Definition:Complex-Valued Function

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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.