358:
473:
201:
2306:"Zur Theorie der analytischen Funktionen mehrerer unabhĂ€ngiger VerĂ€nderlichen, insbesondere ĂŒber die Darstellung derselber durch Reihen welche nach Potentzen einer VerĂ€nderlichen fortschreiten"
2556:
505:
2524:, Teubners Sammlung von LehrbĂŒchern auf dem Gebiet der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen (in German), vol. Bd. XX - 1 (2nd ed.), Leipzig:
693:
560:
590:
637:
50:, 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
1100:. Therefore, the Hartogs's phenomenon is an elementary phenomenon that highlights the difference between the theory of functions of one and several complex variables.
2013:. 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.
663:
1978:
610:
533:
136:
2443:
1856:
1819:
366:
2593:
2177:
2101:
1036:
is identically constant. Since it is zero near infinity, unique continuation applies to show that it is identically zero on some open subset of
132:
2739:
2554:(1932), "Una proprietĂ fondamentale dei campi di olomorfismo di una funzione analitica di una variabile reale e di una variabile complessa",
2050:
1680:
1577:
1489:
1214:
184:
27:
1605:
353:{\displaystyle H_{\varepsilon }=\{z=(z_{1},z_{2})\in \Delta ^{2}:|z_{1}|<\varepsilon \ \ {\text{or}}\ \ 1-\varepsilon <|z_{2}|\}}
2734:
1574:
Lectures on analytic functions of several complex variables â Lectured in 1956â57 at the
Istituto Nazionale di Alta Matematica in Rome
2598:
2533:
2448:
2367:
2190:
2106:
1387:
1561:
Lezioni sulle funzioni analitiche di piĂč variabili complesse â Tenute nel 1956â57 all'Istituto
Nazionale di Alta Matematica in Roma
1054:
and the existence part of Hartog's theorem is proved. Uniqueness is automatic from unique continuation, based on connectedness of
2432:
1859:[Extension of a theorem of Fichera for systems of P.D.E. with constant coefficients, concerning Hartogs's phenomenon],
2020:(1957), "Caratterizzazione della traccia, sulla frontiera di un campo, di una funzione analitica di piĂč variabili complesse",
1455:
1453:
Range, R. Michael (2002), "Extension phenomena in multidimensional complex analysis: correction of the historical record",
2684:
1629:
1379:
180:
172:
108:
2396:(January 12, 1973), "On continuation of regular solutions of partial differential equations with constant coefficients",
140:
2583:
A fundamental property of the domain of holomorphy of an analytic function of one real variable and one complex variable
128:
2679:
1857:"Estensione di un teorema di Fichera relativo al fenomeno di Hartogs per sistemi differenziali a coefficenti costanti"
1162:, p. 284) for a proof (however, in the former reference it is incorrectly stated that the proof is on page 324).
2282:
Sitzungsberichte der Königlich
Bayerischen Akademie der Wissenschaften zu MĂŒnchen, Mathematisch-Physikalische Klasse
1929:
35:
2055:
Characterization of the trace, on the boundary of a domain, of an analytic function of several complex variables
2654:
2504:
2262:
2162:
1643:
2358:, NorthâHolland Mathematical Library, vol. 7 (3rd (Revised) ed.), AmsterdamâLondonâNew YorkâTokyo:
1884:
1847:
478:
1585:
1141:
761:
1369:
1134:, pp. 132â134). In particular, in this last reference on p. 132, the Author explicitly writes :-"
179:), and his ideas were later further explored by Giuliano Bratti. Also the Japanese school of the theory of
1537:
1924:
999:, this (0,1)-form additionally has compact support, so that the Poincaré lemma identifies an appropriate
90:
2310:
1703:
55:
43:
2639:
2489:
2239:
2147:
668:
2066:
Rendiconti Dell' Istituto
Lombardo di Scienze e Lettere. Scienze MatemĂ tiche e Applicazioni, Series A.
538:
2674:
1762:
1684:
1589:
719:
702:
164:
112:
103:: however, the locution "Hartogs's phenomenon" is also used to identify the property of solutions of
51:
39:
2277:
2515:
1639:
Methods of the theory of functions of several complex variables. With a foreword of N.N. Bogolyubov
1413:
104:
94:
2557:
Rendiconti della
Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali
2393:
2022:
Rendiconti della
Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali
1509:
Rendiconti della
Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali
565:
2623:
2473:
2335:
2223:
2131:
2046:
1720:
1637:
1480:
797:— the last in the form that for any smooth and compactly supported differential (0,1)-form
615:
168:
183:
worked much on this topic, with notable contributions by Akira Kaneko. Their approach is to use
2691:
2305:
2278:"Einige Folgerungen aus der Cauchyschen Integralformel bei Funktionen mehrerer VerÀnderlichen."
