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Für nicht ganze Exponenten gilt sie nicht allgemein.",[564,1048,1049],{},"Anwendungen:",[1051,1052,1053,1089,1226],"ul",{},[1054,1055,1056,1059,1060,1088],"li",{},[568,1057,1058],{},"Hohe Potenzen"," komplexer Zahlen berechnen, ohne ",[576,1061,1063,1076],{"className":1062},[579],[576,1064,1066],{"className":1065},[583],[585,1067,1068],{"xmlns":587},[589,1069,1070,1074],{},[592,1071,1072],{},[595,1073,597],{},[599,1075,597],{"encoding":601},[576,1077,1079],{"className":1078,"ariaHidden":606},[605],[576,1080,1082,1085],{"className":1081},[610],[576,1083],{"className":1084,"style":615},[614],[576,1086,597],{"className":1087},[619,620],"-mal auszumultiplizieren.",[1054,1090,1091,1094,1095,1225],{},[568,1092,1093],{},"Vielfachwinkelformeln"," herleiten: Man multipliziert 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i\\sin\\varphi)^n",[576,1140,1142,1173],{"className":1141,"ariaHidden":606},[605],[576,1143,1145,1149,1152,1156,1159,1162,1166,1170],{"className":1144},[610],[576,1146],{"className":1147,"style":1148},[614],"height:1em;vertical-align:-0.25em;",[576,1150,680],{"className":1151},[750],[576,1153,1112],{"className":1154},[1155],"mop",[576,1157],{"className":1158,"style":766},[765],[576,1160,700],{"className":1161},[619,620],[576,1163],{"className":1164,"style":1165},[765],"margin-right:0.2222em;",[576,1167,1120],{"className":1168},[1169],"mbin",[576,1171],{"className":1172,"style":1165},[765],[576,1174,1176,1179,1182,1185,1188,1191,1194],{"className":1175},[610],[576,1177],{"className":1178,"style":1148},[614],[576,1180,697],{"className":1181},[619,620],[576,1183],{"className":1184,"style":766},[765],[576,1186,1125],{"className":1187},[1155],[576,1189],{"className":1190,"style":766},[765],[576,1192,700],{"className":1193},[619,620],[576,1195,1197,1200],{"className":1196},[817],[576,1198,703],{"className":1199},[817],[576,1201,1203],{"className":1202},[776],[576,1204,1206],{"className":1205},[780],[576,1207,1209],{"className":1208},[784],[576,1210,1213],{"className":1211,"style":1212},[788],"height:0.6644em;",[576,1214,1216,1219],{"style":1215},"top:-3.063em;margin-right:0.05em;",[576,1217],{"className":1218,"style":797},[796],[576,1220,1222],{"className":1221},[801,802,803,804],[576,1223,597],{"className":1224},[619,620,804]," mit dem binomischen Lehrsatz aus und vergleicht Real- und Imaginärteil.",[1054,1227,1228,1229,1260],{},"Grundlage für die Berechnung der ",[568,1230,1231,1259],{},[576,1232,1234,1247],{"className":1233},[579],[576,1235,1237],{"className":1236},[583],[585,1238,1239],{"xmlns":587},[589,1240,1241,1245],{},[592,1242,1243],{},[595,1244,597],{},[599,1246,597],{"encoding":601},[576,1248,1250],{"className":1249,"ariaHidden":606},[605],[576,1251,1253,1256],{"className":1252},[610],[576,1254],{"className":1255,"style":615},[614],[576,1257,597],{"className":1258},[619,620],"-ten Wurzeln"," komplexer Zahlen.",[559,1262,1264],{"id":1263},"herleitung","Herleitung",[564,1266,1267,1268,1412,1413,1587,1588,1691],{},"Nach der Eulerschen Formel ist ",[576,1269,1271,1311],{"className":1270},[579],[576,1272,1274],{"className":1273},[583],[585,1275,1276],{"xmlns":587},[589,1277,1278,1308],{},[592,1279,1280,1282,1284,1286,1288,1290,1292,1294,1296,1298],{},[595,1281,1112],{},[678,1283,1115],{},[595,1285,700],{},[678,1287,1120],{},[595,1289,697],{},[595,1291,1125],{},[678,1293,1115],{},[595,1295,700],{},[678,1297,708],{},[673,1299,1300,1302],{},[595,1301,692],{},[592,1303,1304,1306],{},[595,1305,697],{},[595,1307,700],{},[599,1309,1310],{"encoding":601},"\\cos\\varphi + i\\sin\\varphi = e^{i\\varphi}",[576,1312,1314,1339,1370],{"className":1313,"ariaHidden":606},[605],[576,1315,1317,1321,1324,1327,1330,1333,1336],{"className":1316},[610],[576,1318],{"className":1319,"style":1320},[614],"height:0.7778em;vertical-align:-0.1944em;",[576,1322,1112],{"className":1323},[1155],[576,1325],{"className":1326,"style":766},[765],[576,1328,700],{"className":1329},[619,620],[576,1331],{"className":1332,"style":1165},[765],[576,1334,1120],{"className":1335},[1169],[576,1337],{"className":1338,"style":1165},[765],[576,1340,1342,1346,1349,1352,1355,1358,1361,1364,1367],{"className":1341},[610],[576,1343],{"className":1344,"style":1345},[614],"height:0.8623em;vertical-align:-0.1944em;",[576,1347,697],{"className":1348},[619,620],[576,1350],{"className":1351,"style":766},[765],[576,1353,1125],{"className":1354},[1155],[576,1356],{"className":1357,"style":766},[765],[576,1359,700],{"className":1360},[619,620],[576,1362],{"className":1363,"style":849},[765],[576,1365,708],{"className":1366},[853],[576,1368],{"className":1369,"style":849},[765],[576,1371,1373,1377],{"className":1372},[610],[576,1374],{"className":1375,"style":1376},[614],"height:0.8247em;",[576,1378,1380,1383],{"className":1379},[619],[576,1381,692],{"className":1382},[619,620],[576,1384,1386],{"className":1385},[776],[576,1387,1389],{"className":1388},[780],[576,1390,1392],{"className":1391},[784],[576,1393,1395],{"className":1394,"style":1376},[788],[576,1396,1397,1400],{"style":1215},[576,1398],{"className":1399,"style":797},[796],[576,1401,1403],{"className":1402},[801,802,803,804],[576,1404,1406,1409],{"className":1405},[619,804],[576,1407,697],{"className":1408},[619,620,804],[576,1410,700],{"className":1411},[619,620,804],". Mit dem Potenzgesetz ",[576,1414,1416,1460],{"className":1415},[579],[576,1417,1419],{"className":1418},[583],[585,1420,1421],{"xmlns":587},[589,1422,1423,1457],{},[592,1424,1425,1427,1437,1443,1445],{},[678,1426,680],{"stretchy":1109},[673,1428,1429,1431],{},[595,1430,692],{},[592,1432,1433,1435],{},[595,1434,697],{},[595,1436,700],{},[673,1438,1439,1441],{},[678,1440,703],{"stretchy":1109},[595,1442,597],{},[678,1444,708],{},[673,1446,1447,1449],{},[595,1448,692],{},[592,1450,1451,1453,1455],{},[595,1452,697],{},[595,1454,597],{},[595,1456,700],{},[599,1458,1459],{"encoding":601},"(e^{i\\varphi})^n = e^{in\\varphi}",[576,1461,1463,1546],{"className":1462,"ariaHidden":606},[605],[576,1464,1466,1470,1473,1508,1537,1540,1543],{"className":1465},[610],[576,1467],{"className":1468,"style":1469},[614],"height:1.0747em;vertical-align:-0.25em;",[576,1471,680],{"className":1472},[750],[576,1474,1476,1479],{"className":1475},[619],[576,1477,692],{"className":1478},[619,620],[576,1480,1482],{"className":1481},[776],[576,1483,1485],{"className":1484},[780],[576,1486,1488],{"className":1487},[784],[576,1489,1491],{"className":1490,"style":1376},[788],[576,1492,1493,1496],{"style":1215},[576,1494],{"className":1495,"style":797},[796],[576,1497,1499],{"className":1498},[801,802,803,804],[576,1500,1502,1505],{"className":1501},[619,804],[576,1503,697],{"className":1504},[619,620,804],[576,1506,700],{"className":1507},[619,620,804],[576,1509,1511,1514],{"className":1510},[817],[576,1512,703],{"className":1513},[817],[576,1515,1517],{"className":1516},[776],[576,1518,1520],{"className":1519},[780],[576,1521,1523],{"className":1522},[784],[576,1524,1526],{"className":1525,"style":1212},[788],[576,1527,1528,1531],{"style":1215},[576,1529],{"className":1530,"style":797},[796],[576,1532,1534],{"className":1533},[801,802,803,804],[576,1535,597],{"className":1536},[619,620,804],[576,1538],{"className":1539,"style":849},[765],[576,1541,708],{"className":1542},[853],[576,1544],{"className":1545,"style":849},[765],[576,1547,1549,1552],{"className":1548},[610],[576,1550],{"className":1551,"style":1376},[614],[576,1553,1555,1558],{"className":1554},[619],[576,1556,692],{"className":1557},[619,620],[576,1559,1561],{"className":1560},[776],[576,1562,1564],{"className":1563},[780],[576,1565,1567],{"className":1566},[784],[576,1568,1570],{"className":1569,"style":1376},[788],[576,1571,1572,1575],{"style":1215},[576,1573],{"className":1574,"style":797},[796],[576,1576,1578],{"className":1577},[801,802,803,804],[576,1579,1581,1584],{"className":1580},[619,804],[576,1582,925],{"className":1583},[619,620,804],[576,1585,700],{"className":1586},[619,620,804]," und