58:
27:
53:. This plot shows how approaching the essential singularity from different directions yields different behaviors (as opposed to a pole, which, approached from any direction, would be uniformly white).
255:
1815:
1314:
1135:
822:
645:
891:
1081:
1885:
1848:
1753:
1628:
1260:
1020:
768:
591:
1385:
1210:
970:
718:
1508:
364:
169:
1984:
1698:
1659:
451:
390:
1940:
1920:
1588:
1568:
1548:
1528:
1457:
1433:
1413:
1358:
1334:
1179:
1155:
937:
911:
691:
665:
542:
522:
495:
471:
418:
303:
279:
209:
189:
144:
103:
that are especially unmanageable: by definition they fit into neither of the other two categories of singularity that may be dealt with in some manner –
214:
57:
1758:
2119:
1265:
1086:
773:
596:
20:
827:
2030:
1895:
1946:
complex value, except possibly one, infinitely many times. (The exception is necessary; for example, the function
1025:
2142:
1861:
1824:
1703:
1604:
501:
1220:
980:
728:
551:
112:
2163:
1899:
89:
1363:
1188:
948:
696:
1462:
319:
152:
940:
914:
306:
108:
1949:
85:
1664:
1818:
669:
310:
104:
2133:
1633:
1439:
of the
Laurent series is an infinite sum). A related definition is that if there is a point
1891:
258:
100:
427:
8:
2004:
369:
1925:
1905:
1573:
1553:
1533:
1513:
1442:
1418:
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1319:
1164:
1140:
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896:
676:
650:
527:
507:
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403:
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264:
194:
174:
129:
2115:
2086:
1855:
1598:
474:
46:
2076:
2066:
77:
2138:
2071:
2054:
1594:
1436:
1392:
421:
111:. In practice some include non-isolated singularities too; those do not have a
50:
2148:
2157:
2090:
147:
1902:. The latter says that in every neighborhood of an essential singularity
1851:
2081:
26:
2055:"Value-Distribution of the Riemann Zeta-Function Along Its Julia Lines"
498:
124:
1391:
Another way to characterize an essential singularity is that the
61:
Model illustrating essential singularity of a complex function
1661:
has an essential singularity at that point if and only if the
1821:
on the
Riemann sphere has only one essential singularity, at
16:
Location around which a function displays irregular behavior
19:
For essential singularities of real valued functions, see
2114:. Page 920. Alpha Science International, Limited, 2004.
250:{\displaystyle f\colon U\setminus \{a\}\to \mathbb {C} }
1894:
near their essential singularities is described by the
1952:
1928:
1908:
1864:
1827:
1761:
1706:
1667:
1636:
1607:
1576:
1556:
1536:
1516:
1465:
1445:
1435:
has infinitely many negative degree terms (i.e., the
1421:
1401:
1366:
1346:
1322:
1268:
1223:
1191:
1167:
1143:
1089:
1028:
983:
951:
925:
899:
830:
776:
731:
699:
679:
653:
599:
554:
530:
510:
483:
459:
430:
406:
372:
322:
291:
267:
217:
197:
177:
155:
132:
92:
near which the function exhibits striking behavior.
2053:Steuding, Jörn; Suriajaya, Ade Irma (2020-11-01).
