Complex Exponential

Definition¹

The complex exponential of a complex number $z \in \C $ is the complex number

\[ 1 + z + \frac{z^2}{2} + \frac{z^3}{3!} + \cdots = \sum_{n = 1}^{\infty}\frac{z^n}{n!}. \]

Notation

We denote the complex exponential funciton $\exp: \C \to \C $, and so denote the complex exponential of $z \in \C $ by $\exp(z)$, as usual. This overloaded notation is justified by the fact that the complex exponential agrees with the real exponential function on reals. It is also common to denote $\exp(z)$ by $e^z$.

Relation to power series of $\sin$ and $\cos$

It can be shown that

\[ \exp(ix) = \cos(x) + i \sin(x) \quad \text{for all } x \in \R \]

1. Future editions may modify this sheet, as there are several (equivalent) characterizations of the complex exponential function. ↩︎