noch einmal der Eulerschen Formel folgt ",[576,1589,1591,1631],{"className":1590},[579],[576,1592,1594],{"className":1593},[583],[585,1595,1596],{"xmlns":587},[589,1597,1598,1628],{},[592,1599,1600,1602,1604,1606,1608,1610,1612,1614,1616,1618,1620,1622,1624,1626],{},[595,1601,1112],{},[678,1603,1115],{},[678,1605,680],{"stretchy":1109},[595,1607,597],{},[595,1609,700],{},[678,1611,703],{"stretchy":1109},[678,1613,1120],{},[595,1615,697],{},[595,1617,1125],{},[678,1619,1115],{},[678,1621,680],{"stretchy":1109},[595,1623,597],{},[595,1625,700],{},[678,1627,703],{"stretchy":1109},[599,1629,1630],{"encoding":601},"\\cos(n\\varphi) + i\\sin(n\\varphi)",[576,1632,1634,1664],{"className":1633,"ariaHidden":606},[605],[576,1635,1637,1640,1643,1646,1649,1652,1655,1658,1661],{"className":1636},[610],[576,1638],{"className":1639,"style":1148},[614],[576,1641,1112],{"className":1642},[1155],[576,1644,680],{"className":1645},[750],[576,1647,597],{"className":1648},[619,620],[576,1650,700],{"className":1651},[619,620],[576,1653,703],{"className":1654},[817],[576,1656],{"className":1657,"style":1165},[765],[576,1659,1120],{"className":1660},[1169],[576,1662],{"className":1663,"style":1165},[765],[576,1665,1667,1670,1673,1676,1679,1682,1685,1688],{"className":1666},[610],[576,1668],{"className":1669,"style":1148},[614],[576,1671,697],{"className":1672},[619,620],[576,1674],{"className":1675,"style":766},[765],[576,1677,1125],{"className":1678},[1155],[576,1680,680],{"className":1681},[750],[576,1683,597],{"className":1684},[619,620],[576,1686,700],{"className":1687},[619,620],[576,1689,703],{"className":1690},[817],". Ohne Exponentialfunktion lässt sich die Formel durch vollständige Induktion mit den Additionstheoremen beweisen.",[559,1693,1695],{"id":1694},"beispiele","Beispiele",[564,1697,1698,1793,1794,1915,1916,2050],{},[576,1699,1701,1729],{"className":1700},[579],[576,1702,1704],{"className":1703},[583],[585,1705,1706],{"xmlns":587},[589,1707,1708,1726],{},[592,1709,1710,1712,1715,1717,1719],{},[678,1711,680],{"stretchy":1109},[979,1713,1714],{},"1",[678,1716,1120],{},[595,1718,697],{},[673,1720,1721,1723],{},[678,1722,703],{"stretchy":1109},[979,1724,1725],{},"8",[599,1727,1728],{"encoding":601},"(1 + i)^8",[576,1730,1732,1753],{"className":1731,"ariaHidden":606},[605],[576,1733,1735,1738,1741,1744,1747,1750],{"className":1734},[610],[576,1736],{"className":1737,"style":1148},[614],[576,1739,680],{"className":1740},[750],[576,1742,1714],{"className":1743},[619],[576,1745],{"className":1746,"style":1165},[765],[576,1748,1120],{"className":1749},[1169],[576,1751],{"className":1752,"style":1165},[765],[576,1754,1756,1760,1763],{"className":1755},[610],[576,1757],{"className":1758,"style":1759},[614],"height:1.0641em;vertical-align:-0.25em;",[576,1761,697],{"className":1762},[619,620],[576,1764,1766,1769],{"className":1765},[817],[576,1767,703],{"className":1768},[817],[576,1770,1772],{"className":1771},[776],[576,1773,1775],{"className":1774},[780],[576,1776,1778],{"className":1777},[784],[576,1779,1782],{"className":1780,"style":1781},[788],"height:0.8141em;",[576,1783,1784,1787],{"style":1215},[576,1785],{"className":1786,"style":797},[796],[576,1788,1790],{"className":1789},[801,802,803,804],[576,1791,1725],{"className":1792},[619,804],": Es ist ",[576,1795,1797,1819],{"className":1796},[579],[576,1798,1800],{"className":1799},[583],[585,1801,1802],{"xmlns":587},[589,1803,1804,1816],{},[592,1805,1806,1808,1810],{},[595,1807,683],{},[678,1809,708],{},[1811,1812,1813],"msqrt",{},[979,1814,1815],{},"2",[599,1817,1818],{"encoding":601},"r = \\sqrt{2}",[576,1820,1822,1840],{"className":1821,"ariaHidden":606},[605],[576,1823,1825,1828,1831,1834,1837],{"className":1824},[610],[576,1826],{"className":1827,"style":615},[614],[576,1829,683],{"className":1830,"style":761},[619,620],[576,1832],{"className":1833,"style":849},[765],[576,1835,708],{"className":1836},[853],[576,1838],{"className":1839,"style":849},[765],[576,1841,1843,1847],{"className":1842},[610],[576,1844],{"className":1845,"style":1846},[614],"height:1.04em;vertical-align:-0.1328em;",[576,1848,1851],{"className":1849},[619,1850],"sqrt",[576,1852,1855,1906],{"className":1853},[780,1854],"vlist-t2",[576,1856,1858,1901],{"className":1857},[784],[576,1859,1862,1878],{"className":1860,"style":1861},[788],"height:0.9072em;",[576,1863,1867,1871],{"className":1864,"style":1866},[1865],"svg-align","top:-3em;",[576,1868],{"className":1869,"style":1870},[796],"height:3em;",[576,1872,1875],{"className":1873,"style":1874},[619],"padding-left:0.833em;",[576,1876,1815],{"className":1877},[619],[576,1879,1881,1884],{"style":1880},"top:-2.8672em;",[576,1882],{"className":1883,"style":1870},[796],[576,1885,1889],{"className":1886,"style":1888},[1887],"hide-tail","min-width:0.853em;height:1.08em;",[1890,1891,1897],"svg",{"xmlns":1892,"width":1893,"height":1894,"viewBox":1895,"preserveAspectRatio":1896},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","400em","1.08em","0 0 400000 1080","xMinYMin slice",[1898,1899],"path",{"d":1900},"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z",[576,1902,1905],{"className":1903},[1904],"vlist-s","​",[576,1907,1909],{"className":1908},[784],[576,1910,1913],{"className":1911,"style":1912},[788],"height:0.1328em;",[576,1914],{}," und ",[576,1917,1919,1944],{"className":1918},[579],[576,1920,1922],{"className":1921},[583],[585,1923,1924],{"xmlns":587},[589,1925,1926,1941],{},[592,1927,1928,1930,1932],{},[595,1929,700],{},[678,1931,708],{},[1933,1934,1935,1938],"mfrac",{},[595,1936,1937],{},"π",[979,1939,1940],{},"4",[599,1942,1943],{"encoding":601},"\\varphi = \\frac{\\pi}{4}",[576,1945,1947,1966],{"className":1946,"ariaHidden":606},[605],[576,1948,1950,1954,1957,1960,1963],{"className":1949},[610],[576,1951],{"className":1952,"style":1953},[614],"height:0.625em;vertical-align:-0.1944em;",[576,1955,700],{"className":1956},[619,620],[576,1958],{"className":1959,"style":849},[765],[576,1961,708],{"className":1962},[853],[576,1964],{"className":1965,"style":849},[765],[576,1967,1969,1973],{"className":1968},[610],[576,1970],{"className":1971,"style":1972},[614],"height:1.0404em;vertical-align:-0.345em;",[576,1974,1976,1980,2047],{"className":1975},[619],[576,1977],{"className":1978},[750,1979],"nulldelimiter",[576,1981,1983],{"className":1982},[1933],[576,1984,1986,2038],{"className":1985},[780,1854],[576,1987,1989,2035],{"className":1988},[784],[576,1990,1993,2008,2019],{"className":1991,"style":1992},[788],"height:0.6954em;",[576,1994,1996,1999],{"style":1995},"top:-2.655em;",[576,1997],{"className":1998,"style":1870},[796],[576,2000,2002],{"className":2001},[801,802,803,804],[576,2003,2005],{"className":2004},[619,804],[576,2006,1940],{"className":2007},[619,804],[576,2009,2011,2014],{"style":2010},"top:-3.23em;",[576,2012],{"className":2013,"style":1870},[796],[576,2015],{"className":2016,"style":2018},[2017],"frac-line","border-bottom-width:0.04em;",[576,2020,2022,2025],{"style":2021},"top:-3.394em;",[576,2023],{"className":2024,"style":1870},[796],[576,2026,2028],{"className":2027},[801,802,803,804],[576,2029,2031],{"className":2030},[619,804],[576,2032,1937],{"className":2033,"style":2034},[619,620,804],"margin-right:0.0359em;",[576,2036,1905],{"className":2037},[1904],[576,2039,2041],{"className":2040},[784],[576,2042,2045],{"className":2043,"style":2044},[788],"height:0.345em;",[576,2046],{},[576,2048],{"className":2049},[817,1979],", also",[576,2052,2054],{"className":2053},[658],[576,2055,2057,2130],{"className":2056},[579],[576,2058,2060],{"className":2059},[583],[585,2061,2062],{"xmlns":587,"display":667},[589,2063,2064,2127],{},[592,2065,2066,2068,2070,2072,2074,2080,2082,2096,2122,2124],{},[678,2067,680],{"stretchy":1109},[979,2069,1714],{},[678,2071,1120],{},[595,2073,697],{},[673,2075,2076,2078],{},[678,2077,703],{"stretchy":1109},[979,2079,1725],{},[678,2081,708],{},[673,2083,2084,2094],{},[592,2085,2086,2088,2092],{},[678,2087,680],{"fence":606},[1811,2089,2090],{},[979,2091,1815],{},[678,2093,703],{"fence":606},[979,2095,1725],{},[592,2097,2098,2100,2102,2104,2106,2108,2110,2112,2114,2116,2118,2120],{},[678,2099,680],{"fence":606},[595,2101,1112],{},[678,2103,1115],{},[979,2105,1815],{},[595,2107,1937],{},[678,2109,1120],{},[595,2111,697],{},[595,2113,1125],{},[678,2115,1115],{},[979,2117,1815],{},[595,2119,1937],{},[678,2121,703],{"fence":606},[678,2123,708],{},[979,2125,2126],{},"16",[599,2128,2129],{"encoding":601},"(1 + i)^8 = \\left(\\sqrt{2}\\right)^8 \\left(\\cos 2\\pi + i\\sin 2\\pi\\right) = 16",[576,2131,2133,2154,2203,2364],{"className":2132,"ariaHidden":606},[605],[576,2134,2136,2139,2142,2145,2148,2151],{"className":2135},[610],[576,2137],{"className":2138,"style":1148},[614],[576,2140,680],{"className":2141},[750],[576,2143,1714],{"className":2144},[619],[576,2146],{"className":2147,"style":1165},[765],[576,2149,1120],{"className":2150},[1169],[576,2152],{"className":2153,"style":1165},[765],[576,2155,2157,2161,2164,2194,2197,2200],{"className":2156},[610],[576,2158],{"className":2159,"style":2160},[614],"height:1.1141em;vertical-align:-0.25em;",[576,2162,697],{"className":2163},[619,620],[576,2165,2167,2170],{"className":2166},[817],[576,2168,703],{"className":2169},[817],[576,2171,2173],{"className":2172},[776],[576,2174,2176],{"className":2175},[780],[576,2177,2179],{"className":2178},[784],[576,2180,2183],{"className":2181,"style":2182},[788],"height:0.8641em;",[576,2184,2185,2188],{"style":792},[576,2186],{"className":2187,"style":797},[796],[576,2189,2191],{"className":2190},[801,802,803,804],[576,2192,1725],{"className":2193},[619,804],[576,2195],{"className":2196,"style":849},[765],[576,2198,708],{"className":2199},[853],[576,2201],{"className":2202,"style":849},[765],[576,2204,2206,2210,2304,2307,2355,2358,2361],{"className":2205},[610],[576,2207],{"className":2208,"style":2209},[614],"height:2.004em;vertical-align:-0.65em;",[576,2211,2213,2279],{"className":2212},[743],[576,2214,2216,2223,2273],{"className":2215},[743],[576,2217,2219],{"className":2218,"style":752},[750,751],[576,2220,680],{"className":2221},[756,2222],"size2",[576,2224,2226],{"className":2225},[619,1850],[576,2227,2229,2264],{"className":2228},[780,1854],[576,2230,2232,2261],{"className":2231},[784],[576,2233,2236,2248],{"className":2234,"style":2235},[788],"height:0.9561em;",[576,2237,2239,2242],{"className":2238,"style":1866},[1865],[576,2240],{"className":2241,"style":1870},[796],[576,2243,2245],{"className":2244,"style":1874},[619],[576,2246,1815],{"className":2247},[619],[576,2249,2251,2254],{"style":2250},"top:-2.9161em;",[576,2252],{"className":2253,"style":1870},[796],[576,2255,2257],{"className":2256,"style":1888},[1887],[1890,2258,2259],{"xmlns":1892,"width":1893,"height":1894,"viewBox":1895,"preserveAspectRatio":1896},[1898,2260],{"d":1900},[576,2262,1905],{"className":2263},[1904],[576,2265,2267],{"className":2266},[784],[576,2268,2271],{"className":2269,"style":2270},[788],"height:0.0839em;",[576,2272],{},[576,2274,2276],{"className":2275,"style":752},[817,751],[576,2277,703],{"className":2278},[756,2222],[576,2280,2282],{"className":2281},[776],[576,2283,2285],{"className":2284},[780],[576,2286,2288],{"className":2287},[784],[576,2289,2292],{"className":2290,"style":2291},[788],"height:1.354em;",[576,2293,2295,2298],{"style":2294},"top:-3.6029em;margin-right:0.05em;",[576,2296],{"className":2297,"style":797},[796],[576,2299,2301],{"className":2300},[801,802,803,804],[576,2302,1725],{"className":2303},[619,804],[576,2305],{"className":2306,"style":766},[765],[576,2308,2310,2313,2316,2319,2322,2325,2328,2331,2334,2337,2340,2343,2346,2349,2352],{"className":2309},[743],[576,2311,680],{"className":2312,"style":752},[750,751],[576,2314,1112],{"className":2315},[1155],[576,2317],{"className":2318,"style":766},[765],[576,2320,1815],{"className":2321},[619],[576,2323,1937],{"className":2324,"style":2034},[619,620],[576,2326],{"className":2327,"style":1165},[765],[576,2329,1120],{"className":2330},[1169],[576,2332],{"className":2333,"style":1165},[765],[576,2335,697],{"className":2336},[619,620],[576,2338],{"className":2339,"style":766},[765],[576,2341,1125],{"className":2342},[1155],[576,2344],{"className":2345,"style":766},[765],[576,2347,1815],{"className":2348},[619],[576,2350,1937],{"className":2351,"style":2034},[619,620],[576,2353,703],{"className":2354,"style":752},[817,751],[576,2356],{"className":2357,"style":849},[765],[576,2359,708],{"className":2360},[853],[576,2362],{"className":2363,"style":849},[765],[576,2365,2367,2370],{"className":2366},[610],[576,2368],{"className":2369,"style":1013},[614],[576,2371,2126],{"className":2372},[619],[564,2374,2375,1793,2525,1915,2576,2050],{},[576,2376,2378,2412],{"className":2377},[579],[576,2379,2381],{"className":2380},[583],[585,2382,2383],{"xmlns":587},[589,2384,2385,2409],{},[592,2386,2387,2389,2391,2393,2398,2400,2402],{},[678,2388,680],{"stretchy":1109},[979,2390,1714],{},[678,2392,1120],{},[1811,2394,2395],{},[979,2396,2397],{},"3",[685,2399,687],{},[595,2401,697],{},[673,2403,2404,2406],{},[678,2405,703],{"stretchy":1109},[979,2407,2408],{},"6",[599,2410,2411],{"encoding":601},"(1 + \\sqrt{3}\\,i)^6",[576,2413,2415,2436],{"className":2414,"ariaHidden":606},[605],[576,2416,2418,2421,2424,2427,2430,2433],{"className":2417},[610],[576,2419],{"className":2420,"style":1148},[614],[576,2422,680],{"className":2423},[750],[576,2425,1714],{"className":2426},[619],[576,2428],{"className":2429,"style":1165},[765],[576,2431,1120],{"className":2432},[1169],[576,2434],{"className":2435,"style":1165},[765],[576,2437,2439,2443,2490,2493,2496],{"className":2438},[610],[576,2440],{"className":2441,"style":2442},[614],"height:1.1572em;vertical-align:-0.25em;",[576,2444,2446],{"className":2445},[619,1850],[576,2447,2449,2482],{"className":2448},[780,1854],[576,2450,2452,2479],{"className":2451},[784],[576,2453,2455,2467],{"className":2454,"style":1861},[788],[576,2456,2458,2461],{"className":2457,"style":1866},[1865],[576,2459],{"className":2460,"style":1870},[796],[576,2462,2464],{"className":2463,"style":1874},[619],[576,2465,2397],{"className":2466},[619],[576,2468,2469,2472],{"style":1880},[576,2470],{"className":2471,"style":1870},[796],[576,2473,2475],{"className":2474,"style":1888},[1887],[1890,2476,2477],{"xmlns":1892,"width":1893,"height":1894,"viewBox":1895,"preserveAspectRatio":1896},[1898,2478],{"d":1900},[576,2480,1905],{"className":2481},[1904],[576,2483,2485],{"className":2484},[784],[576,2486,2488],{"className":2487,"style":1912},[788],[576,2489],{},[576,2491],{"className":2492,"style":766},[765],[576,2494,697],{"className":2495},[619,620],[576,2497,2499,2502],{"className":2498},[817],[576,2500,703],{"className":2501},[817],[576,2503,2505],{"className":2504},[776],[576,2506,2508],{"className":2507},[780],[576,2509,2511],{"className":2510},[784],[576,2512,2514],{"className":2513,"style":1781},[788],[576,2515,2516,2519],{"style":1215},[576,2517],{"className":2518,"style":797},[796],[576,2520,2522],{"className":2521},[801,802,803,804],[576,2523,2408],{"className":2524},[619,804],[576,2526,2528,2546],{"className":2527},[579],[576,2529,2531],{"className":2530},[583],[585,2532,2533],{"xmlns":587},[589,2534,2535,2543],{},[592,2536,2537,2539,2541],{},[595,2538,683],{},[678,2540,708],{},[979,2542,1815],{},[599,2544,2545],{"encoding":601},"r = 2",[576,2547,2549,2567],{"className":2548,"ariaHidden":606},[605],[576,2550,2552,2555,2558,2561,2564],{"className":2551},[610],[576,2553],{"className":2554,"style":615},[614],[576,2556,683],{"className":2557,"style":761},[619,620],[576,2559],{"className":2560,"style":849},[765],[576,2562,708],{"className":2563},[853],[576,2565],{"className":2566,"style":849},[765],[576,2568,2570,2573],{"className":2569},[610],[576,2571],{"className":2572,"style":1013},[614],[576,2574,1815],{"className":2575},[619],[576,2577,2579,2601],{"className":2578},[579],[576,2580,2582],{"className":2581},[583],[585,2583,2584],{"xmlns":587},[589,2585,2586,2598],{},[592,2587,2588,2590,2592],{},[595,2589,700],{},[678,2591,708],{},[1933,2593,2594,2596],{},[595,2595,1937],{},[979,2597,2397],{},[599,2599,2600],{"encoding":601},"\\varphi = \\frac{\\pi}{3}",[576,2602,2604,2622],{"className":2603,"ariaHidden":606},[605],[576,2605,2607,2610,2613,2616,2619],{"className":2606},[610],[576,2608],{"className":2609,"style":1953},[614],[576,2611,700],{"className":2612},[619,620],[576,2614],{"className":2615,"style":849},[765],[576,2617,708],{"className":2618},[853],[576,2620],{"className":2621,"style":849},[765],[576,2623,2625,2628],{"className":2624},[610],[576,2626],{"className":2627,"style":1972},[614],[576,2629,2631,2634,2693],{"className":2630},[619],[576,2632],{"className":2633},[750,1979],[576,2635,2637],{"className":2636},[1933],[576,2638,2640,2685],{"className":2639},[780,1854],[576,2641,2643,2682],{"className":2642},[784],[576,2644,2646,2660,2668],{"className":2645,"style":1992},[788],[576,2647,2648,2651],{"style":1995},[576,2649],{"className":2650,"style":1870},[796],[576,2652,2654],{"className":2653},[801,802,803,804],[576,2655,2657],{"className":2656},[619,804],[576,2658,2397],{"className":2659},[619,804],[576,2661,2662,2665],{"style":2010},[576,2663],{"className":2664,"style":1870},[796],[576,2666],{"className":2667,"style":2018},[2017],[576,2669,2670,2673],{"style":2021},[576,2671],{"className":2672,"style":1870},[796],[576,2674,2676],{"className":2675},[801,802,803,804],[576,2677,2679],{"className":2678},[619,804],[576,2680,1937],{"className":2681,"style":2034},[619,620,804],[576,2683,1905],{"className":2684},[1904],[576,2686,2688],{"className":2687},[784],[576,2689,2691],{"className":2690,"style":2044},[788],[576,2692],{},[576,2694],{"className":2695},[817,1979],[576,2697,2699],{"className":2698},[658],[576,2700,2702,2773],{"className":2701},[579],[576,2703,2705],{"className":2704},[583],[585,2706,2707],{"xmlns":587,"display":667},[589,2708,2709,2770],{},[592,2710,2711,2713,2715,2717,2721,2723,2725,2731,2733,2739,2765,2767],{},[678,2712,680],{"stretchy":1109},[979,2714,1714],{},[678,2716,1120],{},[1811,2718,2719],{},[979,2720,2397],{},[685,2722,687],{},[595,2724,697],{},[673,2726,2727,2729],{},[678,2728,703],{"stretchy":1109},[979,2730,2408],{},[678,2732,708],{},[673,2734,2735,2737],{},[979,2736,1815],{},[979,2738,2408],{},[592,2740,2741,2743,2745,2747,2749,2751,2753,2755,2757,2759,2761,2763],{},[678,2742,680],{"fence":606},[595,2744,1112],{},[678,2746,1115],{},[979,2748,1815],{},[595,2750,1937],{},[678,2752,1120],{},[595,2754,697],{},[595,2756,1125],{},[678,2758,1115],{},[979,2760,1815],{},[595,2762,1937],{},[678,2764,703],{"fence":606},[678,2766,708],{},[979,2768,2769],{},"64",[599,2771,2772],{"encoding":601},"(1 + \\sqrt{3}\\,i)^6 = 2^6 \\left(\\cos 2\\pi + i\\sin 2\\pi\\right) = 64",[576,2774,2776,2797,2895,2990],{"className":2775,"ariaHidden":606},[605],[576,2777,2779,2782,2785,2788,2791,2794],{"className":2778},[610],[576,2780],{"className":2781,"style":1148},[614],[576,2783,680],{"className":2784},[750],[576,2786,1714],{"className":2787},[619],[576,2789],{"className":2790,"style":1165},[765],[576,2792,1120],{"className":2793},[1169],[576,2795],{"className":2796,"style":1165},[765],[576,2798,2800,2804,2851,2854,2857,2886,2889,2892],{"className":2799},[610],[576,2801],{"className":2802,"style":2803},[614],"height:1.2061em;vertical-align:-0.25em;",[576,2805,2807],{"className":2806},[619,1850],[576,2808,2810,2843],{"className":2809},[780,1854],[576,2811,2813,2840],{"className":2812},[784],[576,2814,2816,2828],{"className":2815,"style":2235},[788],[576,2817,2819,2822],{"className":2818,"style":1866},[1865],[576,2820],{"className":2821,"style":1870},[796],[576,2823,2825],{"className":2824,"style":1874},[619],[576,2826,2397],{"className":2827},[619],[576,2829,2830,2833],{"style":2250},[576,2831],{"className":2832,"style":1870},[796],[576,2834,2836],{"className":2835,"style":1888},[1887],[1890,2837,2838],{"xmlns":1892,"width":1893,"height":1894,"viewBox":1895,"preserveAspectRatio":1896},[1898,2839],{"d":1900},[576,2841,1905],{"className":2842},[1904],[576,2844,2846],{"className":2845},[784],[576,2847,2849],{"className":2848,"style":2270},[788],[576,2850],{},[576,2852],{"className":2853,"style":766},[765],[576,2855,697],{"className":2856},[619,620],[576,2858,2860,2863],{"className":2859},[817],[576,2861,703],{"className":2862},[817],[576,2864,2866],{"className":2865},[776],[576,2867,2869],{"className":2868},[780],[576,2870,2872],{"className":2871},[784],[576,2873,2875],{"className":2874,"style":2182},[788],[576,2876,2877,2880],{"style":792},[576,2878],{"className":2879,"style":797},[796],[576,2881,2883],{"className":2882},[801,802,803,804],[576,2884,2408],{"className":2885},[619,804],[576,2887],{"className":2888,"style":849},[765],[576,2890,708],{"className":2891},[853],[576,2893],{"className":2894,"style":849},[765],[576,2896,2898,2901,2930,2933,2981,2984,2987],{"className":2897},[610],[576,2899],{"className":2900,"style":2160},[614],[576,2902,2904,2907],{"className":2903},[619],[576,2905,1815],{"className":2906},[619],[576,2908,2910],{"className":2909},[776],[576,2911,2913],{"className":2912},[780],[576,2914,2916],{"className":2915},[784],[576,2917,2919],{"className":2918,"style":2182},[788],[576,2920,2921,2924],{"style":792},[576,2922],{"className":2923,"style":797},[796],[576,2925,2927],{"className":2926},[801,802,803,804],[576,2928