1293:
885:
312:
1557:
628:
1373:. A differential algebra is an algebra with the extra operation of derivation (algebraic version of differentiation). Using the derivation operation new equations can be written and their solutions used in
954:
1479:
478:
528:
414:
578:
1104:
361:
206:
2009:
673:
983:
699:
730:
1385:, two special types of transcendental extensions (the logarithm and the exponential) can be added to the field building a tower containing elementary functions.
753:
1057:
1037:
1699:
1663:
1346:
1201:
789:
1425:′ is used.) The derivation captures the properties of differentiation, so that for any two elements of the base field, the derivation is linear
220:
1705:
1493:
2098:
143:
in the 1930s. Many textbooks and dictionaries do not give a precise definition of the elementary functions, and mathematicians differ on it.
1717:
589:
705:. Additionally, certain classes of functions may be obtained by others using the final two rules. For example, the exponential function
1570:. If the base field is over the rationals, care must be taken when extending the field to add the needed transcendental constants.
1864:
986:
110:
891:
1747:
1693:
483:
35:
2159:
2045:
2032:
1431:
1823:
633:
All functions obtained by adding, subtracting, multiplying or dividing a finite number of any of the previous functions
2169:
583:
58:
732:
composed with addition, subtraction, and division provides the hyperbolic functions, while initial composition with
1687:
2164:
424:
78:
1711:
489:
94:
2004:
1681:
1350:
1305:
418:
54:
50:
371:
1329:
under arithmetic operations, root extraction and composition. The elementary functions are closed under
1326:
539:
1988:
1972:
1357:
are defined as the elementary functions and, recursively, the integrals of the
Liouvillian functions.
1370:
1354:
1334:
1168:
1160:
1153:
1062:
702:
640:
317:
125:
102:
98:
2122:
1378:
636:
All functions obtained by root extraction of a polynomial with coefficients in elementary functions
533:
323:
170:
1702: – Says when antiderivatives of elementary functions can be expressed as elementary functions
2104:
2062:
2040:
1920:
1675:
1601:
1309:
1127:
1009:
654:
74:
1856:
1849:
962:
2132:
2094:
1912:
1860:
1753:
1743:
1382:
1313:
1115:
1005:
164:
114:
90:
2086:
2054:
2000:
1984:
1968:
1904:
1180:
678:
140:
136:
132:
106:
62:
1126:, but others allow them. Some have proposed extending the set to include, for example, the
708:
1403:
1374:
1296:
2135:
1879:
735:
1342:
1288:{\displaystyle \mathrm {erf} (x)={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{2}}\,dt,}
1195:
1147:
1119:
1042:
1022:
1016:
880:{\displaystyle {\frac {e^{\tan x}}{1+x^{2}}}\sin \left({\sqrt {1+(\log x)^{2}}}\right)}
211:
70:
1039:, is also elementary as it can be expressed as the composition of a power and root of
2153:
1123:
990:
17:
2108:
1973:"Premier mémoire sur la détermination des intégrales dont la valeur est algébrique"
1485:
1989:"Second mémoire sur la détermination des intégrales dont la valeur est algébrique"
1939:
2090:
2022:
1325:
It follows directly from the definition that the set of elementary functions is
42:
307:{\displaystyle x,\ x^{2},\ {\sqrt {x}}\ (x^{\frac {1}{2}}),\ x^{\frac {2}{3}},}
1330:
1001:
781:
86:
82:
31:
1916:
2140:
1757:
1141:
365:
1552:{\displaystyle \partial (u\cdot v)=\partial u\cdot v+u\cdot \partial v\,.}
2026:
997:
66:
2081:
Davenport, James H. (2007). "What Might "Understand a
Function" Mean?".
2005:"Note sur la détermination des intégrales dont la valeur est algébrique"
1295:
a fact that may not be immediately obvious, but can be proven using the
2066:
1924:
1892:
2085:. Lecture Notes in Computer Science. Vol. 4573. pp. 55–65.
