31:
2288:
852:
1876:
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
90:
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).
1572:
2084:
3201:
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.
2421:
2165:
1341:
2867:
2636:
2504:
523:
2819:
2751:
1984:
1847:
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
1058:
961:
791:
1214:
764:
402:
3040:
1736:
1624:
2957:
2683:
2588:
64:
2983:
2709:
2036:
2010:
1804:
1698:
1144:
1004:
911:
2332:
1655:
1395:
1241:
1085:
1031:
577:
429:
290:
243:
1903:
1874:
1833:
1765:
2893:
2777:
1466:
654:
320:
161:
3082:, 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.
3080:
3060:
2913:
2544:
2524:
1942:
1435:
1415:
1364:
1261:
1192:
1172:
1105:
931:
872:
811:
742:
714:
694:
674:
617:
597:
546:
449:
377:
348:
263:
212:
189:
124:
84:
623:
3108:
3245:
3152:
2173:
3115:
3240:
3125:, a result about the behaviour of bounded holomorphic functions on a line between two other parallel lines in the complex plane.
3095:
816:
3195:
1701:
3122:
1844:
3190:
1487:
2041:
3167:
2337:
2089:
1269:
3185:
2824:
2593:
1367:
549:
2426:
457:
2782:
2714:
1947:
3090:
The maximum modulus principle has many uses in complex analysis, and may be used to prove the following:
1036:
939:
769:
168:
127:
1197:
747:
385:
3000:
3101:
1584:
2918:
2644:
2549:
3213:
3148:
1658:
1627:
1578:
215:
3216:
37:
2962:
2688:
2015:
1989:
1770:
1664:
1110:
970:
877:
103:
2296:
1707:
1633:
1373:
1219:
1063:
1009:
555:
407:
268:
221:
30:
1879:
1850:
1809:
1741:
2872:
2756:
1443:
629:
295:
136:
3104:, a result which in turn has many generalisations and applications in complex analysis.
3065:
3045:
2898:
2529:
2509:
1927:
1420:
1400:
1349:
1246:
1177:
1157:
1090:
916:
857:
796:
727:
699:
679:
659:
626:, which states that a nonconstant holomorphic function maps open sets to open sets: If
602:
582:
531:
434:
362:
333:
248:
197:
174:
131:
109:
69:
3234:
3171:
2994:
1806:
can only have a local minimum (which necessarily has value 0) at an isolated zero of
964:
380:
351:
17:
95:
2086:. We then define the boundary of the closed ball with positive orientation as
3221:
87:
354:
1087:
is not on the boundary, then the maximum modulus principle implies that
164:
3143:
Conway, John B. (1978). Axler, S.; Gehring, F.W.; Ribet, K.A. (eds.).
3042:
is harmonic, it is thus the steady state of a heat flow on the region
2283:{\displaystyle 0\leq \int _{0}^{2\pi }|f(a)|-|f(a+re^{it})|\,dt\leq 0}
357:
2711:. Then one can construct a sequence of distinct points located in
29:
1657:
is a local maximum for this function also, it follows from the
3147:(2 ed.). New York: Springer Science+Business Media, Inc.
3118:, which bounds an analytic function in terms of its real part.
676:, then the image of a sufficiently small open neighborhood of
1146:
also attains the same maximum at any point of the boundary.
