Definition:Complex Function

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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.


Sources

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