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| In mathematics, precisely in the theory of functions of [[several complex variables]], '''Hartogs' 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 the concept of [[isolated singularity]] and [[removable singularity]] coincide for [[analytic function]]s of ''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' lemma''' and '''Hartogs' principle''': in earlier [[Soviet Union|Soviet]] literature,<ref>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>See {{harvtxt|Brown|1936}} and {{harvtxt|Osgood|1929}}.</ref> This property of holomorphic functions of several variables is also called '''Hartogs' phenomenon''': however, the locution "Hartogs' 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>See {{harvtxt|Fichera|1983}} and {{harvtxt|Bratti|1986a}} {{harv|Bratti|1986b}}.</ref>
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| ==Historical note==
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| 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 [[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>Fichera's prof as well as his epoch making paper {{Harv|Fichera|1957}} seem to have been overlooked by many specialists of the [[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>See {{Harvtxt|Bratti|1986a}} {{Harv|Bratti|1986b}}.</ref> Also the [[Japan|Japanese school]] of the theory of [[partial differential operator]]s worked much on this topic, with notable contributions by Akira Kaneko.<ref>See his paper {{Harv|Kaneko|1973}} and the references therein.</ref> Their approach is to use [[Ehrenpreis' fundamental principle]].
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| ==Formal statement==
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| {{EquationRef|1|Theorem 1.1.}} If <math>f</math> is a [[holomorphic function]] on a [[Set (mathematics)|set]] <math>G</math> '''\''' <math>K</math>, where <math>G</math> is an open subset of ℂ''<sup>n</sup>'' (with ''n'' ≥ 2) and <math>K</math> is a compact subset of <math>G</math> such that the [[Complement (set theory)|relative complement]] <math>G</math> '''\''' <math>K</math> is connected, then <math>f</math> can be extended to a unique holomorphic function on <math>G</math>.
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| ==Counterexamples in dimension one==
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| The theorem does not hold when <math>n = 1</math>: to see this, it suffices to consider the function
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| :<math>f(z)=\frac{1}{z}</math>
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| which is clearly holomorphic in ℂ\{<math>0</math>}, but cannot be continued as an holomorphic function on the whole ℂ. Therefore the Hartogs' phenomenon constitutes one elementary phenomenon that emphasizes the difference between the theory of functions of one and several complex variables.
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| | |
| == Notes ==
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| {{reflist|30em}}
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| | |
| ==Historical references==
| |
| *{{Citation
| |
| | last = Fuks
| |
| | first = B. A.
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| | author-link = Boris Abramovich Fuks
| |
| | title = Introduction to the Theory of Analytic Functions of Several Complex Variables
| |
| | place = Providence, RI
| |
| | publisher = [[American Mathematical Society]]
| |
| | series = Translations of Mathematical Monographs
| |
| | volume = 8
| |
| | year = 1963
| |
| | pages = vi+374
| |
| | url = http://books.google.com/books?id=OSlWYzf2FcwC&printsec=frontcover#v=onepage&q&f=true
| |
| | doi =
| |
| | isbn =
| |
| | mr = 0168793
| |
| | zbl = 0138.30902
| |
| }}.
| |
| *{{Citation
| |
| | last = Osgood
| |
| | first = William Fogg
| |
| | author-link = William Fogg Osgood
| |
| | title = Topics in the theory of functions of several complex variables
| |
| | place = New York
| |
| | publisher = [[Dover]]
| |
| | origyear = 1913
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| | year = 1966
| |
| | edition = unabridged and corrected
| |
| | pages = IV+120
| |
| | doi =
| |
| | jfm = 45.0661.02
| |
| | mr = 0201668
| |
| | zbl = 0138.30901
| |
| | isbn =
| |
| }}.
| |
| *{{citation
| |
| | last = Range
| |
| | first = R. Michael
| |
| | title = Extension phenomena in multidimensional complex analysis: correction of the historical record
| |
| | journal = [[The Mathematical Intelligencer]]
| |
| | volume = 24
| |
| | issue = 2
| |
| | year = 2002
| |
| | pages = 4–12
| |
| | doi = 10.1007/BF03024609
| |
| | mr = 1907191
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| }}. An historical paper correcting some inexact historical statements in the theory of [[Several complex variables|holomorphic functions of several variables]], particularly concerning contributions of [[Gaetano Fichera]] and [[Francesco Severi]].
