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 $U$.

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: $\S 1.1$
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): holomorphic