120:
111:
a portion of any line embedded in the complex plane. Images of holomorphic functions can be of real dimension zero (if constant) or two (if non-constant) but never of dimension 1.
714:
757:
739:
167:
29:
78:
The open mapping theorem points to the sharp difference between holomorphy and real-differentiability. On the
135:
is given by the dashed line. Note that all poles are exterior to the open set. The smaller red disk is
692:
697:
430:
377:
102:
72:
52:
8:
373:
315:
735:
334:
171:
17:
183:
751:
702:
95:
33:
727:
361:
119:
79:
60:
456:
will have the same number of roots (counted with multiplicity) in
353:(although this single root may have multiplicity greater than 1).
423:
131:). Black annuli represent poles. The boundary of the open set
107:
101:
The theorem for example implies that a non-constant
337:, and by further decreasing the radius of the disk
202:) has a neighborhood (open disk) which is also in
170:of the complex plane. We have to show that every
749:
329:) is non-constant and holomorphic. The roots of
380:guarantees the existence of a positive minimum
162:is a non-constant holomorphic function and
98:(−1, 1) is the half-open interval [0, 1).
82:, for example, the differentiable function
715:Open mapping theorem (functional analysis)
118:
94:is not an open map, as the image of the
750:
726:
13:
14:
769:
265:> 0 such that the closed disk
686:
63:(i.e. it sends open subsets of
625:) is a subset of the image of
123:Black dots represent zeros of
1:
720:
228:). Then there exists a point
758:Theorems in complex analysis
679:was arbitrary, the function
569:, there exists at least one
349:) has only a single root in
194:), i.e. that every point in
7:
732:Real & Complex Analysis
708:
598:. This means that the disk
10:
774:
693:Maximum modulus principle
644:is an interior point of
360:is a circle and hence a
284:. Consider the function
114:
105:cannot map an open disk
613:The image of the ball
280:is fully contained in
213:Consider an arbitrary
147:
415:the open disk around
378:extreme value theorem
341:, we can assure that
261:is open, we can find
122:
558:|. Thus, for every
494:. This is because
333:are isolated by the
103:holomorphic function
73:invariance of domain
53:holomorphic function
22:open mapping theorem
528:on the boundary of
400:on the boundary of
388:is the minimum of |
374:continuous function
67:to open subsets of
318:of the function.
148:
51:is a non-constant
675:) is open. Since
659:was arbitrary in
372:)| is a positive
765:
744:
698:Rouché's theorem
602:is contained in
431:Rouché's theorem
356:The boundary of
335:identity theorem
18:complex analysis
773:
772:
768:
767:
766:
764:
763:
762:
748:
747:
742:
734:, McGraw-Hill,
723:
711:
689:
667:) we know that
658:
643:
597:
590:
575:
564:
557:
550:
523:
516:
489:
482:
455:
433:, the function
421:
313:
306:
275:
256:
245:
234:
219:
145:
117:
24:states that if
12:
11:
5:
771:
761:
760:
746:
745:
740:
722:
719:
718:
717:
710:
707:
706:
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700:
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685:
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521:
514:
487:
480:
453:
419:
311:
304:
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254:
243:
232:
217:
184:interior point
143:
139:, centered at
116:
113:
71:, and we have
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2:
770:
759:
756:
755:
753:
743:
741:0-07-054234-1
737:
733:
729:
728:Rudin, Walter
725:
724:
716:
713:
712:
704:
703:Schwarz lemma
701:
699:
696:
694:
691:
690:
684:
682:
678:
674:
670:
666:
662:
655:
651:
647:
640:
636:
632:
628:
624:
620:
616:
611:
609:
605:
601:
594:
587:
583:
579:
572:
568:
561:
554:
547:
543:
539:
535:
531:
527:
520:
513:
509:
505:
501:
497:
493:
486:
479:
475:
471:
467:
463:
459:
452:
448:
444:
440:
436:
432:
428:
425:
418:
414:
409:
407:
403:
399:
395:
391:
387:
383:
379:
375:
371:
367:
363:
359:
354:
352:
348:
344:
340:
336:
332:
328:
324:
321:We know that
319:
317:
310:
303:
299:
295:
291:
287:
283:
279:
272:
268:
264:
260:
253:
249:
242:
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231:
227:
223:
216:
211:
209:
205:
201:
197:
193:
189:
185:
181:
177:
173:
169:
165:
161:
157:
153:
142:
138:
134:
130:
126:
121:
112:
110:
109:
104:
99:
97:
96:open interval
93:
89:
85:
81:
76:
74:
70:
66:
62:
58:
54:
50:
46:
42:
38:
35:
34:complex plane
31:
27:
23:
19:
731:
687:Applications
680:
676:
672:
668:
664:
660:
653:
649:
645:
638:
634:
630:
626:
622:
618:
614:
612:
607:
603:
599:
592:
585:
581:
577:
570:
566:
559:
552:
545:
541:
537:
533:
529:
525:
518:
511:
507:
503:
499:
495:
491:
484:
477:
473:
469:
465:
461:
457:
450:
446:
442:
438:
434:
426:
416:
412:
410:
405:
401:
397:
393:
389:
385:
381:
369:
365:
364:, on which |
357:
355:
350:
346:
342:
338:
330:
326:
322:
320:
308:
307:. Note that
301:
297:
293:
289:
285:
281:
277:
276:with radius
270:
266:
262:
258:
251:
247:
240:
236:
229:
225:
221:
214:
212:
207:
203:
199:
195:
191:
187:
179:
175:
163:
159:
155:
151:
149:
140:
136:
132:
128:
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106:
100:
91:
87:
83:
77:
68:
64:
56:
48:
44:
40:
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25:
21:
15:
524:), and for
384:, that is,
362:compact set
721:References
580:such that
411:Denote by
239:such that
683:is open.
652:). Since
408:> 0.
376:, so the
257:). Since
80:real line
752:Category
730:(1966),
709:See also
637:). Thus
483:for any
182:) is an
154: :
61:open map
43: :
396:)| for
269:around
150:Assume
55:, then
32:of the
738:
544:> |
424:radius
168:domain
59:is an
30:domain
20:, the
540:)| ≥
510:) + (
429:. By
422:with
314:is a
172:point
166:is a
115:Proof
28:is a
736:ISBN
610:).
591:) =
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441:) =
404:and
316:root
292:) =
108:onto
90:) =
75:.).
39:and
576:in
565:in
532:, |
490:in
468:):=
460:as
235:in
220:in
210:).
186:of
174:in
16:In
754::
629:,
617:,
551:-
517:-
476:)−
449:)−
300:)−
246:=
158:→
47:→
681:f
677:U
673:U
671:(
669:f
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663:(
661:f
657:0
654:w
650:U
648:(
646:f
642:0
639:w
635:U
633:(
631:f
627:U
623:B
621:(
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615:B
608:B
606:(
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600:D
596:1
593:w
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586:z
584:(
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571:z
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563:1
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546:w
542:e
538:z
536:(
534:g
530:B
526:z
522:1
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515:0
512:w
508:z
506:(
504:g
500:z
498:(
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488:1
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146:.
144:0
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137:B
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127:(
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92:x
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86:(
84:f
69:C
65:U
57:f
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45:U
41:f
37:C
26:U
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