1978:
1934:
1914:
1879:
1842:
1809:
1747:
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1622:
1582:
1562:
1542:
1522:
1502:
1451:
1427:
1407:
1379:
1352:
1328:
1308:
1254:
1204:
1173:
1149:
1129:
1075:
1014:
964:
931:
905:
885:
816:
762:
712:
685:
659:
639:
585:
536:
516:
489:
465:
445:
412:
384:
358:
297:
273:
249:
203:
183:
163:
138:
2052:
1810:{\displaystyle \lim _{z\to 0}{\frac {1}{f(1/z)}}}
2155:
1763:
1708:
1700:has an essential singularity at 0: i.e. neither
1270:
1225:
1091:
1030:
985:
832:
778:
733:
601:
556:
1309:{\displaystyle \lim _{z\to a}{\frac {1}{f(z)}}}
1130:{\displaystyle \lim _{z\to a}{\frac {1}{f(z)}}}
817:{\displaystyle \lim _{z\to a}{\frac {1}{f(z)}}}
640:{\displaystyle \lim _{z\to a}{\frac {1}{f(z)}}}
886:{\displaystyle \lim _{z\to a}|1/f(z)|=\infty }
1076:{\displaystyle \lim _{z\to a}|f(z)|=\infty }
236:
230:
395:
38:, centered on the essential singularity at
2080:
2070:
2059:Computational Methods and Function Theory
1871:
1834:
1614:
243:
157:
56:
25:
2110:Rajendra Kumar Jain, S. R. K. Iyengar;
2156:
1880:{\displaystyle \infty _{\mathbb {C} }}
1858:has a unique essential singularity at
1843:{\displaystyle \infty _{\mathbb {C} }}
1748:{\displaystyle \lim _{z\to 0}{f(1/z)}}
1623:{\displaystyle \infty _{\mathbb {C} }}
2031:"Infinity as an Isolated Singularity"
2002:
497:of the complex plane, and that every
118:
99:is a "left-over" or default group of
2149:Essential Singularity on Planet Math
13:
1866:
1829:
1609:
1255:{\displaystyle \lim _{z\to a}f(z)}
1070:
1015:{\displaystyle \lim _{z\to a}f(z)}
880:
763:{\displaystyle \lim _{z\to a}f(z)}
586:{\displaystyle \lim _{z\to a}f(z)}
14:
2175:
2126:
1898:and by the considerably stronger
227:
21:Classification of discontinuities
2112:Advanced Engineering Mathematics
524:has non-empty intersection with
366:has an essential singularity at
305:if the singularity is neither a
1570:is an essential singularity of
49:, the luminance represents the
2143:Wolfram Demonstrations Project
2046:
2023:
1996:
1973:
1959:
1801:
1787:
1770:
1741:
1727:
1715:
1686:
1672:
1647:
1641:
1491:
1478:
1475:
1469:
1380:{\displaystyle {\frac {1}{f}}}
1300:
1294:
1277:
1249:
1243:
1232:
1205:{\displaystyle {\frac {1}{f}}}
1121:
1115:
1098:
1063:
1059:
1053:
1046:
1037:
1009:
1003:
992:
965:{\displaystyle {\frac {1}{f}}}
873:
869:
863:
848:
839:
808:
802:
785:
757:
751:
740:
713:{\displaystyle {\frac {1}{f}}}
631:
625:
608:
580:
574:
563:
440:
434:
332:
326:
239:
1:
1989:
1986:never takes on the value 0.)
1854:function aside that is not a
1503:{\displaystyle f(z)(z-a)^{n}}
1896:Casorati–Weierstrass theorem
359:{\displaystyle f(z)=e^{1/z}}
164:{\displaystyle \mathbb {C} }
7:
1459:for which no derivative of
10:
2180:
2072:10.1007/s40315-020-00316-x
316:For example, the function
18:
1979:{\displaystyle \exp(1/z)}
45:. The hue represents the
2134:An Essential Singularity
1693:{\displaystyle {f(1/z)}}
1510:converges to a limit as
1022:does not exist (in fact
824:does not exist (in fact
396:Alternative descriptions
2005:"Essential Singularity"
105:removable singularities
1980:
1936:
1916:
1900:Picard's great theorem
1881:
1844:
1811:
1749:
1694:
1655:
1654:{\displaystyle {f(z)}}
1624:
1584:
1564:
1544:
1524:
1504:
1453:
1429:
1409:
1381:
1354:
1330:
1310:
1256:
1206:
1175:
1151:
1131:
1077:
1016:
966:
933:
907:
887:
818:
764:
714:
687:
661:
641:
587:
538:
518:
491:
467:
447:
414:
386:
360:
299:
275:
251:
205:
185:
165:
140:
101:isolated singularities
73:
54:
1981:
1937:
1917:
1892:holomorphic functions
1882:
1845:
1819:Riemann zeta function
1812:
1750:
1695:
1656:
1625:
1585:
1565:
1545:
1525:
1505:
1454:
1430:
1410:
1382:
1355:
1338:essential singularity
1331:
1311:
1257:
1207:
1176:
1152:
1132:
1078:
1017:
967:
934:
908:
888:
819:
765:
715:
688:
670:removable singularity
662:
642:
588:
539:
519:
492:
468:
448:
415:
387:
361:
311:removable singularity
300:
283:essential singularity
276:
252:
206:
186:
166:
141:
97:essential singularity
82:essential singularity
60:
30:Plot of the function
29:
1950:
1926:
1906:
1862:
1825:
1759:
1704:
1665:
1634:
1605:
1574:
1554:
1534:
1514:
1463:
1443:
1419:
1399:
1364:
1344:
1320:
1266:
1221:
1189:
1165:
1141:
1087:
1026:
981:
949:
923:
897:
828:
774:
729:
697:
677:
651:
597:
552:
528:
508:
481:
457:
446:{\displaystyle f(z)}
428:
404:
370:
320:
289:
265:
259:holomorphic function
215:
195:
175:
153:
130:
2107:, McGraw-Hill, 1979
2003:Weisstein, Eric W.