2529:
2363:
2351:
2301:
2273:
1798:
1541:
1533:
1383:
794:
723:
160:
124:
70:
59:
2631:
2607:
2589:
2573:
2565:
2551:
2539:
2481:
2457:
2423:
2405:
2381:
2327:
2319:
2289:
2251:
2231:
2207:
2199:
2139:
2123:
2115:
2081:
2037:
2005:
1987:
1962:
1946:
1938:
1913:
1876:
1839:
1806:
1788:
1770:
1736:
1712:
1676:
1663:
1617:
1564:
1556:
1524:
1516:
1504:
1497:
1464:
1441:
1425:
1401:
2619:
2469:
2419:
2377:
2219:
2077:
2033:
2001:
1958:
1909:
1872:
1835:
1784:
1732:
1659:
1613:
1476:
1437:
1397:
2658:
2635:
2615:
2577:
2569:
2543:
2508:
2485:
2465:
2439:
2427:
2415:
2385:
2373:
2331:
2293:
2266:
2255:
2235:
2215:
2211:
2166:
2143:
2127:
2085:
2073:
2061:
2041:
2029:
2017:
2009:
1997:
1973:
1966:
1954:
1950:
1917:
1905:
1880:
1868:
1843:
1831:
1810:
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1748:
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1528:
1520:
1493:
1472:
1445:
1433:
1429:
1405:
1393:
152:
144:
642:
1766:
2525:
1793:
595:
518:
20:
1507:(1931), "Risoluzione del problema generale di Dirichlet per le funzioni biarmoniche",
2728:
2627:
2477:
2339:
2227:
2173:
2135:
2097:
1893:
1701:(October 1943), "Analytic and meromorphic continuation by means of Green's formula",
1484:
790:
47:
2064:(1983), "Sul fenomeno di Hartogs per gli operatori lineari alle derivate parziali",
1992:
1576:". This book consist of lecture notes from a course held by Francesco Severi at the
1488:. A historical paper correcting some inexact historical statements in the theory of
1672:
1545:
82:
1942:
2519:
1683:, being different from other ones of the same period due to the extensive use of
1373:
1011:; it only remains to show (following the above comments) that it coincides with
2714:
468:{\displaystyle \Delta ^{2}=\{z\in \mathbb {C} ^{2};|z_{1}|<1,|z_{2}|<1\}}
2718:
2708:
2696:
2178:"Ăber einen Hartogs'schen Satz in der Theorie der analytischen Funktionen von
1751:(March 1, 1952), "Partial Differential Equations and Analytic Continuations",
2410:
1375:
Introduction to the Theory of
Analytic Functions of Several Complex Variables
97:. This property of holomorphic functions of several variables is also called
1651:
1304:
in a nonempty open set. To see the nonemptiness, connect an arbitrary point
1802:
1775:
1596:
Struppa, Daniele C. (1988), "The first eighty years of
Hartogs' theorem",
2359:
2258:, the cumulative review of several papers by E. Trost). Available at the
1927:(1936), "On certain analytic continuations and analytic homeomorphisms",
19:"Hartogs' lemma" redirects here. For the lemma on infinite ordinals, see
2704:
1378:, Translations of Mathematical Monographs, vol. 8, Providence, RI:
2611:
2461:
2323:
2203:
2119:
1822:[About an example of Fichera concerning Hartogs's phenomenon],
1724:
1601:
1468:
701:, which lead to the notion of this Hartogs's extension theorem and the
1820:"A proposito di un esempio di Fichera relativo al fenomeno di Hartogs"
1140:), and as the reader shall soon see, the key tool in the proof is the
991:
and identically equal to one on the complement of some compact subset
1647:
1221:
for the correct attribution of many important theorems in this field.
