Definition:Exponential Function/Complex

Definition
The complex exponential function is denoted $\exp$ and can be defined in several ways.

For all definitions of the complex exponential function:


 * The domain of $\exp$ is $\C$.


 * The codomain of $\exp$ is $\C \setminus \left\{ {0}\right\}$, as shown in Image of Complex Exponential Function.

For $z \in \C$, the complex number $\exp z$ is called the exponential of $z$.

Equivalence of Definitions
As shown in Equivalence of Definitions of Complex Exponential the definitions above are equivalent.

Notation
The exponential of $z$, $\exp z$, is frequently written as $e^z$, as in the case of the real exponential.

Also see

 * Basic Properties of Complex Exponential Function