Hartogs's extension theorem

<h2 id="historical-note">Historical note</h2>
The original proof was given by <a href="/facts/Friedrich_Hartogs/abPuQMt1">Friedrich Hartogs</a> in 1906, using <a href="/facts/Cauchy%2527s_integral_formula/4B35nkvV">Cauchy's integral formula</a> for <a href="/facts/Functions_of_several_complex_variables/axwNpV4a">functions of several complex variables</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> Today, usual proofs rely on either the <a href="/facts/Bochner%25E2%2580%2593Martinelli%25E2%2580%2593Koppelman_formula/Vog8qtpk">Bochner–Martinelli–Koppelman formula</a> or the solution of the inhomogeneous <a href="/facts/Cauchy%25E2%2580%2593Riemann_equations/SWkIHflU">Cauchy–Riemann equations</a> with compact support. The latter approach is due to <a href="/facts/Leon_Ehrenpreis/pR1uCfjr">Leon Ehrenpreis</a> who initiated it in the paper (Ehrenpreis 1961). Yet another very simple proof of this result was given by <a href="/facts/Gaetano_Fichera/gMo4k9mu">Gaetano Fichera</a> in the paper (Fichera 1957), by using his solution of the <a href="/facts/Dirichlet_problem/uHErgNCb">Dirichlet problem</a> for <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic functions</a> of several variables and the related concept of CR-function:<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> later he extended the theorem to a certain class of <a href="/facts/Partial_differential_operator/Nr15J0IE">partial differential operators</a> in the paper (Fichera 1983), and his ideas were later further explored by Giuliano Bratti.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> Also the Japanese school of the theory of <a href="/facts/Partial_differential_operator/Nr15J0IE">partial differential operators</a> worked much on this topic, with notable contributions by Akira Kaneko.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> Their approach is to use <a href="/facts/Ehrenpreis%2527s_fundamental_principle/GBxCDX40">Ehrenpreis's fundamental principle</a>.

<h2 id="hartogss-phenomenon">Hartogs's phenomenon</h2>
For example, in two variables, consider the interior domain

H
 
 ε
 
 
 =
 {
 z
 =
 (
 
 z
 
 1
 
 
 ,
 
 z
 
 2
 
 
 )
 ∈
 
 Δ
 
 2
 
 
 :
 
 |
 
 
 z
 
 1
 
 
 
 |
 
 <
 ε
  
  
 
 or
 
  
  
 1
 −
 ε
 <
 
 |
 
 
 z
 
 2
 
 
 
 |
 
 }
 
 
 {\displaystyle H_{\varepsilon }=\{z=(z_{1},z_{2})\in \Delta ^{2}:|z_{1}|<\varepsilon \ \ {\text{or}}\ \ 1-\varepsilon <|z_{2}|\}}

in the two-dimensional polydisk 
 
 
 
 
 Δ
 
 2
 
 
 =
 {
 z
 ∈
 
 
 C
 
 
 2
 
 
 ;
 
 |
 
 
 z
 
 1
 
 
 
 |
 
 <
 1
 ,
 
 |
 
 
 z
 
 2
 
 
 
 |
 
 <
 1
 }
 
 
 {\displaystyle \Delta ^{2}=\{z\in \mathbb {C} ^{2};|z_{1}|<1,|z_{2}|<1\}}
 
 where 
 
 
 
 0
 <
 ε
 <
 1.
 
 
 {\displaystyle 0<\varepsilon <1.}

Theorem Hartogs (1906): Any holomorphic function 
 
 
 
 f
 
 
 {\displaystyle f}
 
 on 
 
 
 
 
 H
 
 ε
 
 
 
 
 {\displaystyle H_{\varepsilon }}
 
 can be analytically continued to 
 
 
 
 
 Δ
 
 2
 
 
 .
 
 
 {\displaystyle \Delta ^{2}.}
 
 Namely, there is a holomorphic function 
 
 
 
 F
 
 
 {\displaystyle F}
 
 on 
 
 
 
 
 Δ
 
 2
 
 
 
 
 {\displaystyle \Delta ^{2}}
 
 such that 
 
 
 
 F
 =
 f
 
 
 {\displaystyle F=f}
 
 on 
 
 
 
 
 H
 
 ε
 
 
 .
 
