# Definition:Rational Function/Complex

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## 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**.

## Sources

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