Definition:Polynomial Function/Complex/Definition 1

Definition
Let $S \subset \C$ be a subset of the complex numbers.

A complex polynomial function on $S$ is a function $f : S \to \C$ for which there exist:
 * a natural number $n \in \N$
 * complex numbers $a_0, \ldots, a_n \in \C$

such that for all $z \in S$:
 * $\map f z = \ds \sum_{k \mathop = 0}^n a_k z^k$

where $\ds \sum$ denotes indexed summation.

Also see

 * Equivalence of Definitions of Complex Polynomial Function