Complex Function is Entire iff it has Everywhere Convergent Power Series

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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:

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


Proof