 
 {\displaystyle H_{\varepsilon }.}

Such a phenomenon is called Hartogs's phenomenon, which lead to the notion of this Hartogs's extension theorem and the <a href="/facts/Domain_of_holomorphy/BHmQtegm">domain of holomorphy</a>.

<h2 id="formal-statement-and-proof">Formal statement and proof</h2>
Let f be a <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic function</a> on a <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> G \ K, where G is an open subset of Cn (n ≥ 2) and K is a compact subset of G. If the <a href="/facts/Complement_(set_theory)/yWUePIYY">complement</a> G \ K is connected, then f can be extended to a unique holomorphic function F on G.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>
Ehrenpreis' proof is based on the existence of smooth <a href="/facts/Bump_function/ExdBtMHN">bump functions</a>, unique continuation of holomorphic functions, and the <a href="/facts/Poincar%25C3%25A9_lemma/uvEUbzUK">Poincaré lemma</a> — the last in the form that for any smooth and compactly supported differential (0,1)-form ω on Cn with ∂ω = 0, there exists a smooth and compactly supported function η on Cn with ∂η = ω. The crucial assumption n ≥ 2 is required for the validity of this Poincaré lemma; if n = 1 then it is generally impossible for η to be compactly supported.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a>
The ansatz for F is φ f − v for smooth functions φ and v on G; such an expression is meaningful provided that φ is identically equal to zero where f is undefined (namely on K). Furthermore, given any holomorphic function on G which is equal to f on some open set, unique continuation (based on connectedness of G \ K) shows that it is equal to f on all of G \ K.
The holomorphicity of this function is identical to the condition ∂v = f ∂φ. For any smooth function φ, the differential (0,1)-form f ∂φ is ∂-closed. Choosing φ to be a smooth function which is identically equal to zero on K and identically equal to one on the complement of some compact subset L of G, this (0,1)-form additionally has compact support, so that the Poincaré lemma identifies an appropriate v of compact support. This defines F as a holomorphic function on G; it only remains to show (following the above comments) that it coincides with f on some open set.
On the set Cn \ L, v is holomorphic since φ is identically constant. Since it is zero near infinity, unique continuation applies to show that it is identically zero on some open subset of G \ L.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a> Thus, on this open subset, F equals f and the existence part of Hartog's theorem is proved. Uniqueness is automatic from unique continuation, based on connectedness of G.

<h2 id="counterexamples-in-dimension-one">Counterexamples in dimension one</h2>
The theorem does not hold when n = 1. To see this, it suffices to consider the function f(z) = z−1, which is clearly holomorphic in C \ {0}, but cannot be continued as a holomorphic function on the whole of C. Therefore, the Hartogs's phenomenon is an elementary phenomenon that highlights the difference between the theory of functions of one and several complex variables.