| |
| *{{Citation
| |
| | last = Severi
| |
| | first = Francesco
| |
| | author-link = Francesco Severi
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| | title = Risoluzione del problema generale di Dirichlet per le funzioni biarmoniche
| |
| | journal = Rendiconti della [[Accademia Nazionale dei Lincei]], Classe di Scienze Fisiche, Matematiche e Naturali
| |
| | series = series 6,
| |
| | volume = 13
| |
| | pages = 795–804
| |
| | year = 1931
| |
| | language = Italian
| |
| | jfm = 57.0393.01
| |
| | zbl = 0002.34202
| |
| }}. This is the first paper where a general solution to the [[Dirichlet problem]] for [[pluriharmonic function]]s is solved for general [[Analytic function|real analyitic data]] on a real analytic [[hypersurface]]. A translation of the title reads as:-"''Solution of the general Dirichlet problem for biharmonic functions''".
| |
| *{{Citation
| |
| | last = Severi
| |
| | first = Francesco
| |
| | author-link = Francesco Severi
| |
| | title = Lezioni sulle funzioni analitiche di più variabili complesse – Tenute nel 1956–57 all'[[Istituto Nazionale di Alta Matematica]] in [[Rome|Roma]]
| |
| | language = Italian
| |
| | place = Padova
| |
| | publisher = CEDAM – Casa Editrice Dott. Antonio Milani
| |
| | year = 1958
| |
| | url =
| |
| | doi =
| |
| | zbl = 0094.28002
| |
| | isbn = }}. A translation of the title is:-"''Lectures on analytic functions of several complex variables – Lectured in 1956–57 at the Istituto Nazionale di Alta Matematica in Rome''". This book consist of lecture notes from a course held by Francesco Severi at the [[Istituto Nazionale di Alta Matematica]] (which at present bears his name), and includes appendices of [[Enzo Martinelli]], [[Giovanni Battista Rizza]] and [[Mario Benedicty]].
| |
| *{{Citation
| |
| | last = Struppa
| |
| | first = Daniele C.
| |
| | author-link =
| |
| | contribution = The first eighty years of Hartogs' theorem
| |
| | series = Seminari di Geometria
| |
| | volume = 1987–1988
| |
| | place = [[Bologna]]
| |
| | pages = 127–209
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| | publisher = [[Università degli Studi di Bologna]]
| |
| | year = 1988
| |
| | url =
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| | jstor =
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| | doi =
| |
| | id =
| |
| | mr = 0973699
| |
| | zbl = 0657.35018
| |
| }}.
| |
| *{{Citation
| |
| | last = Vladimirov
| |
| | first = V. S.
| |
| | author-link = Vasilii Sergeevich Vladimirov
| |
| | editor-last = Ehrenpreis
| |
| | editor-first = L.
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| | editor-link = Leon Ehrenpreis
| |
| | title = Methods of the theory of functions of several complex variables. With a foreword of [[Nikolay Bogolyubov|N.N. Bogolyubov]]
| |
| | place = [[Cambridge, Massachusetts|Cambridge]]-[[London]]
| |
| | publisher = [[MIT Press|The M.I.T. Press]]
| |
| | year = 1966
| |
| | pages = XII+353
| |
| | mr = 0201669
| |
| | zbl = 0125.31904}} ([[Zentralblatt]] review of the original [[Russian language|Russian]] edition). One of the first modern monographs on the theory of [[several complex variables]], being different from other ones of the same period due to the extensive use of [[generalized function]]s.
| |
| | |
| ==References==
| |
| *{{Citation
| |
| | last = Bochner
| |
| | first = Salomon
| |
| | author-link = Salomon Bochner
| |
| | title = Analytic and meromorphic continuation by means of Green's formula
| |
| | journal = [[Annals of Mathematics]]
| |
| | series = Second Series
| |
| | volume = 44
| |
| | issue = 4
| |
| | date = October 1943
| |
| | year = 1943
| |
| | pages = 652–673
| |
| | jstor = 1969103
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| | doi = 10.2307/1969103
| |
| | mr = 0009206
| |
| | zbl = 0060.24206
| |
| }}.
| |
| *{{Citation
| |
| | last = Bochner
| |
| | first = Salomon
| |
| | author-link = Salomon Bochner
| |
| | title = Partial Differential Equations and Analytic Continuations
| |
| | journal = [[PNAS]]
| |
| | volume = 38
| |
| | issue = 3
| |
| | date = March 1, 1952
| |
| | pages = 227–230
| |
| | doi = 10.1073/pnas.38.3.227
| |
| | mr = 0050119
| |
| | zbl = 0046.09902
| |
| }}.