,2408],{"className":2929},[619,804],[576,2931],{"className":2932,"style":766},[765],[576,2934,2936,2939,2942,2945,2948,2951,2954,2957,2960,2963,2966,2969,2972,2975,2978],{"className":2935},[743],[576,2937,680],{"className":2938,"style":752},[750,751],[576,2940,1112],{"className":2941},[1155],[576,2943],{"className":2944,"style":766},[765],[576,2946,1815],{"className":2947},[619],[576,2949,1937],{"className":2950,"style":2034},[619,620],[576,2952],{"className":2953,"style":1165},[765],[576,2955,1120],{"className":2956},[1169],[576,2958],{"className":2959,"style":1165},[765],[576,2961,697],{"className":2962},[619,620],[576,2964],{"className":2965,"style":766},[765],[576,2967,1125],{"className":2968},[1155],[576,2970],{"className":2971,"style":766},[765],[576,2973,1815],{"className":2974},[619],[576,2976,1937],{"className":2977,"style":2034},[619,620],[576,2979,703],{"className":2980,"style":752},[817,751],[576,2982],{"className":2983,"style":849},[765],[576,2985,708],{"className":2986},[853],[576,2988],{"className":2989,"style":849},[765],[576,2991,2993,2996],{"className":2992},[610],[576,2994],{"className":2995,"style":1013},[614],[576,2997,2769],{"className":2998},[619],[564,3000,3001,3002,3053],{},"Doppelwinkelformeln aus ",[576,3003,3005,3023],{"className":3004},[579],[576,3006,3008],{"className":3007},[583],[585,3009,3010],{"xmlns":587},[589,3011,3012,3020],{},[592,3013,3014,3016,3018],{},[595,3015,597],{},[678,3017,708],{},[979,3019,1815],{},[599,3021,3022],{"encoding":601},"n = 2",[576,3024,3026,3044],{"className":3025,"ariaHidden":606},[605],[576,3027,3029,3032,3035,3038,3041],{"className":3028},[610],[576,3030],{"className":3031,"style":615},[614],[576,3033,597],{"className":3034},[619,620],[576,3036],{"className":3037,"style":849},[765],[576,3039,708],{"className":3040},[853],[576,3042],{"className":3043,"style":849},[765],[576,3045,3047,3050],{"className":3046},[610],[576,3048],{"className":3049,"style":1013},[614],[576,3051,1815],{"className":3052},[619],":",[576,3055,3057],{"className":3056},[658],[576,3058,3060,3165],{"className":3059},[579],[576,3061,3063],{"className":3062},[583],[585,3064,3065],{"xmlns":587,"display":667},[589,3066,3067,3162],{},[592,3068,3069,3071,3073,3075,3077,3079,3081,3083,3085,3087,3089,3091,3093,3095,3097,3099,3101,3103,3105,3107,3109,3115,3117,3127,3129,3132,3142,3144,3146,3148,3150,3152,3154,3156,3158,3160],{},[595,3070,1112],{},[678,3072,1115],{},[979,3074,1815],{},[595,3076,700],{},[678,3078,1120],{},[595,3080,697],{},[595,3082,1125],{},[678,3084,1115],{},[979,3086,1815],{},[595,3088,700],{},[678,3090,708],{},[678,3092,680],{"stretchy":1109},[595,3094,1112],{},[678,3096,1115],{},[595,3098,700],{},[678,3100,1120],{},[595,3102,697],{},[595,3104,1125],{},[678,3106,1115],{},[595,3108,700],{},[673,3110,3111,3113],{},[678,3112,703],{"stretchy":1109},[979,3114,1815],{},[678,3116,708],{},[673,3118,3119,3125],{},[592,3120,3121,3123],{},[595,3122,1112],{},[678,3124,1115],{},[979,3126,1815],{},[595,3128,700],{},[678,3130,3131],{},"−",[673,3133,3134,3140],{},[592,3135,3136,3138],{},[595,3137,1125],{},[678,3139,1115],{},[979,3141,1815],{},[595,3143,700],{},[678,3145,1120],{},[979,3147,1815],{},[595,3149,697],{},[595,3151,1125],{},[678,3153,1115],{},[595,3155,700],{},[595,3157,1112],{},[678,3159,1115],{},[595,3161,700],{},[599,3163,3164],{"encoding":601},"\\cos 2\\varphi + i\\sin 2\\varphi = (\\cos\\varphi + i\\sin\\varphi)^2 = \\cos^2\\varphi - \\sin^2\\varphi + 2i\\sin\\varphi\\cos\\varphi",[576,3166,3168,3196,3229,3256,3315,3366,3419],{"className":3167,"ariaHidden":606},[605],[576,3169,3171,3175,3178,3181,3184,3187,3190,3193],{"className":3170},[610],[576,3172],{"className":3173,"style":3174},[614],"height:0.8389em;vertical-align:-0.1944em;",[576,3176,1112],{"className":3177},[1155],[576,3179],{"className":3180,"style":766},[765],[576,3182,1815],{"className":3183},[619],[576,3185,700],{"className":3186},[619,620],[576,3188],{"className":3189,"style":1165},[765],[576,3191,1120],{"className":3192},[1169],[576,3194],{"className":3195,"style":1165},[765],[576,3197,3199,3202,3205,3208,3211,3214,3217,3220,3223,3226],{"className":3198},[610],[576,3200],{"className":3201,"style":1345},[614],[576,3203,697],{"className":3204},[619,620],[576,3206],{"className":3207,"style":766},[765],[576,3209,1125],{"className":3210},[1155],[576,3212],{"className":3213,"style":766},[765],[576,3215,1815],{"className":3216},[619],[576,3218,700],{"className":3219},[619,620],[576,3221],{"className":3222,"style":849},[765],[576,3224,708],{"className":3225},[853],[576,3227],{"className":3228,"style":849},[765],[576,3230,3232,3235,3238,3241,3244,3247,3250,3253],{"className":3231},[610],[576,3233],{"className":3234,"style":1148},[614],[576,3236,680],{"className":3237},[750],[576,3239,1112],{"className":3240},[1155],[576,3242],{"className":3243,"style":766},[765],[576,3245,700],{"className":3246},[619,620],[576,3248],{"className":3249,"style":1165},[765],[576,3251,1120],{"className":3252},[1169],[576,3254],{"className":3255,"style":1165},[765],[576,3257,3259,3262,3265,3268,3271,3274,3277,3306,3309,3312],{"className":3258},[610],[576,3260],{"className":3261,"style":2160},[614],[576,3263,697],{"className":3264},[619,620],[576,3266],{"className":3267,"style":766},[765],[576,3269,1125],{"className":3270},[1155],[576,3272],{"className":3273,"style":766},[765],[576,3275,700],{"className":3276},[619,620],[576,3278,3280,3283],{"className":3279},[817],[576,3281,703],{"className":3282},[817],[576,3284,3286],{"className":3285},[776],[576,3287,3289],{"className":3288},[780],[576,3290,3292],{"className":3291},[784],[576,3293,3295],{"className":3294,"style":2182},[788],[576,3296,3297,3300],{"style":792},[576,3298],{"className":3299,"style":797},[796],[576,3301,3303],{"className":3302},[801,802,803,804],[576,3304,1815],{"className":3305},[619,804],[576,3307],{"className":3308,"style":849},[765],[576,3310,708],{"className":3311},[853],[576,3313],{"className":3314,"style":849},[765],[576,3316,3318,3322,3351,3354,3357,3360,3363],{"className":3317},[610],[576,3319],{"className":3320,"style":3321},[614],"height:1.0585em;vertical-align:-0.1944em;",[576,3323,3325,3328],{"className":3324},[1155],[576,3326,1112],{"className":3327},[1155],[576,3329,3331],{"className":3330},[776],[576,3332,3334],{"className":3333},[780],[576,3335,3337],{"className":3336},[784],[576,3338,3340],{"className":3339,"style":2182},[788],[576,3341,3342,3345],{"style":792},[576,3343],{"className":3344,"style":797},[796],[576,3346,3348],{"className":3347},[801,802,803,804],[576,3349,1815],{"className":3350},[619,804],[576,3352],{"className":3353,"style":766},[765],[576,3355,700],{"className":3356},[619,620],[576,3358],{"className":3359,"style":1165},[765],[576,3361,3131],{"className":3362},[1169],[576,3364],{"className":3365,"style":1165},[765],[576,3367,3369,3373,3404,3407,3410,3413,3416],{"className":3368},[610],[576,3370],{"className":3371,"style":3372},[614],"height:1.0663em;vertical-align:-0.1944em;",[576,3374,3376,3379],{"className":3375},[1155],[576,3377,1125],{"className":3378},[1155],[576,3380,3382],{"className":3381},[776],[576,3383,3385],{"className":3384},[780],[576,3386,3388],{"className":3387},[784],[576,3389,3392],{"className":3390,"style":3391},[788],"height:0.8719em;",[576,3393,3395,3398],{"style":3394},"top:-3.1208em;margin-right:0.05em;",[576,3396],{"className":3397,"style":797},[796],[576,3399,3401],{"className":3400},[801,802,803,804],[576,3402,1815],{"className":3403},[619,804],[576,3405],{"className":3406,"style":766},[765],[576,3408,700],{"className":3409},[619,620],[576,3411],{"className":3412,"style":1165},[765],[576,3414,1120],{"className":3415},[1169],[576,3417],{"className":3418,"style":1165},[765],[576,3420,3422,3425,3428,3431,3434,3437,3440,3443,3446,3449,3452],{"className":3421},[610],[576,3423],{"className":3424,"style":1345},[614],[576,3426,1815],{"className":3427},[619],[576,3429,697],{"className":3430},[619,620],[576,3432],{"className":3433,"style":766},[765],[576,3435,1125],{"className":3436},[1155],[576,3438],{"className":3439,"style":766},[765],[576,3441,700],{"className":3442},[619,620],[576,3444],{"className":3445,"style":766},[765],[576,3447,1112],{"className":3448},[1155],[576,3450],{"className":3451,"style":766},[765],[576,3453,700],{"className":3454},[619,620],[564,3456,3457,3458,1915,3631,3733,3734,3785,3786,3984],{},"Der