<h2 id="notes">Notes</h2>

<h3>Historical references</h3>
<ul><li>Fuks, B. A. (1963), <a href="https://books.google.com/books?id=OSlWYzf2FcwC">Introduction to the Theory of Analytic Functions of Several Complex Variables</a>, Translations of Mathematical Monographs, vol. 8, Providence, RI: <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>, pp. vi+374, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780821886441, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0168793">0168793</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0138.30902">0138.30902</a> {{citation}}: ISBN / Date incompatibility (help).</li>
<li><a href="/facts/William_Fogg_Osgood/1mTufCMj">Osgood, William Fogg</a> (1966) [1913], Topics in the theory of functions of several complex variables (unabridged and corrected ed.), New York: <a href="/facts/Dover/Ru9mNrrw">Dover</a>, pp. IV+120, <a href="/facts/JFM_(identifier)/P6rFxKKx">JFM</a> <a href="https://zbmath.org/?format=complete&q=an:45.0661.02">45.0661.02</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0201668">0201668</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0138.30901">0138.30901</a>.</li>
<li>Range, R. Michael (2002), "Extension phenomena in multidimensional complex analysis: correction of the historical record", <a href="/facts/The_Mathematical_Intelligencer/tr4EnU3C">The Mathematical Intelligencer</a>, 24 (2): 4–12, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF03024609">10.1007/BF03024609</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1907191">1907191</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:120531925">120531925</a>. A historical paper correcting some inexact historical statements in the theory of <a href="/facts/Function_of_several_complex_variables/axwNpV4a">holomorphic functions of several variables</a>, particularly concerning contributions of <a href="/facts/Gaetano_Fichera/gMo4k9mu">Gaetano Fichera</a> and <a href="/facts/Francesco_Severi/LRqMFxVM">Francesco Severi</a>.</li>
<li><a href="/facts/Francesco_Severi/LRqMFxVM">Severi, Francesco</a> (1931), "Risoluzione del problema generale di Dirichlet per le funzioni biarmoniche", Rendiconti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, series 6 (in Italian), 13: 795–804, <a href="/facts/JFM_(identifier)/P6rFxKKx">JFM</a> <a href="https://zbmath.org/?format=complete&q=an:57.0393.01">57.0393.01</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0002.34202">0002.34202</a>. This is the first paper where a general solution to the <a href="/facts/Dirichlet_problem/uHErgNCb">Dirichlet problem</a> for <a href="/facts/Pluriharmonic_function/qh3nKWXP">pluriharmonic functions</a> is given for general <a href="/facts/Analytic_function/JhL3z8FP">real analytic data</a> on a real analytic <a href="/facts/Hypersurface/cTvtLVsV">hypersurface</a>. A translation of the title reads as:-"Solution of the general Dirichlet problem for biharmonic functions".</li>
<li><a href="/facts/Francesco_Severi/LRqMFxVM">Severi, Francesco</a> (1958), Lezioni sulle funzioni analitiche di più variabili complesse – Tenute nel 1956–57 all'Istituto Nazionale di Alta Matematica in Roma (in Italian), Padova: CEDAM – Casa Editrice Dott. Antonio Milani, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0094.28002">0094.28002</a>. 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 <a href="/facts/Istituto_Nazionale_di_Alta_Matematica/u9kyPs1e">Istituto Nazionale di Alta Matematica</a> (which at present bears his name), and includes appendices of <a href="/facts/Enzo_Martinelli/8aKLx9wu">Enzo Martinelli</a>, <a href="/facts/Giovanni_Battista_Rizza/wYtUucce">Giovanni Battista Rizza</a> and Mario Benedicty.</li>
<li>Struppa, Daniele C. (1988), "The first eighty years of Hartogs' theorem", Seminari di Geometria 1987–1988, <a href="/facts/Bologna/glE9rlrW">Bologna</a>: <a href="/facts/Universit%25C3%25A0_degli_Studi_di_Bologna/UMwF3Aca">Università degli Studi di Bologna</a> – Dipartimento di Matematica, pp. 127–209, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0973699">0973699</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0657.35018">0657.35018</a>.</li>