3062:. Suppose a strict maximum was attained on the interior of
2993:
A physical interpretation of this principle comes from the
847:{\displaystyle f\colon {\overline {D}}\to \mathbb {C} }
936:
This follows from the first version as follows. Since
622:
This statement can be viewed as a special case of the
3068:
3048:
3003:
2965:
2921:
2901:
2875:
2827:
2785:
2759:
2717:
2691:
2647:
2596:
2552:
2532:
2512:
2429:
2340:
2299:
2176:
2092:
2044:
2018:
1992:
1950:
1930:
1882:
1853:
1812:
1773:
1744:
1710:
1667:
1636:
1587:
1490:
1446:
1423:
1403:
1376:
1352:
1272:
1249:
1222:
1200:
1180:
1160:
1113:
1093:
1066:
1039:
1012:
973:
942:
919:
880:
860:
819:
799:
772:
750:
730:
702:
682:
662:
632:
605:
585:
558:
534:
460:
437:
410:
388:
365:
336:
298:
271:
251:
224:
200:
177:
139:
112:
72:
40:
2506:. This also holds for all balls of radius less than
2821:is closed, the sequence converges to some point in
1767:is constant as well. Similar reasoning shows that
913:attains a maximum at some point of the boundary of
3074:
3054:
3034:
2977:
2951:
2907:
2887:
2861:
2813:
2771:
2745:
2703:
2677:
2630:
2582:
2538:
2518:
2498:
2415:
2326:
2282:
2159:
2078:
2030:
2004:
1978:
1936:
1897:
1868:
1827:
1798:
1759:
1730:
1692:
1649:
1618:
1566:
1477:Using the maximum principle for harmonic functions
1460:
1429:
1409:
1389:
1358:
1335:
1255:
1235:
1208:
1186:
1166:
1138:
1099:
1079:
1052:
1025:
998:
955:
925:
905:
866:
846:
805:
785:
758:
736:
708:
688:
668:
648:
611:
591:
571:
540:
517:
443:
423:
396:
371:
342:
314:
284:
257:
237:
206:
183:
155:
118:
78:
58:
2167:. Invoking Cauchy's integral formula, we obtain
854:is a continuous function that is holomorphic on
1567:{\displaystyle \log f(z)=\ln |f(z)|+i\arg f(z)}
744:is a bounded nonempty connected open subset of
1440:Proof: Apply the maximum modulus principle to
265:there exist other points arbitrarily close to
2079:{\displaystyle {\overline {B}}(a,r)\subset D}
8:
2416:{\displaystyle |f(a)|-|f(a+re^{it})|\geq 0}
2160:{\displaystyle \gamma (t)=a+re^{it},t\in }
1336:{\displaystyle 0<|f(z_{0})|\leq |f(z)|}
3067:
3047:
3027:
3010:
3002:
2964:
2920:
2900:
2874:
2862:{\displaystyle {\overline {B}}(a,r)\in D}
2828:
2826:
2786:
2784:
2758:
2718:
2716:
2690:
2646:
2631:{\displaystyle z\in {\overline {B}}(a,r)}
2603:
2595:
2551:
2531:
2511:
2491:
2479:
2455:
2447:
2430:
2428:
2402:
2390:
2366:
2358:
2341:
2339:
2298:
2267:
2262:
2250:
2226:
2218:
2201:
2192:
2187:
2175:
2121:
2091:
2045:
2043:
2017:
1991:
1951:
1949:
1929:
1881:
1852:
1811:
1791:
1774:
1772:
1743:
1709:
1685:
1668:
1666:
1641:
1635:
1611:
1594:
1586:
1535:
1518:
1489:
1450:
1445:
1422:
1402:
1381:
1375:
1351:
1328:
1311:
1303:
1294:
1279:
1271:
1248:
1227:
1221:
1202:
1201:
1199:
1179:
1159:
1131:
1114:
1112:
1092:
1071:
1065:
1040:
1038:
1017:
1011:
991:
974:
972:
943:
941:
918:
898:
881:
879:
859:
840:
839:
826:
818:
798:
773:
771:
752:
751:
749:
729:
701:
681:
661:
641:
633:
631:
604:
584:
563:
557:
533:
510:
493:
485:
476:
461:
459:
436:
415:
409:
390:
389:
387:
364:
335:
307:
299:
297:
276:
270:
250:
229:
223:
199:
176:
148:
140:
138:
111:
71:
39:
3176:(2nd ed.). Oxford University Press.
27:Mathematical theorem in complex analysis
3135:
967:and nonempty, the continuous function
2499:{\displaystyle |f(a)|=|f(a+re^{it})|}
518:{\displaystyle |f(z_{0})|\geq |f(z)|}
7:
3111:, an extension to unbounded domains.