1714: – Analytic function that does not satisfy a polynomial equation
2058:
1908:
1369:, or a function in elementary form, is considered in the context of
623:{\displaystyle \operatorname {arsinh} x,\ \operatorname {arcosh} x,}
1684: – Mathematical formula involving a given set of operations
1742:(3rd ed.). Houston, Tex.: Publish or Perish. p. 359.
1893:"Algebraic Properties of the Elementary Functions of Analysis"
1690: – Study of Galois symmetry groups of differential fields
1720: – Formula that visually represents itself when graphed
647:
Certain elementary functions of a single complex variable
643:
a finite number of any of the previously listed functions
1880:
Weisstein, Eric W. "Elementary
Function." From MathWorld
949:{\displaystyle -i\log \left(x+i{\sqrt {1-x^{2}}}\right)}
1065:
1496:
1434:
1204:
1045:
1025:
965:
894:
792:
738:
711:
681:
657:
592:
542:
492:
427:
374:
326:
223:
173:
1474:{\displaystyle \partial (u+v)=\partial u+\partial v}
1822:Subbotin, Igor Ya.; Bilotskii, N. N. (March 2008).
1848:
1551:
1473:
1287:
1098:
1051:
1031:
977:
948:
879:
747:
724:
693:
667:
622:
572:
522:
472:
408:
355:
306:
200:
1824:"Algorithms and Fundamental Concepts of Calculus"
139:treatment of elementary functions was started by
124:All elementary functions are continuous on their
2010:Journal für die reine und angewandte Mathematik
1940:"A new elementary function for our curricula?"
755:instead provides the trigonometric functions.
135:in a series of papers from 1833 to 1841. An
8:
1696: – System of arithmetic in proof theory
1409:for example) together with a derivation map
1337:. Importantly, the elementary functions are
2083:Towards Mechanized Mathematical Assistants
1831:Journal of Research in Innovative Teaching
1794:
1782:
1770:
1700:Liouville's theorem (differential algebra)
1421:is a new function. Sometimes the notation
763:Examples of elementary functions include:
156:Elementary functions of a single variable
1545:
1495:
1433:
1275:
1267:
1259:
1249:
1244:
1228:
1205:
1203:
1088:
1082:
1074:
1066:
1064:
1044:
1024:
964:
933:
921:
893:
865:
841:
822:
799:
793:
791:
737:
716:
710:
680:
658:
656:
591:
541:
491:
473:{\displaystyle \sin x,\ \cos x,\ \tan x,}
426:
394:
373:
347:
331:
325:
290:
266:
249:
237:
222:
172:
131:Elementary functions were introduced by
2043:(1972). "Integration in finite terms".
1730:
1122:or discontinuous functions such as the
1377:of the algebra. By starting with the
523:{\displaystyle \arcsin x,\ \arccos x,}
1947:Australian Senior Mathematics Journal
7:
1817:
1815:
1806:
1706:Tarski's high school algebra problem
1133:Some examples of functions that are
409:{\displaystyle \log x,\ \log _{a}x}
1539:
1518:
1497:
1465:
1456:
1435:
1365:The mathematical definition of an
1212:
1209:
1206:
573:{\displaystyle \sinh x,\ \cosh x,}
25:
1718:Tupper's self-referential formula
1993:Journal de l'École Polytechnique
1977:Journal de l'École Polytechnique
1114:Many mathematicians exclude non-
1099:{\textstyle |x|={\sqrt {x^{2}}}}
2126:at Encyclopaedia of Mathematics
1897:American Journal of Mathematics
1851:Ordinary Differential Equations
484:Inverse trigonometric functions
1694:Elementary function arithmetic
1512:
1500:
1450:
1438:
1222:
1216:
1075:
1067:
959:The last function is equal to
862:
849:
277:
259:
36:Elementary function arithmetic
34:. For the logical system, see
30:For the complexity class, see
1:
2046:American Mathematical Monthly
1678: – Mathematical function
1402:(rational functions over the
356:{\displaystyle e^{x},\ a^{x}}
201:{\displaystyle 2,\ \pi ,\ e,}
1708: – Mathematical problem
1577:of a differential extension
1333:. They are not closed under
584:Inverse hyperbolic functions
65:) that is defined as taking
2091:10.1007/978-3-540-73086-6_5
668:{\displaystyle {\sqrt {z}}}
2186:
1688:Differential Galois theory
639:All functions obtained by
29:
1891:Risch, Robert H. (1979).
