Complex Exponential Function is Entire

Theorem
Let $\exp: \C \to \C$ be the complex exponential function.

Then $\exp$ is entire.

Proof
By the definition of the complex exponential function, $\exp$ admits a power series expansion about $0$:


 * $\displaystyle \exp z = \sum_{n \mathop = 0}^\infty \frac {z^n} {n!}$

By Complex Function is Entire iff it has Everywhere Convergent Power Series, to show that $\exp$ is entire it suffices to show that this series is everywhere convergent.

Note that this power series is of the form:


 * $\displaystyle \sum_{n \mathop = 0}^\infty \frac {\paren {z - \xi}^n} {n!}$

with $\xi = 0$.

Therefore, by Radius of Convergence of Power Series over Factorial: Complex Case, we have that the former power series is everywhere convergent.

Hence the result.