Definition:Rational Function/Complex

Definition
Let $P: \C \to \C$ and $Q: \C \to \C$ be polynomial functions on the set of complex numbers.

Let $S$ be the set $\C$ from which all the roots of $Q$ have been removed.

That is:
 * $S = \C \setminus \set {z \in \C: \map Q z = 0}$

Then the equation $y = \dfrac {\map P z} {\map Q z}$ defines a function from $S$ to $\C$.

Such a function is a rational (algebraic) function.

Also known as
Such a function is also known as a rational transformation.