1738:Spivak, Michael. (1994).
978:{\displaystyle \arccos x}
1581:of a differential field
1335:limits and infinite sums
1110:Non-elementary functions
1855:. Dover. 1985. p.
1712:Transcendental function
1306:nonelementary integrals
1120:absolute value function
1017:absolute value function
774:Multiplication, e.g. (2
419:Trigonometric functions
1938:Stewart, Seán (2005).
1682:Closed-form expression
1553:
1475:
1351:nonelementary integral
1289:
1100:
1053:
1033:
979:
950:
881:
749:
726:
695:
694:{\displaystyle \log z}
669:
624:
574:
524:
474:
410:
357:
308:
202:
2136:"Elementary function"
1554:
1476:
1355:Liouvillian functions
1290:
1154:Liouvillian functions
1101:
1054:
1034:
980:
951:
882:
750:
727:
725:{\displaystyle e^{z}}
696:
670:
625:
575:
525:
475:
411:
358:
318:Exponential functions
309:
203:
105:functions, and their
27:Mathematical function
2160:Differential algebra
2124:Elementary functions
2028:Differential Algebra
1995:. tome XIV: 149–193.
1979:. tome XIV: 124–148.
1494:
1486:Leibniz product rule
1432:
1371:differential algebra
1361:Differential algebra
1202:
1169:logarithmic integral
1063:
1043:
1023:
963:
892:
790:
736:
709:
679:
655:
590:
540:
534:Hyperbolic functions
490:
425:
372:
324:
221:
171:
18:Elementary functions
2041:Rosenlicht, Maxwell
1664:Liouville's theorem
1650: / a for
1587:elementary function
1367:elementary function
1347:Liouville's theorem
1254:
1010:algebraic functions
212:Rational powers of
47:elementary function
2170:Types of functions
2133:Weisstein, Eric W.
1676:Algebraic function
1549:
1484:and satisfies the
1471:
1390:differential field
1383:rational functions
1310:Dirichlet integral
1285:
1240:
1128:Lambert W function
1116:analytic functions
1096:
1049:
1029:
1006:rational functions
975:
946:
877:
759:Composite examples
748:{\displaystyle iz}
745:
722:
691:
665:
620:
570:
520:
470:
406:
353:
304:
198:
165:Constant functions
2100:978-3-540-73083-5
2001:Liouville, Joseph
1985:Liouville, Joseph
1969:Liouville, Joseph
1566:is a constant if
1314:elliptic integral
1238:
1237:
1094:
1052:{\displaystyle x}
1032:{\displaystyle x}
1012:are elementary.
