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 $U$.

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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 $U$ of $\C$ as an analytic (complex) function.

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

We also say that $f$ is complex-differentiable in $U$.

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