Definition:Complex-Valued Function

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 range 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.