1023:
751:
1018:{\displaystyle m={\frac {1}{2\pi i}}\int _{\vert z-z_{0}\vert =\rho }{\frac {f'(z)}{f(z)}}\,dz=\lim _{k\to \infty }{\frac {1}{2\pi i}}\int _{\vert z-z_{0}\vert =\rho }{\frac {f'_{k}(z)}{f_{k}(z)}}\,dz=\lim _{k\to \infty }N_{k}}
718:
271:
606:
1028:
In the above step, we were able to interchange the integral and the limit because of the uniform convergence of the integrand. We have shown that
1221:
614:
17:
200:
1127:
1165:
1205:
1200:
179:
The theorem does not guarantee that the result will hold for arbitrary disks. Indeed, if one chooses a disk such that
157:
320:
39:
1074:
1195:
47:
184:
54:
51:
570:
742:
376:
1161:
1123:
1158:
Complex analysis. An introduction to the theory of analytic functions of one complex variable
1146:
Complex analysis. An introduction to the theory of analytic functions of one complex variable
1183:
1115:
31:
1173:
307:
has exactly one zero in the disk corresponding to the real value 1 − (1/
1215:
342:} a sequence of holomorphic functions which converge uniformly on compact subsets of
84:
58:
1160:, International Series in Pure and Applied Mathematics (3rd ed.), McGraw-Hill,
408:
be an analytic function on an open subset of the complex plane with a zero of order
1153:
1141:
1148:, International Series in Pure and Applied Mathematics (2nd ed.), McGraw-Hill
27:
1190:, second edition (Oxford University Press, 1939; reprinted 1985), p. 119.
57:
functions with that of their corresponding limit. The theorem is named after
324:
188:
77:
43:
16:
This article is about a theorem in complex analysis. For other uses, see
426:} is a sequence of functions converging uniformly on compact subsets to
713:{\displaystyle {\frac {f_{k}'(z)}{f_{k}(z)}}\to {\frac {f'(z)}{f(z)}}.}
608:
uniformly on the disc, and hence we have another uniform convergence:
266:{\displaystyle f_{n}(z)=z-1+{\frac {1}{n}},\qquad z\in \mathbb {C} }
495:) converges uniformly on the disc we have chosen, we can find
187:, the theorem fails. An explicit example is to consider the
76:} be a sequence of holomorphic functions on a connected
1091:
1089:
754:
617:
573:
203:
365:
is either identically zero or also is nowhere zero.
1017:
712:
600:
265:
990:
862:
383:that converge uniformly on compact subsets of
319:Hurwitz's theorem is used in the proof of the
118: > 0 and for sufficiently large
8:
1180:. Springer-Verlag, New York, New York, 1978.
919:
900:
803:
784:
526:on the circle, ensuring that the quotient
1110:
1108:
1009:
993:
979:
961:
937:
930:
913:
899:
877:
865:
851:
814:
797:
783:
761:
753:
670:
649:
625:
618:
616:
578:
572:
259:
258:
238:
208:
202:
160:. Furthermore, these zeroes converge to
1099:
1095:
1085:
23:Limit of roots of sequence of functions
567:. By Weierstrass's theorem we have
442:) ≠ 0 in 0 < |
7:
1178:Functions of One Complex Variable I
1000:
872:
14:
323:, and also has the following two
723:Denoting the number of zeros of
95:which is not constantly zero on
251:
141:zeroes in the disk defined by |
30:and in particular the field of
997:
973:
967:
952:
946:
869:
845:
839:
831:
825:
701:
695:
687:
681:
667:
661:
655:
640:
634:
587:
335:be a connected, open set and {
220:
214:
1:
327:as an immediate consequence:
288: − 1. The function
276:which converges uniformly to
38:is a theorem associating the
1222:Theorems in complex analysis
601:{\displaystyle f_{k}'\to f'}
434: > 0 such that
194:and the sequence defined by
114:then for every small enough
55:locally uniformly convergent
1201:Encyclopedia of Mathematics
1194:Solomentsev, E.D. (2001) ,
1043: → ∞. Since the
453:| ≤ ρ. Choose δ such that |
83:that converge uniformly on
1238:
548:) is well defined for all
387:to a holomorphic function
346:to a holomorphic function
91:to a holomorphic function
15:
395:is univalent or constant.
357:is nonzero everywhere in
379:on a connected open set
296:) contains no zeroes in
1188:The Theory of Functions
321:Riemann mapping theorem
1061:for large enough
1019:
714:
602:
267:
1020:
715:
603:
268:
1050:are integer valued,
752:
615:
571:
419:, and suppose that {
201:
103:has a zero of order
945:
741:, we may apply the
633:
586:
377:univalent functions
375:} is a sequence of
126:(depending on
1015:
1004:
933:
876:
743:argument principle
710:
621:
598:
574:
461:)| >
263:
183:has zeroes on its
1196:"Hurwitz theorem"
1116:Gamelin, Theodore
989:
977:
893:
861:
849:
777:
734:) in the disk by
705:
665:
246:
152:| <
36:Hurwitz's theorem
18:Hurwitz's theorem
1229:
1208:
1184:E. C. Titchmarsh
1170:
1154:Ahlfors, Lars V.
