Definition:Exponential Function/Complex/Real Functions

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


 * $\exp z := e^x \left({ \cos \left({y}\right) + i \sin \left({y}\right) }\right)$

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

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

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