49:
1533:
1418:
727:
1256:
965:
can be multiple-valued over some domains. The principal value of some of these functions can be obtained by decomposing the function into simpler ones whereby the principal value of the simple functions are straightforward to obtain.
1067:
1766:
1316:
578:
1462:
924:
1304:
512:
789:
1697:
1630:
206:
1491:
1593:
1149:
270:
367:
313:
435:
407:
1100:
546:
462:
1120:
827:
336:
1464:
Sometimes a branch cut is introduced so that negative real numbers are not in the domain of the square root function and eliminating the possibility that
1524:, etc.) can be defined in terms of logarithms and their principal values can be defined in terms of the principal values of the logarithm.
984:
35:
181:. For example, 4 has two square roots: 2 and −2; of these the positive root, 2, is considered the principal root and is denoted as
1713:
113:
1812:
85:
1122:(inclusive), so we take this to be the principal value of the argument, and we write the argument function on this branch
92:
1413:{\displaystyle \mathrm {pv} {\sqrt {z}}=\exp \left({\frac {\mathrm {pv} \log z}{2}}\right)={\sqrt {r}}\,e^{i\phi /2}}
805:), a single-valued component of the multiple-valued log function. When the focus is on a single branch, sometimes a
132:
722:{\displaystyle \log {z}=\ln {|z|}+i\left(\mathrm {arg} \ z\right)=\ln {|z|}+i\left(\mathrm {Arg} \ z+2\pi k\right)}
66:
1537:
99:
962:
797:
162:
70:
1429:
809:
is used; in this case removing the non-positive real numbers from the domain of the function and eliminating
81:
891:
381:
1834:
1269:
471:
756:
1664:
1600:
1251:{\displaystyle \mathrm {pv} \log {z}=\mathrm {Log} \,z=\ln {|z|}+i\left(\mathrm {Arg} \,z\right).}
1079:
is intrinsically multivalued. One often defines the argument of some complex number to be between
184:
1467:
59:
1563:
751:
170:
166:
24:
239:
341:
285:
106:
412:
386:
1082:
934:
528:
444:
158:
1105:
812:
558:
But this has a consequence that may be surprising in comparison of real valued functions:
8:
837:
975:
321:
372:
However, there are other solutions, which is evidenced by considering the position of
1808:
1549:
1424:
841:
216:
1776:
150:
1802:
227:
1828:
377:
16:
Values along one branch of a multivalued function so that it is single-valued
1781:
1495:
1307:
178:
174:
146:
858:, and along this branch, the values the function takes are known as the
806:
48:
1062:{\displaystyle \log {z}=\ln {|z|}+i\left(\mathrm {arg} \ z\right).}
733:
1553:
1143:, we obtain the principal value of the logarithm, and we write
1636:
1541:
1532:
1761:{\displaystyle ({\tfrac {-\pi }{2}},{\tfrac {\pi }{2}}].}
956:
1741:
1721:
1496:
Inverse trigonometric and inverse hyperbolic functions
836:. With this branch cut, the single-branch function is
1716:
1667:
1603:
1566:
1470:
1432:
1319:
1272:
1152:
1108:
1085:
987:
894:
815:
759:
581:
531:
474:
447:
415:
389:
344:
324:
288:
242:
187:
1804:
A First Course in
Complex Analysis with Applications
525:. It becomes clear that we can add any multiple of
73:. Unsourced material may be challenged and removed.
1760:
1691:
1624:
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1485:
1456:
1412:
1298:
1250:
1114:
1094:
1061:
918:
821:
783:
721:
540:
506:
456:
429:
401:
361:
330:
307:
264:
200:
548:to our initial solution to obtain all values for
1826:
37:Inverse trigonometric function § Principal value
1635:For example, many computing systems include an
1807:. Jones & Bartlett Learning. p. 166.
1800:
881:is multiple-valued, the principal branch of
441:initially, but if we rotate further another
1512:, etc.) and inverse hyperbolic functions (
1388:
1295:
1236:
1186:
903:
133:Learn how and when to remove this message
1801:Zill, Dennis; Shanahan, Patrick (2009).
