20:
2277:
841:
1865:
is maximized to any other point in the domain, while being totally contained within the domain. Thus the existence of a maximum value implies that all the values in the domain are the same, thus
79:
centered at the origin (shown in blue). As predicted by the theorem, the maximum of the modulus cannot be inside of the disk (so the highest value on the red surface is somewhere along its edge).
1561:
2073:
3190:
Conway, John B. (1978). Axler, S.; Gehring, F.W.; Ribet, K.A. (eds.). Functions of One
Complex Variable I (2 ed.). New York: Springer Science+Business Media, Inc. ISBN 978-1-4612-6314-2.
2410:
2154:
1330:
2856:
2625:
2493:
512:
2808:
2740:
1973:
1836:
to "force" all points within overlapping open disks to assume the same value as the maximum. The disks are laid such that their centers form a polygonal path from the value where
1047:
950:
780:
1203:
753:
391:
3029:
1725:
1613:
2946:
2672:
2577:
53:
2972:
2698:
2025:
1999:
1793:
1687:
1133:
993:
900:
2321:
1644:
1384:
1230:
1074:
1020:
566:
418:
279:
232:
1892:
1863:
1822:
1754:
2882:
2766:
1455:
643:
309:
150:
3071:, the heat at this maximum would be dispersing to the points around it, which would contradict the assumption that this represents the steady state of a system.
3069:
3049:
2902:
2533:
2513:
1931:
1424:
1404:
1353:
1250:
1181:
1161:
1094:
920:
861:
800:
731:
703:
683:
663:
606:
586:
535:
438:
366:
337:
252:
201:
178:
113:
73:
612:
3097:
3234:
3141:
2162:
3104:
3229:
3114:, a result about the behaviour of bounded holomorphic functions on a line between two other parallel lines in the complex plane.
3084:
805:
3184:
1690:
3111:
1833:
3179:
1476:
2030:
3156:
2326:
2078:
1258:
3174:
2813:
2582:
1356:
538:
2415:
446:
2771:
2703:
1936:
3079:
The maximum modulus principle has many uses in complex analysis, and may be used to prove the following:
1025:
928:
758:
157:
116:
1186:
736:
374:
2989:
3090:
1573:
2907:
2633:
2538:
3202:
3137:
1647:
1616:
1567:
204:
3205:
26:
2951:
2677:
2004:
1978:
1759:
1653:
1099:
959:
866:
92:
2285:
1696:
1622:
1362:
1208:
1052:
998:
544:
396:
257:
210:
19:
1868:
1839:
1798:
1730:
2861:
2745:
1432:
618:
284:
125:
3093:, a result which in turn has many generalisations and applications in complex analysis.
3054:
3034:
2887:
2518:
2498:
1916:
1409:
1389:
1338:
1235:
1166:
1146:
1079:
905:
846:
785:
716:
688:
668:
648:
615:, which states that a nonconstant holomorphic function maps open sets to open sets: If
591:
571:
520:
423:
351:
322:
237:
186:
163:
120:
98:
58:
3223:
3160:
2983:
1795:
can only have a local minimum (which necessarily has value 0) at an isolated zero of
953:
369:
340:
84:
2075:. We then define the boundary of the closed ball with positive orientation as
3210:
76:
343:
1076:
is not on the boundary, then the maximum modulus principle implies that
153:
3132:
Conway, John B. (1978). Axler, S.; Gehring, F.W.; Ribet, K.A. (eds.).
3031:
is harmonic, it is thus the steady state of a heat flow on the region
2272:{\displaystyle 0\leq \int _{0}^{2\pi }|f(a)|-|f(a+re^{it})|\,dt\leq 0}
346:
2700:. Then one can construct a sequence of distinct points located in
18:
1646:
is a local maximum for this function also, it follows from the
3136:(2 ed.). New York: Springer Science+Business Media, Inc.
3107:, which bounds an analytic function in terms of its real part.
665:, then the image of a sufficiently small open neighborhood of
1135:
also attains the same maximum at any point of the boundary.