| |
| *{{Citation
| |
| | last = Bratti
| |
| | first = Giuliano
| |
| | title = A proposito di un esempio di Fichera relativo al fenomeno di Hartogs
| |
| | journal = Rendiconti della Accademia Nazionale delle Scienze Detta dei XL
| |
| | series = serie 5,
| |
| | volume = X
| |
| | issue = 1
| |
| | pages = 241–246
| |
| | year = 1986a
| |
| | language = Italian. English [[Abstract (summary)|summary]]
| |
| | url = http://www.accademiaxl.it/Biblioteca/Pubblicazioni/browser.php?VoceID=2020
| |
| | doi =
| |
| | mr = 0879111
| |
| | zbl = 0646.35007
| |
| }}. A translation of the title reads as:-"''About an example of Fichera concerning Hartogs' phenomenon''".
| |
| *{{Citation
| |
| | last = Bratti
| |
| | first = Giuliano
| |
| | title = Estensione di un teorema di Fichera relativo al fenomeno di Hartogs per sistemi differenziali a coefficenti costanti
| |
| | journal = Rendiconti della Accademia Nazionale delle Scienze Detta dei XL
| |
| | series = serie 5
| |
| | volume = X
| |
| | issue = 1
| |
| | pages = 255–259
| |
| | year = 1986b
| |
| | language = Italian. English summary
| |
| | url = http://www.accademiaxl.it/Biblioteca/Pubblicazioni/browser.php?VoceID=2023
| |
| | doi =
| |
| | mr = 0879114
| |
| | zbl = 0646.35008
| |
| }}. An English translation of the title reads as:-"''Extension of a theorem of Fichera for systems of P.D.E. with constant coefficients, concerning Hartogs' phenomenon''".
| |
| *{{Citation
| |
| | last = Bratti
| |
| | first = Giuliano
| |
| | title = Su di un teorema di Hartogs
| |
| | journal = Rendiconti del Seminario Matematico della Università di Padova
| |
| | volume = 79
| |
| | pages = 59–70
| |
| | year = 1988
| |
| | language = Italian
| |
| | url = http://www.numdam.org/item?id=RSMUP_1988__79__59_0
| |
| | doi =
| |
| | mr = 964020
| |
| | zbl = 0657.46033
| |
| }}. An English translation of the title reads as:-"''On a theorem of Hartogs''".
| |
| *{{Citation
| |
| | last = Brown
| |
| | first = Arthur B.
| |
| | author-link = Arthur Barton Brown
| |
| | title = On certain analytic continuations and analytic homeomorphisms
| |
| | journal = [[Duke Mathematical Journal]]
| |
| | volume = 2
| |
| | pages = 20–28
| |
| | year = 1936
| |
| | url = http://projecteuclid.org/euclid.dmj/1077489338
| |
| | doi = 10.1215/S0012-7094-36-00203-X
| |
| | jfm = 62.0396.02
| |
| | mr = 1545903
| |
| | zbl = 0013.40701
| |
| }}
| |
| *{{Citation
| |
| | last = Ehrenpreis
| |
| | first = Leon
| |
| | author-link = Leon Ehrenpreis
| |
| | title = A new proof and an extension of Hartog's theorem
| |
| | journal = [[Bulletin of the American Mathematical Society]]
| |
| | volume = 67
| |
| | pages = 507–509
| |
| | year = 1961
| |
| | doi = 10.1090/S0002-9904-1961-10661-7
| |
| | mr = 0131663
| |
| | zbl = 0099.07801
| |
| }}. A fundamental paper in the theory of Hrtogs' phenomenon. The typographical error in the title is reproduced in as it is appears in the original version of the paper.
| |
| *{{Citation
| |
| | last = Fichera
| |
| | first = Gaetano
| |
| | author-link = Gaetano Fichera
| |
| | title = Caratterizzazione della traccia, sulla frontiera di un campo, di una funzione analitica di più variabili complesse
| |
| | journal = Rendiconti della [[Accademia Nazionale dei Lincei]], Classe di Scienze Fisiche, Matematiche e Naturali
| |
| | series = series 8,
| |
| | volume = 22
| |
| | issue = 6
| |
| | pages = 706–715
| |
| | year = 1957
| |
| | language = Italian
| |
| | mr = 0093597
| |
| | zbl = 0106.05202
| |
| }}. An epoch-making paper in the theory of [[CR-function]]s, where the Dirichlet problem for [[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''".