2814:{\displaystyle {\overline {B}}(a,r)}
2746:{\displaystyle {\overline {B}}(a,r)}
1979:{\displaystyle {\overline {B}}(a,r)}
3145:Functions of One Complex Variable I
2641:Now consider the constant function
1838:
350:be a holomorphic function on some
25:
1908:
1839:Using Gauss's mean value theorem
1006:attains a maximum at some point
2753:where the holomorphic function
1909:Using Cauchy's Integral Formula
1053:{\displaystyle {\overline {D}}}
956:{\displaystyle {\overline {D}}}
786:{\displaystyle {\overline {D}}}
3096:fundamental theorem of algebra
3028:
3024:
3018:
3011:
2946:
2940:
2931:
2925:
2850:
2838:
2808:
2796:
2740:
2728:
2672:
2666:
2657:
2651:
2625:
2613:
2577:
2571:
2562:
2556:
2492:
2488:
2463:
2456:
2448:
2444:
2438:
2431:
2403:
2399:
2374:
2367:
2359:
2355:
2349:
2342:
2321:
2306:
2263:
2259:
2234:
2227:
2219:
2215:
2209:
2202:
2154:
2139:
2102:
2096:
2067:
2055:
1973:
1961:
1892:
1886:
1863:
1857:
1822:
1816:
1792:
1788:
1782:
1775:
1754:
1748:
1725:
1719:
1686:
1682:
1676:
1669:
1612:
1608:
1602:
1595:
1561:
1555:
1536:
1532:
1526:
1519:
1506:
1500:
1329:
1325:
1319:
1312:
1304:
1300:
1287:
1280:
1132:
1128:
1122:
1115:
992:
988:
982:
975:
899:
895:
889:
882:
836:
642:
634:
511:
507:
501:
494:
486:
482:
469:
462:
404:and taking complex values. If
308:
300:
149:
141:
53:
47:
1:
1843:Another proof works by using
1700:is constant. Then, using the
3246:Theorems in complex analysis
3123:Hadamard three-lines theorem
2833:
2791:
2723:
2608:
2050:
1956:
1209:{\displaystyle \mathbb {C} }
1045:
948:
831:
778:
759:{\displaystyle \mathbb {C} }
397:{\displaystyle \mathbb {C} }
167:that is strictly within the
3217:"Maximum Modulus Principle"
3191:Encyclopedia of Mathematics
3186:"Maximum-modulus principle"
3184:E. D. Solomentsev (2001) ,
3109:Phragmén–Lindelöf principle
3035:{\displaystyle \log |f(z)|}
1986:(a closed ball centered at
1154:For a holomorphic function
656:attains a local maximum at
3262:
3116:Borel–Carathéodory theorem
1845:Gauss's mean value theorem
1619:{\displaystyle \ln |f(z)|}
2952:{\displaystyle f(z)=f(a)}
2678:{\displaystyle g(z)=f(a)}
2583:{\displaystyle f(z)=f(a)}
1481:One can use the equality
1150:Minimum modulus principle
100:maximum modulus principle
34:A plot of the modulus of
1702:Cauchy–Riemann equations
1174:on a connected open set
163:cannot exhibit a strict
3241:Mathematical principles
3173:The Theory of Functions
2989:Physical interpretation
2895:vanishes everywhere in
194:In other words, either
59:{\displaystyle \cos(z)}
3076:
3056:
3036:
2979:
2978:{\displaystyle z\in D}
2953:
2909:
2889:
2863:
2815:
2773:
2747:
2705:
2704:{\displaystyle z\in D}
2679:
2632:
2584:
2540:
2520:
2500:
2417:
2328:
2284:
2161:
2080:
2032:
2031:{\displaystyle r>0}
2006:
2005:{\displaystyle a\in D}
1980:
1944:is open, there exists
1938:
1899:
1870:
1829:
1800:
1799:{\displaystyle |f(z)|}
1761:
1732:
1694:
1693:{\displaystyle |f(z)|}
1651:
1620:
1568:
1462:
1431:
1411:
1391:
1360:
1337:
1257:
1237:
1210:
1188:
1168:
1140:
1139:{\displaystyle |f(z)|}
1101:
1081:
1054:
1027:
1000:
999:{\displaystyle |f(z)|}
957:
927:
907:
906:{\displaystyle |f(z)|}
868:
848:
807:
787:
760:
738:
710:
690:
670:
650:
613:
593:
573:
542:
519:
445:
425:
398:
373:
344:
316:
286:
259:
239:
208:
185:
157:
120:
91:
80:
60:
3077:
3057:
3037:
2980:
2954:
2910:
2890:
2864:
2816:
2774:
2748:
2706:
2680:
2633:
2585:
2541:
2521:
2501:
2418:
2329:
2327:{\displaystyle t\in }
2285:
2162:
2081:
2033:
2007:
1981:
1939:
1900:
1871:
1830:
1801:
1762:
1733:
1731:{\displaystyle f'(z)}
1695:
1652:
1650:{\displaystyle z_{0}}
1621:
1569:
1463:
1432:
1412:
1392:
1390:{\displaystyle z_{0}}
1361:
1338:
1258:
1238:
1236:{\displaystyle z_{0}}
1211:
1189:
1169:
1141:
1102:
1082:
1080:{\displaystyle z_{0}}
1055:
1028:
1026:{\displaystyle z_{0}}
1001:
958:
928:
908:
869:
849:
808:
788:
761:
739:
711:
691:
671:
651:
614:
594:
574:
572:{\displaystyle z_{0}}
543:
520:
446:
426:
424:{\displaystyle z_{0}}
399:
374:
345:
322:takes larger values.