<li><a href="/facts/Vasilii_Sergeevich_Vladimirov/NjVW9QlR">Vladimirov, V. S.</a> (1966), <a href="/facts/Leon_Ehrenpreis/pR1uCfjr">Ehrenpreis, L.</a> (ed.), <a href="/facts/Nikolay_Bogolyubov/39w1trXl">Methods of the theory of functions of several complex variables. With a foreword of N.N. Bogolyubov</a>, <a href="/facts/Cambridge%2c_Massachusetts/k9UMIJ7U">Cambridge</a>-<a href="/facts/London/KozCCsy9">London</a>: <a href="/facts/MIT_Press/eTerUM3c">The M.I.T. Press</a>, pp. XII+353, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0201669">0201669</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0125.31904">0125.31904</a> (<a href="/facts/Zentralblatt/P6rFxKKx">Zentralblatt</a> review of the original <a href="/facts/Russian_language/ETXrTgHV">Russian</a> edition). One of the first modern monographs on the theory of <a href="/facts/Function_of_several_complex_variables/axwNpV4a">several complex variables</a>, being different from other ones of the same period due to the extensive use of <a href="/facts/Generalized_function/fCM4VsyT">generalized functions</a>.</li></ul>
<h3>Scientific references</h3>
<ul><li><a href="/facts/Salomon_Bochner/n35gmr7h">Bochner, Salomon</a> (October 1943), "Analytic and meromorphic continuation by means of Green's formula", <a href="/facts/Annals_of_Mathematics/CUYktUqD">Annals of Mathematics</a>, Second Series, 44 (4): 652–673, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1969103">10.2307/1969103</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1969103">1969103</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0009206">0009206</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0060.24206">0060.24206</a>.</li>
<li><a href="/facts/Salomon_Bochner/n35gmr7h">Bochner, Salomon</a> (March 1, 1952), "Partial Differential Equations and Analytic Continuations", <a href="/facts/PNAS/w75CThX0">PNAS</a>, 38 (3): 227–230, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1952PNAS...38..227B">1952PNAS...38..227B</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1073%2Fpnas.38.3.227">10.1073/pnas.38.3.227</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0050119">0050119</a>, <a href="/facts/PMC_(identifier)/dX1zMt71">PMC</a> <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1063536">1063536</a>, <a href="/facts/PMID_(identifier)/JlHAvMHt">PMID</a> <a href="https://pubmed.ncbi.nlm.nih.gov/16589083">16589083</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0046.09902">0046.09902</a>.</li>
<li>Bratti, Giuliano (1986a), <a href="https://web.archive.org/web/20110726235834/http://www.accademiaxl.it/Biblioteca/Pubblicazioni/browser.php?VoceID=2020">"A proposito di un esempio di Fichera relativo al fenomeno di Hartogs"</a> [About an example of Fichera concerning Hartogs's phenomenon], Rendiconti della Accademia Nazionale delle Scienze Detta dei XL, serie 5 (in Italian and English), X (1): 241–246, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0879111">0879111</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0646.35007">0646.35007</a>, archived from <a href="http://www.accademiaxl.it/Biblioteca/Pubblicazioni/browser.php?VoceID=2020">the original</a> on 2011-07-26</li>
<li>Bratti, Giuliano (1986b), <a href="https://web.archive.org/web/20110726235922/http://www.accademiaxl.it/Biblioteca/Pubblicazioni/browser.php?VoceID=2023">"Estensione di un teorema di Fichera relativo al fenomeno di Hartogs per sistemi differenziali a coefficenti costanti"</a> [Extension of a theorem of Fichera for systems of P.D.E. with constant coefficients, concerning Hartogs's phenomenon], Rendiconti della Accademia Nazionale delle Scienze Detta dei XL, serie 5 (in Italian and English), X (1): 255–259, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0879114">0879114</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0646.35008">0646.35008</a>, archived from <a href="http://www.accademiaxl.it/Biblioteca/Pubblicazioni/browser.php?VoceID=2023">the original</a> on 2011-07-26</li>
