Show that $f(z) = z^n$ is an entire function

Question

Show that $f(z) = z^n$ is an entire function where $n = 0,1,2...$

I can show that $z$, $z^2$ and $z^3$ are entire functions but what about $z^n$ how do you crack that one.

Answer 1

Hint: the product of two entire function is entire (product rule).

Now use induction for your problem.

Answer 2

Hint. For $z \neq z_0$ and $n \ge 2$, we have

$$\frac{z^n - z_0^n}{z - z_0} = \sum_{k = 0}^{n-1}z^{n-1-k}z_0^k.$$