# Definition:Complex Function

## Definition

A complex function is a function whose domain and codomain are subsets of the set of complex numbers $\C$.

### Independent Variable

Let $f: \C \to \C$ be a complex function.

Let $\map f z = w$.

Then $z$ is referred to as an independent variable (of $f$).

### Dependent Variable

Let $f: \C \to \C$ be a complex function.

Let $\map f z = w$.

Then $w$ is referred to as the dependent variable (of $f$).

## Examples

### Square Function

Let $f: \C \to \C$ be the function defined as:

$\forall z \in \C: \map f z = z^2$

This is a complex function.

### Imaginary Part

Let $f: \C \to \C$ be the function defined as:

$\forall z \in \C: \map f z = \map \Im z$

where $\map \Im z$ denotes the imaginary part of $z$.

$f$ is a complex function whose image is the set of real numbers $\R$.

### Principal Argument

Let $f: \C \to \C$ be the function defined as:

$\forall z \in \C: \map f z = \Arg z$

where $\Arg z$ denotes the principal argument of $z$.

$f$ is a complex function whose image is the set of real numbers $\R$.

## Also see

• Results about complex functions can be found here.