<li>Bratti, Giuliano (1988), <a href="http://www.numdam.org/item?id=RSMUP_1988__79__59_0">"Su di un teorema di Hartogs"</a> [On a theorem of Hartogs], Rendiconti del Seminario Matematico della Università di Padova (in Italian), 79: 59–70, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0964020">0964020</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0657.46033">0657.46033</a></li>
<li>Brown, Arthur B. (1936), "On certain analytic continuations and analytic homeomorphisms", <a href="/facts/Duke_Mathematical_Journal/S6wn8HYp">Duke Mathematical Journal</a>, 2: 20–28, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1215%2FS0012-7094-36-00203-X">10.1215/S0012-7094-36-00203-X</a>, <a href="/facts/JFM_(identifier)/P6rFxKKx">JFM</a> <a href="https://zbmath.org/?format=complete&q=an:62.0396.02">62.0396.02</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1545903">1545903</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0013.40701">0013.40701</a></li>
<li><a href="/facts/Leon_Ehrenpreis/pR1uCfjr">Ehrenpreis, Leon</a> (1961), "A new proof and an extension of Hartog's theorem", <a href="/facts/Bulletin_of_the_American_Mathematical_Society/8ER5KY5z">Bulletin of the American Mathematical Society</a>, 67 (5): 507–509, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1090%2FS0002-9904-1961-10661-7">10.1090/S0002-9904-1961-10661-7</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0131663">0131663</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0099.07801">0099.07801</a>. 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.</li>
<li><a href="/facts/Gaetano_Fichera/gMo4k9mu">Fichera, Gaetano</a> (1957), "Caratterizzazione della traccia, sulla frontiera di un campo, di una funzione analitica di più variabili complesse", Rendiconti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, series 8 (in Italian), 22 (6): 706–715, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0093597">0093597</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0106.05202">0106.05202</a>. An epoch-making paper in the theory of CR-functions, where the Dirichlet problem for <a href="/facts/Function_of_several_complex_variables/axwNpV4a">analytic functions of several complex variables</a> 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".</li>
<li><a href="/facts/Gaetano_Fichera/gMo4k9mu">Fichera, Gaetano</a> (1983), "Sul fenomeno di Hartogs per gli operatori lineari alle derivate parziali", Rendiconti Dell' Istituto Lombardo di Scienze e Lettere. Scienze Matemàtiche e Applicazioni, Series A. (in Italian), 117: 199–211, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0848259">0848259</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0603.35013">0603.35013</a>. An English translation of the title reads as:-"Hartogs phenomenon for certain linear partial differential operators".</li>
<li><a href="/facts/Rudolf_Fueter/qyPgUXwD">Fueter, Rudolf</a> (1939–1940), <a href="https://web.archive.org/web/20111002073042/http://retro.seals.ch/digbib/en/view?rid=comahe-001:1939-1940:12::10">"Über einen Hartogs'schen Satz"</a> [On a theorem of Hartogs], <a href="/facts/Commentarii_Mathematici_Helvetici/ddjErBn3">Commentarii Mathematici Helvetici</a> (in German), 12 (1): 75–80, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fbf01620640">10.1007/bf01620640</a>, <a href="/facts/JFM_(identifier)/P6rFxKKx">JFM</a> <a href="https://zbmath.org/?format=complete&q=an:65.0363.03">65.0363.03</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:120266425">120266425</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0022.05802">0022.05802</a>, archived from <a href="http://retro.seals.ch/digbib/en/view?rid=comahe-001:1939-1940:12::10">the original</a> on 2011-10-02, retrieved 2011-01-16. Available at the <a href="http://retro.seals.ch/digbib/home">SEALS Portal</a> <a href="https://web.archive.org/web/20121110040541/http://retro.seals.ch/digbib/home">Archived</a> 2012-11-10 at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a>.</li>
