\begin{definition}[Definition:Complex Transformation]
A '''complex transformation''' is a mapping on the complex plane $f: \C \to \C$ which is specifically ''not'' a multifunction.
Let $z = x + i y$ be a complex variable.
Let $w = u + i v = \map f z$.
Then $w$ can be expressed as:
:$u + i v = \map f {x + i y}$
such that:
:$u = \map u {x, y}$
and:
:$v = \map v {x, y}$
are real functions of two variables.
Thus a point $P = \tuple {x, y}$ in the complex plane is '''transformed''' to a point $P' = \tuple {\map u {x, y}, \map v {x, y} }$ by $f$.
Thus $P'$ is the image of $P$ under $f$.
\end{definition}