2651:
2501:
2259:
2184:[On a theorem of Hartogs in the theory of analytic functions of
2159:
2091:
Hartogs phenomenon for certain linear partial differential operators
1716:
2344:
2053:
is solved for general data. A translation of the title reads as:-"
1563:(in Italian), Padova: CEDAM â Casa Editrice Dott. Antonio Milani,
1550:
Solution of the general
Dirichlet problem for biharmonic functions
1421:
1094:
but cannot be continued as a holomorphic function on the whole of
942:
The holomorphicity of this function is identical to the condition
1332:
may have many connected components, but the component containing
69:
complex variables. A first version of this theorem was proved by
2444:"Sopra una dimostrazione di R. Fueter per un teorema di Hartogs"
1753:
16:
Singularities of holomorphic functions extend infinitely outward
1861:
Rendiconti della Accademia Nazionale delle Scienze Detta dei XL
1824:
Rendiconti della Accademia Nazionale delle Scienze Detta dei XL
1898:
Rendiconti del Seminario Matematico della UniversitĂ di Padova
1679:
edition). One of the first modern monographs on the theory of
1580:(which at present bears his name), and includes appendices of
1418:
Topics in the theory of functions of several complex variables
987:
to be a smooth function which is identically equal to zero on
151:). Yet another very simple proof of this result was given by
1976:(1961), "A new proof and an extension of Hartog's theorem",
195:
For example, in two variables, consider the interior domain
2446:[On a proof by R. Fueter of a theorem of Hartogs],
1532:. This is the first paper where a general solution to the
1213:) seem to have been overlooked by many specialists of the
911:
open set, unique continuation (based on connectedness of
817:, there exists a smooth and compactly supported function
2356:
An Introduction to Complex Analysis in Several Variables
847:
is required for the validity of this Poincaré lemma; if
793:, unique continuation of holomorphic functions, and the
1158:, p. 153), which refers the reader to the book of
1122:
and its description in various historical surveys by
789:
Ehrenpreis' proof is based on the existence of smooth
171:: later he extended the theorem to a certain class of
671:
645:
618:
598:
568:
541:
521:
481:
369:
204:
42:
of several variables. Informally, it states that the
1073:. To see this, it suffices to consider the function
143:
with compact support. The latter approach is due to
1209:
Fichera's proof as well as his epoch making paper (
899:). Furthermore, given any holomorphic function on
687:
657:
631:
604:
584:
554:
527:
499:
467:
352:
2581:. An English translation of the title reads as:-"
2089:. An English translation of the title reads as:-"
887:; such an expression is meaningful provided that
777:can be extended to a unique holomorphic function
46:of the singularities of such functions cannot be
1608:â Dipartimento di Matematica, pp. 127â209,
1215:theory of functions of several complex variables
167:of several variables and the related concept of
2051:analytic functions of several complex variables
1420:(unabridged and corrected ed.), New York:
1322:via a line. The intersection of the line with
2692:"failure of Hartogs' theorem in one dimension"
1979:Bulletin of the American Mathematical Society
99:
8:
462:
383:
347:
218:
1492:, particularly concerning contributions of
1490:holomorphic functions of several variables
1155:
148:
2409:
2045:. An epoch-making paper in the theory of
1991:
1792:
1774:
1272:
1260:
1114:
1112:
676:
670:
644:
623:
617:
597:
573:
567:
546:
540:
520:
480:
451:
445:
436:
422:
416:
407:
398:
394:
393:
374:
368:
342:
336:
327:
304:
287:
281:
272:
263:
247:
234:
209:
203:
135:. Today, usual proofs rely on either the
1548:. A translation of the title reads as:-"
592:Namely, there is a holomorphic function
1235:
1231:
1210:
1197:
1193:
1189:
1137:
1131:
1119:
1108:
512:
500:{\displaystyle 0<\varepsilon <1.}
176:
156:
2596:[About a theorem of Hartogs],
1248:
1176:
1136:As it is pointed out in the title of (
1127:
1123:
133:functions of several complex variables
2594:"A proposito d'un teorema di Hartogs"
1578:Istituto Nazionale di Alta Matematica
1218:
1172:
139:or the solution of the inhomogeneous
7:
1863:, serie 5 (in Italian and English),
1826:, serie 5 (in Italian and English),
1159:
854:then it is generally impossible for
137:BochnerâMartinelliâKoppelman formula
2104:[On a theorem of Hartogs],
1896:[On a theorem of Hartogs],
891:is identically equal to zero where
2521:Lehrbuch der Funktionentheorie. II
2049:, where the Dirichlet problem for
1572:. A translation of the title is:-"
1087:, which is clearly holomorphic in
620:
570:
371:
260:
185:Ehrenpreis's fundamental principle
115:satisfying Hartogs-type theorems.
