# Definition:Holomorphic Function/Complex Plane

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## 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 $U$**.

## 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** as a **(complex) analytic function**.

By Holomorphic Function is Analytic, the two are equivalent.

## Sources

- 1964: Murray R. Spiegel:
*Theory and Problems of Complex Variables*: Chapter $3$: Complex Differentiation and The Cauchy-Riemann Equations: Analytic Functions - 2001: Christian Berg:
*Kompleks funktionsteori*: $\S 1.1$