Complex Exponential Function is Entire
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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 $0$:
- $\ds \exp z = \sum_{n \mathop = 0}^\infty \frac {z^n} {n!}$
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:
- $\ds \sum_{n \mathop = 0}^\infty \frac {\paren {z - \xi}^n} {n!}$
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.
$\blacksquare$
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): entire function
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): entire function