3051:. Suppose a strict maximum was attained on the interior of
2982:
A physical interpretation of this principle comes from the
836:{\displaystyle f\colon {\overline {D}}\to \mathbb {C} }
925:
This follows from the first version as follows. Since
611:
This statement can be viewed as a special case of the
3057:
3037:
2992:
2954:
2910:
2890:
2864:
2816:
2774:
2748:
2706:
2680:
2636:
2585:
2541:
2521:
2501:
2418:
2329:
2288:
2165:
2081:
2033:
2007:
1981:
1939:
1919:
1871:
1842:
1801:
1762:
1733:
1699:
1656:
1625:
1576:
1479:
1435:
1412:
1392:
1365:
1341:
1261:
1238:
1211:
1189:
1169:
1149:
1102:
1082:
1055:
1028:
1001:
962:
931:
908:
869:
849:
808:
788:
761:
739:
719:
691:
671:
651:
621:
594:
574:
547:
523:
449:
426:
399:
377:
354:
325:
287:
260:
240:
213:
189:
166:
128:
101:
61:
29:
2495:. This also holds for all balls of radius less than
2810:is closed, the sequence converges to some point in
1756:is constant as well. Similar reasoning shows that
902:attains a maximum at some point of the boundary of
3063:
3043:
3023:
2966:
2940:
2896:
2876:
2850:
2802:
2760:
2734:
2692:
2666:
2619:
2571:
2527:
2507:
2487:
2404:
2315:
2271:
2148:
2067:
2019:
1993:
1967:
1925:
1886:
1857:
1816:
1787:
1748:
1719:
1681:
1638:
1607:
1555:
1466:Using the maximum principle for harmonic functions
1449:
1418:
1398:
1378:
1347:
1324:
1244:
1224:
1197:
1175:
1155:
1127:
1088:
1068:
1041:
1014:
987:
944:
914:
894:
855:
835:
794:
774:
747:
725:
697:
677:
657:
637:
600:
580:
560:
529:
506:
432:
412:
385:
360:
331:
303:
273:
246:
226:
195:
172:
144:
107:
67:
47:
2156:. Invoking Cauchy's integral formula, we obtain
843:is a continuous function that is holomorphic on
1556:{\displaystyle \log f(z)=\ln |f(z)|+i\arg f(z)}
733:is a bounded nonempty connected open subset of
1429:Proof: Apply the maximum modulus principle to
254:there exist other points arbitrarily close to
2068:{\displaystyle {\overline {B}}(a,r)\subset D}
8:
2405:{\displaystyle |f(a)|-|f(a+re^{it})|\geq 0}
2149:{\displaystyle \gamma (t)=a+re^{it},t\in }
1325:{\displaystyle 0<|f(z_{0})|\leq |f(z)|}
3056:
3036:
3016:
2999:
2991:
2953:
2909:
2889:
2863:
2851:{\displaystyle {\overline {B}}(a,r)\in D}
2817:
2815:
2775:
2773:
2747:
2707:
2705:
2679:
2635:
2620:{\displaystyle z\in {\overline {B}}(a,r)}
2592:
2584:
2540:
2520:
2500:
2480:
2468:
2444:
2436:
2419:
2417:
2391:
2379:
2355:
2347:
2330:
2328:
2287:
2256:
2251:
2239:
2215:
2207:
2190:
2181:
2176:
2164:
2110:
2080:
2034:
2032:
2006:
1980:
1940:
1938:
1918:
1870:
1841:
1800:
1780:
1763:
1761:
1732:
1698:
1674:
1657:
1655:
1630:
1624:
1600:
1583:
1575:
1524:
1507:
1478:
1439:
1434:
1411:
1391:
1370:
1364:
1340:
1317:
1300:
1292:
1283:
1268:
1260:
1237:
1216:
1210:
1191:
1190:
1188:
1168:
1148:
1120:
1103:
1101:
1081:
1060:
1054:
1029:
1027:
1006:
1000:
980:
963:
961:
932:
930:
907:
887:
870:
868:
848:
829:
828:
815:
807:
787:
762:
760:
741:
740:
738:
718:
690:
670:
650:
630:
622:
620:
593:
573:
552:
546:
522:
499:
482:
474:
465:
450:
448:
425:
404:
398:
379:
378:
376:
353:
324:
296:
288:
286:
265:
259:
239:
218:
212:
188:
165:
137:
129:
127:
100:
60:
28:
3165:(2nd ed.). Oxford University Press.
16:Mathematical theorem in complex analysis
3124:
956:and nonempty, the continuous function
2488:{\displaystyle |f(a)|=|f(a+re^{it})|}
507:{\displaystyle |f(z_{0})|\geq |f(z)|}
7:
3100:, an extension to unbounded domains.