<li><a href="/facts/Rudolf_Fueter/qyPgUXwD">Fueter, Rudolf</a> (1941–1942), <a href="https://web.archive.org/web/20111002073152/http://retro.seals.ch/digbib/en/view?rid=comahe-002:1941-1942:14::21">"Über einen Hartogs'schen Satz in der Theorie der analytischen Funktionen von n komplexen Variablen"</a> [On a theorem of Hartogs in the theory of analytic functions of n complex variables], <a href="/facts/Commentarii_Mathematici_Helvetici/ddjErBn3">Commentarii Mathematici Helvetici</a> (in German), 14 (1): 394–400, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fbf02565627">10.1007/bf02565627</a>, <a href="/facts/JFM_(identifier)/P6rFxKKx">JFM</a> <a href="https://zbmath.org/?format=complete&q=an:68.0175.02">68.0175.02</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0007445">0007445</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:122750611">122750611</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0027.05703">0027.05703</a>, archived from <a href="http://retro.seals.ch/digbib/en/view?rid=comahe-002:1941-1942:14::21">the original</a> on 2011-10-02, retrieved 2011-01-16 (see also <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0060.24505">0060.24505</a>, the cumulative review of several papers by E. Trost). Available at the <a href="http://retro.seals.ch/digbib/home">SEALS Portal</a> <a href="https://web.archive.org/web/20121110040541/http://retro.seals.ch/digbib/home">Archived</a> 2012-11-10 at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a>.</li>
<li><a href="/facts/Friedrich_Hartogs/abPuQMt1">Hartogs, Fritz</a> (1906), <a href="https://archive.org/stream/bub_gb_N-sAAAAAYAAJ#page/n229/mode/1up">"Einige Folgerungen aus der Cauchyschen Integralformel bei Funktionen mehrerer Veränderlichen."</a>, Sitzungsberichte der Königlich Bayerischen Akademie der Wissenschaften zu München, Mathematisch-Physikalische Klasse (in German), 36: 223–242, <a href="/facts/JFM_(identifier)/P6rFxKKx">JFM</a> <a href="https://zbmath.org/?format=complete&q=an:37.0443.01">37.0443.01</a>.</li>
<li><a href="/facts/Friedrich_Hartogs/abPuQMt1">Hartogs, Fritz</a> (1906a), <a href="http://www.digizeitschriften.de/dms/img/?PPN=GDZPPN002260913">"Zur Theorie der analytischen Funktionen mehrerer unabhängiger Veränderlichen, insbesondere über die Darstellung derselber durch Reihen welche nach Potentzen einer Veränderlichen fortschreiten"</a>, <a href="/facts/Mathematische_Annalen/GPhGaonc">Mathematische Annalen</a> (in German), 62: 1–88, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01448415">10.1007/BF01448415</a>, <a href="/facts/JFM_(identifier)/P6rFxKKx">JFM</a> <a href="https://zbmath.org/?format=complete&q=an:37.0444.01">37.0444.01</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:122134517">122134517</a>. Available at the <a href="http://www.digizeitschriften.de/">DigiZeitschriften</a>.</li>
<li><a href="/facts/Lars_H%25C3%25B6rmander/mxyNnmRA">Hörmander, Lars</a> (1990) [1966], An Introduction to Complex Analysis in Several Variables, North–Holland Mathematical Library, vol. 7 (3rd (Revised) ed.), Amsterdam–London–New York–Tokyo: <a href="/facts/Elsevier/jkPSAEmR">North-Holland</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-444-88446-7, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1045639">1045639</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0685.32001">0685.32001</a>.</li>
