204:
91:
313:
357:
464:
435:
484:
401:
381:
232:
144:
Starting with the extra hypothesis that the function is continuous (or bounded), the theorem is much easier to prove and in this form is known as
606:
541:
517:
43:
601:
156:
162:
20:
141:). Therefore, 'separate analyticity' and 'analyticity' are coincident notions, in the theory of several complex variables.
567:
49:
240:
562:
557:
207:
585:
46:. Roughly speaking, it states that a 'separately analytic' function is continuous. More precisely, if
408:
404:
119:
360:
19:"Hartogs's theorem" redirects here. For the theorem on extensions of holomorphic functions, see
577:
537:
527:
513:
321:
211:
145:
94:
39:
507:
495:
138:
440:
414:
469:
386:
366:
217:
24:
595:
531:
234:
will necessarily be continuous. A counterexample in two dimensions is given by
31:
581:
126:
152:
486:
to include the origin and have the function be continuous there.)
466:
are not equal, so there is no way to extend the definition of
199:{\displaystyle f\colon {\textbf {R}}^{n}\to {\textbf {R}}}
509:
Theory of
Analytic Functions of Several Complex Variables
502:, AMS Chelsea Publishing, Providence, Rhode Island, 1992.
533:
472:
443:
417:
389:
369:
324:
243:
220:
165:
114:, while the other variables are held constant, then
52:
86:{\displaystyle F:{\textbf {C}}^{n}\to {\textbf {C}}}
214:) in each variable separately, it is not true that
478:
458:
429:
395:
375:
351:
307:
226:
198:
85:
308:{\displaystyle f(x,y)={\frac {xy}{x^{2}+y^{2}}}.}
586:Creative Commons Attribution/Share-Alike License
8:
500:Function Theory of Several Complex Variables
137:-variable sense (i.e. that locally it has a
133:is then in fact an analytic function in the
23:. For the theorem on infinite ordinals, see
578:Hartogs's theorem on separate analyticity
471:
442:
416:
388:
368:
323:
293:
280:
265:
242:
219:
190:
189:
180:
174:
173:
164:
77:
76:
67:
61:
60:
51:
576:This article incorporates material from
7:
191:
175:
78:
62:
14:
512:. American Mathematical Society.
359:, this function has well-defined
159:. If we assume that a function
536:(3rd ed.), North Holland,
506:Fuks, Boris Abramovich (1963).
584:, which is licensed under the
340:
328:
259:
247:
186:
73:
1:
403:at the origin, but it is not
151:There is no analogue of this
607:Theorems in complex analysis
563:Encyclopedia of Mathematics
38:is a fundamental result of
21:Hartogs's extension theorem
623:
18:
602:Several complex variables
318:If in addition we define
44:several complex variables
407:at origin. (Indeed, the
352:{\displaystyle f(0,0)=0}
93:is a function which is
480:
460:
431:
397:
377:
353:
309:
228:
200:
87:
481:
461:
432:
398:
378:
354:
310:
229:
201:
129:is that the function
88:
470:
459:{\displaystyle x=-y}
441:
415:
387:
367:
322:
241:
218:
163:
50:
16:Mathematical theorem
430:{\displaystyle x=y}
361:partial derivatives
120:continuous function
476:
456:
427:
393:
373:
349:
305:
224:
196:
83:
558:"Hartogs theorem"
543:978-1-493-30273-4
519:978-1-4704-4428-0
479:{\displaystyle f}
396:{\displaystyle y}
376:{\displaystyle x}
300:
227:{\displaystyle f}
193:
177:
97:in each variable
80:
64:
42:in the theory of
40:Friedrich Hartogs
36:Hartogs's theorem
614:
571:
546:
523:
496:Steven G. Krantz
485:
483:
482:
477:
465:
463:
462:
457:
436:
434:
433:
428:
411:along the lines
402:
400:
399:
394:
382:
380:
379:
374:
358:
356:
355:
350:
314:
312:
311:
306:
301:
299:
298:
297:
285:
284:
274:
266:
233:
231:
230:
225:
205:
203:
202:
197:
195:
194:
185:
184:
179:
178:
139:Taylor expansion
92:
90:
89:
84:
82:
81:
72:
71:
66:
65:
622:
621:
617:
616:
615:
613:
612:
611:
592:
591:
556:
553:
544:
528:Hörmander, Lars
526:
520:
505:
492:
468:
467:
439:
438:
413:
412:
385:
384:
365:
364:
320:
319:
289:
276:
275:
267:
239:
238:
216:
215:
172:
161:
160:
105:
59:
48:
47:
28:
17:
12:
11:
5:
620:
618:
610:
609:
604:
594:
593:
573:
572:
552:
551:External links
549:
548:
547:
542:
524:
518:
503:
491:
488:
475:
455:
452:
449:
446:
426:
423:
420:
392:
372:
348:
345:
342:
339:
336:
333:
330:
327:
316:
315:
304:
296:
292:
288:
283:
279:
273:
270:
264:
261:
258:
255:
252:
249:
246:
223:
208:differentiable
188:
183:
171:
168:
157:real variables
146:Osgood's lemma
101:
75:
70:
58:
55:
25:Hartogs number
15:
13:
10:
9:
6:
4:
3:
2:
619:
608:
605:
603:
600:
599:
597:
590:
589:
587:
583:
579:
569:
565:
564:
559:
555:
554:
550:
545:
539:
535:
534:
529:
525:
521:
515:
511:
510:
504:
501:
497:
494:
493:
489:
487:
473:
453:
450:
447:
444:
424:
421:
418:
410:
406:
390:
370:
362:
346:
343:
337:
334:
331:
325:
302:
294:
290:
286:
281:
277:
271:
268:
262:
256:
253:
250:
244:
237:
236:
235:
221:
213:
209:
181:
169:
166:
158:
154:
149:
147:
142:
140:
136:
132:
128:
123:
121:
117:
113:
109:
104:
100:
96:
68:
56:
53:
45:
41:
37:
33:
26:
22:
575:
574:
561:
532:
508:
499:
317:
150:
143:
134:
130:
124:
115:
111:
107:
106:, 1 ≤
102:
98:
35:
29:
32:mathematics
596:Categories
582:PlanetMath
490:References
405:continuous
568:EMS Press
530:(1990) ,
451:−
210:(or even
187:→
170::
127:corollary
74:→
212:analytic
110:≤
95:analytic
570:, 2001
153:theorem
540:
516:
409:limits
118:is a
538:ISBN
514:ISBN
437:and
383:and
155:for
580:on
363:in
206:is
30:In
598::
566:,
560:,
498:.
148:.
125:A
122:.
34:,
588:.
522:.
474:f
454:y
448:=
445:x
425:y
422:=
419:x
391:y
371:x
347:0
344:=
341:)
338:0
335:,
332:0
329:(
326:f
303:.
295:2
291:y
287:+
282:2
278:x
272:y
269:x
263:=
260:)
257:y
254:,
251:x
248:(
245:f
222:f
192:R
182:n
176:R
167:f
135:n
131:F
116:F
112:n
108:i
103:i
99:z
79:C
69:n
63:C
57::
54:F
27:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.