317:
287:
285:{\displaystyle z_{0}}
260:
245:inside the domain of
240:
238:{\displaystyle z_{0}}
209:
186:
158:
121:
81:
61:
33:
3066:
3046:
3001:
2963:
2919:
2899:
2873:
2825:
2783:
2757:
2715:
2689:
2645:
2594:
2550:
2530:
2510:
2427:
2338:
2297:
2174:
2090:
2042:
2016:
1990:
1948:
1928:
1898:{\displaystyle f(z)}
1880:
1869:{\displaystyle f(z)}
1851:
1828:{\displaystyle f(z)}
1810:
1771:
1760:{\displaystyle f(z)}
1742:
1708:
1665:
1634:
1585:
1488:
1444:
1421:
1401:
1374:
1350:
1270:
1247:
1220:
1198:
1178:
1158:
1111:
1091:
1064:
1037:
1010:
971:
940:
917:
878:
858:
817:
797:
770:
748:
728:
700:
680:
660:
630:
624:open mapping theorem
603:
583:
556:
532:
458:
435:
408:
386:
363:
334:
296:
269:
249:
222:
218:, or, for any point
198:
175:
137:
128:holomorphic function
110:
70:
38:
2888:{\displaystyle f-g}
2772:{\displaystyle g-f}
2200:
1738:= 0, and thus that
1461:{\displaystyle 1/f}
696:cannot be open, so
649:{\displaystyle |f|}
315:{\displaystyle |f|}
156:{\displaystyle |f|}
3214:Weisstein, Eric W.
3072:
3052:
3032:
2975:
2949:
2905:
2885:
2859:
2811:
2769:
2743:
2701:
2675:
2628:
2580:
2536:
2516:
2496:
2413:
2324:
2280:
2183:
2157:
2076:
2028:
2002:
1976:
1934:
1895:
1866:
1825:
1796:
1757:
1728:
1690:
1647:
1616:
1579:natural logarithms
1564:
1472:Sketches of proofs
1458:
1427:
1407:
1387:
1356:
1333:
1253:
1233:
1206:
1184:
1164:
1136:
1097:
1077:
1050:
1023:
996:
953:
923:
903:
864:
844:
803:
793:be the closure of
783:
756:
734:
706:
686:
666:
646:
609:
589:
569:
538:
515:
441:
421:
394:
369:
340:
312:
282:
255:
235:
204:
181:
153:
116:
92:
76:
56:
3168:Titchmarsh, E. C.
3154:978-1-4612-6314-2
3075:{\displaystyle D}
3055:{\displaystyle D}
2997:. That is, since
2908:{\displaystyle D}
2836:
2794:
2726:
2611:
2539:{\displaystyle a}
2519:{\displaystyle r}
2053:
1959:
1937:{\displaystyle D}
1659:maximum principle
1628:harmonic function
1430:{\displaystyle D}
1410:{\displaystyle f}
1359:{\displaystyle z}
1256:{\displaystyle D}
1187:{\displaystyle D}
1167:{\displaystyle f}
1100:{\displaystyle f}
1048:
951:
926:{\displaystyle D}
867:{\displaystyle D}
834:
806:{\displaystyle D}
781:
737:{\displaystyle D}
720:Related statement
709:{\displaystyle f}
689:{\displaystyle z}
669:{\displaystyle z}
612:{\displaystyle D}
592:{\displaystyle f}
541:{\displaystyle z}
444:{\displaystyle D}
372:{\displaystyle D}
343:{\displaystyle f}
258:{\displaystyle f}
216:constant function
207:{\displaystyle f}
184:{\displaystyle f}
119:{\displaystyle f}
79:{\displaystyle z}
18:Maximal principle
16:(Redirected from
3253:
3227:
3226:
3198:
3179:(See chapter 5.)