1531:
1457:{\displaystyle -\pi <\phi \leq \pi .}
1827:
957:Principal values of standard functions
565:does not have one definite value. For
275:Now, for example, say we wish to find
34:for arcsines, arccosines, etc. , see
23:in describing improper integrals, see
969:
173:. A simple case arises in taking the
1129:(with the leading capital A). Using
71:adding citations to reliable sources
42:
1527:
919:{\displaystyle \mathrm {pv} \,f(z)}
13:
1355:
1352:
1324:
1321:
1232:
1229:
1226:
1182:
1179:
1176:
1157:
1154:
1041:
1038:
1035:
899:
896:
692:
689:
686:
635:
632:
629:
14:
1846:
1500:Inverse trigonometric functions (
409:. We can rotate counterclockwise
468:again. So, we can conclude that
161:are the values along one chosen
47:
1299:{\displaystyle z=re^{i\phi }\,}
865:
746:is the (principal) argument of
507:{\displaystyle i(\pi /2+2\pi )}
58:needs additional citations for
1794:
1752:
1717:
1683:
1668:
1619:
1604:
1582:
1567:
1261:
1209:
1201:
1018:
1010:
913:
907:
795:determines what is known as a
784:{\displaystyle (-\pi ,\ \pi ]}
778:
760:
669:
661:
612:
604:
501:
478:
279:. This means we want to solve
1:
1787:
1692:{\displaystyle (-\pi ,\pi ].}
210:
1625:{\displaystyle (-\pi ,\pi ]}
847:The branch corresponding to
201:{\displaystyle {\sqrt {4}}.}
7:
1770:
1486:{\displaystyle \phi =\pi .}
1306:the principal value of the
10:
1851:
844:everywhere in its domain.
29:
18:
1588:{\displaystyle [0,2\pi )}
1661:will be in the interval
1649:function. The value of
829:as a possible value for
437:radians from 1 to reach
265:{\displaystyle e^{w}=z.}
30:For the use of the term
19:For the use of the term
1550:complex number argument
1548:The principal value of
362:{\displaystyle i\pi /2}
308:{\displaystyle e^{w}=i}
226:. It is defined as the
1762:
1693:
1626:
1589:
1545:
1487:
1458:
1414:
1300:
1252:
1116:
1096:
1063:
920:
823:
785:
750:defined to lie in the
723:
542:
508:
458:
431:
430:{\displaystyle \pi /2}
403:
402:{\displaystyle \arg i}
380:and in particular its
363:
332:
309:
266:
202:
25:Cauchy principal value
1763:
1694:
1651:atan2(imaginary_part(
1627:
1590:
1535:
1488:
1459:
1415:
1301:
1266:For a complex number
1253:
1117:
1097:
1095:{\displaystyle -\pi }
1064:
974:We have examined the
921:
824:
786:
724:
543:
541:{\displaystyle 2\pi }
509:
459:
457:{\displaystyle 2\pi }
432:
404:
364:
333:
310:
267:
203:
1714:
1665:
1601:
1597:values in the range
1564:
1560:values in the range
1468:
1430:
1317:
1270:
1150:
1115:{\displaystyle \pi }
1106:
1083:
985:
963:elementary functions
892:
822:{\displaystyle \pi }
813:
757:
579:
529:
472:
445:
413:
387:
342:
322:
286:
240:
185:
159:multivalued function
67:improve this article
1556:can be defined as:
1758:
1750:
1735:
1689:
1622:
1585:
1546:
1483:
1454:
1410:
1296:
1248:
1112:
1092:
1059:
976:logarithm function
970:Logarithm function
953:is single-valued.
916:
819:
781:
719:
538:
504:
454:
427:
399:
359:
328:
305:
262:
198:
1814:978-0-7637-5772-4
1749:
1734:
1386:
1372:
1333:
1047:
774:
698:
641:
331:{\displaystyle w}
217:complex logarithm
193:
143:
142:
135:
117:
82:"Principal value"
1842:
1835:Complex analysis
1819:
1818:
1798:
1777:Principal branch
1767:
1765:
1764:
1759:
1751:
1742:
1736:
1730:
1722:
1710:is typically in
1709:
1698:
1696:
1695:
1690:
1660:
1647:
1631:
1629:
1628:
1623:
1594:
1592:
1591:
1586:
1528:Complex argument