85:literature, it is also called the
73:, and as such it is known also as
14:
2599:Commentarii Mathematici Helvetici
2449:Commentarii Mathematici Helvetici
2191:Commentarii Mathematici Helvetici
2107:Commentarii Mathematici Helvetici
1606:UniversitĂ degli Studi di Bologna
1003:of compact support. This defines
688:{\displaystyle H_{\varepsilon }.}
562:can be analytically continued to
2398:Proceedings of the Japan Academy
1062:Counterexamples in dimension one
555:{\displaystyle H_{\varepsilon }}
363:in the two-dimensional polydisk
159:), by using his solution of the
123:The original proof was given by
2102:"Ăber einen Hartogs'schen Satz"
1993:10.1090/S0002-9904-1961-10661-7
1598:Seminari di Geometria 1987â1988
1066:The theorem does not hold when
147:who initiated it in the paper (
1456:The Mathematical Intelligencer
965:, the differential (0,1)-form
452:
437:
423:
408:
343:
328:
288:
273:
253:
227:
181:partial differential operators
173:partial differential operators
89:, acknowledging later work by
26:In the theory of functions of
1:
1943:10.1215/S0012-7094-36-00203-X
1894:"Su di un teorema di Hartogs"
1380:American Mathematical Society
1336:gives a continuous path from
1251:) and the references therein.
1046:. Thus, on this open subset,
1007:as a holomorphic function on
2740:Theorems in complex analysis
921:) shows that it is equal to
697:Such a phenomenon is called
585:{\displaystyle \Delta ^{2}.}
2680:Encyclopedia of Mathematics
1284:Any connected component of
858:to be compactly supported.
632:{\displaystyle \Delta ^{2}}
515:: Any holomorphic function
32:Hartogs's extension theorem
2756:
1855:Bratti, Giuliano (1986b),
1818:Bratti, Giuliano (1986a),
1118:See the original paper of
961:. For any smooth function
709:Formal statement and proof
190:
18:
2735:Several complex variables
2715:Proof of Hartogs' theorem
2560:, series 6 (in Italian),
2024:, series 8 (in Italian),
1930:Duke Mathematical Journal
1892:Bratti, Giuliano (1988),
1681:several complex variables
1511:, series 6 (in Italian),
840:. The crucial assumption
129:Cauchy's integral formula
34:is a statement about the
28:several complex variables
2188:complex variables],
1130:, pp. 111â115) and
895:is undefined (namely on
141:CauchyâRiemann equations
2673:Chirka, E. M. (2001) ,
1675:review of the original
1586:Giovanni Battista Rizza
1538:pluriharmonic functions
1142:Cauchy integral formula
756:is a compact subset of
2411:10.3792/pja/1195519488
689:
659:
633:
606:
586:
556:
529:
501:
469:
354:
2528:, pp. VIII+307,
2311:Mathematische Annalen
1776:10.1073/pnas.38.3.227
1704:Annals of Mathematics
1692:Scientific references
1685:generalized functions
1540:is given for general
1363:Historical references
1032:is holomorphic since
875:for smooth functions
739:is an open subset of
690:
660:
634:
607:
587:
557:
530:
502:
470:
355:
165:holomorphic functions
113:convolution equations
56:removable singularity
40:holomorphic functions
2182:komplexen Variablen"
1654:, pp. XII+353,
1414:Osgood, William Fogg
720:holomorphic function
703:domain of holomorphy
699:Hartogs's phenomenon
669:
643:
616:
596:
566:
539:
519:
479:
367:
202:
191:Hartogs's phenomenon
109:partial differential
100:Hartogs's phenomenon
87:OsgoodâBrown theorem
52:isolated singularity
2650:. Available at the
2500:. Available at the
2343:. Available at the
2158:. Available at the
1767:1952PNAS...38..227B
1544:on a real analytic
1424:, pp. IV+120,
1382:, pp. vi+374,
1126:, pp. 56â59),
773:is connected, then
658:{\displaystyle F=f}
95:William Fogg Osgood
91:Arthur Barton Brown
79:Hartogs's principle
2657:2012-11-10 at the
2612:10.1007/bf02565650
2507:2012-11-10 at the
2462:10.1007/bf02565649
2324:10.1007/BF01448415
2265:2012-11-10 at the
2204:10.1007/bf02565627
2165:2012-11-10 at the
2120:10.1007/bf01620640
1542:real analytic data
1469:10.1007/BF03024609
1015:on some open set.