939:
871:
829:
663:
607:
557:
507:
457:
442:
389:
342:
298:
285:
274:
258:
254:
248:
232:
191:
182:
16:(Redirected from
2177:
2165:Computer algebra
2146:
2145:
2112:
2070:
2036:
2018:
1996:
1980:
1955:
1954:
1944:
1935:
1929:
1928:
1888:
1882:
1877:
1871:
1870:
1854:
1845:
1839:
1838:
1828:
1819:
1810:
1804:
1798:
1792:
1786:
1780:
1774:
1768:
1762:
1761:
1735:
1593:if the function
1568:∂h = 0
1558:
1556:
1555:
1550:
1480:
1478:
1477:
1472:
1308:, including the
1294:
1292:
1291:
1286:
1274:
1273:
1272:
1271:
1253:
1248:
1239:
1233:
1229:
1215:
1105:
1103:
1102:
1097:
1095:
1093:
1092:
1083:
1078:
1070:
1058:
1056:
1055:
1050:
1038:
1036:
1035:
1030:
989:, in the entire
984:
982:
981:
976:
955:
953:
952:
947:
945:
941:
940:
938:
937:
922:
886:
884:
883:
878:
876:
872:
870:
869:
842:
830:
828:
827:
826:
810:
809:
794:
777:
770:
767:Addition, e.g. (
754:
752:
751:
746:
731:
729:
728:
723:
721:
720:
700:
698:
697:
692:
674:
672:
671:
666:
664:
659:
650:
629:
627:
626:
621:
605:
579:
577:
576:
571:
555:
529:
527:
526:
521:
505:
479:
477:
476:
471:
455:
440:
415:
413:
412:
407:
399:
398:
387:
362:
360:
359:
354:
352:
351:
340:
336:
335:
313:
311:
310:
305:
300:
299:
291:
283:
276:
275:
267:
256:
255:
250:
246:
242:
241:
230:
215:
207:
205:
204:
199:
189:
180:
159:
141:Joseph Fels Ritt
133:Joseph Liouville
21:
2185:
2184:
2180:
2179:
2178:
2176:
2175:
2174:
2150:
2149:
2131:
2130:
2119:
2101:
2080:
2077:
2075:Further reading
2059:10.2307/2318066
2039:
2021:
1999:
1983:
1967:
1964:
1959:
1958:
1942:
1937:
1936:
1932:
1909:10.2307/2373917
1890:
1889:
1885:
1878:
1874:
1867:
1847:
1846:
1842:
1826:
1821:
1820:
1813:
1805:
1801:
1795:Liouville 1833c
1793:
1789:
1783:Liouville 1833b
1781:
1777:
1771:Liouville 1833a
1769:
1765:
1750:
1737:
1736:
1732:
1727:
1672:
1492:
1491:
1430:
1429:
1401:
1363:
1331:differentiation
1323:
1297:Risch algorithm
1263:
1255:
1200:
1199:
1152:non-elementary
1112:
1084:
1061:
1060:
1041:
1040:
1021:
1020:
961:
960:
929:
911:
907:
890:
889:
861:
837:
818:
811:
795:
788:
787:
775:
768:
761:
734:
733:
712:
707:
706:
677:
676:
653:
652:
648:
588:
587:
538:
537:
488:
487:
423:
422:
390:
370:
369:
343:
327:
322:
321:
286:
262:
233:
219:
218:
213:
169:
168:
157:
154:
149:
39:
28:
23:
22:
15:
12:
11:
5:
2183:
2181:
2173:
2172:
2167:
2162:
2152:
2151:
2148:
2147:
2128:
2118:
2117:External links
2115:
2114:
2113:
2099:
2076:
2073:
2072:
2071:
2053:(9): 963–972.
2037:
2019:
1997:
1981:
1963:
1960:
1957:
1956:
1930:
1903:(4): 743–759.