3177:
3159:
3158:
3140:
3081:
3079:
3078:
3073:
3061:
3059:
3058:
3053:
3041:
3039:
3038:
3033:
3031:
3014:
2984:
2982:
2981:
2976:
2958:
2956:
2955:
2950:
2914:
2912:
2911:
2906:
2894:
2892:
2891:
2886:
2868:
2866:
2865:
2860:
2837:
2829:
2820:
2818:
2817:
2812:
2795:
2787:
2778:
2776:
2775:
2770:
2752:
2750:
2749:
2744:
2727:
2719:
2710:
2708:
2707:
2702:
2684:
2682:
2681:
2676:
2637:
2635:
2634:
2629:
2612:
2604:
2589:
2587:
2586:
2581:
2545:
2543:
2542:
2537:
2525:
2523:
2522:
2517:
2505:
2503:
2502:
2497:
2495:
2487:
2486:
2459:
2451:
2434:
2422:
2420:
2419:
2414:
2406:
2398:
2397:
2370:
2362:
2345:
2333:
2331:
2330:
2325:
2289:
2287:
2286:
2281:
2266:
2258:
2257:
2230:
2222:
2205:
2199:
2191:
2166:
2164:
2163:
2158:
2129:
2128:
2085:
2083:
2082:
2077:
2054:
2046:
2037:
2035:
2034:
2029:
2011:
2009:
2008:
2003:
1985:
1983:
1982:
1977:
1960:
1952:
1943:
1941:
1940:
1935:
1921:
1920:
1916:
1904:
1902:
1901:
1896:
1875:
1873:
1872:
1867:
1834:
1832:
1831:
1826:
1805:
1803:
1802:
1797:
1795:
1778:
1766:
1764:
1763:
1758:
1737:
1735:
1734:
1729:
1718:
1699:
1697:
1696:
1691:
1689:
1672:
1656:
1654:
1653:
1648:
1646:
1645:
1625:
1623:
1622:
1617:
1615:
1598:
1573:
1571:
1570:
1565:
1539:
1522:
1467:
1465:
1464:
1459:
1454:
1436:
1434:
1433:
1428:
1416:
1414:
1413:
1408:
1396:
1394:
1393:
1388:
1386:
1385:
1365:
1363:
1362:
1357:
1342:
1340:
1339:
1334:
1332:
1315:
1307:
1299:
1298:
1283:
1262:
1260:
1259:
1254:
1242:
1240:
1239:
1234:
1232:
1231:
1215:
1213:
1212:
1207:
1205:
1193:
1191:
1190:
1185:
1173:
1171:
1170:
1165:
1145:
1143:
1142:
1137:
1135:
1118:
1107:is constant, so
1106:
1104:
1103:
1098:
1086:
1084:
1083:
1078:
1076:
1075:
1059:
1057:
1056:
1051:
1049:
1041:
1032:
1030:
1029:
1024:
1022:
1021:
1005:
1003:
1002:
997:
995:
978:
962:
960:
959:
954:
952:
944:
932:
930:
929:
924:
912:
910:
909:
904:
902:
885:
873:
871:
870:
865:
853:
851:
850:
845:
843:
835:
827:
812:
810:
809:
804:
792:
790:
789:
784:
782:
774:
765:
763:
762:
757:
755:
743:
741:
740:
735:
715:
713:
712:
707:
695:
693:
692:
687:
675:
673:
672:
667:
655:
653:
652:
647:
645:
637:
618:
616:
615:
610:
598:
596:
595:
590:
578:
576:
575:
570:
568:
567:
547:
545:
544:
539:
524:
522:
521:
516:
514:
497:
489:
481:
480:
465:
450:
448:
447:
442:
430:
428:
427:
422:
420:
419:
403:
401:
400:
395:
393:
378:
376:
375:
370:
349:
347:
346:
341:
326:Formal statement
321:
319:
318:
313:
311:
303:
291:
289:
288:
283:
281:
280:
264:
262:
261:
256:
244:
242:
241:
236:
234:
233:
213:
211:
210:
205:
190:
188:
187:
182:
162:
160:
159:
154:
152:
144:
125:
123:
122:
117:
104:complex analysis
85:
83:
82:
77:
65:
63:
62:
57:
21:
3261:
3260:
3256:
3255:
3254:
3252:
3251:
3250:
3231:
3230:
3212:
3211:
3208:
3183:
3166:
3163:
3162:
3155:
3142:
3141:
3137:
3132:
3102:Schwarz's lemma
3088:
3064:
3063:
3044:
3043:
2999:
2998:
2991:
2961:
2960:
2917:
2916:
2897:
2896:
2871:
2870:
2823:
2822:
2781:
2780:
2755:
2754:
2713:
2712:
2687:
2686:
2643:
2642:
2592:
2591:
2548:
2547:
2528:
2527:
2508:
2507:
2475:
2425:
2424:
2386:
2336:
2335:
2295:
2294:
2246:
2172:
2171:
2117:
2088:
2087:
2040:
2039:
2014:
2013:
1988:
1987:
1946:
1945:
1926:
1925:
1922:
1918:
1914:
1912:
1911:
1878:
1877:
1849:
1848:
1841:
1808:
1807:
1769:
1768:
1740:
1739:
1711:
1706:
1705:
1663:
1662:
1637:
1632:
1631:
1583:
1582:
1581:to deduce that
1486:
1485:
1479:
1474:
1442:
1441:
1419:
1418:
1417:is constant on
1399:
1398:
1377:
1372:
1371:
1348:
1347:
1290:
1268:
1267:
1245:
1244:
1223:
1218:
1217:
1196:
1195:
1176:
1175:
1156:
1155:
1152:
1109:
1108:
1089:
1088:
1067:
1062:
1061:
1035:
1034:
1013:
1008:
1007:
969:
968:
938:
937:
915:
914:
876:
875:
856:
855:
815:
814:
813:. Suppose that
795:
794:
768:
767:
746:
745:
726:
725:
722:
698:
697:
678:
677:
658:
657:
628:
627:
601:
600:
599:is constant on
581:
580:
559:
554:
553:
530:
529:
472:
456:
455:
433:
432:
411:
406:
405:
384:
383:
361:
360:
332:
331:
328:
294:
293:
272:
267:
266:
247:
246:
225:
220:
219:
196:
195:
173:
172:
135:
134:
108:
107:
106:states that if
68:
67:
36:
35:
28:
23:
22:
15:
12:
11:
5:
3259:
3257:
3249:
3248:
3243:
3233:
3232:
3229:
3228:
3207:
3206:External links
3204:
3203:
3202:
3199:
3181:
3161:
3160:
3153:
3134:
3133:
3131:
3128:
3127:
3126:
3119:
3112:
3105:
3099:
3087:
3084:
3071:
3051:
3030:
3026:
3023:
3020:
3017:
3013:
3009:
3006:
2990:
2987:
2974:
2971:
2968:
2948:
2945:
2942:
2939:
2936:
2933:
2930:
2927:
2924:
2915:which implies
2904:
2884:
2881:
2878:
2858:
2855:
2852:
2849:
2846:
2843:
2840:
2835:
2832:
2810:
2807:
2804:
2801:
2798:
2793:
2790:
2768:
2765:
2762:
2742:
2739:
2736:
2733:
2730:
2725:
2722:
2700:
2697:
2694:
2674:
2671:
2668:
2665:
2662:
2659:
2656:
2653:
2650:
2627:
2624:
2621:
2618:
2615:
2610:
2607:
2602:
2599:
2579:
2576:
2573:
2570:
2567:
2564:
2561:
2558:
2555:
2535:
2515:
2494:
2490:
2485:
2482:
2478:
2474:
2471:
2468:
2465:
2462:
2458:
2454:
2450:
2446:
2443:
2440:
2437:
2433:
2412:
2409:
2405:
2401:
2396:
2393:
2389:
2385:
2382:
2379:
2376:
2373:
2369:
2365:
2361:
2357:
2354:
2351:
2348:
2344:
2323:
2320:
2317:
2314:
2311:
2308:
2305:
2302:
2291:
2290:
2279:
2276:
2273:
2270:
2265:
2261:
2256:
2253:
2249:
2245:
2242:
2239:
2236:
2233:
2229:
2225:
2221:
2217:
2214:
2211:
2208:
2204:
2198:
2195:
2190:
2186:
2182:
2179:
2156:
2153:
2150:
2147:
2144:
2141:
2138:
2135:
2132:
2127:
2124:
2120:
2116:
2113:
2110:
2107:
2104:
2101:
2098:
2095:
2075:
2072:
2069:
2066:
2063:
2060:
2057:
2052:
2049:
2027:
2024:
2021:
2001:
1998:
1995:
1975:
1972:
1969:
1966:
1963:
1958:
1955:
1933:
1910:
1907:
1894:
1891:
1888:
1885:
1865:
1862:
1859:
1856:
1840:
1837:
1824:
1821:
1818:
1815:
1794:
1790:
1787:
1784:
1781:
1777:
1756:
1753:
1750:
1747:
1727:
1724:
1721:
1717:
1714:
1688:
1684:
1681:
1678:
1675:
1671:
1644:
1640:
1614:
1610:
1607:
