Definition:Exponential Function/Complex/Real Functions

Definition
Let $\exp: \C \to \C \setminus \set 0$ denote the (complex) exponential function. The exponential function can be defined by the real exponential, sine and cosine functions:


 * $\exp z := e^x \paren {\cos y + i \sin y}$

where $z = x + i y$ with $x, y \in \R$.

Here, $e^x$ denotes the real exponential function, which must be defined first.

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

Also see

 * Equivalence of Definitions of Complex Exponential Function