17:
981:
666:
1243:
1297:
1194:
1140:
1398:
170:
611:
252:
790:
220:
1083:
712:
103:
1431:
735:
133:
1037:
1007:
876:
1358:
1331:
757:
686:
574:
549:
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502:
479:
447:
295:
68:
409:
361:
916:
896:
850:
830:
810:
381:
335:
315:
275:
190:
924:
1039:, i.e. constructing a global holomorphic function which admits no extension from non-extendable functions defined only locally. This is called the
1056:
616:
1202:
1248:
29:
1145:
1407:
is always solvable in a domain of holomorphy; this is also true, with additional topological assumptions, for the second
1485:
1099:
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138:
420:
583:
225:
766:
199:
527:
1059:
691:
77:
717:
112:
1016:
986:
1421:
855:
1336:
1309:
411:
case, every open set is a domain of holomorphy: we can define a holomorphic function with zeros
742:
671:
559:
534:
509:
487:
464:
426:
280:
53:
41:
1055:
using methods from functional analysis and partial differential equations (a consequence of
1300:
255:
37:
8:
416:
388:
340:
1469:
901:
881:
835:
815:
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412:
366:
320:
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260:
175:
1426:
1044:
450:
1052:
16:
976:{\displaystyle 1\Leftrightarrow 2,3\Leftrightarrow 4,1\Rightarrow 4,3\Rightarrow 5}
1408:
1404:
193:
1436:
1010:
1479:
1040:
552:
1048:
25:
1465:
1245:
is an ascending sequence of domains of holomorphy, then their union
661:{\displaystyle S_{n}\rightarrow S,\partial S_{n}\rightarrow \Gamma }
737:
cannot be "touched from inside" by a sequence of analytic surfaces)
48:
1464:
This article incorporates material from Domain of holomorphy on
1238:{\displaystyle \Omega _{1}\subseteq \Omega _{2}\subseteq \dots }
36:
is a domain which is maximal in the sense that there exists a
1292:{\displaystyle \Omega =\bigcup _{n=1}^{\infty }\Omega _{n}}
1452:, AMS Chelsea Publishing, Providence, Rhode Island, 1992.
1366:
1339:
1312:
1251:
1205:
1189:{\displaystyle \Omega =\bigcap _{j=1}^{n}\Omega _{j}}
1148:
1102:
1062:
1019:
989:
927:
904:
884:
858:
838:
818:
798:
769:
745:
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619:
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429:
391:
369:
343:
323:
303:
283:
263:
228:
202:
178:
141:
115:
80:
56:
1142:are domains of holomorphy, then their intersection
1392:
1352:
1325:
1291:
1237:
1188:
1134:
1077:
1031:
1001:
975:
910:
890:
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844:
824:
804:
784:
751:
729:
706:
680:
660:
605:
568:
543:
518:
496:
473:
441:
423:for a domain of definition of its reciprocal. For
403:
375:
355:
329:
309:
289:
269:
246:
214:
184:
164:
127:
97:
62:
1477:
1470:Creative Commons Attribution/Share-Alike License
1135:{\displaystyle \Omega _{1},\dots ,\Omega _{n}}
1393:{\displaystyle \Omega _{1}\times \Omega _{2}}
1450:Function Theory of Several Complex Variables
898:cannot be extended to any neighbourhood of
449:this is no longer true, as it follows from
165:{\displaystyle V\subset {\mathbb {C} }^{n}}
109:if there do not exist non-empty open sets
481:the following conditions are equivalent:
151:
84:
456:
15:
1013:). The main difficulty lies in proving
613:of analytic compact surfaces such that
1478:
606:{\displaystyle S_{n}\subseteq \Omega }
247:{\displaystyle U\subset \Omega \cap V}
785:{\displaystyle x\in \partial \Omega }
215:{\displaystyle V\not \subset \Omega }
1299:is also a domain of holomorphy (see
419:of the domain, which must then be a
297:there exists a holomorphic function
13:
1381:
1368:
1341:
1314:
1280:
1274:
1252:
1220:
1207:
1177:
1149:
1123:
1104:
1078:{\displaystyle {\bar {\partial }}}
1066:
865:
779:
776:
746:
724:
721:
707:{\displaystyle S\subseteq \Omega }
701:
675:
655:
639:
600:
563:
538:
513:
491:
468:
284:
235:
209:
122:
98:{\displaystyle {\mathbb {C} }^{n}}
57:
14:
1497:
1457:Introduction to Complex Analysis
1360:are domains of holomorphy, then
730:{\displaystyle \partial \Omega }
128:{\displaystyle U\subset \Omega }
28:, in the theory of functions of
1196:is also a domain of holomorphy.
