partial derivatives). I.e. the derivative is equal regardless of the order of differentiation.

 

[Kruum]

Are you forgetting that u,x,y=u,y,x (the commas indicate partial derivatives). I.e. the derivative is equal regardless of the order of differentiation.

No, I'm not. But you're tip doesn't ring a bell.

Edit: I'll take that back! Got it now! Thanks for the help! How can I be so stupid...

 

[Dick]

Differentiate the first equation with respect to x and the second one with respect to y. The first equation has u,xx in it and the second one has -u,yy. And they are equal to the same thing. So they are equal to each other.

 

[Kruum]

Differentiate the first equation with respect to x and the second one with respect to y. The first equation has u,xx in it and the second one has -u,yy. And they are equal to the same thing. So they are equal to each other.

I solved it just before you replied. I can't believe I couldn't see that. Thanks a bunch!