# Definition:Complex-Valued Function

## Definition

Let $f: S \to T$ be a function.

Let $S_1 \subseteq S$ such that $f \left({S_1}\right) \subseteq \C$.

Then $f$ is defined as **complex-valued on $S_1$**.

That is, $f$ is defined as **complex-valued on $S_1$** if the image of $S_1$ under $f$ lies entirely within the set of complex numbers $\C$.

A **complex-valued function** is a function $f: S \to \C$ whose codomain is the set of complex numbers $\C$.

That is $f$ is complex-valued iff it is complex-valued over its entire domain.

## Note

Compare real-valued function.

Note that as $\R \subseteq \C$, it is *technically* correct to refer to a real-valued function as complex-valued.