# Definition:Holomorphic Function

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## Definition

### Complex Function

Let $U \subseteq \C$ be an open set.

Let $f : U \to \C$ be a complex function.

Then $f$ is **holomorphic in $U$** if and only if $f$ is differentiable at each point of $U$.

We also say that $f$ is **complex-differentiable in $U$**.

### Vector-Valued Function

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## Also defined as

A **holomorphic function** is sometimes defined as continuously differentiable.

By Holomorphic Function is Continuously Differentiable, the two are equivalent.

## Also known as

Some authors refer to a **holomorphic function** on an open set of $\C$ as a **(complex) analytic function**.

This is because, by Holomorphic Function is Analytic, they are equivalent.