704:
259:
147:
351:
447:
486:
506:
593:
176:
71:
1288:
is rational; however, different choices differ only by multiplication by a root of unity, which gets absorbed into the factor ζ.
1411:
1441:
1377:
280:
1395:
1387:
379:
1365:
1228:
452:
1369:
524:
1476:
1205:
1471:
373:
1456:
1421:
39:
8:
1197:
491:
1437:
1407:
1373:
1399:
1357:
1417:
699:{\displaystyle \log _{p}(1+x)=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}x^{n}}{n}},}
576:, but there are multiple such roots and there is no canonical choice among them.
43:
21:
1403:
360:-adic series converge if and only if the summands tend to zero, and since the
1465:
825:) by imposing that it continues to satisfy this last property and setting log
17:
945:) = 0. In fact, there is an extension of the logarithm from |
254:{\displaystyle \exp _{p}(z)=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}.}
1019:, then their sum is too and we have the usual addition formula: exp
142:{\displaystyle \exp(z)=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}.}
46:. As in the complex case, it has an inverse function, named the
364:! in the denominator of each summand tends to make them large
152:
Entirely analogously, one defines the exponential function on
1280:
as above, there is a choice of a root involved in writing
1392:
Number theory, Volume I: Tools and
Diophantine equations
596:
494:
455:
382:
283:
179:
74:
534:-adic analogue. This is because the power series exp
759: < 1 satisfying the usual property log
698:
500:
480:
441:
345:
253:
141:
1463:
1231:— can be used instead which converges on |
1227:. A modified exponential function — the
1184:is a rational number and ζ is a root of unity.
264:However, unlike exp which converges on all of
1211:Another major difference to the situation in
1013:are both in the radius of convergence for exp
835:) = 0. Specifically, every element
161:, the completion of the algebraic closure of
579:
372:is needed in the numerator. It follows from
878:a rational number, ζ a root of unity, and |
1366:London Mathematical Society Student Texts
1215:is that the domain of convergence of exp
1204: = 1. This is a corollary of
346:{\displaystyle |z|_{p}<p^{-1/(p-1)}.}
1457:p-adic exponential and p-adic logarithm
1356:
519:-adic exponential is sometimes denoted
442:{\displaystyle |z|_{p}<p^{-1/(p-1)}}
1464:
1427:
1308:
1157:The roots of the Iwasawa logarithm log
1398:, vol. 239, New York: Springer,
1386:
1344:
1332:
1320:
732: < 1 and so defines the
888: < 1, in which case log
551:. It is possible to choose a number
481:{\displaystyle {\frac {z^{n}}{n!}}}
13:
1187:Note that there is no analogue in
644:
221:
109:
65:is defined by the infinite series
61:The usual exponential function on
14:
1488:
1450:
1221:is much smaller than that of log
816:(the set of nonzero elements of
1338:
1326:
1314:
1302:
1270:
1251:
1167:) are exactly the elements of
935:to emphasize the choice of log
662:
652:
622:
610:
434:
422:
393:
384:
335:
323:
294:
285:
199:
193:
87:
81:
1:
1396:Graduate Texts in Mathematics
1295:
1000:
56:
955: < 1 to all of
368:-adically, a small value of
38:-adic analogue of the usual
7:
274:only converges on the disc
10:
1493:
1370:Cambridge University Press
793:can be extended to all of
32:-adic exponential function
1428:Robert, Alain M. (2000),
1404:10.1007/978-0-387-49923-9
1244:
1064:are nonzero elements of
931:is sometimes called the
737:-adic logarithm function
583:-adic logarithm function
1229:Artin–Hasse exponential
544:) does not converge at
978:for each choice of log
700:
648:
502:
482:
443:
347:
255:
225:
143:
113:
949: − 1|
882: − 1|
753: − 1|
701:
628:
503:
483:
444:
348:
256:
205:
144:
93:
1347:, Proposition 4.4.45
1323:, Proposition 4.4.44
1257:or a 4th root of exp
1206:Strassmann's theorem
1108:in the domain of exp
908:). This function on
594:
492:
453:
380:
281:
177:
72:
40:exponential function
1241: < 1.