983:-closed. Choosing
903:which is equal to
685:
655:
629:
602:
582:
552:
525:
497:
465:
350:
2675:"Hartogs theorem"
2590:Severi, Francesco
2552:Severi, Francesco
2345:DigiZeitschriften
1707:, Second Series,
1630:Vladimirov, V. S.
1557:Severi, Francesco
1534:Dirichlet problem
1505:Severi, Francesco
1318:to some point of
1092: \ {0},
605:{\displaystyle F}
528:{\displaystyle f}
314:
311:
307:
303:
300:
161:Dirichlet problem
125:Friedrich Hartogs
71:Friedrich Hartogs
60:analytic function
2747:
2705:Hartogs' theorem
2701:
2687:
2649:
2648:
2647:
2638:, archived from
2580:
2546:
2499:
2498:
2497:
2488:, archived from
2440:Martinelli, Enzo
2430:
2413:
2388:
2342:
2296:
2249:
2248:
2247:
2238:, archived from
2187:
2181:
2157:
2156:
2155:
2146:, archived from
2088:
2062:Fichera, Gaetano
2044:
2018:Fichera, Gaetano
2012:
1995:
1974:Ehrenpreis, Leon
1969:
1925:Brown, Arthur B.
1920:
1888:
1883:, archived from
1851:
1846:, archived from
1813:
1796:
1778:
1749:Bochner, Salomon
1743:
1699:Bochner, Salomon
1670:
1652:The M.I.T. Press
1624:
1571:
1531:
1498:Francesco Severi
1487:
1448:
1408:
1351:
1349:
1339:
1335:
1331:
1321:
1317:
1307:
1303:
1293:
1282:
1276:
1270:
1264:
1263:, Theorem 2.3.2.
1258:
1252:
1245:
1239:
1228:
1222:
1207:
1201:
1186:
1180:
1169:
1163:
1156:Vladimirov (1966
1154:See for example
1152:
1146:
1116:
1099:
1093:
1086:
1072:
1057:
1053:
1049:
1045:
1035:
1031:
1027:
1014:
1010:
1006:
1002:
998:
994:
990:
986:
982:
981:
976:
972:
964:
960:
956:
946:
938:
924:
920:
906:
902:
898:
894:
890:
886:
882:
878:
874:
864:
857:
853:
846:
839:
831:
826:
820:
816:
811:
806:
800:
784:
780:
776:
772:
759:
755:
751:
744:
738:
734:
717:
694:
692:
691:
686:
681:
680:
664:
662:
661:
656:
638:
636:
635:
630:
628:
627:
611:
609:
608:
603:
591:
589:
588:
583:
578:
577:
561:
559:
558:
553:
551:
550:
534:
532:
531:
526:
506:
504:
503:
498:
474:
472:
471:
466:
455:
450:
449:
440:
426:
421:
420:
411:
403:
402:
397:
379:
378:
359:
357:
356:
351:
346:
341:
340:
331:
312:
309:
308:
305:
301:
298:
291:
286:
285:
276:
268:
267:
252:
251:
239:
238:
214:
213:
68:
2755:
2754:
2750:
2749:
2748:
2746:
2745:
2744:
2725:
2724:
2690:
2672:
2669:
2664:
2659:Wayback Machine
2645:
2643:
2588:
2550:
2536:
2514:
2509:Wayback Machine
2495:
2493:
2438:
2431:, available at
2392:
2370:
2352:Hörmander, Lars
2350:
2300:
2272:
2267:Wayback Machine
2245:
2243:
2185:
2179:
2172:
2167:Wayback Machine
2153:
2151:
2096:
2060:
2016:
1972:
1923:
1891:
1854:
1817:
1747:
1717:10.2307/1969103
1697:
1694:
1628:
1595:
1590:Mario Benedicty
1582:Enzo Martinelli
1555:
1503:
1494:Gaetano Fichera
1452:
1412:
1390:
1368:
1365:
1359:
1354:
1341:
1337:
1333:
1323:
1319:
1309:
1305:
1295:
1294:must intersect
1285:
1283:
1279:
1271:
1267:
1259:
1255:
1247:See his paper (
1246:
1242:
1229:
1225:
1208:
1204:
1187:
1183:
1170:
1166:
1153:
1149:
1117:
1110:
1106:
1095:
1088:
1074:
1067:
1064:
1055:
1051:
1047:
1037:
1033:
1029:
1019:
1012:
1008:
1004:
1000:
996:
992:
988:
984:
979:
978:
970:
966:
962:
954:
944:
943:
930:
922:
912:
904:
900:
896:
892:
888:
884:
880:
876:
866:
862:
861:The ansatz for
855:
848:
841:
829:
828:
822:
818:
809:
808:
802:
798:
782:
778:
774:
764:
757:
753:
746:
740:
736:
726:
715:
711:
672:
667:
666:
641:
640:
619:
614:
613:
594:
593:
569:
564:
563:
542:
537:
536:
517:
516:
477:
476:
441:
412:
392:
370:
365:
364:
332:
277:
259:
243:
230:
205:
200:
199:
193:
153:Gaetano Fichera
149:Ehrenpreis 1961
145:Leon Ehrenpreis
127:in 1906, using
121:
119:Historical note
75:Hartogs's lemma
63:
24:
17:
12:
11:
5:
2753:
2751:
2743:
2742:
2737:
2727:
2726:
2723:
2722:
2712:
2702:
2688:
2668:
2667:External links
2665:
2663:
2662:
2606:(1): 350â352,
2602:(in Italian),
2586:
2548:
2534:
2512:
2456:(1): 340â349,
2452:(in Italian),
2436:
2433:Project Euclid
2390:
2368:
2348:
2302:Hartogs, Fritz
2298:
2274:Hartogs, Fritz
2270:
2198:(1): 394â400,
2174:Fueter, Rudolf
2170:
2098:Fueter, Rudolf
2094:
2068:(in Italian),
2058:
2028:(6): 706â715,
2014:
1986:(5): 507â509,
1970:
1921:
1900:(in Italian),
1889:
1867:(1): 255â259,
1852:
1830:(1): 241â246,
1815:
1761:(3): 227â230,
1745:
1711:(4): 652â673,
1693:
1690:
1689:
1688:
1634:Ehrenpreis, L.
1626:
1593:
1553:
1501:
1450:
1410:
1388:
1364:
1361:
1360:
1358:
1355:
1353:
1352:
1277:
1273:Hörmander 1990
1265:
1261:Hörmander 1990
1253:
1240:
1232:Bratti (1986a)
1223:
1202:
1194:Bratti (1986a)
1190:Fichera (1983)
1181:
1164:
1147:
1120:Hartogs (1906)
1107:
1105:
1102:
1063:
1060:
795:Poincaré lemma
791:bump functions
787:
786:
710:
707:
684:
679:
675:
654:
651:
648:
626:
622:
601:
581:
576:
572:
549:
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524:
513:Hartogs (1906)
496:
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192:
189:
175:in the paper (
155:in the paper (
120:
117:
21:Hartogs number
15:
13:
10:
9:
6:
4:
3:
2:
2752:
2741:
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2693:
2689:
2686:
2682:
2681:
2676:
2671:
2670:
2666:
2660:
2656:
2653:
2642:on 2011-10-02
2641:
2637:
2633:
2629:
2625:
2621:
2617:
2613:
2609:
2605:
2601:
2600:
2595:
2592:(1942â1943),
2591:
2587:
2584:
2579:
2575:
2571:
2567:
2563:
2559:
2558:
2553:
2549:
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2541:
2537:
2535:9780828401821
2531:
2527:
2526:B. G. Teubner
2523:
2522:
2517:
2516:Osgood, W. F.
2513:
2510:
2506:
2503:
2492:on 2011-10-02
2491:
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2483:
2479:
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2442:(1942â1943),
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2437:
2434:
2429:
2425:
2421:
2417:
2412:
2407:
2403:
2399:
2395:
2394:Kaneko, Akira
2391:
2387:
2383:
2379:
2375:
2371:
2369:0-444-88446-7
2365:
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2360:North-Holland
2357:
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2314:(in German),
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2242:on 2011-10-02
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2176:(1941â1942),
2175:
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2161:
2150:on 2011-10-02
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2019:
2015:
2011:
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1899:
1895:
1890:
1887:on 2011-07-26
1886:
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1853:
1850:on 2011-07-26
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1389:9780821886441
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1367:
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1362:
1356:
1348:
1345: \
1344:
1330:
1327: \
1326:
1316:
1313: \
1312:
1302:
1299: \
1298:
1292:
1289: \
1288:
1281:
1278:
1275:, p. 30.
