# Complex Function is Entire iff it has Everywhere Convergent Power Series

## Theorem

Let $f: \C \to \C$ be a complex function.

Then $f$ is an **entire function** if and only if $f$ can be given by an everywhere convergent power series:

- $\displaystyle f: \C \to \C: f \left({z}\right) = \sum_{n \mathop = 0}^\infty a_n z^n; \quad \lim_{n \mathop \to \infty} \sqrt [n] {\left|{a_n}\right|} = 0$