Editing Holomorphic function (section)

== Terminology ==
The term ''holomorphic'' was introduced in 1875 by [[Charles Auguste Briot|Charles Briot]] and [[Jean-Claude Bouquet]], two of [[Augustin-Louis Cauchy]]'s students, and derives from the Greek [[wikt:ὅλος|ὅλος]] (''hólos'') meaning "whole", and [[wikt:μορφή|μορφή]] (''morphḗ'') meaning "form" or "appearance" or "type", in contrast to the term ''[[meromorphic function|meromorphic]]'' derived from [[wikt:μέρος|μέρος]] (''méros'') meaning "part". A holomorphic function resembles an [[entire function]] ("whole") in a [[domain (mathematical analysis)|domain]] of the complex plane while a meromorphic function (defined to mean holomorphic except at certain isolated [[Zeros and poles|poles]]), resembles a rational fraction ("part") of entire functions in a domain of the complex plane.<ref>The original French terms were ''holomorphe'' and ''méromorphe''. {{pb}}
{{cite book |authorlink1= Charles Auguste Briot |last1=Briot |first1=Charles Auguste |authorlink2=Jean-Claude Bouquet |last2=Bouquet |first2=Jean-Claude |date=1875 |title=Théorie des fonctions elliptiques |edition=2nd |publisher=Gauthier-Villars |chapter=§15 fonctions holomorphes |chapter-url=https://archive.org/details/thoriedesfonct00briouoft/page/14/ |pages=14–15 |quote=Lorsqu'une fonction est continue, monotrope, et a une dérivée, quand la variable se meut dans une certaine partie du plan, nous dirons qu'elle est ''holomorphe'' dans cette partie du plan. Nous indiquons par cette dénomination qu'elle est semblable aux fonctions entières qui jouissent de ces propriétés dans toute l'étendue du plan. [...] ¶ Une fraction rationnelle admet comme pôles les racines du dénominateur; c'est une fonction holomorphe dans toute partie du plan qui ne contient aucun de ses pôles. ¶ Lorsqu'une fonction est holomorphe dans une partie du plan, excepté en certains pôles, nous dirons qu'elle est ''méromorphe'' dans cette partie du plan, c'est-à-dire semblable aux fractions rationnelles. |trans-quote=When a function is continuous, [[Monodromy|monotropic]], and has a derivative, when the variable moves in a certain part of the [complex] plane, we say that it is ''holomorphic'' in that part of the plane. We mean by this name that it resembles [[entire function]]s which enjoy these properties in the full extent of the plane.&nbsp;[...] ¶ A rational fraction admits as [[zeros and poles|poles]] the [[zeros and poles|roots]] of the denominator; it is a holomorphic function in all that part of the plane which does not contain any poles. ¶ When a function is holomorphic in part of the plane, except at certain poles, we say that it is ''meromorphic'' in that part of the plane, that is to say it resembles rational fractions.}} {{pb}} {{cite book |authorlink1=James Harkness (mathematician) |first1=James |last1=Harkness |authorlink2=Frank Morley |first2=Frank |last2=Morley |date=1893 |chapter=5. Integration |chapter-url=https://archive.org/details/treatiseontheory00harkrich/page/n176/ |title=A Treatise on the Theory of Functions |publisher=Macmillan |page=161}}</ref> Cauchy had instead used the term ''synectic''.<ref>Briot & Bouquet had previously also adopted Cauchy’s term ''synectic'' (''synectique'' in French), in the 1859 first edition of their book. {{pb}} {{cite book |authorlink1= Charles Auguste Briot |last1=Briot |first1=Charles Auguste |authorlink2=Jean-Claude Bouquet |last2=Bouquet |first2=Jean-Claude |date=1859 |title= Théorie des fonctions doublement périodiques |publisher= Mallet-Bachelier |chapter=§10 |chapter-url=https://archive.org/details/fonctsdoublement00briorich/page/n37/ |page=11 }}</ref>

Today, the term "holomorphic function" is sometimes preferred to "analytic function". An important result in complex analysis is that every holomorphic function is complex analytic, a fact that does not follow obviously from the definitions. The term "analytic" is however also in wide use.