Definition:Exponential Function/Complex/Power Series Expansion

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Definition

Let $\exp: \C \to \C \setminus \set 0$ denote the (complex) exponential function.

The exponential function can be defined as a (complex) power series:

\(\ds \exp z\) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \frac {z^n} {n!}\)
\(\ds \) \(=\) \(\ds 1 + \frac z {1!} + \frac {z^2} {2!} + \frac {z^3} {3!} + \cdots + \frac {z^n} {n!} + \cdots\)


The complex number $\exp z$ is called the exponential of $z$.


Also see


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