1274:
1269:
1266:
1262:
1257:
1254:
1250:
1244:
1241:
1237:
1233:
1227:
1224:
1220:
1216:
1212:
1206:
1203:
1199:
1195:
1191:
1185:
1182:
1178:
1177:Osgood (1929)
1174:
1168:
1165:
1161:
1157:
1151:
1148:
1144:
1143:
1139:
1133:
1132:Struppa (1988
1129:
1125:
1121:
1115:
1113:
1109:
1103:
1101:
1098:
1091:
1085:
1081:
1077:
1070:
1061:
1059:
1044:
1041: \
1040:
1026:
1023: \
1022:
1016:
975:
969:
959:
953:
949:
940:
937:
934: \
933:
928:
919:
916: \
915:
910:
873:
869:
859:
851:
844:
838:
834:
825:
814:
805:
796:
792:
771:
768: \
767:
763:
749:
743:
733:
730: \
729:
725:
721:
713:
712:
708:
706:
704:
700:
695:
682:
677:
673:
652:
649:
646:
624:
599:
579:
574:
547:
543:
522:
514:
511:
507:
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138:
134:
130:
126:
118:
116:
114:
110:
106:
102:
101:
96:
92:
88:
84:
81:: in earlier
80:
76:
72:
66:
61:
57:
53:
49:
45:
41:
37:
36:singularities
33:
29:
22:
2695:
2678:
2652:SEALS Portal
2644:, retrieved
2640:the original
2603:
2597:
2582:
2561:
2555:
2520:
2502:SEALS Portal
2494:, retrieved
2490:the original
2453:
2447:
2404:(1): 17â19,
2401:
2397:
2355:
2315:
2309:
2285:
2281:
2260:SEALS Portal
2244:, retrieved
2240:the original
2195:
2189:
2160:SEALS Portal
2152:, retrieved
2148:the original
2114:(1): 75â80,
2111:
2105:
2090:
2069:
2065:
2054:
2047:CR-functions
2025:
2021:
1983:
1977:
1934:
1928:
1901:
1897:
1885:the original
1864:
1860:
1848:the original
1827:
1823:
1758:
1752:
1708:
1702:
1673:Zentralblatt
1638:
1597:
1573:
1560:
1549:
1546:hypersurface
1512:
1508:
1460:
1454:
1417:
1374:
1346:
1342:
1328:
1324:
1314:
1310:
1300:
1296:
1290:
1286:
1280:
1268:
1256:
1243:
1236:Bratti 1986b
1226:
1219:Range (2002)
1211:Fichera 1957
1205:
1198:Bratti 1986b
1184:
1173:Brown (1936)
1167:
1150:
1138:Hartogs 1906
1135:
1128:Severi (1958
1124:Osgood (1966
1096:
1089:
1083:
1079:
1075:
1068:
1065:
1042:
1038:
1024:
1020:
1017:
973:
967:
957:
951:
947:
941:
935:
931:
926:
917:
913:
908:
871:
867:
860:
849:
842:
836:
832:
823:
812:
803:
788:
769:
765:
747:
741:
731:
727:
698:
696:
509:
508:
362:
194:
177:Fichera 1983
157:Fichera 1957
122:
98:
86:
78:
74:
64:
54:is always a
31:
25:
2564:: 487â490,
2288:: 223â242,
2072:: 199â211,
1515:: 795â804,
1463:(2): 4â12,
1370:Fuks, B. A.
