# Definition:Complex Transformation

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## 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