# 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 an **analytic (complex) function**.

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

This theorem requires a proof.In particular: holomorphic iff analyticYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

Sometimes the term **regular function** can be seen, which means the same thing.

## Also see

- Results about
**holomorphic functions**can be found**here**.