# Definition:Holomorphic Function/Complex Plane

## Definition

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 $U$**.

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## Also defined as

A **holomorphic function** is sometimes defined as continuously differentiable on the open set $U$.

By Holomorphic Function is Continuously Differentiable, the two definitions 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.

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

## Also see

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

## Sources

- 1964: Murray R. Spiegel:
*Theory and Problems of Complex Variables*... (next): Chapter $3$: Complex Differentiation and The Cauchy-Riemann Equations: Analytic Functions - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**holomorphic** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**holomorphic function** - 2001: Christian Berg:
*Kompleks funktionsteori*: $\S 1.1$ - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**holomorphic function** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**holomorphic** - 2021: Richard Earl and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(6th ed.) ... (previous) ... (next):**holomorphic**