1149:
1142:Ahlfors, Lars V.
1134:
1133:
1120:Complex Analysis
1112:
1103:
1093:
1075:Rouché's theorem
1024:
1022:
1021:
1016:
1014:
1013:
1003:
978:
976:
966:
965:
955:
941:
931:
929:
928:
918:
917:
894:
892:
878:
875:
850:
848:
834:
824:
815:
813:
812:
802:
801:
778:
776:
762:
719:
717:
716:
711:
706:
704:
690:
680:
671:
666:
664:
654:
653:
643:
629:
619:
607:
605:
604:
599:
597:
582:
300:; however, each
272:
270:
269:
264:
262:
247:
239:
213:
212:
171: → ∞.
32:complex analysis
1237:
1236:
1232:
1231:
1230:
1228:
1227:
1226:
1212:
1211:
1193:
1168:
1152:
1140:
1137:
1130:
1114:
1113:
1106:
1098:, p. 176,
1094:
1087:
1083:
1071:
1055:
1048:
1033:
1005:
957:
956:
932:
909:
895:
882:
835:
817:
816:
793:
779:
766:
750:
749:
739:
728:
691:
673:
672:
645:
644:
620:
613:
612:
590:
569:
568:
562:
552:on the circle |
542:
531:
510:)| ≥
504:
489:
479:
469:on the circle |
452:
424:
418:
402:
373:
355:
340:
317:
305:
204:
199:
198:
177:
166:
151:
135:
113:
74:
67:
24:
21:
12:
11:
5:
1235:
1233:
1225:
1224:
1214:
1213:
1210:
1209:
1191:
1181:
1174:John B. Conway
1171:
1166:
1150:
1136:
1135:
1129:978-0387950693
1128:
1104:
1084:
1082:
1079:
1078:
1077:
1070:
1067:
1053:
1046:
1031:
1026:
1025:
1012:
1008:
1002:
999:
996:
992:
988:
985:
982:
975:
972:
969:
964:
960:
954:
951:
948:
944:
940:
936:
927:
924:
921:
916:
912:
908:
905:
902:
898:
891:
888:
885:
881:
874:
871:
868:
864:
860:
857:
854:
847:
844:
841:
838:
833:
830:
827:
823:
820:
811:
808:
805:
800:
796:
792:
789:
786:
782:
775:
772:
769:
765:
760:
757:
737:
726:
721:
720:
709:
703:
700:
697:
694:
689:
686:
683:
679:
676:
669:
663:
660:
657:
652:
648:
642:
639:
636:
632:
628:
624:
596:
593:
589:
585:
581:
577:
563:| =
560:
540:
529:
502:
487:
480:| =
477:
450:
422:
416:
401:
398:
397:
396:
391:, then either
371:
366:
353:
338:
316:
313:
303:
274:
273:
261:
257:
254:
250:
245:
242:
237:
234:
231:
228:
225:
222:
219:
216:
211:
207:
176:
173:
164:
149:
137:has precisely
133:
111:
72:
66:
63:
22:
13:
10:
9:
6:
4:
3:
2:
1234:
1223:
1220:
1219:
1217:
1207:
1203:
1202:
1197:
1192:
1189:
1185:
1182:
1179:
1175:
1172:
1169:
1163:
1159:
1155:
1151:
1147:
1143:
1139:
1138:
1131:
1125:
1121:
1117:
1111:
1109:
1105:
1102:, p. 178
1101:
1097:
1092:
1090:
1086:
1080:
1076:
1073:
1072:
1068:
1066:
1064:
1060:
1056:
1049:
1042:
1038:
1035: →
1034:
1010:
1006:
994:
986:
983:
980:
970:
962:
958:
949:
942:
938:
934:
925:
922:
914:
910:
906:
903:
896:
889:
886:
883:
879:
866:
858:
855:
852:
842:
836:
828:
821:
818:
809:
806:
798:
794:
790:
787:
780:
773:
770:
767:
763:
758:
755:
748:
747:
746:
744:
740:
733:
729:
707:
698:
692:
684:
677:
674:
658:
650:
646:
637:
630:
626:
622:
611:
610:
609:
594:
591:
583:
579:
575:
566:
559:
556: −
555:
551:
547:
543:
536:
532:
525:
521:
518: ≥
517:
514:/2 for every
513:
509:
505:
498:
494:
490:
483:
476:
473: −
472:
468:
464:
460:
456:
449:
446: −
445:
441:
437:
433:
429:
425:
415:
411:
407:
399:
394:
390:
386:
382:
378:
374:
367:
364:
360:
356:
349:
345:
341:
334:
330:
329:
328:
326:
322:
314:
312:
310:
306:
299:
295:
291:
287:
283:
279:
255:
252:
248:
243:
240:
235:
232:
229:
226:
223:
217:
209:
205:
197:
196:
195:
193:
190:
186:
182:
174:
172:
170:
163:
159:
155:
148:
145: −
144:
140:
136:
129:
125:
122: ∈
121:
117:
110:
106:
102:
98:
94:
90:
86:
82:
79:
75:
64:
62:
60:
59:Adolf Hurwitz
56:
53:
49:
45:
41:
37:
33:
29:
19:
1199:
1187:
1177:
1157:
1145:
1122:. Springer.