| |
| *{{Citation
| |
| | last = Fichera
| |
| | first = Gaetano
| |
| | author-link = Gaetano Fichera
| |
| | title = Sul fenomeno di Hartogs per gli operatori lineari alle derivate parziali
| |
| | journal = Rendiconti dell' Istituto Lombardo di Scienze e Lettere. Scienze Matemàtiche e Applicazioni, Series A.
| |
| | volume = 117
| |
| | pages = 199–211
| |
| | year = 1983
| |
| | language = Italian
| |
| | doi =
| |
| | mr = 0848259
| |
| | zbl = 0603.35013
| |
| }}. An English translation of the title reads as:-"''Hartogs phenomenon for certain linear partial differential operators''".
| |
| *{{Citation
| |
| | last = Fueter
| |
| | first = Rudolf
| |
| | author-link = Rudolf Fueter
| |
| | title = Über einen Hartogs'schen Satz
| |
| | journal = [[Commentarii Mathematici Helvetici]]
| |
| | volume = 12
| |
| | issue = 1
| |
| | pages = 75–80
| |
| | year = 1939-1940
| |
| | language = German
| |
| | url = http://retro.seals.ch/digbib/en/view?rid=comahe-001:1939-1940:12::10
| |
| | doi = 10.5169/seals-12795
| |
| | jfm = 65.0363.03
| |
| | mr =
| |
| | zbl = 0022.05802
| |
| }}. Available at the [http://retro.seals.ch/digbib/home SEALS Portal]. An English translation of the title reads as:-"''On a theorem of Hartogs''".
| |
| *{{Citation
| |
| | last = Fueter
| |
| | first = Rudolf
| |
| | author-link = Rudolf Fueter
| |
| | title = Über einen Hartogs'schen Satz in der Theorie der analytischen Funktionen von <math>n</math> komplexen Variablen
| |
| | journal = [[Commentarii Mathematici Helvetici]]
| |
| | volume = 14
| |
| | issue = 1
| |
| | pages = 394–400
| |
| | year = 1941-1942
| |
| | language = German
| |
| | url = http://retro.seals.ch/digbib/en/view?rid=comahe-002:1941-1942:14::21
| |
| | doi = 10.5169/seals-14312
| |
| | jfm = 68.0175.02
| |
| | mr = 0007445
| |
| | zbl = 0027.05703
| |
| }} (see also {{Zbl|0060.24505}}, the cumulative review of several papers by E. Trost). Available at the [http://retro.seals.ch/digbib/home SEALS Portal]. An English translation of the title reads as:-"''On a theorem of Hartogs in the theory of analytic functions of <math>n</math> complex variables''".
| |
| *{{Citation
| |
| | last = Hartogs
| |
| | first = Fritz
| |
| | author-link = Friedrich Hartogs
| |
| | title = Einige Folgerungen aus der ''Cauchy''schen Integralformel bei Funktionen mehrerer Veränderlichen.
| |
| | journal = Sitzungsberichte der Königlich Bayerischen Akademie der Wissenschaften zu München, Mathematisch-Physikalische Klasse
| |
| | language = German
| |
| | volume = 36
| |
| | pages = 223–242
| |
| | year = 1906
| |
| | url =
| |
| | doi =
| |
| | jfm = 37.0443.01
| |
| }}.
| |
| *{{Citation
| |
| | last = Hartogs
| |
| | first = Fritz
| |
| | author-link = Friedrich Hartogs
| |
| | title = Zur Theorie der analytischen Funktionen mehrerer unabhängiger Veränderlichen, insbesondere über die Darstellung derselber durch Reihen welche nach Potentzen einer Veränderlichen fortschreiten
| |
| | journal = [[Mathematische Annalen]]
| |
| | language = German
| |
| | volume = 62
| |
| | pages = 1–88
| |
| | year = 1906a
| |
| | url = http://www.digizeitschriften.de/dms/img/?PPN=GDZPPN002260913
| |
| | doi = 10.1007/BF01448415
| |
| | jfm = 37.0444.01
| |
| }}. Available at the [http://www.digizeitschriften.de/ DigiZeitschriften].
| |
| *{{Citation
| |
| | last = Hörmander
| |
| | first = Lars
| |
| | author-link = Lars Hörmander
| |
| | title = An Introduction to Complex Analysis in Several Variables
| |
| | place = Amsterdam–London–New York–Tokyo
| |
| | publisher = [[Elsevier|North-Holland]]
| |
| | origyear = 1966
| |
| | year = 1990
| |
| | series = North–Holland Mathematical Library
| |
| | volume = 7
| |
| | edition = 3rd (Revised)
| |
| | url =
| |
| | doi =
| |
| | mr = 1045639
| |
| | zbl = 0685.32001
| |
| | isbn = 0-444-88446-7
| |
| }}.