385:{\displaystyle z=0}
1976:
1932:
1912:
1877:
1840:
1807:
1777:
1745:
1722:
1690:
1651:
1620:
1580:
1560:
1540:
1520:
1500:
1449:
1425:
1405:
1377:
1350:
1326:
1306:
1284:
1252:
1239:
1202:
1171:
1147:
1127:
1105:
1073:
1044:
1012:
999:
962:
929:
903:
883:
846:
814:
792:
760:
747:
710:
683:
657:
637:
615:
583:
570:
534:
514:
487:
463:
453:is not defined at
443:
424:, and assume that
410:
382:
356:
295:
271:
247:
201:
181:
161:
136:
119:Formal description
74:
55:
2103:Lars V. Ahlfors;
1935:{\displaystyle f}
1915:{\displaystyle a}
1856:rational function
1805:
1762:
1707:
1599:point at infinity
1583:{\displaystyle f}
1563:{\displaystyle a}
1543:{\displaystyle a}
1523:{\displaystyle z}
1452:{\displaystyle a}
1428:{\displaystyle a}
1408:{\displaystyle f}
1375:
1353:{\displaystyle f}
1329:{\displaystyle a}
1304:
1269:
1224:
1200:
1174:{\displaystyle f}
1150:{\displaystyle a}
1125:
1090:
1029:
984:
960:
932:{\displaystyle f}
906:{\displaystyle a}
831:
812:
777:
732:
708:
686:{\displaystyle f}
660:{\displaystyle a}
635:
600:
555:
537:{\displaystyle U}
517:{\displaystyle a}
490:{\displaystyle U}
466:{\displaystyle a}
413:{\displaystyle a}
298:{\displaystyle f}
274:{\displaystyle a}
204:{\displaystyle U}
191:be an element of
184:{\displaystyle a}
139:{\displaystyle U}
2171:
2164:Complex analysis
2105:Complex Analysis
2095:
2094:
2084:
2074:
2050:
2044:
2043:
2041:
2040:
2035:
2027:
2021:
2020:
2018:
2016:
2000:
1985:
1983:
1982:
1977:
1969:
1941:
1939:
1938:
1933:
1921:
1919:
1918:
1913:
1890:The behavior of
1886:
1884:
1883:
1878:
1876:
1875:
1874:
1850:. Indeed, every
1849:
1847:
1846:
1841:
1839:
1838:
1837:
1816:
1814:
1813:
1808:
1806:
1804:
1797:
1779:
1776:
1754:
1752:
1751:
1746:
1744:
1737:
1721:
1699:
1697:
1696:
1691:
1689:
1682:
1660:
1658:
1657:
1652:
1650:
1629:
1627:
1626:
1621:
1619:
1618:
1617:
1589:
1587:
1586:
1581:
1569:
1567:
1566:
1561:
1549:
1547:
1546:
1541:
1529:
1527:
1526:
1521:
1509:
1507:
1506:
1501:
1499:
1498:
1458:
1456:
1455:
1450:
1434:
1432:
1431:
1426:
1414:
1412:
1411:
1406:
1386:
1384:
1383:
1378:
1376:
1368:
1359:
1357:
1356:
1351:
1335:
1333:
1332:
1327:
1315:
1313:
1312:
1307:
1305:
1303:
1286:
1283:
1261:
1259:
1258:
1253:
1238:
1211:
1209:
1208:
1203:
1201:
1193:
1180:
1178:
1177:
1172:
1156:
1154:
1153:
1148:
1136:
1134:
1133:
1128:
1126:
1124:
1107:
1104:
1082:
1080:
1079:
1074:
1066:
1049:
1043:
1021:
1019:
1018:
1013:
998:
971:
969:
968:
963:
961:
953:
938:
936:
935:
930:
912:
910:
909:
904:
892:
890:
889:
884:
876:
859:
851:
845:
823:
821:
820:
815:
813:
811:
794:
791:
769:
767:
766:
761:
746:
719:
717:
716:
711:
709:
701:
692:
690:
689:
684:
666:
664:
663:
658:
646:
644:
643:
638:
636:
634:
617:
614:
592:
590:
589:
584:
569:
543:
541:
540:
535:
523:
521:
520:
515:
496:
494:
493:
488:
472:
470:
469:
464:
452:
450:
449:
444:
419:
417:
416:
411:
391:
389:
388:
383:
365:
363:
362:
357:
355:
354:
350:
304:
302:
301:
296:
285:of the function
280:
278:
277:
272:
256:
254:
253:
248:
246:
210:
208:
207:
202:
190:
188:
187:
182:
170:
168:
167:
162:
160:
145:
143:
142:
137:
78:complex analysis
72:
47:complex argument
44:
37:
2179:
2178:
2174:
2173:
2172:
2170:
2169:
2168:
2154:
2153:
2139:Stephen Wolfram
2129:
2124:
2099:
2098:
2051:
2047:
2038:
2036:
2033:
2029:
2028:
2024:
2014:
2012:
2001:
1997:
1992:
1965:
1951:
1948:
1947:
1927:
1924:
1923:
1922:, the function
1907:
1904:
1903:
1870:
1869:
1865:
1863:
1860:
1859:
1833:
1832:
1828:
1826:
1823:
1822:
1793:
1783:
1778:
1766:
1760:
1757:
1756:
1733:
1723:
1711:
1705:
1702:
1701:
1678:
1668:
1666:
1663:
1662:
1637:
1635:
1632:
1631:
1630:, the function
1613:
1612:
1608:
1606:
1603:
1602:
1575:
1572:
1571:
1555:
1552:
1551:
1535:
1532:
1531:
1515:
1512:
1511:
1494:
1490:
1464:
1461:
1460:
1444:
1441:
1440:
1420:
1417:
1416:
1400:
1397:
1396:
1367:
1365:
1362:
1361:
1345:
1342:
1341:
1321:
1318:
1317:
1290:
1285:
1273:
1267:
1264:
1263:
1228:
1222:
1219:
1218:
1192:
1190:
1187:
1186:
1166:
1163:
1162:
1142:
1139:
1138:
1111:
1106:
1094:
1088:
1085:
1084:
1062:
1045:
1033:
1027:
1024:
1023:
988:
982:
979:
978:
952:
950:
947:
946:
924:
921:
920:
898:
895:
894:
872:
855:
847:
835:
829:
826:
825:
798:
793:
781:
775:
772:
771:
736:
730:
727:
726:
700:
698:
695:
694:
678:
675:
674:
652:
649:
648:
621:
616:
604:
598:
595:
594:
559:
553:
550:
549:
529:
526:
525:
509:
506:
505:
482:
479:
478:
477:in some region
458:
455:
454:
429:
426:
425:
405:
402:
401:
398:
371:
368:
367:
346:
342:
338:
321:
318:
317:
290:
287:
286:
266:
263:
262:
242:
216:
213:
212:
196:
193:
192:
176:
173:
172:
156:
154:
151:
150:
131:
128:
127:
121:
62:
39:
31:
24:
17:
12:
11:
5:
2177:
2167:
2166:
2152:
2151:
2146:
2128:
2127:External links
2125:
2123:
2122:
2108:
2100:
2097:
2096:
2065:(3): 389–401.
2045:
2022:
1994:
1993:
1991:
1988:
1975:
1972:
1968:
1964:
1961:
1958:
1955:
1931:
1911:
1873:
1868:
1836:
1831:
1803:
1800:
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1685:
1681:
1677:
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1671:
1649:
1646:
1643:
1640:
1616:
1611:
1595:Riemann sphere
1579:
1559:
1539:
1519:
1497:
1493:
1489:
1486:
1483:
1480:
1477:
1474:
1471:
1468:
1448:
1437:principal part
1424:
1404:
1393:Laurent series
1389:
1388:
1374:
1371:
1349:
1325:
1302:
1299:
1296:
1293:
1289:
1282:
1279:
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1272:
1251:
1248:
1245:
1242:
1237:
1234:
1231:
1227:
1214:
1213:
1199:
1196:
1170:
1146:
1123:
1120:
1117:
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1110:
1103:
1100:
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1048:
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1039:
1036:
1032:
1011:
1008:
1005:
1002:
997:
994:
991:
987:
977:Similarly, if
974:
973:
959:
956:
928:
902:
882:
879:
875:
871:
868:
865:
862:
858:
854:
850:
844:
841:
838:
834:
810:
807:
804:
801:
797:
790:
787:
784:
780:
759:
756:
753:
750:
745:
742:
739:
735:
722:
721:
707:
704:
682:
656:
633:
630:
627:
624:
620:
613:
610:
607:
603:
582:
579:
576:
573:
568:
565:
562:
558:
533:
513:
486:
462:
442:
439:
436:
433:
422:complex number
409:
397:
394:
381:
378:
375:
353:
349:
345:
341:
337:
334:
331:
328:
325:
294:
270:
245:
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1990:References
67:= exp(1/(6
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