1883:
1872:
1865:
1840:
1811:
1799:
1787:
1775:
1763:
1748:
1729:
1728:
1726:
1723:
1722:
1721:
1715:
1709:
1703:
1697:
1691:
1685:
1679:
1671:
1668:
1660:
1659:
1636:
1609:
1560:
1559:
1548:
1544:
1541:
1538:
1535:
1532:
1529:
1526:
1523:
1520:
1517:
1514:
1511:
1508:
1505:
1502:
1499:
1482:
1481:
1470:
1467:
1464:
1461:
1458:
1455:
1452:
1449:
1446:
1443:
1440:
1437:
1413: → ∂
1399:
1362:
1359:
1345:, as shown by
1340:
1322:
1319:
1318:
1317:
1302:
1301:
1300:
1284:
1281:
1278:
1270:
1266:
1262:
1258:
1252:
1247:
1243:
1236:
1232:
1227:
1224:
1221:
1218:
1214:
1211:
1208:
1196:error function
1192:
1150:
1148:gamma function
1144:
1111:
1108:
1091:
1087:
1081:
1077:
1073:
1069:
1048:
1028:
987:inverse cosine
974:
971:
968:
957:
956:
944:
936:
932:
928:
925:
920:
917:
914:
910:
906:
903:
900:
897:
887:
875:
868:
864:
860:
857:
854:
851:
848:
845:
840:
836:
833:
825:
821:
817:
814:
808:
805:
802:
798:
785:
779:
772:
760:
757:
744:
741:
719:
715:
690:
687:
684:
662:
645:
644:
637:
634:
631:
619:
616:
613:
610:
604:
601:
598:
595:
581:
569:
566:
563:
560:
554:
551:
548:
545:
531:
519:
516:
513:
510:
504:
501:
498:
495:
481:
469:
466:
463:
460:
454:
451:
448:
445:
439:
436:
433:
430:
416:
405:
402:
397:
393:
386:
383:
380:
377:
363:
350:
346:
339:
334:
330:
315:
303:
297:
294:
289:
282:
279:
273:
270:
265:
261:
253:
245:
240:
236:
229:
226:
209:
197:
194:
188:
185:
179:
176:
153:
152:Basic examples
150:
148:
145:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2182:
2171:
2168:
2166:
2163:
2161:
2158:
2157:
2155:
2143:
2142:
2137:
2134:
2129:
2127:
2125:
2121:
2120:
2116:
2110:
2106:
2102:
2096:
2092:
2088:
2084:
2079:
2078:
2074:
2068:
2064:
2060:
2056:
2052:
2048:
2047:
2042:
2038:
2034:
2030:
2029:
2024:
2020:
2016:
2012:
2011:
2006:
2002:
1998:
1994:
1990:
1986:
1982:
1978:
1974:
1970:
1966:
1965:
1961:
1952:
1948:
1941:
1934:
1931:
1926:
1922:
1918:
1914:
1910:
1906:
1902:
1898:
1894:
1887:
1884:
1881:
1876:
1873:
1868:
1866:0-486-64940-7
1862:
1858:
1853:
1852:
1844:
1841:
1836:
1832:
1825:
1818:
1816:
1812:
1808:
1803:
1800:
1796:
1791:
1788:
1784:
1779:
1776:
1772:
1767:
1764:
1759:
1755:
1751:
1745:
1741:
1734:
1731:
1724:
1719:
1716:
1713:
1710:
1707:
1704:
1701:
1698:
1695:
1692:
1689:
1686:
1683:
1680:
1677:
1674:
1673:
1669:
1667:
1665:
1657:
1653:
1649:
1645:
1641:
1637:
1634:
1630:
1626:
1622:
1618:
1614:
1610:
1607:
1603:
1599:
1598:
1597:
1596:
1592:
1588:
1584:
1580:
1576:
1571:
1569:
1565:
1546:
1542:
1536:
1533:
1530:
1527:
1524:
1521:
1515:
1509:
1506:
1503:
1490:
1489:
1488:
1487:
1468:
1462:
1459:
1453:
1447:
1444:
1441:
1428:
1427:
1426:
1424:
1420:
1416:
1412:
1408:
1405:
1398:
1394:
1391:
1386:
1384:
1380:
1376:
1372:
1368:
1360:
1358:
1356:
1352:
1348:
1344:
1341:closed under
1338:
1336:
1332:
1328:
1320:
1315:
1311:
1307:
1303:
1298:
1282:
1279:
1276:
1268:
1264:
1260:
1256:
1250:
1245:
1241:
1234:
1230:
1225:
1219:
1197:
1193:
1190:
1186:
1182:
1178:
1174:
1170:
1166:
1162:
1158:
1157:
1155:
1151:
1149:
1145:
1143:
1140:
1139:
1138:
1136:
1131:
1129:
1125:
1124:step function
1121:
1117:
1109:
1107:
1089:
1085:
1079:
1071:
1046:
1026:
1018:
1013:
1011:
1007:
1003:
999:
994:
992:
991:complex plane
988:
972:
969:
966:
942:
934:
930:
926:
923:
918:
915:
912:
908:
904:
901:
898:
895:
888:
873:
866:
858:
855:
852:
846:
843:
838:
834:
831:
823:
819:
815:
812:
806:
803:
800:
796:
786:
783:
780:
773:
766:
765:
764:
758:
756:
742:
739:
717:
713:
704:
688:
685:
682:
660:
642:
638:
635:
632:
617:
614:
611:
608:
602:
599:
596:
593:
585:
582:
567:
564:
561:
558:
552:
549:
546:
543:
535:
532:
517:
514:
511:
508:
502:
499:
496:
493:
485:
482:
467:
464:
461:
458:
452:
449:
446:
443:
437:
434:
431:
428:
420:
417:
403:
400:
395:
391:
384:
381:
378:
375:
367:
364:
348:
344:
337:
332:
328:
319:
316:
301:
295:
292:
287:
280:
271:
268:
263:
251:
243:
238:
234:
227:
224:
216:
210:
195:
192:
186:
183:
177:
174:
166:
163:
162:
161:
151:
146:
144:
142:
138:
134:
129:
127:
122:
120:
116:
112:
108:
104:
100:
96:
95:trigonometric
92:
88:
84:
80:
76:
72:
68:
64:
60:
56:
52:
48:
44:
37:
33:
19:
2139:
2123:
2082:
2050:
2044:
2027:
2023:Ritt, Joseph
2014:
2008:
1992:
1976:
1950:
1946:
1933:
1900:
1896:
1886:
1875:
1850:
1843:
1834:
1830:
1802:
1790:
1778:
1766:
1739:
1733:
1661:
1655:
1651:
1647:
1643:
1642:, that is, ∂
1639:
1632:
1628:
1624:
1620:
1616:
1615:, that is, ∂
1612:
1605:
1594:
1590:
1586:
1582:
1578:
1574:
1572:
1567:
1563:
1561:
1483:
1422:
1418:
1414:
1410:
1406:
1396:
1392:
1389:
1387:
1366:
1364:
1324:
1191:) integrals.
1188:
1184:
1176:
1172:
1164:
1156:, including
1137:elementary:
1134:
1132:
1118:such as the
1113:
1014:
995:
958:
762:
646:
155:
130:
123:
118:
79:compositions
53:of a single
46:
40:
1837:(1): 82–94.
1613:exponential
1573:A function
1562:An element
1395:is a field
1343:integration
1161:exponential
1019:, for real
1002:polynomials
703:multivalued
103:exponential
57:(typically
43:mathematics
2154:Categories
2017:: 347–359.
1962:References
1953:(2): 8–26.
1749:0914098896
1662:(see also
1417:. (Here ∂
1375:extensions
782:Polynomial
651:, such as
366:Logarithms
99:hyperbolic
87:polynomial
32:ELEMENTARY
2141:MathWorld
2003:(1833c).
1987:(1833b).
1971:(1833a).
1917:0002-9327
1807:Ritt 1950
1640:logarithm
1602:algebraic
1540:∂
1537:⋅
1525:⋅
1519:∂
1507:⋅
1498:∂
1466:∂
1457:∂
1436:∂
1404:rationals
1261:−
1242:∫
1235:π
1142:tetration
998:monomials
970:
927:−
905:
896:−
856:
835:
804:
784:functions
701:, may be
686:
641:composing
612:
597:
562:
547:
512:
497:
462:
447:
432:
401:
379:
184:π
160:include:
137:algebraic
2025:(1950).