1604:
1601:
1597:
1593:
1590:
1575:
1574:
1563:
1560:
1557:
1554:
1551:
1548:
1545:
1542:
1538:
1534:
1531:
1528:
1525:
1521:
1517:
1514:
1511:
1508:
1505:
1502:
1499:
1496:
1493:
1478:
1475:
1473:
1470:
1457:
1453:
1449:
1426:
1406:
1384:
1380:
1355:
1344:
1343:
1331:
1327:
1324:
1321:
1318:
1314:
1310:
1306:
1302:
1297:
1293:
1289:
1286:
1282:
1278:
1275:
1252:
1243:is a point in
1230:
1226:
1204:
1183:
1163:
1151:
1148:
1134:
1130:
1127:
1124:
1121:
1117:
1096:
1074:
1070:
1047:
1044:
1020:
1016:
994:
990:
987:
984:
981:
977:
950:
947:
922:
901:
897:
894:
891:
888:
884:
863:
842:
838:
833:
830:
825:
822:
802:
780:
777:
754:
733:
721:
718:
705:
685:
665:
644:
640:
636:
608:
588:
566:
562:
537:
526:
525:
513:
509:
506:
503:
500:
496:
492:
488:
484:
479:
475:
471:
468:
464:
440:
431:is a point in
418:
414:
392:
368:
339:
327:
324:
310:
306:
302:
279:
275:
254:
232:
228:
203:
180:
151:
147:
143:
115:
75:
55:
52:
49:
46:
43:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3258:
3247:
3244:
3242:
3239:
3238:
3236:
3224:
3223:
3218:
3215:
3210:
3209:
3205:
3200:
3197:
3193:
3192:
3187:
3182:
3180:
3175:
3174:
3169:
3165:
3164:
3156:
3150:
3146:
3139:
3136:
3129:
3124:
3120:
3117:
3113:
3110:
3106:
3103:
3100:
3097:
3093:
3092:
3091:
3085:
3083:
3069:
3049:
3021:
3015:
3007:
3004:
2996:
2995:heat equation
2988:
2986:
2972:
2969:
2966:
2943:
2937:
2934:
2928:
2922:
2902:
2882:
2879:
2876:
2869:. This means
2856:
2853:
2847:
2844:
2841:
2830:
2805:
2802:
2799:
2788:
2779:vanishes. As
2766:
2763:
2760:
2737:
2734:
2731:
2720:
2698:
2695:
2692:
2669:
2663:
2660:
2654:
2648:
2639:
2622:
2619:
2616:
2605:
2600:
2597:
2574:
2568:
2565:
2559:
2553:
2546:. Therefore,
2533:
2513:
2483:
2480:
2476:
2472:
2469:
2466:
2460:
2452:
2441:
2435:
2410:
2407:
2394:
2391:
2387:
2383:
2380:
2377:
2371:
2363:
2352:
2346:
2318:
2315:
2312:
2309:
2303:
2300:
2277:
2274:
2271:
2268:
2254:
2251:
2247:
2243:
2240:
2237:
2231:
2223:
2212:
2206:
2196:
2193:
2188:
2184:
2180:
2177:
2170:
2169:
2168:
2151:
2148:
2145:
2142:
2136:
2133:
2130:
2125:
2122:
2118:
2114:
2111:
2108:
2105:
2099:
2093:
2073:
2070:
2064:
2061:
2058:
2047:
2025:
2022:
2019:
1999:
1996:
1993:
1970:
1967:
1964:
1953:
1931:
1917:
1906:
1905:is constant.
1889:
1883:
1860:
1854:
1846:
1836:
1819:
1813:
1785:
1779:
1751:
1745:
1722:
1715:
1712:
1704:we show that
1703:
1679:
1673:
1660:
1642:
1638:
1629:
1605:
1599:
1591:
1588:
1580:
1558:
1552:
1549:
1546:
1543:
1540:
1529:
1523:
1515:
1512:
1509:
1503:
1497:
1494:
1491:
1484:
1483:
1482:
1476:
1471:
1469:
1455:
1451:
1447:
1438:
1424:
1404:
1382:
1378:
1369:
1353:
1322:
1316:
1308:
1295:
1291:
1284:
1276:
1273:
1266:
1265:
1264:
1250:
1228:
1224:
1181:
1161:
1149:
1147:
1125:
1119:
1094:
1072:
1068:
1042:
1018:
1014:
985:
979:
966:
945:
934:
920:
892:
886:
861:
828:
823:
820:
800:
775:
731:
724:Suppose that
719:
717:
716:is constant.