862:can be written as
787:. The function log
696:
498:
478:
439:
374:Legendre's formula
343:
251:
139:
1413:978-0-387-49922-2
1358:Cassels, J. W. S.
1130:)) = 1+
1083:) = log
1033:) = exp
933:Iwasawa logarithm
898:) = log
769:) = log
691:
587:The power series
501:{\displaystyle 0}
476:
246:
134:
1484:
1446:
1424:
1383:
1348:
1342:
1336:
1330:
1324:
1318:
1312:
1306:
1289:
1274:
1268:
1267:
1255:
1198:Euler's identity
1092: + log
977:
976:
975:
972:
964:
961:
930:
929:
928:
925:
917:
914:
861:
860:
859:
856:
848:
845:
815:
814:
813:
810:
802:
799:
778: + log
705:
703:
702:
697:
692:
687:
686:
685:
676:
675:
650:
647:
642:
606:
605:
575:
550:
507:
505:
504:
499:
487:
485:
484:
479:
477:
475:
467:
466:
457:
448:
446:
445:
440:
438:
437:
421:
402:
401:
396:
387:
356:This is because
352:
350:
349:
344:
339:
338:
322:
303:
302:
297:
288:
260:
258:
257:
252:
247:
245:
237:
236:
227:
224:
219:
189:
188:
148:
146:
145:
140:
135:
133:
125:
124:
115:
112:
107:
1492:
1491:
1487:
1486:
1485:
1483:
1482:
1481:
1462:
1461:
1453:
1444:
1414:
1380:
1352:
1351:
1343:
1339:
1331:
1327:
1319:
1315:
1307:
1303:
1298:
1293:
1292:
1275:
1271:
1262:
1260:
1256:
1252:
1247:
1240:
1226:
1220:
1195:
1175:
1162:
1150:)) =
1145:
1139:
1125:
1119:
1113:
1097:
1088:
1078:
1072:
1048:
1038:
1024:
1018:
1003:
996:
983:
973:
968:
966:
965:
962:
957:
956:
954:
940:
926:
921:
919:
918:
915:
910:
909:
903:
893:
887:
857:
852:
850:
849:
846:
841:
840:
830:
824:
811:
806:
804:
803:
800:
795:
794:
792:
783:
774:
764:
758:
744:
731:
721:
681:
677:
665:
661:
651:
649:
643:
632:
601:
597:
595:
592:
591:
585:
570:
564:
559:-th root of exp
545:
539:
493:
490:
489:
468:
462:
458:
456:
454:
451:
450:
417:
410:
406:
397:
392:
391:
383:
381:
378:
377:
318:
311:
307:
298:
293:
292:
284:
282:
279:
278:
273:
238:
232:
228:
226:
220:
209:
184:
180:
178:
175:
174:
169:
160:
126:
120:
116:
114:
108:
97:
73:
70:
69:
59:
51:-adic logarithm
44:complex numbers
20:, particularly
12:
11:
5:
1490:
1480:
1479:
1477:P-adic numbers
1474:
1460:
1459:
1452:
1451:External links
1449:
1448:
1447:
1442:
1434:-adic Analysis
1425:
1412:
1384:
1378:
1355:Chapter 12 of
1350:
1349:
1337:
1325:
1313:
1300:
1299:
1297:
1294:
1291:
1290:
1269:
1258:
1249:
1248:
1246:
1243:
1236:
1222:
1216:
1191:
1171:
1158:
1141:
1135:
1121:
1115:
1109:
1093:
1084:
1074:
1068:
1044:
1034:
1020:
1014:
1002:
999:
992:
979:
950:
936:
899:
889:
883:
826:
820:
788:
779:
770:
760:
754:
740:
727:
717:
709:converges for
707:
706:
695:
690:
684:
680:
674:
671:
668:
664:
660:
657:
654:
646:
641:
638:
635:
631:
627:
624:
621:
618:
615:
612:
609:
604:
600:
584:
578:
560:
535:
530:itself has no
497:
474:
471:
465:
461:
436:
433:
430:
427:
424:
420:
416:
413:
409:
405:
400:
395:
390:
386:
354:
353:
342:
337:
334:
331:
328:
325:
321:
317:
314:
310:
306:
301:
296:
291:
287:
269:
262:
261:
250:
244:
241:
235:
231:
223:
218:
215:
212:
208:
204:
201:
198:
195:
192:
187:
183:
165:
156:
150:
149:
138:
132:
129:
123:
119:
111:
106:
103:
100:
96:
92:
89:
86:
83:
80:
77:
58:
55:
25:-adic analysis
9:
6:
4:
3:
2:
1489:
1478:
1475:
1473:
1470:
1469:
1467:
1458:
1455:
1454:
1445:
1443:0-387-98669-3
1439:
1435:
1431:
1426:
1423:
1419:
1415:
1409:
1405:
1401:
1397:
1393:
1389:
1385:
1381:
1379:0-521-31525-5
1375:
1371:
1367:
1363:
1359:
1354:
1353:
1346:
1341:
1334:
1329:
1322:
1317:
1311:, p. 252
1310:
1305:
1301:
1287:
1283:
1279:
1276:In factoring
1273:
1265:
1254:
1250:
1242:
1239:
1234:
1230:
1225:
1219:
1214:
1209:
1207:
1203:
1199:
1194:
1190:
1185:
1183:
1179:
1174:
1170:
1166:
1161:
1155:
1153:
1149:
1144:
1138:
1133:
1129:
1124:
1118:
1114:, we have exp
1112:
1107:
1102:
1100:
1096:
1091:
1087:
1082:
1077:
1071:
1067:
1063:
1059:
1056:Similarly if
1054:
1052:
1047:
1042:
1037:
1032:
1029: +
1028:
1023:
1017:
1012:
1008:
998:
995:
991:
987:
982:
971:
960:
953:
948:
944:
939:
934:
924:
913:
907:
902:
897:
892:
886:
881:
877:
873:
869:
866: =
865:
855:
844:
838:
834:
829:
823:
819:
809:
798:
791:
786:
782:
777:
773:
768:
763:
757:
752:
748:
743:
738:
736:
730:
725:
720:
716:
712:
693:
688:
682:
678:
672:
669:
666:
658:
655:
639:
636:
633:
629:
625:
619:
616:
613:
607:
602:
598:
590:
589:
588:
582:
577:
573:
568:
563:
558:
554:
548:
543:
538:
533:
529:
528:
522:
518:
515:Although the
513:
511:
495:
472:
469:
463:
459:
431:
428:
425:
418:
414:
411:
407:
403:
398:
388:
375:
371:
367:
363:
359:
340:
332:
329:
326:
319:
315:
312:
308:
304:
299:
289:
277:
276:
275:
272:
267:
248:
242:
239:
233:
229:
216:
213:
210:
206:
202:
196:
190:
185:
181:
173:
172:
171:
168:
164:
159:
155:
136:
130:
127:
121:
117:
104:
101:
98:
94:
90:
84:
78:
75:
68:
67:
66:
64:
54:
52:
50:
45:
41:
37:
33:
31:
26:
24:
19:
1472:Exponentials
1436:, Springer,
1433:
1430:A Course in
1429:
1391:
1388:Cohen, Henri
1362:Local fields
1361:
1340:
1328:
1316:
1304:
1285:
1281:
1277:
1272:
1263:
1253:
1237:
1232:
1223:
1217:
1212:
1210:
1201:
1192:
1188:
1186:
1181:
1177:
1176:of the form
1172:
1168:
1164:
1159:
1156:
1151:
1147:
1142:
1136:
1131:
1127:
1122:
1116:
1110:
1105:
1103:
1098:
1094:
1089:
1085:
1080:
1075:
1069:
1065:
1061:
1057:
1055:
1050:
1045:
1040:
1035:
1030:
1026:
1021:
1015:
1010:
1006:
1004:
993:
989:
985:
980:
969:
958:
951:
946:
942:
937:
932:
922:
911:
905:
900:
895:
890:
884:
879:
875:
871:
867:
863:
853:
842:
836:
832:
827:
821:
817:
807:
796:
789:
784:
780:
775:
771:
766:
761:
755:
750:
746:
741:
734:
733:
728:
723:
722:satisfying |
718:
714:
710:
708:
586:
580:
571:
566:
561:
556:
552:
546:
541:
536:
531:
526:
520:
516:
514:
509:
369:
365:
361:
357:
355:
270:
265:
263:
166:
162:
157:
153:
151:
62:
60:
48:
47:
35:
29:
28:
22:
15:
1309:Robert 2000
512:-adically.