1119:
1100:Ahlfors 1978
1096:Ahlfors 1966
1062:
1058:
1051:
1044:
1040:
1036:
1029:
1027:
735:
731:
724:
722:
564:
557:
553:
549:
545:
538:
534:
527:
523:
519:
515:
511:
507:
500:
496:
492:
485:
481:
474:
470:
466:
462:
458:
454:
447:
443:
439:
435:
431:
430:. Fix some
427:
420:
413:
409:
405:
403:
392:
388:
384:
380:
369:
362:
358:
351:
347:
343:
336:
332:
318:
315:Applications
308:
301:
297:
293:
289:
285:
281:
277:
275:
191:
180:
178:
168:
161:
158:multiplicity
156:, including
153:
146:
142:
138:
131:
127:
123:
119:
115:
108:
104:
100:
96:
92:
88:
80:
70:
68:
35:
25:
1057:must equal
499:such that |
325:corollaries
87:subsets of
48:holomorphic
28:mathematics
1167:0070006571
1081:References
522:and every
350:. If each
1206:EMS Press
1001:∞
998:→
926:ρ
907:−
897:∫
887:π
873:∞
870:→
810:ρ
791:−
781:∫
771:π
668:→
588:→
284:) =
256:∈
230:−
189:unit disk
65:Statement
1216:Category
1156:(1978),
1144:(1966),
1118:(2001).
1069:See also
943:′
822:′
745:to find
678:′
631:′
595:′
584:′
484:. Since
185:boundary
167:as
78:open set
44:sequence
361:, then
175:Remarks
85:compact
52:compact
1164:
1126:
40:zeroes
400:Proof
99:. If
69:Let {
42:of a
1162:ISBN
1124:ISBN
465:for
404:Let
368:If {
331:Let
130:),
1039:as
991:lim
863:lim
412:at
311:).
107:at
46:of
34:,
26:In
1218::
1204:,
1198:,
1186:,
1176:.
1107:^
1088:^
1065:.
537:)/
533:′(
61:.
50:,
1132:.
1063:k
1059:m
1054:k
1052:N
1047:k
1045:N
1041:k
1037:m
1032:k
1030:N
1011:k
1007:N
995:k
987:=
984:z
981:d
974:)
971:z
968:(
963:k
959:f
953:)
950:z
947:(
939:k
935:f
923:=
920:|
915:0
911:z
904:z
901:|
890:i
884:2
880:1
867:k
859:=
856:z
853:d
846:)
843:z
840:(
837:f
832:)
829:z
826:(
819:f
807:=
804:|
799:0
795:z
788:z
785:|
774:i
768:2
764:1
759:=
756:m
738:k
736:N
732:z
730:(
727:k
725:f
708:.
702:)
699:z
696:(
693:f
688:)
685:z
682:(
675:f
662:)
659:z
656:(
651:k
647:f
641:)
638:z
635:(
627:k
623:f
592:f
580:k
576:f
565:ρ
561:0
558:z
554:z
550:z
546:z
544:(
541:k
539:f
535:z
530:k
528:f
524:z
520:N
516:k
512:δ
508:z
506:(
503:k
501:f
497:N
493:z
491:(
488:k
486:f
482:ρ
478:0
475:z
471:z
467:z
463:δ
459:z
457:(
455:f
451:0
448:z
444:z
440:z
438:(
436:f
432:ρ
428:f
423:n
421:f
417:0
414:z
410:m
406:f
393:f
389:f
385:G
381:G
372:n
370:f
363:f
359:G
354:n
352:f
348:f
344:G
339:n
337:f
333:G
309:n
304:n
302:f
298:D
294:z
292:(
290:f
286:z
282:z
280:(
278:f
260:C
253:z
249:,
244:n
241:1
236:+
233:1
227:z
224:=
221:)
218:z
215:(
210:n
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181:f
169:k
165:0
162:z
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150:0
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143:z
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134:k
132:f
128:ρ
124:N
120:k
116:ρ
112:0
109:z
105:m
101:f
97:G
93:f
89:G
81:G
73:k
71:f
20:.
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