\begin{definition}[Definition:Exponential Function/Complex/Real Functions]
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 $z$'''.
\end{definition}