<li>Kaneko, Akira (January 12, 1973), "On continuation of regular solutions of partial differential equations with constant coefficients", Proceedings of the Japan Academy, 49 (1): 17–19, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.3792%2Fpja%2F1195519488">10.3792/pja/1195519488</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0412578">0412578</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0265.35008">0265.35008</a>, available at <a href="http://projecteuclid.org/DPubS?Service=UI&version=1.0&verb=Display&handle=euclid">Project Euclid</a>.</li>
<li><a href="/facts/Enzo_Martinelli/8aKLx9wu">Martinelli, Enzo</a> (1942–1943), <a href="https://web.archive.org/web/20111002072948/http://retro.seals.ch/digbib/en/view?rid=comahe-002%3A1942-1943%3A15%3A%3A26">"Sopra una dimostrazione di R. Fueter per un teorema di Hartogs"</a> [On a proof by R. Fueter of a theorem of Hartogs], <a href="/facts/Commentarii_Mathematici_Helvetici/ddjErBn3">Commentarii Mathematici Helvetici</a> (in Italian), 15 (1): 340–349, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fbf02565649">10.1007/bf02565649</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0010729">0010729</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:119960691">119960691</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0028.15201">0028.15201</a>, archived from <a href="http://retro.seals.ch/digbib/en/view?rid=comahe-002:1942-1943:15::26">the original</a> on 2011-10-02, retrieved 2011-01-16. Available at the <a href="http://retro.seals.ch/digbib/home">SEALS Portal</a> <a href="https://web.archive.org/web/20121110040541/http://retro.seals.ch/digbib/home">Archived</a> 2012-11-10 at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a>.</li>
<li><a href="/facts/William_Fogg_Osgood/1mTufCMj">Osgood, W. F.</a> (1929), <a href="https://books.google.com/books?id=1pSzLtN4Qp4C">Lehrbuch der Funktionentheorie. II</a>, Teubners Sammlung von Lehrbüchern auf dem Gebiet der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen (in German), vol. Bd. XX - 1 (2nd ed.), Leipzig: <a href="/facts/Teubner_Verlag/L96LC0Su">B. G. Teubner</a>, pp. VIII+307, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780828401821, <a href="/facts/JFM_(identifier)/P6rFxKKx">JFM</a> <a href="https://zbmath.org/?format=complete&q=an:55.0171.02">55.0171.02</a> {{citation}}: ISBN / Date incompatibility (help).</li>
<li><a href="/facts/Francesco_Severi/LRqMFxVM">Severi, Francesco</a> (1932), "Una proprietà fondamentale dei campi di olomorfismo di una funzione analitica di una variabile reale e di una variabile complessa", Rendiconti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, series 6 (in Italian), 15: 487–490, <a href="/facts/JFM_(identifier)/P6rFxKKx">JFM</a> <a href="https://zbmath.org/?format=complete&q=an:58.0352.05">58.0352.05</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0004.40702">0004.40702</a>. 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".</li>
<li><a href="/facts/Francesco_Severi/LRqMFxVM">Severi, Francesco</a> (1942–1943), <a href="https://web.archive.org/web/20111002073401/http://retro.seals.ch/digbib/en/view?rid=comahe-002:1942-1943:15::27">"A proposito d'un teorema di Hartogs"</a> [About a theorem of Hartogs], <a href="/facts/Commentarii_Mathematici_Helvetici/ddjErBn3">Commentarii Mathematici Helvetici</a> (in Italian), 15 (1): 350–352, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fbf02565650">10.1007/bf02565650</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0010730">0010730</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:120514642">120514642</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0028.15301">0028.15301</a>, archived from <a href="http://retro.seals.ch/digbib/en/view?rid=comahe-002:1942-1943:15::27">the original</a> on 2011-10-02, retrieved 2011-06-25. Available at the <a href="http://retro.seals.ch/digbib/home">SEALS Portal</a> <a href="https://web.archive.org/web/20121110040541/http://retro.seals.ch/digbib/home">Archived</a> 2012-11-10 at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a>.</li></ul>