40:on this domain which cannot be
1468:, which is licensed under the
1069:
1032:{\displaystyle 5\Rightarrow 1}
1023:
1002:{\displaystyle 1\Rightarrow 3}
993:
967:
955:
943:
931:
652:
630:
1:
1442:
1090:
871:{\displaystyle U\cap \Omega }
1455:Boris Vladimirovich Shabat,
1432:solution of the Levi problem
792:there exist a neighbourhood
7:
1415:
1353:{\displaystyle \Omega _{2}}
1326:{\displaystyle \Omega _{1}}
74:-dimensional complex space
20:The sets in the definition.
10:
1502:
1400:is a domain of holomorphy.
1047:) and was first solved by
983:are standard results (for
1486:Several complex variables
504:is a domain of holomorphy
30:several complex variables
752:{\displaystyle \Omega }
681:{\displaystyle \Gamma }
569:{\displaystyle \Omega }
544:{\displaystyle \Omega }
519:{\displaystyle \Omega }
497:{\displaystyle \Omega }
474:{\displaystyle \Omega }
442:{\displaystyle n\geq 2}
290:{\displaystyle \Omega }
63:{\displaystyle \Omega }
1394:
1354:
1327:
1293:
1278:
1239:
1190:
1175:
1136:
1079:
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872:
846:
826:
806:
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753:
731:
708:
682:
662:
607:
570:
545:
528:holomorphically convex
520:
498:
475:
443:
405:
377:
357:
331:
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271:
248:
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166:
129:
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21:
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1004:
978:
913:
893:
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847:
827:
807:
787:
754:
732:
709:
683:
663:
608:
580:- for every sequence
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521:
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457:Equivalent conditions
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406:
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332:
312:
292:
272:
249:
217:
187:
167:
130:
100:
65:
44:to a bigger domain.
19:
1422:Behnke–Stein theorem
1364:
1337:
1310:
1301:Behnke-Stein theorem
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987:
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256:holomorphic function
254:such that for every
226:
200:
176:
139:
113:
107:domain of holomorphy
78:
54:
38:holomorphic function
34:domain of holomorphy
761:local Levi property
404:{\displaystyle n=1}
356:{\displaystyle f=g}
1448:Steven G. Krantz.
1390:
1350:
1323:
1289:
1235:
1186:
1132:
1075:
1029:
999:
973:
908:
888:
868:
842:
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763:- for every point
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415:everywhere on the
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287:
267:
244:
212:
182:
162:
125:
95:
60:
22:
1427:Levi pseudoconvex
1072:
911:{\displaystyle x}
891:{\displaystyle f}
845:{\displaystyle f}
825:{\displaystyle x}
805:{\displaystyle U}
376:{\displaystyle U}
330:{\displaystyle V}
310:{\displaystyle g}
270:{\displaystyle f}
185:{\displaystyle V}
1493:
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1111:
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421:natural boundary
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1223:
1219:
1210:
1206:
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1201:
1200:
1180:
1176:
1170:
1159:
1147:
1144:
1143:
1126:
1122:
1107:
1103:
1101:
1098:
1097:
1093:
1064:
1063:
1061:
1058:
1057:
1018:
1015:
1014:
988:
985:
984:
926:
923:
922:
903:
900:
899:
883:
880:
879:
857:
854:
853:
852:holomorphic on
837:
834:
833:
817:
814:
813:
797:
794:
793:
768:
765:
764:
744:
741:
740:
719:
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693:
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669:
646:
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591:
587:
585:
582:
581:
561:
558:
557:
536:
533:
532:
511:
508:
507:
489:
486:
485:
466:
463:
462:
459:
428:
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424:
390:
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368:
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364:
342:
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338:
322:
319:
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302:
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298:
282:
279:
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227:
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223:
201:
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174:
173:
156:
150:
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140:
137:
136:
114:
111:
110:
89:
83:
82:
81:
79:
76:
75:
55:
52:
51:
12:
11:
5:
1499:
1489:
1488:
1461:
1460:
1453:
1444:
1441:
1440:
1439:
1437:Stein manifold
1434:
1429:
1424:
1417:
1414:
1413:
1412:
1409:Cousin problem
1405:Cousin problem
1401:
1387:
1383:
1379:
1374:
1370:
1347:
1343:
1320:
1316:
1304:
1286:
1282:
1276:
1271:
1268:
1265:
1261:
1257:
1254:
1234:
1231:
1226:
1222:
1218:
1213:
1209:
1197:
1183:
1179:
1173:
1168:
1165:
1162:
1158:
1154:
1151:
1129:
1125:
1121:
1118:
1115:
1110:
1106:
1092:
1089:
1071:
1068:
1053:Lars Hörmander
1051:, and then by
1028:
1025:
1022:
998:
995:
992:
972:
969:
966:
963:
960:
957:
954:
951:
948:
945:
942:
939:
936:
933:
930:
919:
918:
907:
887:
867:
864:
861:
841:
821:
801:
781:
778:
775:
772:
748:
738:
726:
723:
703:
700:
697:
677:
657:
654:
649:
645:
641:
638:
635:
632:
627:
623:
602:
599:
594:
590:
565:
555:
540:
530:
515:
505:
493:
470:
458:
455:
451:Hartogs' lemma
438:
435:
432:
400:
397:
394:
372:
352:
349:
346:
326:
306:
286:
266:
243:
240:
237:
234:
231:
211:
208:
205:
181:
159:
153:
147:
144:
124:
121:
118:
92:
86:
59:
47:Formally, an
9:
6:
4:
3:
2:
1498:
1487:
1484:
1483:
1481:
1474:
1473:
1471:
1467:
1458:
1454:
1451:
1447:
1446:
1438:
1435:
1433:
1430:
1428:
1425:
1423:
1420:
1419:
1410:
1406:
1402:
1385:
1377:
1372:
1345:
1318:
1305:
1302:
1284:
1269:
1266:
1263:
1259:
1255:
1232:
1229:
1224:
1216:
1211:
1198:
1181:
1171:
1166:
1163:
1160:
1156:
1152:
1127:
1119:
1116:
1113:
1108:
1095:
1094:
1088:
1086:
1054:
1050:
1046:
1042:
1026:
1020:
1012:
996:
990:
970:
964:
961:
958:
952:
949:
946:
940:
937:
934:
928:
921:Implications
905:
885:
862:
859:
839:
819:
799:
773:
770:
762:
739:
698:
695:
668:for some set
647:
643:
636:
633:
625:
621:
597:
592:
588:
579:
556:
554:
531:
529:
506:
484:
483:
482:
461:For a domain
454:
452:
436:
433:
430:
422:
418:
414:
398:
395:
392:
383:
370:
350:
347:
344:
324:
304:
264:
257:
241:
238:
232:
229:
206:
203:
195:
179:
157:
145:
142:
119:
116:
108:
90:
73:
50:
45:
43:
39:
35:
31:
27:
18:
1463:
1462:
1456:
1449:
1041:Levi problem
920:
760:
577:
553:pseudoconvex
460:
413:accumulating
384:
106:
105:is called a
71:
46:
33:
23:
1459:, AMS, 1992
1049:Kiyoshi Oka
1011:Oka's lemma
578:Levi convex
26:mathematics
1466:PlanetMath
1443:References
1403:The first
1091:Properties
1045:E. E. Levi
878:such that
1382:Ω
1378:×
1369:Ω
1342:Ω
1315:Ω
1281:Ω
1275:∞
1260:⋃
1253:Ω
1233:…
1230:⊆
1221:Ω
1217:⊆
1208:Ω
1178:Ω
1157:⋂
1150:Ω
1124:Ω
1117:…
1105:Ω
1070:¯
1067:∂
1024:⇒
994:⇒
968:⇒
956:⇒
944:⇔
932:⇔
866:Ω
863:∩
780:Ω
777:∂
774:∈
747:Ω
725:Ω
722:∂
702:Ω
699:⊆
676:Γ
656:Γ
653:→
640:∂
631:→
601:Ω
598:⊆
564:Ω
539:Ω
514:Ω
492:Ω
469:Ω
434:≥
285:Ω
239:∩
236:Ω
233:⊂
210:Ω
194:connected
146:⊂
123:Ω
120:⊂
58:Ω
1480:Category
1416:See also
1085:-problem
688:we have
417:boundary
207:⊄
49:open set
42:extended
1043:(after
385:In the
70:in the
1009:, see
172:where
1333:and
337:with
222:and
135:and
832:and
759:has
32:, a
1306:If
1199:If
1096:If
1087:).
812:of
576:is
551:is
526:is
453:.
363:on
317:on
277:on
196:,
192:is
24:In
1482::
1303:).
1472:.
1411:.
1386:2
1373:1
1346:2
1319:1
1285:n
1270:1
1267:=
1264:n
1256:=
1225:2
1212:1
1182:j
1172:n
1167:1
1164:=
1161:j
1153:=
1128:n
1120:,
1114:,
1109:1
1027:1
1021:5
997:3
991:1
971:5
965:3
962:,
959:4
953:1
950:,
947:4
941:3
938:,
935:2
929:1
906:x
886:f
860:U
840:f
820:x
800:U
771:x
714:(
696:S
648:n
644:S
637:,
634:S
626:n
622:S
593:n
589:S
437:2
431:n
399:1
396:=
393:n
371:U
351:g
348:=
345:f
325:V
305:g
265:f
242:V
230:U
204:V
180:V
158:n
152:C
143:V
117:U
91:n
85:C
72:n
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
