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:
\(\ds \forall z \in \C: \, \) | \(\ds \exp z\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac {z^n} {n!}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds 1 + \frac z {1!} + \frac {z^2} {2!} + \frac {z^3} {3!} + \cdots + \frac {z^n} {n!} + \cdots\) |
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
- 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: $\S 1.5$