1758:31441929
1740:Calculus
1670:See also
147:Examples
107:inverses
91:rational
83:finitely
71:products
55:variable
51:function
2109:8049737
2067:2318066
1925:2373917
1321:Closure
1181:Fresnel
126:domains
109:(e.g.,
63:complex
2107:
2097:
2065:
1923:
1915:
1863:
1756:
1746:
1611:is an
1585:is an
1353:. The
1349:, see
1327:closed
1304:other
1179:) and
985:, the
967:arccos
609:arcosh
606:
594:arsinh
556:
509:arccos
506:
494:arcsin
456:
441:
388:
341:
284:
257:
247:
231:
190:
181:
111:arcsin
101:, and
2105:S2CID
2063:JSTOR
1943:(PDF)
1921:JSTOR
1827:(PDF)
1725:Notes
1638:is a
1604:over
1589:over
1379:field
117:, or
85:many
75:roots
49:is a
45:, an
2095:ISBN
1913:ISSN
1861:ISBN
1754:OCLC
1744:ISBN
1635:, or
1627:for
1608:, or
1312:and
1194:the
1187:and
1159:the
1146:the
1015:The
1008:and
996:All
675:and
630:etc.
580:etc.
559:cosh
544:sinh
530:etc.
480:etc.
314:etc.
208:etc.
77:and
67:sums
59:real
2087:doi
2055:doi
2033:AMS
1905:doi
1901:101
1646:= ∂
1600:is
1381:of
1339:not
1175:or
1167:),
1135:not
902:log
853:log
832:sin
801:tan
771:+1)
683:log
459:tan
444:cos
429:sin
392:log
376:log
121:).
115:log
81:of
61:or
41:In
2156::
2138:.
2103:.
2093:.
2061:.
2051:79
2049:.
2031:.
2015:10
2013:.
2007:.
1991:.
1975:.
1951:19
1949:.
1945:.
1919:.
1911:.
1899:.
1895:.
1859:.
1857:17
1833:.
1829:.
1814:^
1752:.
1666:)
1654:∈
1631:∈
1619:=
1388:A
1198:,
1177:li
1173:Li
1165:Ei
1130:.
1106:.
1059::
1004:,
1000:,
993:.
586::
536::
486::
421::
368::
320::
217::
167::
128:.
113:,
97:,
93:,
89:,
73:,
69:,
2144:.
2111:.
2089::
2069:.
2057::
2035:.
1927:.
1907::
1869:.
1835:1
1809:.
1797:.
1785:.
1773:.
1760:.
1658:.
1656:F
1652:a
1648:a
1644:u
1633:F
1629:a
1625:a
1623:∂
1621:u
1617:u
1606:F
1595:u
1591:F
1583:F
1579:F
1575:u
1564:h
1547:.
1543:v
1534:u
1531:+
1528:v
1522:u
1516:=
1513:)
1510:v
1504:u
1501:(
1469:v
1463:+
1460:u
1454:=
1451:)
1448:v
1445:+
1442:u
1439:(
1423:u
1419:u
1415:u
1411:u
1407:Q
1400:0
1397:F
1393:F
1316:.
1299:.
1283:,
1280:t
1277:d
1269:2
1265:t
1257:e
1251:x
1246:0
1231:2
1226:=
1223:)
1220:x
1217:(
1213:f
1210:r
1207:e
1189:C
1185:S
1183:(
1171:(
1163:(
1090:2
1086:x
1080:=
1076:|
1072:x
1068:|
1047:x
1027:x
973:x
943:)
935:2
931:x
924:1
919:i
916:+
913:x
909:(
899:i
874:)
867:2
863:)
859:x
850:(
847:+
844:1
839:(
824:2
820:x
816:+
813:1
807:x
797:e
778:)
776:x
769:x
743:z
740:i
718:z
714:e
689:z
661:z
649:z
618:,
615:x
603:,
600:x
568:,
565:x
553:,
550:x
518:,
515:x
503:,
500:x
468:,
465:x
453:,
450:x
438:,
435:x
404:x
396:a
385:,
382:x
349:x
345:a
338:,
333:x
329:e
302:,
296:3
293:2
288:x
281:,
278:)
272:2
269:1
264:x
260:(
252:x
244:,
239:2
235:x
228:,
225:x
214:x
196:,
193:e
187:,
178:,
175:2
158:x
119:x
38:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.