Definition:Complex Transformation

Definition
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 $w = f \left({z}\right)$.

Let $w = u + i v = f \left({z}\right)$.

Then $w$ can be expressed as:
 * $u + i v = f \left({x + i y}\right)$

such that:
 * $u = u \left({x, y}\right)$

and:
 * $v = v \left({x, y}\right)$

are real functions of two variables.

Thus a point $P = \left({x, y}\right)$ in the complex plane is transformed to a point $P' = \left({u \left({x, y}\right), v \left({x, y}\right)}\right)$ by $f$.

Thus $P'$ is the image of $P$ under $f$.

Also known as
When the context is clear, a complex transformation is often referred to as a transformation.

Some sources use the term mapping function, which borrows from the generic terminology.