Definition:Exponential Function/Complex/Power Series Expansion

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:

\(\ds \forall z \in \C: \, \)

\(\ds \exp z\)

\(=\)

\(\ds \sum_{n \mathop = 0}^\infty \frac {z^n} {n!}\)

\(\ds \)

\(=\)

\(\ds 1 + \frac z {1!} + \frac {z^2} {2!} + \frac {z^3} {3!} + \cdots + \frac {z^n} {n!} + \cdots\)

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

Exponential Series

The power series expansion of the exponential function:

$\map \exp z = 1 + \dfrac z {1!} + \dfrac {z^2} {2!} + \dfrac {z^3} {3!} + \cdots + \dfrac {z^n} {n!} + \cdots$

is known as the exponential series.

Also see

• Equivalence of Definitions of Complex Exponential Function

• Results about the exponential function can be found here.

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations
• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.1$. Introduction: $(4.1)$
• 2001: Christian Berg: Kompleks funktionsteori: $\S 1.5$