# Definition:Holomorphic Function

## Contents

## Definition

### Complex Function

Let $U\subset\C$ 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 $U$.

### Vector-Valued Function

## Also defined as

A **holomorphic function** is sometimes defined as continuously differentiable. By Goursat's Theorem, 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.

## Also see

## Sources

- 1964: Murray R. Spiegel:
*Theory and Problems of Complex Variables*: Chapter $1$: Complex Differentiation and The Cauchy-Riemann Equations: $\S$ Analytic Functions