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.