Vergleich der Real- und Imaginärteile liefert ",[576,3459,3461,3509],{"className":3460},[579],[576,3462,3464],{"className":3463},[583],[585,3465,3466],{"xmlns":587},[589,3467,3468,3506],{},[592,3469,3470,3472,3474,3476,3478,3480,3490,3492,3494,3504],{},[595,3471,1112],{},[678,3473,1115],{},[979,3475,1815],{},[595,3477,700],{},[678,3479,708],{},[673,3481,3482,3488],{},[592,3483,3484,3486],{},[595,3485,1112],{},[678,3487,1115],{},[979,3489,1815],{},[595,3491,700],{},[678,3493,3131],{},[673,3495,3496,3502],{},[592,3497,3498,3500],{},[595,3499,1125],{},[678,3501,1115],{},[979,3503,1815],{},[595,3505,700],{},[599,3507,3508],{"encoding":601},"\\cos 2\\varphi = \\cos^2\\varphi - \\sin^2\\varphi",[576,3510,3512,3539,3590],{"className":3511,"ariaHidden":606},[605],[576,3513,3515,3518,3521,3524,3527,3530,3533,3536],{"className":3514},[610],[576,3516],{"className":3517,"style":3174},[614],[576,3519,1112],{"className":3520},[1155],[576,3522],{"className":3523,"style":766},[765],[576,3525,1815],{"className":3526},[619],[576,3528,700],{"className":3529},[619,620],[576,3531],{"className":3532,"style":849},[765],[576,3534,708],{"className":3535},[853],[576,3537],{"className":3538,"style":849},[765],[576,3540,3542,3546,3575,3578,3581,3584,3587],{"className":3541},[610],[576,3543],{"className":3544,"style":3545},[614],"height:1.0085em;vertical-align:-0.1944em;",[576,3547,3549,3552],{"className":3548},[1155],[576,3550,1112],{"className":3551},[1155],[576,3553,3555],{"className":3554},[776],[576,3556,3558],{"className":3557},[780],[576,3559,3561],{"className":3560},[784],[576,3562,3564],{"className":3563,"style":1781},[788],[576,3565,3566,3569],{"style":1215},[576,3567],{"className":3568,"style":797},[796],[576,3570,3572],{"className":3571},[801,802,803,804],[576,3573,1815],{"className":3574},[619,804],[576,3576],{"className":3577,"style":766},[765],[576,3579,700],{"className":3580},[619,620],[576,3582],{"className":3583,"style":1165},[765],[576,3585,3131],{"className":3586},[1169],[576,3588],{"className":3589,"style":1165},[765],[576,3591,3593,3596,3625,3628],{"className":3592},[610],[576,3594],{"className":3595,"style":3372},[614],[576,3597,3599,3602],{"className":3598},[1155],[576,3600,1125],{"className":3601},[1155],[576,3603,3605],{"className":3604},[776],[576,3606,3608],{"className":3607},[780],[576,3609,3611],{"className":3610},[784],[576,3612,3614],{"className":3613,"style":3391},[788],[576,3615,3616,3619],{"style":3394},[576,3617],{"className":3618,"style":797},[796],[576,3620,3622],{"className":3621},[801,802,803,804],[576,3623,1815],{"className":3624},[619,804],[576,3626],{"className":3627,"style":766},[765],[576,3629,700],{"className":3630},[619,620],[576,3632,3634,3670],{"className":3633},[579],[576,3635,3637],{"className":3636},[583],[585,3638,3639],{"xmlns":587},[589,3640,3641,3667],{},[592,3642,3643,3645,3647,3649,3651,3653,3655,3657,3659,3661,3663,3665],{},[595,3644,1125],{},[678,3646,1115],{},[979,3648,1815],{},[595,3650,700],{},[678,3652,708],{},[979,3654,1815],{},[595,3656,1125],{},[678,3658,1115],{},[595,3660,700],{},[595,3662,1112],{},[678,3664,1115],{},[595,3666,700],{},[599,3668,3669],{"encoding":601},"\\sin 2\\varphi = 2\\sin\\varphi\\cos\\varphi",[576,3671,3673,3700],{"className":3672,"ariaHidden":606},[605],[576,3674,3676,3679,3682,3685,3688,3691,3694,3697],{"className":3675},[610],[576,3677],{"className":3678,"style":1345},[614],[576,3680,1125],{"className":3681},[1155],[576,3683],{"className":3684,"style":766},[765],[576,3686,1815],{"className":3687},[619],[576,3689,700],{"className":3690},[619,620],[576,3692],{"className":3693,"style":849},[765],[576,3695,708],{"className":3696},[853],[576,3698],{"className":3699,"style":849},[765],[576,3701,3703,3706,3709,3712,3715,3718,3721,3724,3727,3730],{"className":3702},[610],[576,3704],{"className":3705,"style":1345},[614],[576,3707,1815],{"className":3708},[619],[576,3710],{"className":3711,"style":766},[765],[576,3713,1125],{"className":3714},[1155],[576,3716],{"className":3717,"style":766},[765],[576,3719,700],{"className":3720},[619,620],[576,3722],{"className":3723,"style":766},[765],[576,3725,1112],{"className":3726},[1155],[576,3728],{"className":3729,"style":766},[765],[576,3731,700],{"className":3732},[619,620],". Ebenso folgt aus ",[576,3735,3737,3755],{"className":3736},[579],[576,3738,3740],{"className":3739},[583],[585,3741,3742],{"xmlns":587},[589,3743,3744,3752],{},[592,3745,3746,3748,3750],{},[595,3747,597],{},[678,3749,708],{},[979,3751,2397],{},[599,3753,3754],{"encoding":601},"n = 3",[576,3756,3758,3776],{"className":3757,"ariaHidden":606},[605],[576,3759,3761,3764,3767,3770,3773],{"className":3760},[610],[576,3762],{"className":3763,"style":615},[614],[576,3765,597],{"className":3766},[619,620],[576,3768],{"className":3769,"style":849},[765],[576,3771,708],{"className":3772},[853],[576,3774],{"className":3775,"style":849},[765],[576,3777,3779,3782],{"className":3778},[610],[576,3780],{"className":3781,"style":1013},[614],[576,3783,2397],{"className":3784},[619]," die Formel ",[576,3787,3789,3845],{"className":3788},[579],[576,3790,3792],{"className":3791},[583],[585,3793,3794],{"xmlns":587},[589,3795,3796,3842],{},[592,3797,3798,3800,3802,3804,3806,3808,3818,3820,3822,3824,3826,3828,3830,3840],{},[595,3799,1112],{},[678,3801,1115],{},[979,3803,2397],{},[595,3805,700],{},[678,3807,708],{},[673,3809,3810,3816],{},[592,3811,3812,3814],{},[595,3813,1112],{},[678,3815,1115],{},[979,3817,2397],{},[595,3819,700],{},[678,3821,3131],{},[979,3823,2397],{},[595,3825,1112],{},[678,3827,1115],{},[595,3829,700],{},[673,3831,3832,3838],{},[592,3833,3834,3836],{},[595,3835,1125],{},[678,3837,1115],{},[979,3839,1815],{},[595,3841,700],{},[599,3843,3844],{"encoding":601},"\\cos 3\\varphi = \\cos^3\\varphi - 