18:mathematics
1466:Categories
1345:Cohen 2007
1333:Cohen 2007
1321:Cohen 2007
1296:References
1001:Properties
57:Definition
1335:, §4.4.11
1261:(4), for
1180:·ζ where
656:−
645:∞
630:∑
608:
488:tends to
429:−
412:−
330:−
313:−
222:∞
207:∑
191:
110:∞
95:∑
79:
1390:(2007),
1360:(1986).
1073:then log
555:to be a
376:that if
1422:2312337
1134:and log
749:) for |
525:number
42:on the
1440:
1420:
1410:
1376:
1284:since
974:
963:
927:
916:
858:
847:
812:
801:
569:) for
523:, the
27:, the
1245:Notes
988:) in
874:with
449:then
268:, exp
170:, by
34:is a
1438:ISBN
1408:ISBN
1374:ISBN
1140:(exp
1120:(log
1104:For
1060:and
1043:)exp
1009:and
404:<
305:<
1400:doi
1266:= 2
1196:of
1126:(1+
1053:).
1005:If
870:·ζ·
839:of
739:log
713:in
599:log
574:≠2
549:= 1
182:exp
76:exp
16:In
1468::
1418:MR
1416:,
1406:,
1394:,
1372:.
1368:.
1364:.
1208:.
1200:,
1154:.
1101:.
1081:zw
997:.
767:zw
508:,
53:.
1432:p
1402::
1382:.
1286:r
1282:p
1278:w
1264:p
1259:2
1238:p
1235:|
1233:z
1224:p
1218:p
1213:C
1202:e
1193:p
1189:C
1182:r
1178:p
1173:p
1169:C
1165:z
1163:(
1160:p
1152:z
1148:z
1146:(
1143:p
1137:p
1132:z
1128:z
1123:p
1117:p
1111:p
1106:z
1099:w
1095:p
1090:z
1086:p
1079:(
1076:p
1070:p
1066:C
1062:w
1058:z
1051:w
1049:(
1046:p
1041:z
1039:(
1036:p
1031:w
1027:z
1025:(
1022:p
1016:p
1011:w
1007:z
994:p
990:C
986:p
984:(
981:p
970:p
967:Ă—
959:C
952:p
947:z
943:p
941:(
938:p
923:p
920:Ă—
912:C
906:z
904:(
901:p
896:w
894:(
891:p
885:p
880:z
876:r
872:z
868:p
864:w
854:p
851:Ă—
843:C
837:w
833:p
831:(
828:p
822:p
818:C
808:p
805:Ă—
797:C
790:p
785:w
781:p
776:z
772:p
765:(
762:p
756:p
751:z
747:z
745:(
742:p
735:p
729:p
726:|
724:x
719:p
715:C
711:x
694:,
689:n
683:n
679:x
673:1
670:+
667:n
663:)
659:1
653:(
640:1
637:=
634:n
626:=
623:)
620:x
617:+
614:1
611:(
603:p
581:p
572:p
567:p
565:(
562:p
557:p
553:e
547:x
542:x
540:(
537:p
532:p
527:e
521:e
517:p
510:p
496:0
473:!
470:n
464:n
460:z
435:)
432:1
426:p
423:(
419:/
415:1
408:p
399:p
394:|
389:z
385:|
370:z
366:p
362:n
358:p
341:.
336:)
333:1
327:p
324:(
320:/
316:1
309:p
300:p
295:|
290:z
286:|
271:p
266:C
249:.
243:!
240:n
234:n
230:z
217:0
214:=
211:n
203:=
200:)
197:z
194:(
186:p
167:p
163:Q
158:p
154:C
137:.
131:!
128:n
122:n
118:z
105:0
102:=
99:n
91:=
88:)
85:z
82:(
63:C
49:p
36:p
30:p
23:p
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