1523:
1519:
1515:
1511:
1507:
1503:
1492:
1490:
1489:
1484:
1463:
1461:
1460:
1455:
1419:
1417:
1416:
1411:
1409:
1408:
1404:
1387:
1382:
1377:
1373:
1368:
1358:
1349:
1334:
1329:
1327:
1305:
1303:
1302:
1297:
1294:
1293:
1257:
1255:
1254:
1249:
1244:
1240:
1235:
1213:
1212:
1204:
1185:
1171:
1160:
1142:
1135:
1128:
1121:
1119:
1118:
1113:
1102:(exclusive) and
1101:
1099:
1098:
1093:
1078:
1068:
1066:
1065:
1060:
1055:
1051:
1045:
1044:
1022:
1021:
1013:
998:
952:
940:
932:
925:
923:
922:
917:
902:
884:
880:
860:principal values
856:principal branch
854:is known as the
853:
835:
828:
826:
825:
820:
794:
791:. Each value of
790:
788:
787:
782:
772:
749:
745:
738:
728:
726:
725:
720:
718:
714:
696:
695:
673:
672:
664:
649:
645:
639:
638:
616:
615:
607:
592:
571:
564:
554:
547:
545:
544:
539:
524:
513:
511:
510:
505:
488:
467:
463:
461:
460:
455:
440:
436:
434:
433:
428:
423:
408:
406:
405:
400:
375:
368:
366:
365:
360:
355:
337:
335:
334:
329:
314:
312:
311:
306:
298:
297:
278:
271:
269:
268:
263:
252:
251:
232:
225:
207:
205:
204:
199:
194:
189:
169:, so that it is
155:principal values
151:complex analysis
138:
131:
127:
124:
118:
116:
75:
51:
43:
1850:
1849:
1845:
1844:
1843:
1841:
1840:
1839:
1825:
1824:
1823:
1822:
1815:
1799:
1795:
1790:
1773:
1740:
1723:
1720:
1715:
1712:
1711:
1700:
1699:In comparison,
1666:
1663:
1662:
1650:
1637:
1602:
1599:
1598:
1565:
1562:
1561:
1530:
1521:
1517:
1513:
1509:
1505:
1501:
1498:
1469:
1466:
1465:
1431:
1428:
1427:
1400:
1393:
1389:
1381:
1351:
1350:
1348:
1344:
1328:
1320:
1318:
1315:
1314:
1286:
1282:
1271:
1268:
1267:
1264:
1225:
1224:
1220:
1208:
1200:
1199:
1175:
1167:
1153:
1151:
1148:
1147:
1137:
1130:
1123:
1107:
1104:
1103:
1084:
1081:
1080:
1073:
1034:
1033:
1029:
1017:
1009:
1008:
994:
986:
983:
982:
972:
961:Complex valued
959:
942:
938:
930:
895:
893:
890:
889:
882:
871:
870:In general, if
868:
848:
830:
814:
811:
810:
792:
758:
755:
754:
747:
740:
736:
685:
684:
680:
668:
660:
659:
628:
627:
623:
611:
603:
602:
588:
580:
577:
576:
566:
559:
549:
530:
527:
526:
519:
518:a solution for
484:
473:
470:
469:
465:
446:
443:
442:
438:
419:
414:
411:
410:
388:
385:
384:
373:
369:is a solution.
351:
343:
340:
339:
323:
320:
319:
293:
289:
287:
284:
283:
276:
247:
243:
241:
238:
237:
230:
220:
213:
188:
186:
183:
182:
149:, specifically
139:
128:
122:
119:
76:
74:
64:
52:
41:
32:principal value
28:
21:principal value
17:
12:
11:
5:
1848:
1838:
1837:
1821:
1820:
1813:
1792:
1791:
1789:
1786:
1785:
1784:
1779:
1772:
1769:
1757:
1754:
1748:
1745:
1739:
1733:
1729:
1726:
1719:
1688:
1685:
1682:
1679:
1676:
1673:
1670:
1633:
1632:
1621:
1618:
1615:
1612:
1609:
1606:
1595:
1584:
1581:
1578:
1575:
1572:
1569:
1536:comparison of
1529:
1526:
1497:
1494:
1482:
1479:
1476:
1473:
1453:
1450:
1447:
1444:
1441:
1438:
1435:
1421:
1420:
1407:
1403:
1399:
1396:
1392:
1385:
1380:
1376:
1371:
1367:
1364:
1361:
1357:
1354:
1347:
1343:
1340:
1337:
1332:
1326:
1323:
1292:
1289:
1285:
1281:
1278:
1275:
1263:
1260:
1259:
1258:
1247:
1243:
1239:
1234:
1231:
1228:
1223:
1219:
1216:
1211:
1207:
1203:
1198:
1195:
1192:
1189:
1184:
1181:
1178:
1174:
1170:
1166:
1163:
1159:
1156:
1111:
1091:
1088:
1070:
1069:
1058:
1054:
1050:
1043:
1040:
1037:
1032:
1028:
1025:
1020:
1016:
1012:
1007:
1004:
1001:
997:
993:
990:
978:above, i.e.,
971:
968:
958:
955:
