Holomorphic

Part of speech: adjective

Definitions

1. A mathematical function can be labeled in this way if it is differentiable in the complex sense across an entire neighborhood surrounding each point in its domain
2. This term applies to functions that maintain complex differentiability throughout every point in a specified region within their domain
3. A function is referred to by this term when it exhibits complex differentiability at every point within a given area of its domain

Etymology: "Holomorphic" is a compound built from two Greek-derived elements: "holo-", from Greek "holos" meaning "whole" or "entire", and "-morphic", from Greek "morphe" meaning "form" or "shape", giving roughly "wholly formed" or "entire in form." The word is a technical coinage from mathematics, so the classical roots describe the idea rather than reflecting any ancient usage of the term itself. The "whole" in holomorphic points to the mathematical property it names. A function that is complex-differentiable at every point across a region is, in effect, fully characterised by its behaviour there, and the prefix signals that this property holds throughout the entire area rather than at isolated points. The term therefore distinguishes a region-wide property from the weaker, local condition of being differentiable at a single point.

Synonyms: analytic, complex, smooth, differentiable, continuous