2803:{\displaystyle {\overline {B}}(a,r)}
2735:{\displaystyle {\overline {B}}(a,r)}
1968:{\displaystyle {\overline {B}}(a,r)}
3134:Functions of One Complex Variable I
2630:Now consider the constant function
1827:
339:be a holomorphic function on some
14:
1897:
1828:Using Gauss's mean value theorem
995:attains a maximum at some point
2742:where the holomorphic function
1898:Using Cauchy's Integral Formula
1042:{\displaystyle {\overline {D}}}
945:{\displaystyle {\overline {D}}}
775:{\displaystyle {\overline {D}}}
3085:fundamental theorem of algebra
3017:
3013:
3007:
3000:
2935:
2929:
2920:
2914:
2839:
2827:
2797:
2785:
2729:
2717:
2661:
2655:
2646:
2640:
2614:
2602:
2566:
2560:
2551:
2545:
2481:
2477:
2452:
2445:
2437:
2433:
2427:
2420:
2392:
2388:
2363:
2356:
2348:
2344:
2338:
2331:
2310:
2295:
2252:
2248:
2223:
2216:
2208:
2204:
2198:
2191:
2143:
2128:
2091:
2085:
2056:
2044:
1962:
1950:
1881:
1875:
1852:
1846:
1811:
1805:
1781:
1777:
1771:
1764:
1743:
1737:
1714:
1708:
1675:
1671:
1665:
1658:
1601:
1597:
1591:
1584:
1550:
1544:
1525:
1521:
1515:
1508:
1495:
1489:
1318:
1314:
1308:
1301:
1293:
1289:
1276:
1269:
1121:
1117:
1111:
1104:
981:
977:
971:
964:
888:
884:
878:
871:
825:
631:
623:
500:
496:
490:
483:
475:
471:
458:
451:
393:and taking complex values. If
297:
289:
138:
130:
42:
36:
1:
1832:Another proof works by using
1689:is constant. Then, using the
3235:Theorems in complex analysis
3112:Hadamard three-lines theorem
2822:
2780:
2712:
2597:
2039:
1945:
1198:{\displaystyle \mathbb {C} }
1034:
937:
820:
767:
748:{\displaystyle \mathbb {C} }
386:{\displaystyle \mathbb {C} }
156:that is strictly within the
3206:"Maximum Modulus Principle"
3180:Encyclopedia of Mathematics
3175:"Maximum-modulus principle"
3173:E. D. Solomentsev (2001) ,
3098:Phragmén–Lindelöf principle
3024:{\displaystyle \log |f(z)|}
1975:(a closed ball centered at
1143:For a holomorphic function
645:attains a local maximum at
3251:
3105:Borel–Carathéodory theorem
1834:Gauss's mean value theorem
1608:{\displaystyle \ln |f(z)|}
2941:{\displaystyle f(z)=f(a)}
2667:{\displaystyle g(z)=f(a)}
2572:{\displaystyle f(z)=f(a)}
1470:One can use the equality
1139:Minimum modulus principle
89:maximum modulus principle
23:A plot of the modulus of
1691:Cauchy–Riemann equations
1163:on a connected open set
152:cannot exhibit a strict
3230:Mathematical principles
3162:The Theory of Functions
2978:Physical interpretation
2884:vanishes everywhere in
183:In other words, either
48:{\displaystyle \cos(z)}
3065:
3045:
3025:
2968:
2967:{\displaystyle z\in D}
2942:
2898:
2878:
2852:
2804:
2762:
2736:
2694:
2693:{\displaystyle z\in D}
2668:
2621:
2573:
2529:
2509:
2489:
2406:
2317:
2273:
2150:
2069:
2021:
2020:{\displaystyle r>0}
1995:
1994:{\displaystyle a\in D}
1969:
1933:is open, there exists
1927:
1888:
1859:
1818:
1789:
1788:{\displaystyle |f(z)|}
1750:
1721:
1683:
1682:{\displaystyle |f(z)|}
1640:
1609:
1557:
1451:
1420:
1400:
1380:
1349:
1326:
1246:
1226:
1199:
1177:
1157:
1129:
1128:{\displaystyle |f(z)|}
1090:
1070:
1043:
1016:
989:
988:{\displaystyle |f(z)|}
946:
916:
896:
895:{\displaystyle |f(z)|}
857:
837:
796:
776:
749:
727:
699:
679:
659:
639:
602:
582:
562:
531:
508:
434:
414:
387:
362:
333:
305:
275:
248:
228:
197:
174:
146:
109:
80:
69:
49:
3066:
3046:
3026:
2969:
2943:
2899:
2879:
2853:
2805:
2763:
2737:
2695:
2669:
2622:
2574:
2530:
2510:
2490:
2407:
2318:
2316:{\displaystyle t\in }
2274:
2151:
2070:
2022:
1996:
1970:
1928:
1889:
1860:
1819:
1790:
1751:
1722:
1720:{\displaystyle f'(z)}
1684:
1641:
1639:{\displaystyle z_{0}}
1610:
1558:
1452:
1421:
1401:
1381:
1379:{\displaystyle z_{0}}
1350:
1327:
1247:
1227:
1225:{\displaystyle z_{0}}
1200:
1178:
1158:
1130:
1091:
1071:
1069:{\displaystyle z_{0}}
1044:
1017:
1015:{\displaystyle z_{0}}
990:
947:
917:
897:
858:
838:
797:
777:
750:
728:
700:
680:
660:
640:
603:
583:
563:
561:{\displaystyle z_{0}}
532:
509:
435:
415:
413:{\displaystyle z_{0}}
388:
363:
334:
311:takes larger values.
