How would I go about using the Cauchy-Riemann equations to show that $\displaystyle f(z) = z^n$ is an entire function for any n using polar coordinates?
$\displaystyle f(z) = r^ne^{i\theta n} $
Cauchy-Riemann is $\displaystyle u_r = \frac{1}{r}v_{\theta} $ and $\displaystyle v_r = \frac{-1}{r}u_{\theta}$
I split f(z) up like so:
$\displaystyle u(r,\theta) = r^n \cos(\theta n) $
$\displaystyle v(r,\theta) = r^n \sin(\theta n) $
I'm stuck on what to do next... am I allowed to differentiate as I would over R?