Definition:Exponential Function/Complex/Real Functions

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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