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In the theory of functions of [[Function of several complex variables|several complex variables]], '''Hartogs's extension theorem''' is a statement about the [[Singularity (mathematics)|singularities]] of [[holomorphic function]]s of several variables. Informally, it states that the [[Support (mathematics)|support]] of the singularities of such functions cannot be [[compact space|compact]], therefore the singular set of a function of several complex variables must (loosely speaking) 'go off to infinity' in some direction. More precisely, it shows that an [[isolated singularity]] is always a [[removable singularity]] for any [[analytic function]] of {{math|''n'' > 1}} complex variables. A first version of this theorem was proved by [[Friedrich Hartogs]],<ref name="hartogs">See the original paper of {{Harvtxt|Hartogs|1906}} and its description in various historical surveys by {{harvtxt|Osgood|1963|pp=56–59}}, {{harvtxt|Severi|1958|pp=111–115}} and {{harvtxt|Struppa|1988|pp=132–134}}. In particular, in this last reference on p. 132, the Author explicitly writes :-"''As it is pointed out in the title of {{harv|Hartogs|1906}}, and as the reader shall soon see, the key tool in the proof is the [[Cauchy integral formula]]''".</ref> and as such it is known also as '''Hartogs's lemma''' and '''Hartogs's principle''': in earlier [[Soviet Union|Soviet]] literature,<ref group=note>See for example {{harvtxt|Vladimirov|1966|p=153}}, which refers the reader to the book of {{harvtxt|Fuks|1963|p=284}} for a proof (however, in the former reference it is incorrectly stated that the proof is on page 324).</ref> it is also called '''Osgood–Brown theorem''', acknowledging later work by [[Arthur Barton Brown]] and [[William Fogg Osgood]].<ref group=note>See {{harvtxt|Brown|1936}} and {{harvtxt|Osgood|1929}}.</ref> This property of holomorphic functions of several variables is also called '''[[#Hartogs's phenomenon|Hartogs's phenomenon]]''': however, the locution "Hartogs's phenomenon" is also used to identify the property of solutions of [[System of equations|systems]] of [[partial differential equation|partial differential]] or [[convolution operator|convolution equation]]s satisfying Hartogs type theorems.<ref group=note>See {{harvtxt|Fichera|1983}} and {{harvtxt|Bratti|1986a}} {{harv|Bratti|1986b}}.</ref>
==Historical note==
The original proof was given by [[Friedrich Hartogs]] in 1906, using [[Cauchy's integral formula]] for [[functions of several complex variables]].<ref name="hartogs"/> Today, usual proofs rely on either the [[Bochner–Martinelli–Koppelman formula]] or the solution of the inhomogeneous [[Cauchy–Riemann equations]] with compact support. The latter approach is due to [[Leon Ehrenpreis]] who initiated it in the paper {{Harv|Ehrenpreis|1961}}. Yet another very simple proof of this result was given by [[Gaetano Fichera]] in the paper {{Harv|Fichera|1957}}, by using his solution of the [[Dirichlet problem]] for [[holomorphic function]]s of several variables and the related concept of [[CR-function]]:<ref group=note>Fichera's prof as well as his epoch making paper {{Harv|Fichera|1957}} seem to have been overlooked by many specialists of the [[Function of several complex variables|theory of functions of several complex variables]]: see {{Harvtxt|Range|2002}} for the correct attribution of many important theorems in this field.</ref> later he extended the theorem to a certain class of [[partial differential operator]]s in the paper {{Harv|Fichera|1983}}, and his ideas were later further explored by Giuliano Bratti.<ref group=note>See {{Harvtxt|Bratti|1986a}} {{Harv|Bratti|1986b}}.</ref> Also the Japanese school of the theory of [[partial differential operator]]s worked much on this topic, with notable contributions by Akira Kaneko.<ref group=note>See his paper {{Harv|Kaneko|1973}} and the references therein.</ref> Their approach is to use [[Ehrenpreis's fundamental principle]].
==Hartogs's phenomenon==
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| mr = 1907191
| s2cid = 120531925
}}. <ref group=note>A historical paper correcting some inexact historical statements in the theory of [[Function of several complex variables|holomorphic functions of several variables]], particularly concerning contributions of [[Gaetano Fichera]] and [[Francesco Severi]].</ref>
*{{Citation
| last = Severi
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| zbl = 0099.07801
| doi-access = free
}}.<ref group=note> A fundamental paper in the theory of Hartogs's phenomenon. The typographical error in the title is reproduced as it appears in the original version of the paper.</ref>
*{{Citation
| last = Fichera
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| mr = 0093597
| zbl = 0106.05202
}}.<ref group=note> An epoch-making paper in the theory of [[CR-function]]s, where the Dirichlet problem for [[Function of several complex variables|analytic functions of several complex variables]] is solved for general data. A translation of the title reads as:-"''Characterization of the trace, on the boundary of a ___domain, of an analytic function of several complex variables''".</ref>
*{{Citation
| last = Fichera
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| jfm = 58.0352.05
| zbl = 0004.40702
}}.<ref group=note>An English translation of the title reads as:-"''A fundamental property of the ___domain of holomorphy of an analytic function of one real variable and one complex variable''".</ref>
*{{Citation
| last = Severi
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