703:
683:
663:
638:
625:
620:
606:
586:
564:
560:
551:
535:
504:
498:
490:
477:
473:
466:
454:
453:
452:
438:
416:
412:
382:
381:complex plane
366:
359:
356:
353:
337:
325:
323:
304:
277:
273:
252:
230:
226:
217:
214:is locally a
201:
192:
178:
170:
166:
145:
133:
129:
113:
105:
101:
97:
89:
73:
66:(in red) for
50:
44:
41:
32:
19:
3220:
3189:
3178:
3172:
3144:
3138:
3089:
3086:Applications
2992:
2640:
2526:centered at
2292:
2038:) such that
2012:with radius
1923:
1842:
1577:for complex
1576:
1480:
1439:
1368:neighborhood
1345:
1153:
935:
723:
621:
550:neighborhood
527:
329:
193:
99:
93:
451:such that
130:, then the
96:mathematics
3235:Categories
3130:References
1263:such that
3222:MathWorld
3196:EMS Press
3008:
2970:∈
2880:−
2854:∈
2834:¯
2792:¯
2764:−
2724:¯
2696:∈
2609:¯
2601:∈
2408:≥
2364:−
2319:π
2304:∈
2275:≤
2224:−
2197:π
2185:∫
2181:≤
2152:π
2137:∈
2094:γ
2071:⊂
2051:¯
1997:∈
1957:¯
1592:
1550:
1516:
1495:
1309:≤
1046:¯
949:¯
837:→
832:¯
824::
779:¯
491:≥
352:connected
292:at which
88:unit disk
45:
3170:(1939).
2959:for all
2685:for all
2590:for all
2293:For all
1716:′
1630:. Since
1366:in some
1346:for all
548:in some
528:for all
1397:, then
965:compact
874:. Then
579:, then
379:of the
165:maximum
132:modulus
86:in the
3151:
1913:": -->
1661:that
766:. Let
358:subset
169:domain
98:, the
2423:, so
1626:is a
1216:, if
1060:. If
126:is a
3149:ISBN
3121:The
3114:The
3107:The
3094:The
2023:>
1915:edit
1277:<
355:open
330:Let
3005:log
1924:As
1547:arg
1492:log
1370:of
1194:of
1033:of
963:is
552:of
191:.
171:of
102:in
94:In
42:cos
3237::
3219:.
3194:,
3188:,
2985:.
2638:.
2334:,
1835:.
1589:ln
1513:ln
1468:.
1437:.
933:.
619:.
3225:.
3157:.
3098:.
3070:D
3050:D
3029:|
3025:)
3022:z
3019:(
3016:f
3012:|
2973:D
2967:z
2947:)
2944:a
2941:(
2938:f
2935:=
2932:)
2929:z
2926:(
2923:f
2903:D
2883:g
2877:f
2857:D
2851:)
2848:r
2845:,
2842:a
2839:(
2831:B
2809:)
2806:r
2803:,
2800:a
2797:(
2789:B
2767:f
2761:g
2741:)
2738:r
2735:,
2732:a
2729:(
2721:B
2699:D
2693:z
2673:)
2670:a
2667:(
2664:f
2661:=
2658:)
2655:z
2652:(
2649:g
2626:)
2623:r
2620:,
2617:a
2614:(
2606:B
2598:z
2578:)
2575:a
2572:(
2569:f
2566:=
2563:)
2560:z
2557:(
2554:f
2534:a
2514:r
2493:|
2489:)
2484:t
2481:i
2477:e
2473:r
2470:+
2467:a
2464:(
2461:f
2457:|
2453:=
2449:|
2445:)
2442:a
2439:(
2436:f
2432:|
2411:0
2404:|
2400:)
2395:t
2392:i
2388:e
2384:r
2381:+
2378:a
2375:(
2372:f
2368:|
2360:|
2356:)
2353:a
2350:(
2347:f
2343:|
2322:]
2316:2
2313:,
2310:0
2307:[
2301:t
2278:0
2272:t
2269:d
2264:|
2260:)
2255:t
2252:i
2248:e
2244:r
2241:+
2238:a
2235:(
2232:f
2228:|
2220:|
2216:)
2213:a
2210:(
2207:f
2203:|
2194:2
2189:0
2178:0
2155:]
2149:2
2146:,
2143:0
2140:[
2134:t
2131:,
2126:t
2123:i
2119:e
2115:r
2112:+
2109:a
2106:=
2103:)
2100:t
2097:(
2074:D
2068:)
2065:r
2062:,
2059:a
2056:(
2048:B
2026:0
2020:r
2000:D
1994:a
1974:)
1971:r
1968:,
1965:a
1962:(
1954:B
1932:D
1919:]
1893:)
1890:z
1887:(
1884:f
1864:)
1861:z
1858:(
1855:f
1823:)
1820:z
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231:0
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150:|
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74:z
54:)
51:z
48:(
20:)
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