<h2 id="external-links">External links</h2>
<ul><li>Chirka, E. M. (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Hartogs_theorem">"Hartogs theorem"</a>, <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li>
<li><a href="https://planetmath.org/FailureOfHartogsTheoremInOneDimension">"failure of Hartogs' theorem in one dimension"</a>. <a href="/facts/PlanetMath/wphPCtLf">PlanetMath</a>.</li>
<li><a href="https://planetmath.org/HartogsTheorem">Hartogs' theorem</a> at <a href="/facts/PlanetMath/wphPCtLf">PlanetMath</a>.</li>
<li><a href="https://planetmath.org/ProofOfHartogsTheorem">Proof of Hartogs' theorem</a> at <a href="/facts/PlanetMath/wphPCtLf">PlanetMath</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">See the original paper of Hartogs (1906) and its description in various historical surveys by Osgood (1966, pp. 56–59), Severi (1958, pp. 111–115) and 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 (Hartogs 1906), and as the reader shall soon see, the key tool in the proof is the Cauchy integral formula". - Hartogs, Fritz (1906), "Einige Folgerungen aus der Cauchyschen Integralformel bei Funktionen mehrerer Veränderlichen.", Sitzungsberichte der Königlich Bayerischen Akademie der Wissenschaften zu München, Mathematisch-Physikalische Klasse (in German), 36: 223–242, JFM 37.0443.01 <a href="https://archive.org/stream/bub_gb_N-sAAAAAYAAJ#page/n229/mode/1up" target="_blank">https://archive.org/stream/bub_gb_N-sAAAAAYAAJ#page/n229/mode/1up</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">See for example Vladimirov (1966, p. 153), which refers the reader to the book of Fuks (1963, p. 284) for a proof (however, in the former reference it is incorrectly stated that the proof is on page 324). - Vladimirov, V. S. (1966), Ehrenpreis, L. (ed.), Methods of the theory of functions of several complex variables. With a foreword of N.N. Bogolyubov, Cambridge-London: The M.I.T. Press, pp. XII+353, MR 0201669, Zbl 0125.31904 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0201669" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0201669</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">See Brown (1936) and Osgood (1929). - Brown, Arthur B. (1936), "On certain analytic continuations and analytic homeomorphisms", Duke Mathematical Journal, 2: 20–28, doi:10.1215/S0012-7094-36-00203-X, JFM 62.0396.02, MR 1545903, Zbl 0013.40701 <a href="https://doi.org/10.1215%2FS0012-7094-36-00203-X" target="_blank">https://doi.org/10.1215%2FS0012-7094-36-00203-X</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">See Fichera (1983) and Bratti (1986a) (Bratti 1986b). - Fichera, Gaetano (1983), "Sul fenomeno di Hartogs per gli operatori lineari alle derivate parziali", Rendiconti Dell' Istituto Lombardo di Scienze e Lettere. Scienze Matemàtiche e Applicazioni, Series A. (in Italian), 117: 199–211, MR 0848259, Zbl 0603.35013 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0848259" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0848259</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">See the original paper of Hartogs (1906) and its description in various historical surveys by Osgood (1966, pp. 56–59), Severi (1958, pp. 111–115) and 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 (Hartogs 1906), and as the reader shall soon see, the key tool in the proof is the Cauchy integral formula". - Hartogs, Fritz (1906), "Einige Folgerungen aus der Cauchyschen Integralformel bei Funktionen mehrerer Veränderlichen.", Sitzungsberichte der Königlich Bayerischen Akademie der Wissenschaften zu München, Mathematisch-Physikalische Klasse (in German), 36: 223–242, JFM 37.0443.01 <a href="https://archive.org/stream/bub_gb_N-sAAAAAYAAJ#page/n229/mode/1up" target="_blank">https://archive.org/stream/bub_gb_N-sAAAAAYAAJ#page/n229/mode/1up</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Fichera's proof as well as his epoch making paper (Fichera 1957) seem to have been overlooked by many specialists of the theory of functions of several complex variables: see Range (2002) for the correct attribution of many important theorems in this field. - Fichera, Gaetano (1957), "Caratterizzazione della traccia, sulla frontiera di un campo, di una funzione analitica di più variabili complesse", Rendiconti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, series 8 (in Italian), 22 (6): 706–715, MR 0093597, Zbl 0106.05202 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0093597" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0093597</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">See Bratti (1986a) (Bratti 1986b). - Bratti, Giuliano (1986a), "A proposito di un esempio di Fichera relativo al fenomeno di Hartogs" [About an example of Fichera concerning Hartogs's phenomenon], Rendiconti della Accademia Nazionale delle Scienze Detta dei XL, serie 5 (in Italian and English), X (1): 241–246, MR 0879111, Zbl 0646.35007, archived from the original on 2011-07-26 <a href="https://web.archive.org/web/20110726235834/http://www.accademiaxl.it/Biblioteca/Pubblicazioni/browser.php?VoceID=2020" target="_blank">https://web.archive.org/web/20110726235834/http://www.accademiaxl.it/Biblioteca/Pubblicazioni/browser.php?VoceID=2020</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">See his paper (Kaneko 1973) and the references therein. - Kaneko, Akira (January 12, 1973), "On continuation of regular solutions of partial differential equations with constant coefficients", Proceedings of the Japan Academy, 49 (1): 17–19, doi:10.3792/pja/1195519488, MR 0412578, Zbl 0265.35008 <a href="https://doi.org/10.3792%2Fpja%2F1195519488" target="_blank">https://doi.org/10.3792%2Fpja%2F1195519488</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Hörmander 1990, Theorem 2.3.2. - Hörmander, Lars (1990) [1966], An Introduction to Complex Analysis in Several Variables, North–Holland Mathematical Library, vol. 7 (3rd (Revised) ed.), Amsterdam–London–New York–Tokyo: North-Holland, ISBN 0-444-88446-7, MR 1045639, Zbl 0685.32001 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1045639" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=1045639</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Hörmander 1990, p. 30. - Hörmander, Lars (1990) [1966], An Introduction to Complex Analysis in Several Variables, North–Holland Mathematical Library, vol. 7 (3rd (Revised) ed.), Amsterdam–London–New York–Tokyo: North-Holland, ISBN 0-444-88446-7, MR 1045639, Zbl 0685.32001 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1045639" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=1045639</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Any connected component of Cn \ L must intersect G \ L in a nonempty open set. To see the nonemptiness, connect an arbitrary point p of Cn \ L to some point of L via a line. The intersection of the line with Cn \ L may have many connected components, but the component containing p gives a continuous path from p into G \ L. <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
</ol>

Hartogs's extension theorem open-in-new

Hartogs's extension theorem