# 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.

## Also see

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

- Results about
**complex-valued functions**can be found**here**.

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