Is this an entire function?

Question

Is $f(z)=|z^k|$ [where k is an odd natural number] an entire function?

I can see that for k=1 it is not.(It is not even real differentiable) But what about let's say k=3?

Answer 1

The only real valued entire functions are constants as you can see from C - R equations.

Answer 2

The derivative of $|z^k| = |z|^k$ "should be" $$ k |z|^{k-1}\;\frac{d}{dz} |z| . $$ At any point where $|z|\ne 0$, use the fact that $|z|$ is not differentiable to conclude that $|z|^k$ is also not differentiable.