| |
| *{{Citation
| |
| | last = Kaneko
| |
| | first = Akira
| |
| | author-link = Akira Kaneko
| |
| | title = On continuation of regular solutions of partial differential equations with constant coefficients
| |
| | journal = Proceedings of the Japan Academy
| |
| | volume = 49
| |
| | issue = 1
| |
| | pages = 17–19
| |
| | date = January 12, 1973
| |
| | url = http://projecteuclid.org/euclid.pja/1195519488
| |
| | doi = 10.3792/pja/1195519488
| |
| | mr = 0412578
| |
| | zbl = 0265.35008
| |
| }}, available at [http://projecteuclid.org/DPubS?Service=UI&version=1.0&verb=Display&handle=euclid Project Euclid].
| |
| *{{Citation
| |
| | last = Martinelli
| |
| | first = Enzo
| |
| | author-link = Enzo Martinelli
| |
| | title = Sopra una dimostrazione di R. Fueter per un teorema di Hartogs
| |
| | journal = [[Commentarii Mathematici Helvetici]]
| |
| | volume = 15
| |
| | issue = 1
| |
| | pages = 340–349
| |
| | year = 1942-1943
| |
| | language = Italian
| |
| | url = http://retro.seals.ch/digbib/en/view?rid=comahe-002:1942-1943:15::26
| |
| | doi = 10.5169/seals-14896
| |
| | mr = 0010729
| |
| | zbl = 0028.15201
| |
| }}. Available at the [http://retro.seals.ch/digbib/home SEALS Portal]. An English translation of the title reads as:-"''On a proof by R. Fueter of a theorem of Hartogs''".
| |
| *{{Citation
| |
| | last = Osgood
| |
| | first = W. F.
| |
| | author-link = William Fogg Osgood
| |
| | title = Lehrbuch der Funktionentheorie. II
| |
| | place = Leipzig
| |
| | publisher = [[Teubner Verlag|B. G. Teubner]]
| |
| | series = Teubners Sammlung von Lehrbüchern auf dem Gebiet der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen
| |
| | volume = Bd. XX - 1
| |
| | year = 1929
| |
| | pages = VIII+307
| |
| | language = German
| |
| | edition = 2nd
| |
| | url = http://books.google.com/books?id=1pSzLtN4Qp4C&printsec=frontcover&hl=it#v=onepage&q&f=true
| |
| | doi =
| |
| | jfm = 55.0171.02}}
| |
| *{{Citation
| |
| | last = Severi
| |
| | first = Francesco
| |
| | author-link = Francesco Severi
| |
| | title = Una proprietà fondamentale dei campi di olomorfismo di una funzione analitica di una variabile reale e di una variabile complessa
| |
| | journal = [[Rendiconti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali]]
| |
| | series = series 6,
| |
| | volume = 15
| |
| | pages = 487–490
| |
| | year = 1932
| |
| | language = Italian
| |
| | jfm = 58.0352.05
| |
| | zbl = 0004.40702
| |
| }}. 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''".
| |
| *{{Citation
| |
| | last = Severi
| |
| | first = Francesco
| |
| | author-link = Francesco Severi
| |
| | title = A proposito d'un teorema di Hartogs
| |
| | journal = [[Commentarii Mathematici Helvetici]]
| |
| | volume = 15
| |
| | issue = 1
| |
| | pages = 350–352
| |
| | year = 1942-1943
| |
| | language = Italian
| |
| | url =http://retro.seals.ch/digbib/en/view?rid=comahe-002:1942-1943:15::27
| |
| | doi = 10.5169/seals-14897
| |
| | mr = 0010730
| |
| | zbl = 0028.15301
| |
| }}. Available at the [http://retro.seals.ch/digbib/home SEALS Portal]. An English translation of the title reads as:-"''About a theorem of Hartogs''".
| |
| | |
| ==External links==
| |
| *{{springer
| |
| | title=Hartogs theorem
| |
| | id= h/h046650
| |
| | last= Chirka
| |
| | first= E. M.
| |
| }}
| |
| *{{planetmath reference|id=10242|title=Failure of Hartogs' theorem in one dimension (counterexample)}}
| |
| *{{PlanetMath|urlname=HartogsTheorem|title=Hartogs' theorem}}
| |
| *{{planetmath reference|id=10238|title=Proof of Hartogs' theorem}}
| |
| | |
| [[Category:Several complex variables]]
| |
| [[Category:Theorems in complex analysis]]
| |