929:such that for
927:
926:
915:
912:
909:
906:
901:
898:
867:
864:
818:
780:
777:
771:
768:
765:
762:
730:
729:
717:
713:
710:
707:
704:
701:
694:
691:
688:
683:
679:
676:
671:
667:
663:
658:
655:
652:
648:
644:
637:
634:
631:
626:
622:
619:
614:
610:
606:
601:
598:
595:
591:
587:
584:
537:
534:
503:
500:
497:
494:
491:
487:
483:
480:
477:
453:
450:
426:
422:
418:
398:
395:
392:
358:
354:
350:
347:
327:
316:
315:
304:
301:
296:
292:
273:
272:
261:
258:
255:
250:
246:
228:complex number
212:
209:
197:
192:
177:of a positive
141:
140:
55:
53:
46:
15:
9:
6:
4:
3:
2:
1847:
1836:
1833:
1832:
1830:
1816:
1810:
1806:
1805:
1797:
1793:
1783:
1780:
1778:
1775:
1774:
1768:
1755:
1746:
1743:
1737:
1731:
1727:
1724:
1708:
1704:
1686:
1680:
1677:
1674:
1671:
1658:
1655:), real_part(
1654:
1648:
1645:
1641:
1616:
1613:
1610:
1607:
1596:
1579:
1576:
1573:
1570:
1559:
1558:
1557:
1555:
1551:
1543:
1539:
1534:
1525:
1493:
1480:
1477:
1474:
1471:
1451:
1448:
1445:
1442:
1439:
1436:
1433:
1426:
1405:
1401:
1397:
1394:
1390:
1383:
1378:
1374:
1369:
1365:
1362:
1359:
1345:
1341:
1338:
1335:
1330:
1313:
1312:
1311:
1309:
1290:
1287:
1283:
1279:
1276:
1273:
1245:
1241:
1237:
1221:
1217:
1214:
1205:
1196:
1193:
1190:
1187:
1172:
1168:
1164:
1161:
1146:
1145:
1144:
1141:
1134:
1127:
1109:
1089:
1086:
1077:
1056:
1052:
1048:
1030:
1026:
1023:
1014:
1005:
1002:
999:
995:
991:
988:
981:
980:
979:
977:
967:
964:
954:
950:
946:
936:
910:
904:
888:
887:
886:
878:
874:
863:
861:
857:
851:
845:
843:
839:
834:
816:
808:
804:
800:
799:
775:
769:
766:
763:
753:
744:
735:
715:
711:
708:
705:
702:
699:
681:
677:
674:
665:
656:
653:
650:
646:
642:
624:
620:
617:
608:
599:
596:
593:
589:
585:
582:
575:
574:
573:
570:
563:
556:
553:
535:
532:
523:
517:
498:
495:
492:
489:
485:
481:
475:
451:
448:
424:
420:
416:
396:
393:
390:
383:
379:
378:complex plane
370:
356:
352:
348:
345:
325:
302:
299:
294:
290:
282:
281:
280:
259:
256:
253:
248:
244:
236:
235:
234:
229:
224:
218:
215:Consider the
208:
195:
190:
180:
176:
172:
171:single-valued
168:
164:
160:
156:
152:
148:
137:
134:
126:
115:
112:
108:
105:
101:
98:
94:
91:
87:
84: –
83:
79:
78:Find sources:
72:
68:
62:
61:
56:This article
54:
50:
45:
44:
39:
38:
33:
26:
22:
1803:
1796:
1782:Branch point
1706:
1702:
1656:
1652:
1643:
1639:
1634:
1552:measured in
1547:
1499:
1422:
1265:
1139:
1132:
1125:
1075:
1071:
973:
960:
948:
944:
928:
876:
872:
869:
866:General case
859:
855:
849:
846:
832:
802:
796:
742:
731:
568:
561:
557:
551:
521:
515:
371:
338:. The value
317:
274:
222:
214:
154:
144:
129:
120:
110:
103:
96:
89:
77:
65:Please help
60:verification
57:
36:
31:
20:
1308:square root
1262:Square root
1136:instead of
885:is denoted
179:real number
175:square root
147:mathematics
1788:References
838:continuous
807:branch cut
572:, we have
233:such that
211:Motivation
123:March 2023
93:newspapers
1744:π
1728:π
1725:−
1681:π
1675:π
1672:−
1617:π
1611:π
1608:−
1580:π
1544:functions
1478:π
1472:ϕ
1449:π
1446:≤
1443:ϕ
1437:π
1434:−
1398:ϕ
1363:
1342:
1291:ϕ
1197:
1165:
1110:π
1090:π
1087:−
1006:
992:
817:π
776:π
767:π
764:−
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452:π
417:π
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219:function
1829:Category
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318:for
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