1249:Kaneko 1973
1018:On the set
169:CR-function
2729:Categories
2719:PlanetMath
2709:PlanetMath
2697:PlanetMath
2646:2011-06-25
2636:0028.15301
2578:0004.40702
2570:58.0352.05
2544:55.0171.02
2496:2011-01-16
2486:0028.15201
2428:0265.35008
2386:0685.32001
2332:37.0444.01
2294:37.0443.01
2256:0060.24505
2250:(see also
2246:2011-01-16
2236:0027.05703
2212:68.0175.02
2154:2011-01-16
2144:0022.05802
2128:65.0363.03
2086:0603.35013
2042:0106.05202
2010:0099.07801
1967:0013.40701
1951:62.0396.02
1918:0657.46033
1881:0646.35008
1844:0646.35007
1811:0046.09902
1741:0060.24206
1668:0125.31904
1622:0657.35018
1569:0094.28002
1529:0002.34202
1521:57.0393.01
1446:0138.30901
1430:45.0661.02
1406:0138.30902
1357:References
1160:Fuks (1963
762:complement
639:such that
2685:EMS Press
2628:120514642
2478:119960691
2354:(1990) ,
2340:122134517
2304:(1906a),
2228:122750611
2136:120266425
1937:: 20â28,
1904:: 59â70,
1644:Cambridge
1485:120531925
1416:(1966) ,
760:. If the
678:ε
621:Δ
571:Δ
548:ε
489:ε
390:∈
372:Δ
322:ε
319:−
296:ε
261:Δ
257:∈
211:ε
2655:Archived
2518:(1929),
2505:Archived
2318:: 1â88,
2276:(1906),
2263:Archived
2163:Archived
1803:16589083
1632:(1966),
1559:(1958),
1372:(1963),
868:φ f
735:, where
58:for any
2620:0010730
2470:0010729
2420:0412578
2378:1045639
2220:0007445
2078:0848259
2034:0093597
2002:0131663
1959:1545903
1910:0964020
1873:0879114
1836:0879111
1794:1063536
1785:0050119
1763:Bibcode
1733:0009206
1725:1969103
1677:Russian
1660:0201669
1636:(ed.),
1614:0973699
1602:Bologna
1477:1907191
1438:0201668
1398:0168793
1050:equals
510:Theorem
105:systems
48:compact
44:support
2634:
2626:
2618:
2576:
2568:
2542:
2532:
2484:
2476:
2468:
2426:
2418:
2384:
2376:
2366:
2338:
2330:
2292:
2254:
2234:
2226:
2218:
2210:
2142:
2134:
2126:
2084:
2076:
2040:
2032:
2008:
2000:
1965:
1957:
1949:
1916:
1908:
1879:
1871:
1842:
1834:
1809:
1801:
1791:
1783:
1739:
1731:
1723:
1666:
1658:
1648:London
1620:
1612:
1567:
1527:
1519:
1483:
1475:
1444:
1436:
1428:
1404:
1396:
1386:
1217:: see
1034:φ
985:φ
974:φ
963:φ
958:φ
889:φ
877:φ
856:η
837:ω
833:η
819:η
813:ω
799:ω
752:) and
475:where
313:
310:
302:
299:
83:Soviet
67:> 1
2624:S2CID
2474:S2CID
2336:S2CID
2224:S2CID
2132:S2CID
1721:JSTOR
1481:S2CID
1422:Dover
1340:into
1104:Notes
827:with
807:with
722:on a
718:be a
2530:ISBN
2364:ISBN
1799:PMID
1754:PNAS
1588:and
1536:for
1496:and
1384:ISBN
1230:See
1192:and
1188:See
1175:and
1171:See
1082:) =
909:some
879:and
714:Let
665:on
492:<
486:<
457:<
428:<
325:<
293:<
163:for
131:for
93:and
77:and
2717:at
2707:at
2632:Zbl
2608:doi
2574:Zbl
2566:JFM
2540:JFM
2482:Zbl
2458:doi
2424:Zbl
2406:doi
2382:Zbl
2328:JFM
2320:doi
2290:JFM
2252:Zbl
2232:Zbl
2208:JFM
2200:doi
2140:Zbl
2124:JFM
2116:doi
2082:Zbl
2070:117
2038:Zbl
2006:Zbl
1988:doi
1963:Zbl
1947:JFM
1939:doi
1914:Zbl
1877:Zbl
1840:Zbl
1807:Zbl
1789:PMC
1771:doi
1737:Zbl
1713:doi
1664:Zbl
1618:Zbl
1565:Zbl
1525:Zbl
1517:JFM
1465:doi
1442:Zbl
1426:JFM
1402:Zbl
1308:of
1071:= 1
995:of
977:is
929:of
927:all
925:on
907:on
883:on
865:is
852:= 1
845:â„ 2
821:on
815:= 0
801:on
781:on
750:â„ 2
724:set
612:on
535:on
111:or
107:of
62:of
38:of
2731::
2694:.
2683:,
2677:,
2630:,
2622:,
2616:MR
2614:,
2604:15
2585:".
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2414:,
2402:49
2400:,
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2362:,
2334:,
2326:,
2316:62
2308:,
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2280:,
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2222:,
2216:MR
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2057:".
2036:,
2030:MR
2026:22
2004:,
1998:MR
1996:,
1984:67
1982:,
1961:,
1955:MR
1953:,
1945:,
1933:,
1912:,
1906:MR
1902:79
1875:,
1869:MR
1838:,
1832:MR
1805:,
1797:,
1787:,
1781:MR
1779:,
1769:,
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1757:,
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1719:,
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1650::
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