306:
276:
274:{\displaystyle z_{0}}
249:
234:inside the domain of
229:
227:{\displaystyle z_{0}}
198:
175:
147:
110:
70:
50:
22:
3055:
3035:
2990:
2952:
2908:
2888:
2862:
2814:
2772:
2746:
2704:
2678:
2634:
2583:
2539:
2519:
2499:
2416:
2327:
2286:
2163:
2079:
2031:
2005:
1979:
1937:
1917:
1887:{\displaystyle f(z)}
1869:
1858:{\displaystyle f(z)}
1840:
1817:{\displaystyle f(z)}
1799:
1760:
1749:{\displaystyle f(z)}
1731:
1697:
1654:
1623:
1574:
1477:
1433:
1410:
1390:
1363:
1339:
1259:
1236:
1209:
1187:
1167:
1147:
1100:
1080:
1053:
1026:
999:
960:
929:
906:
867:
847:
806:
786:
759:
737:
717:
689:
669:
649:
619:
613:open mapping theorem
592:
572:
545:
521:
447:
424:
397:
375:
352:
323:
285:
258:
238:
211:
207:, or, for any point
187:
164:
126:
117:holomorphic function
99:
59:
27:
2877:{\displaystyle f-g}
2761:{\displaystyle g-f}
2189:
1727:= 0, and thus that
1450:{\displaystyle 1/f}
685:cannot be open, so
638:{\displaystyle |f|}
304:{\displaystyle |f|}
145:{\displaystyle |f|}
3203:Weisstein, Eric W.
3061:
3041:
3021:
2964:
2938:
2894:
2874:
2848:
2800:
2758:
2732:
2690:
2664:
2617:
2569:
2525:
2505:
2485:
2402:
2313:
2269:
2172:
2146:
2065:
2017:
1991:
1965:
1923:
1884:
1855:
1814:
1785:
1746:
1717:
1679:
1636:
1605:
1568:natural logarithms
1553:
1461:Sketches of proofs
1447:
1416:
1396:
1376:
1345:
1322:
1242:
1222:
1195:
1173:
1153:
1125:
1086:
1066:
1039:
1012:
985:
942:
912:
892:
853:
833:
792:
782:be the closure of
772:
745:
723:
695:
675:
655:
635:
598:
578:
558:
527:
504:
430:
410:
383:
358:
329:
301:
271:
244:
224:
193:
170:
142:
105:
81:
65:
45:
3157:Titchmarsh, E. C.
3143:978-1-4612-6314-2
3064:{\displaystyle D}
3044:{\displaystyle D}
2986:. That is, since
2897:{\displaystyle D}
2825:
2783:
2715:
2600:
2528:{\displaystyle a}
2508:{\displaystyle r}
2042:
1948:
1926:{\displaystyle D}
1648:maximum principle
1617:harmonic function
1419:{\displaystyle D}
1399:{\displaystyle f}
1348:{\displaystyle z}
1245:{\displaystyle D}
1176:{\displaystyle D}
1156:{\displaystyle f}
1089:{\displaystyle f}
1037:
940:
915:{\displaystyle D}
856:{\displaystyle D}
823:
795:{\displaystyle D}
770:
726:{\displaystyle D}
709:Related statement
698:{\displaystyle f}
678:{\displaystyle z}
658:{\displaystyle z}
601:{\displaystyle D}
581:{\displaystyle f}
530:{\displaystyle z}
433:{\displaystyle D}
361:{\displaystyle D}
332:{\displaystyle f}
247:{\displaystyle f}
205:constant function
196:{\displaystyle f}
173:{\displaystyle f}
108:{\displaystyle f}
68:{\displaystyle z}
3242:
3216:
3215:
3187:
3168:(See chapter 5.)
