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 $w = u + i v = \map f z$.

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 $f$.

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.

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: Transformations