3\\cos\\varphi\\sin^2\\varphi",[576,3846,3848,3875,3925],{"className":3847,"ariaHidden":606},[605],[576,3849,3851,3854,3857,3860,3863,3866,3869,3872],{"className":3850},[610],[576,3852],{"className":3853,"style":3174},[614],[576,3855,1112],{"className":3856},[1155],[576,3858],{"className":3859,"style":766},[765],[576,3861,2397],{"className":3862},[619],[576,3864,700],{"className":3865},[619,620],[576,3867],{"className":3868,"style":849},[765],[576,3870,708],{"className":3871},[853],[576,3873],{"className":3874,"style":849},[765],[576,3876,3878,3881,3910,3913,3916,3919,3922],{"className":3877},[610],[576,3879],{"className":3880,"style":3545},[614],[576,3882,3884,3887],{"className":3883},[1155],[576,3885,1112],{"className":3886},[1155],[576,3888,3890],{"className":3889},[776],[576,3891,3893],{"className":3892},[780],[576,3894,3896],{"className":3895},[784],[576,3897,3899],{"className":3898,"style":1781},[788],[576,3900,3901,3904],{"style":1215},[576,3902],{"className":3903,"style":797},[796],[576,3905,3907],{"className":3906},[801,802,803,804],[576,3908,2397],{"className":3909},[619,804],[576,3911],{"className":3912,"style":766},[765],[576,3914,700],{"className":3915},[619,620],[576,3917],{"className":3918,"style":1165},[765],[576,3920,3131],{"className":3921},[1169],[576,3923],{"className":3924,"style":1165},[765],[576,3926,3928,3931,3934,3937,3940,3943,3946,3949,3978,3981],{"className":3927},[610],[576,3929],{"className":3930,"style":3372},[614],[576,3932,2397],{"className":3933},[619],[576,3935],{"className":3936,"style":766},[765],[576,3938,1112],{"className":3939},[1155],[576,3941],{"className":3942,"style":766},[765],[576,3944,700],{"className":3945},[619,620],[576,3947],{"className":3948,"style":766},[765],[576,3950,3952,3955],{"className":3951},[1155],[576,3953,1125],{"className":3954},[1155],[576,3956,3958],{"className":3957},[776],[576,3959,3961],{"className":3960},[780],[576,3962,3964],{"className":3963},[784],[576,3965,3967],{"className":3966,"style":3391},[788],[576,3968,3969,3972],{"style":3394},[576,3970],{"className":3971,"style":797},[796],[576,3973,3975],{"className":3974},[801,802,803,804],[576,3976,1815],{"className":3977},[619,804],[576,3979],{"className":3980,"style":766},[765],[576,3982,700],{"className":3983},[619,620],".",[559,3986,3988],{"id":3987},"typische-fehler","Typische Fehler",[1051,3990,3991,4331,4563],{},[1054,3992,3993,3994,4022,4023,4174,4175,4250,4251,3984],{},"Den Betrag vergessen oder mit ",[576,3995,3997,4010],{"className":3996},[579],[576,3998,4000],{"className":3999},[583],[585,4001,4002],{"xmlns":587},[589,4003,4004,4008],{},[592,4005,4006],{},[595,4007,597],{},[599,4009,597],{"encoding":601},[576,4011,4013],{"className":4012,"ariaHidden":606},[605],[576,4014,4016,4019],{"className":4015},[610],[576,4017],{"className":4018,"style":615},[614],[576,4020,597],{"className":4021},[619,620]," multiplizieren. Der Betrag wird potenziert: ",[576,4024,4026,4056],{"className":4025},[579],[576,4027,4029],{"className":4028},[583],[585,4030,4031],{"xmlns":587},[589,4032,4033,4053],{},[592,4034,4035,4049,4051],{},[673,4036,4037,4047],{},[592,4038,4039,4041,4045],{},[678,4040,680],{"fence":606},[1811,4042,4043],{},[979,4044,1815],{},[678,4046,703],{"fence":606},[979,4048,1725],{},[678,4050,708],{},[979,4052,2126],{},[599,4054,4055],{"encoding":601},"\\left(\\sqrt{2}\\right)^8 = 16",[576,4057,4059,4165],{"className":4058,"ariaHidden":606},[605],[576,4060,4062,4066,4156,4159,4162],{"className":4061},[610],[576,4063],{"className":4064,"style":4065},[614],"height:1.4612em;vertical-align:-0.35em;",[576,4067,4069,4131],{"className":4068},[743],[576,4070,4072,4078,4125],{"className":4071},[743],[576,4073,4075],{"className":4074,"style":752},[750,751],[576,4076,680],{"className":4077},[756,757],[576,4079,4081],{"className":4080},[619,1850],[576,4082,4084,4117],{"className":4083},[780,1854],[576,4085,4087,4114],{"className":4086},[784],[576,4088,4090,4102],{"className":4089,"style":1861},[788],[576,4091,4093,4096],{"className":4092,"style":1866},[1865],[576,4094],{"className":4095,"style":1870},[796],[576,4097,4099],{"className":4098,"style":1874},[619],[576,4100,1815],{"className":4101},[619],[576,4103,4104,4107],{"style":1880},[576,4105],{"className":4106,"style":1870},[796],[576,4108,4110],{"className":4109,"style":1888},[1887],[1890,4111,4112],{"xmlns":1892,"width":1893,"height":1894,"viewBox":1895,"preserveAspectRatio":1896},[1898,4113],{"d":1900},[576,4115,1905],{"className":4116},[1904],[576,4118,4120],{"className":4119},[784],[576,4121,4123],{"className":4122,"style":1912},[788],[576,4124],{},[576,4126,4128],{"className":4127,"style":752},[817,751],[576,4129,703],{"className":4130},[756,757],[576,4132,4134],{"className":4133},[776],[576,4135,4137],{"className":4136},[780],[576,4138,4140],{"className":4139},[784],[576,4141,4144],{"className":4142,"style":4143},[788],"height:1.1112em;",[576,4145,4147,4150],{"style":4146},"top:-3.3601em;margin-right:0.05em;",[576,4148],{"className":4149,"style":797},[796],[576,4151,4153],{"className":4152},[801,802,803,804],[576,4154,1725],{"className":4155},[619,804],[576,4157],{"className":4158,"style":849},[765],[576,4160,708],{"className":4161},[853],[576,4163],{"className":4164,"style":849},[765],[576,4166,4168,4171],{"className":4167},[610],[576,4169],{"className":4170,"style":1013},[614],[576,4172,2126],{"className":4173},[619],", nicht 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oder ",[576,4492,4494,4518],{"className":4493},[579],[576,4495,4497],{"className":4496},[583],[585,4498,4499],{"xmlns":587},[589,4500,4501,4515],{},[592,4502,4503,4513],{},[673,4504,4505,4511],{},[592,4506,4507,4509],{},[595,4508,1112],{},[678,4510,1115],{},[595,4512,597],{},[595,4514,700],{},[599,4516,4517],{"encoding":601},"\\cos^n\\varphi",[576,4519,4521],{"className":4520,"ariaHidden":606},[605],[576,4522,4524,4528,4557,4560],{"className":4523},[610],[576,4525],{"className":4526,"style":4527},[614],"height:0.8588em;vertical-align:-0.1944em;",[576,4529,4531,4534],{"className":4530},[1155],[576,4532,1112],{"className":4533},[1155],[576,4535,4537],{"className":4536},[776],[576,4538,4540],{"className":4539},[780],[576,4541,4543],{"className":4542},[784],[576,4544,4546],{"className":4545,"style":1212},[788],[576,4547,4548,4551],{"style":1215},[576,4549],{"className":4550,"style":797},[796],[576,4552,4554],{"className":4553},[801,802,803,804],[576,4555,597],{"className":4556},[619,620,804],[576,4558],{"className":4559,"style":766},[765],[576,4561,700],{"className":4562},[619,620],[1054,4564,4565,4566,3984],{},"Die Potenz in algebraischer Form gliedweise bilden: ",[576,4567,4569,4617],{"className":4568},[579],[576,4570,4572],{"className":4571},[583],[585,4573,4574],{"xmlns":587},[589,4575,4576,4614],{},[592,4577,4578,4580,4582,4584,4586,4592,4596,4602,4604,4610,4612],{},[678,4579,680],{"stretchy":1109},[979,4581,1714],{},[678,4583,1120],{},[595,4585,697],{},[673,4587,4588,4590],{},[678,4589,703],{"stretchy":1109},[979,4591,1725],{},[678,4593,4595],{"mathvariant":4594},"normal","≠",[673,4597,4598,4600],{},[979,4599,1714],{},[979,4601,1725],{},[678,4603,1120],{},[673,4605,4606,4608],{},[595,4607,697],{},[979,4609,1725],{},[678,4611,708],{},[979,4613,1815],{},[599,4615,4616],{"encoding":601},"(1 + i)^8 \\neq 1^8 + i^8 = 