3166:
3148:
3147:
3129:
3070:
3068:
3067:
3062:
3050:
3048:
3047:
3042:
3030:
3028:
3027:
3022:
3020:
3003:
2973:
2971:
2970:
2965:
2947:
2945:
2944:
2939:
2903:
2901:
2900:
2895:
2883:
2881:
2880:
2875:
2857:
2855:
2854:
2849:
2826:
2818:
2809:
2807:
2806:
2801:
2784:
2776:
2767:
2765:
2764:
2759:
2741:
2739:
2738:
2733:
2716:
2708:
2699:
2697:
2696:
2691:
2673:
2671:
2670:
2665:
2626:
2624:
2623:
2618:
2601:
2593:
2578:
2576:
2575:
2570:
2534:
2532:
2531:
2526:
2514:
2512:
2511:
2506:
2494:
2492:
2491:
2486:
2484:
2476:
2475:
2448:
2440:
2423:
2411:
2409:
2408:
2403:
2395:
2387:
2386:
2359:
2351:
2334:
2322:
2320:
2319:
2314:
2278:
2276:
2275:
2270:
2255:
2247:
2246:
2219:
2211:
2194:
2188:
2180:
2155:
2153:
2152:
2147:
2118:
2117:
2074:
2072:
2071:
2066:
2043:
2035:
2026:
2024:
2023:
2018:
2000:
1998:
1997:
1992:
1974:
1972:
1971:
1966:
1949:
1941:
1932:
1930:
1929:
1924:
1910:
1909:
1905:
1893:
1891:
1890:
1885:
1864:
1862:
1861:
1856:
1823:
1821:
1820:
1815:
1794:
1792:
1791:
1786:
1784:
1767:
1755:
1753:
1752:
1747:
1726:
1724:
1723:
1718:
1707:
1688:
1686:
1685:
1680:
1678:
1661:
1645:
1643:
1642:
1637:
1635:
1634:
1614:
1612:
1611:
1606:
1604:
1587:
1562:
1560:
1559:
1554:
1528:
1511:
1456:
1454:
1453:
1448:
1443:
1425:
1423:
1422:
1417:
1405:
1403:
1402:
1397:
1385:
1383:
1382:
1377:
1375:
1374:
1354:
1352:
1351:
1346:
1331:
1329:
1328:
1323:
1321:
1304:
1296:
1288:
1287:
1272:
1251:
1249:
1248:
1243:
1231:
1229:
1228:
1223:
1221:
1220:
1204:
1202:
1201:
1196:
1194:
1182:
1180:
1179:
1174:
1162:
1160:
1159:
1154:
1134:
1132:
1131:
1126:
1124:
1107:
1096:is constant, so
1095:
1093:
1092:
1087:
1075:
1073:
1072:
1067:
1065:
1064:
1048:
1046:
1045:
1040:
1038:
1030:
1021:
1019:
1018:
1013:
1011:
1010:
994:
992:
991:
986:
984:
967:
951:
949:
948:
943:
941:
933:
921:
919:
918:
913:
901:
899:
898:
893:
891:
874:
862:
860:
859:
854:
842:
840:
839:
834:
832:
824:
816:
801:
799:
798:
793:
781:
779:
778:
773:
771:
763:
754:
752:
751:
746:
744:
732:
730:
729:
724:
704:
702:
701:
696:
684:
682:
681:
676:
664:
662:
661:
656:
644:
642:
641:
636:
634:
626:
607:
605:
604:
599:
587:
585:
584:
579:
567:
565:
564:
559:
557:
556:
536:
534:
533:
528:
513:
511:
510:
505:
503:
486:
478:
470:
469:
454:
439:
437:
436:
431:
419:
417:
416:
411:
409:
408:
392:
390:
389:
384:
382:
367:
365:
364:
359:
338:
336:
335:
330:
315:Formal statement
310:
308:
307:
302:
300:
292:
280:
278:
277:
272:
270:
269:
253:
251:
250:
245:
233:
231:
230:
225:
223:
222:
202:
200:
199:
194:
179:
177:
176:
171:
151:
149:
148:
143:
141:
133:
114:
112:
111:
106:
93:complex analysis
74:
72:
71:
66:
54:
52:
51:
46:
3250:
3249:
3245:
3244:
3243:
3241:
3240:
3239:
3220:
3219:
3201:
3200:
3197:
3172:
3155:
3152:
3151:
3144:
3131:
3130:
3126:
3121:
3091:Schwarz's lemma
3077:
3053:
3052:
3033:
3032:
2988:
2987:
2980:
2950:
2949:
2906:
2905:
2886:
2885:
2860:
2859:
2812:
2811:
2770:
2769:
2744:
2743:
2702:
2701:
2676:
2675:
2632:
2631:
2581:
2580:
2537:
2536:
2517:
2516:
2497:
2496:
2464:
2414:
2413:
2375:
2325:
2324:
2284:
2283:
2235:
2161:
2160:
2106:
2077:
2076:
2029:
2028:
2003:
2002:
1977:
1976:
1935:
1934:
1915:
1914:
1911:
1907:
1903:
1901:
1900:
1867:
1866:
1838:
1837:
1830:
1797:
1796:
1758:
1757:
1729:
1728:
1700:
1695:
1694:
1652:
1651:
1626:
1621:
1620:
1572:
1571:
1570:to deduce that
1475:
1474:
1468:
1463:
1431:
1430:
1408:
1407:
1406:is constant on
1388:
1387:
1366:
1361:
1360:
1337:
1336:
1279:
1257:
1256:
1234:
1233:
1212:
1207:
1206:
1185:
1184:
1165:
1164:
1145:
1144:
1141:
1098:
1097:
1078:
1077:
1056:
1051:
1050:
1024:
1023:
1002:
997:
996:
958:
957:
927:
926:
904:
903:
865:
864:
845:
844:
804:
803:
802:. Suppose that
784:
783:
757:
756:
735:
734:
715:
714:
711:
687:
686:
667:
666:
647:
646:
617:
616:
590:
589:
588:is constant on
570:
569:
548:
543:
542:
519:
518:
461:
445:
444:
422:
421:
400:
395:
394:
373:
372:
350:
349:
321:
320:
317:
283:
282:
261:
256:
255:
236:
235:
214:
209:
208:
185:
184:
162:
161:
124:
123:
97:
96:
95:states that if
57:
56:
25:
24:
17:
12:
11:
5:
3248:
3246:
3238:
3237:
3232:
3222:
3221:
3218:
3217:
3196:
3195:External links
3193:
3192:
3191:
3188:
3170:
3150:
3149:
3142:
3123:
3122:
3120:
3117:
3116:
3115:
3108:
3101:
3094:
3088:
3076:
3073:
3060:
3040:
3019:
3015:
3012:
3009:
3006:
3002:
2998:
2995:
2979:
2976:
2963:
2960:
2957:
2937:
2934:
2931:
2928:
2925:
2922:
2919:
2916:
2913:
2904:which implies
2893:
2873:
2870:
2867:
2847:
2844:
2841:
2838:
2835:
2832:
2829:
2824:
2821:
2799:
2796:
2793:
2790:
2787:
2782:
2779:
2757:
2754:
2751:
2731:
2728:
2725:
2722:
2719:
2714:
2711:
2689:
2686:
2683:
2663:
2660:
2657:
2654:
2651:
2648:
2645:
2642:
2639:
2616:
2613:
2610:
2607:
2604:
2599:
2596:
2591:
2588:
2568:
2565:
2562:
2559:
2556:
2553:
2550:
2547:
2544:
2524:
2504:
2483:
2479:
2474:
2471:
2467:
2463:
2460:
2457:
2454:
2451:
2447:
2443:
2439:
2435:
2432:
2429:
2426:
2422:
2401:
2398:
2394:
2390:
2385:
2382:
2378:
2374:
2371:
2368:
2365:
2362:
2358:
2354:
2350:
2346:
2343:
2340:
2337:
2333:
2312:
2309:
2306:
2303:
2300:
2297:
2294:
2291:
2280:
2279:
2268:
2265:
2262:
2259:
2254:
2250:
2245:
2242:
2238:
2234:
2231:
2228:
2225:
2222:
2218:
2214:
2210:
2206:
2203:
2200:
2197:
2193:
2187:
2184:
2179:
2175:
2171:
2168:
2145:
2142:
2139:
2136:
2133:
2130:
2127:
2124:
2121:
2116:
2113:
2109:
2105:
2102:
2099:
2096:
2093:
2090:
2087:
2084:
2064:
2061:
2058:
2055:
2052:
2049:
2046:
2041:
2038:
2016:
2013:
2010:
1990:
1987:
1984:
1964:
1961:
1958:
1955:
1952:
1947:
1944:
1922:
1899:
1896:
1883:
1880:
1877:
1874:
1854:
1851:
1848:
1845:
1829:
1826:
1813:
1810:
1807:
1804:
1783:
1779:
1776:
1773:
1770:
1766:
1745:
1742:
1739:
1736:
1716:
1713:
1710:
1706:
1703:
1677:
1673:
1670:
1667:
1664:
1660:
1633:
1629:
1603:
1599:
