I have the following question which I am supposed to use the Cauchy-Riemann equations to prove:
Let u and v have continuous second derivatives satisfying:
$$\frac{\delta u}{\delta x} = \frac{\delta v}{\delta y} \\\frac{\delta u}{\delta y} = -\frac{\delta v}{\delta x}$$
Show that u and v are harmonic functions. (Recall that if a harmonic function f(x,y) satisfies $F_{xx}+F_{yy} = 0$).
I understand the given equations, I've read through the Wikipedia article about them, and I know what a harmonic function is, but I do not understand this question. It states that $u$ and $v$ have second derivatives, but my understanding of the Cauchy-Riemann equations is that they tell us when a function is continuously differentiable.
I am not looking for someone to do it for me, but if anyone can give a hint as to where to start, or clarify the hint in parenthesis (which does appear to be a sentence fragment) I would appreciate it. Thank you!
Since $u$ and $v$'s second derivative satisfies the Cauchy-Riemann equations, their first derivatives do too:
$$u_{x} = v_{y} \qquad u_{y} = -v_{x}$$
Now differentiate rearrange the equations and use the equality of mixed partials.