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 known as

A polynomial function is often simply called polynomial.

Some sources refer to it as a rational integral function.

Also see

• Equivalence of Definitions of Complex Polynomial Function

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: The Elementary Functions: $1$