Definition:Holomorphic Function/Complex Plane
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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.
- 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