1596:
1593:
1590:
1586:
1582:
1579:
1564:
1563:
1552:
1549:
1546:
1543:
1540:
1537:
1534:
1531:
1527:
1523:
1520:
1517:
1514:
1510:
1506:
1503:
1500:
1497:
1494:
1491:
1488:
1485:
1482:
1467:
1464:
1462:
1459:
1446:
1442:
1438:
1415:
1395:
1373:
1369:
1344:
1333:
1332:
1320:
1316:
1313:
1310:
1307:
1303:
1299:
1295:
1291:
1286:
1282:
1278:
1275:
1271:
1267:
1264:
1241:
1232:is a point in
1219:
1215:
1193:
1172:
1152:
1140:
1137:
1123:
1119:
1116:
1113:
1110:
1106:
1085:
1063:
1059:
1036:
1033:
1009:
1005:
983:
979:
976:
973:
970:
966:
939:
936:
911:
890:
886:
883:
880:
877:
873:
852:
831:
827:
822:
819:
814:
811:
791:
769:
766:
743:
722:
710:
707:
694:
674:
654:
633:
629:
625:
597:
577:
555:
551:
526:
515:
514:
502:
498:
495:
492:
489:
485:
481:
477:
473:
468:
464:
460:
457:
453:
429:
420:is a point in
407:
403:
381:
357:
328:
316:
313:
299:
295:
291:
268:
264:
243:
221:
217:
192:
169:
140:
136:
132:
104:
64:
44:
41:
38:
35:
32:
15:
13:
10:
9:
6:
4:
3:
2:
3247:
3236:
3233:
3231:
3228:
3227:
3225:
3213:
3212:
3207:
3204:
3199:
3198:
3194:
3189:
3186:
3182:
3181:
3176:
3171:
3169:
3164:
3163:
3158:
3154:
3153:
3145:
3139:
3135:
3128:
3125:
3118:
3113:
3109:
3106:
3102:
3099:
3095:
3092:
3089:
3086:
3082:
3081:
3080:
3074:
3072:
3058:
3038:
3010:
3004:
2996:
2993:
2985:
2984:heat equation
2977:
2975:
2961:
2958:
2955:
2932:
2926:
2923:
2917:
2911:
2891:
2871:
2868:
2865:
2858:. This means
2845:
2842:
2836:
2833:
2830:
2819:
2794:
2791:
2788:
2777:
2768:vanishes. As
2755:
2752:
2749:
2726:
2723:
2720:
2709:
2687:
2684:
2681:
2658:
2652:
2649:
2643:
2637:
2628:
2611:
2608:
2605:
2594:
2589:
2586:
2563:
2557:
2554:
2548:
2542:
2535:. Therefore,
2522:
2502:
2472:
2469:
2465:
2461:
2458:
2455:
2449:
2441:
2430:
2424:
2399:
2396:
2383:
2380:
2376:
2372:
2369:
2366:
2360:
2352:
2341:
2335:
2307:
2304:
2301:
2298:
2292:
2289:
2266:
2263:
2260:
2257:
2243:
2240:
2236:
2232:
2229:
2226:
2220:
2212:
2201:
2195:
2185:
2182:
2177:
2173:
2169:
2166:
2159:
2158:
2157:
2140:
2137:
2134:
2131:
2125:
2122:
2119:
2114:
2111:
2107:
2103:
2100:
2097:
2094:
2088:
2082:
2062:
2059:
2053:
2050:
2047:
2036:
2014:
2011:
2008:
1988:
1985:
1982:
1959:
1956:
1953:
1942:
1920:
1906:
1895:
1894:is constant.
1878:
1872:
1849:
1843:
1835:
1825:
1808:
1802:
1774:
1768:
1740:
1734:
1711:
1704:
1701:
1693:we show that
1692:
1668:
1662:
1649:
1631:
1627:
1618:
1594:
1588:
1580:
1577:
1569:
1547:
1541:
1538:
1535:
1532:
1529:
1518:
1512:
1504:
1501:
1498:
1492:
1486:
1483:
1480:
1473:
1472:
1471:
1465:
1460:
1458:
1444:
1440:
1436:
1427:
1413:
1393:
1371:
1367:
1358:
1342:
1311:
1305:
1297:
1284:
1280:
1273:
1265:
1262:
1255:
1254:
1253:
1239:
1217:
1213:
1170:
1150:
1138:
1136:
1114:
1108:
1083:
1061:
1057:
1031:
1007:
1003:
974:
968:
955:
934:
923:
909:
881:
875:
850:
817:
812:
809:
789:
764:
720:
713:Suppose that
708:
706:
705:is constant.
692:
672:
652:
627:
614:
609:
595:
575:
553:
549:
540:
524:
493:
487:
479:
466:
462:
455:
443:
442:
441:
427:
405:
401:
371:
370:complex plane
355:
348:
345:
342:
326:
314:
312:
293:
266:
262:
241:
219:
215:
206:
203:is locally a
190:
181:
167:
159:
155:
134:
122:
118:
102:
94:
90:
86:
78:
62:
55:(in red) for
39:
33:
30:
21:
3209:
3178:
3167:
3161:
3133:
3127:
3078:
3075:Applications
2981:
2629:
2515:centered at
2281:
2027:) such that
2001:with radius
1912:
1831:
1566:for complex
1565:
1469:
1428:
1357:neighborhood
1334:
1142:
924:
712:
610:
539:neighborhood
516:
318:
182:
88:
82:
440:such that
119:, then the
85:mathematics
3224:Categories
3119:References
1252:such that
3211:MathWorld
3185:EMS Press
2997:
2959:∈
2869:−
2843:∈
2823:¯
2781:¯
2753:−
2713:¯
2685:∈
2598:¯
2590:∈
2397:≥
2353:−
2308:π
2293:∈
2264:≤
2213:−
2186:π
2174:∫
2170:≤
2141:π
2126:∈
2083:γ
2060:⊂
2040:¯
1986:∈
1946:¯
1581:
1539:
1505:
1484:
1298:≤
1035:¯
938:¯
826:→
821:¯
813::
768:¯
480:≥
341:connected
281:at which
77:unit disk
34:
3159:(1939).
2948:for all
2674:for all
2579:for all
2282:For all
1705:′
1619:. Since
1355:in some
1335:for all
537:in some
517:for all
1386:, then
954:compact
863:. Then
568:, then
368:of the
154:maximum
121:modulus
75:in the
3140:
1902:": -->
1650:that
755:. Let
347:subset
158:domain
87:, the
2412:, so
1615:is a
1205:, if
1049:. If
115:is a
3138:ISBN
3110:The
3103:The
3096:The
3083:The
2012:>
1904:edit
1266:<
344:open
319:Let
2994:log
1913:As
1536:arg
1481:log
1359:of
1183:of
1022:of
952:is
541:of
180:.
160:of
91:in
83:In
31:cos
3226::
3208:.
3183:,
3177:,
2974:.
2627:.
2323:,
1824:.
1578:ln
1502:ln
1457:.
1426:.
922:.
608:.
3214:.
3146:.
3087:.
3059:D
3039:D
3018:|
3014:)
3011:z
3008:(
3005:f
3001:|
2962:D
2956:z
2936:)
2933:a
2930:(
2927:f
2924:=
2921:)
2918:z
2915:(
2912:f
2892:D
2872:g
2866:f
2846:D
2840:)
2837:r
2834:,
2831:a
2828:(
2820:B
2798:)
2795:r
2792:,
2789:a
2786:(
2778:B
2756:f
2750:g
2730:)
2727:r
2724:,
2721:a
2718:(
2710:B
2688:D
2682:z
2662:)
2659:a
2656:(
2653:f
2650:=
2647:)
2644:z
2641:(
2638:g
2615:)
2612:r
2609:,
2606:a
2603:(
2595:B
2587:z
2567:)
2564:a
2561:(
2558:f
2555:=
2552:)
2549:z
2546:(
2543:f
2523:a
2503:r
2482:|
2478:)
2473:t
2470:i
2466:e
2462:r
2459:+
2456:a
2453:(
2450:f
2446:|
2442:=
2438:|
2434:)
2431:a
2428:(
2425:f
2421:|
2400:0
2393:|
2389:)
2384:t
2381:i
2377:e
2373:r
2370:+
2367:a
2364:(
2361:f
2357:|
2349:|
2345:)
2342:a
2339:(
2336:f
2332:|
2311:]
2305:2
2302:,
2299:0
2296:[
2290:t
2267:0
2261:t
2258:d
2253:|
2249:)
2244:t
2241:i
2237:e
2233:r
2230:+
2227:a
2224:(
2221:f
2217:|
2209:|
2205:)
2202:a
2199:(
2196:f
2192:|
2183:2
2178:0
2167:0
2144:]
2138:2
2135:,
2132:0
2129:[
2123:t
2120:,
2115:t
2112:i
2108:e
2104:r
2101:+
2098:a
2095:=
2092:)
2089:t
2086:(
2063:D
2057:)
2054:r
2051:,
2048:a
2045:(
2037:B
2015:0
2009:r
1989:D
1983:a
1963:)
1960:r
1957:,
1954:a
1951:(
1943:B
1921:D
1908:]
1882:)
1879:z
1876:(
1873:f
1853:)
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