2",[576,4618,4620,4641,4729,4774,4818],{"className":4619,"ariaHidden":606},[605],[576,4621,4623,4626,4629,4632,4635,4638],{"className":4622},[610],[576,4624],{"className":4625,"style":1148},[614],[576,4627,680],{"className":4628},[750],[576,4630,1714],{"className":4631},[619],[576,4633],{"className":4634,"style":1165},[765],[576,4636,1120],{"className":4637},[1169],[576,4639],{"className":4640,"style":1165},[765],[576,4642,4644,4647,4650,4679,4682,4726],{"className":4643},[610],[576,4645],{"className":4646,"style":1759},[614],[576,4648,697],{"className":4649},[619,620],[576,4651,4653,4656],{"className":4652},[817],[576,4654,703],{"className":4655},[817],[576,4657,4659],{"className":4658},[776],[576,4660,4662],{"className":4661},[780],[576,4663,4665],{"className":4664},[784],[576,4666,4668],{"className":4667,"style":1781},[788],[576,4669,4670,4673],{"style":1215},[576,4671],{"className":4672,"style":797},[796],[576,4674,4676],{"className":4675},[801,802,803,804],[576,4677,1725],{"className":4678},[619,804],[576,4680],{"className":4681,"style":849},[765],[576,4683,4685,4719,4723],{"className":4684},[853],[576,4686,4688],{"className":4687},[853],[576,4689,4692],{"className":4690},[619,4691],"vbox",[576,4693,4696],{"className":4694},[4695],"thinbox",[576,4697,4700,4704,4715],{"className":4698},[4699],"rlap",[576,4701],{"className":4702,"style":4703},[614],"height:0.8889em;vertical-align:-0.1944em;",[576,4705,4708],{"className":4706},[4707],"inner",[576,4709,4711],{"className":4710},[619],[576,4712,4714],{"className":4713},[853],"",[576,4716],{"className":4717},[4718],"fix",[576,4720],{"className":4721},[765,4722],"nobreak",[576,4724,708],{"className":4725},[853],[576,4727],{"className":4728,"style":849},[765],[576,4730,4732,4736,4765,4768,4771],{"className":4731},[610],[576,4733],{"className":4734,"style":4735},[614],"height:0.8974em;vertical-align:-0.0833em;",[576,4737,4739,4742],{"className":4738},[619],[576,4740,1714],{"className":4741},[619],[576,4743,4745],{"className":4744},[776],[576,4746,4748],{"className":4747},[780],[576,4749,4751],{"className":4750},[784],[576,4752,4754],{"className":4753,"style":1781},[788],[576,4755,4756,4759],{"style":1215},[576,4757],{"className":4758,"style":797},[796],[576,4760,4762],{"className":4761},[801,802,803,804],[576,4763,1725],{"className":4764},[619,804],[576,4766],{"className":4767,"style":1165},[765],[576,4769,1120],{"className":4770},[1169],[576,4772],{"className":4773,"style":1165},[765],[576,4775,4777,4780,4809,4812,4815],{"className":4776},[610],[576,4778],{"className":4779,"style":1781},[614],[576,4781,4783,4786],{"className":4782},[619],[576,4784,697],{"className":4785},[619,620],[576,4787,4789],{"className":4788},[776],[576,4790,4792],{"className":4791},[780],[576,4793,4795],{"className":4794},[784],[576,4796,4798],{"className":4797,"style":1781},[788],[576,4799,4800,4803],{"style":1215},[576,4801],{"className":4802,"style":797},[796],[576,4804,4806],{"className":4805},[801,802,803,804],[576,4807,1725],{"className":4808},[619,804],[576,4810],{"className":4811,"style":849},[765],[576,4813,708],{"className":4814},[853],[576,4816],{"className":4817,"style":849},[765],[576,4819,4821,4824],{"className":4820},[610],[576,4822],{"className":4823,"style":1013},[614],[576,4825,1815],{"className":4826},[619],{"title":4828,"searchDepth":4829,"depth":4829,"links":4830},"",2,[4831,4832,4833,4834],{"id":561,"depth":4829,"text":562},{"id":1263,"depth":4829,"text":1264},{"id":1694,"depth":4829,"text":1695},{"id":3987,"depth":4829,"text":3988},[4836,4841,4845,4848,4850,4853,4855,4857,4859,4861,4863,4865,4868,4871,4874,4877],{"lhs":4837,"rhs":4838,"where":4839},"re((r 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xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>r\u003C\u002Fmi>\u003Cmo>>\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">r > 0,\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.5782em;vertical-align:-0.0391em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">r\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">0\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>φ\u003C\u002Fmi>\u003Cmo>∈\u003C\u002Fmo>\u003Cmi mathvariant=\"double-struck\">R\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\varphi \\in \\mathbb{R},\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.7335em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">φ\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∈\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8833em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathbb\">R\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>n\u003C\u002Fmi>\u003Cmo>∈\u003C\u002Fmo>\u003Cmi mathvariant=\"double-struck\">Z\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">n \\in \\mathbb{Z}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.5782em;vertical-align:-0.0391em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∈\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6889em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathbb\">Z\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",null,"md","de-moivre","\\bigl(r(\\cos\\varphi + i\\sin\\varphi)\\bigr)^n = r^n\\bigl(\\cos(n\\varphi) + i\\sin(n\\varphi)\\bigr)","\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmo fence=\"true\" stretchy=\"true\" minsize=\"1.2em\" maxsize=\"1.2em\">(\u003C\u002Fmo>\u003Cmi>r\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>cos\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmi>φ\u003C\u002Fmi>\u003Cmo>+\u003C\u002Fmo>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>sin\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmi>φ\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmsup>\u003Cmo fence=\"true\" stretchy=\"true\" minsize=\"1.2em\" maxsize=\"1.2em\">)\u003C\u002Fmo>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsup>\u003Cmo>=\u003C\u002Fmo>\u003Cmsup>\u003Cmi>r\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsup>\u003Cmo fence=\"true\" stretchy=\"true\" minsize=\"1.2em\" maxsize=\"1.2em\">(\u003C\u002Fmo>\u003Cmi>cos\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>n\u003C\u002Fmi>\u003Cmi>φ\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>+\u003C\u002Fmo>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>sin\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>n\u003C\u002Fmi>\u003Cmi>φ\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo fence=\"true\" stretchy=\"true\" minsize=\"1.2em\" maxsize=\"1.2em\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\bigl(r(\\cos\\varphi + i\\sin\\varphi)\\bigr)^n = r^n\\bigl(\\cos(n\\varphi) + i\\sin(n\\varphi)\\bigr)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1.2em;vertical-align:-0.35em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">\u003Cspan class=\"delimsizing size1\">(\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">r\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mop\">cos\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">φ\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1.2543em;vertical-align:-0.35em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">i\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">sin\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">φ\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mclose\">\u003Cspan class=\"mclose\">\u003Cspan class=\"delimsizing size1\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.9043em;\">\u003Cspan style=\"top:-3.3029em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1.2em;vertical-align:-0.35em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">r\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7144em;\">\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">\u003Cspan class=\"delimsizing size1\">(\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mop\">cos\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">φ\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1.2em;vertical-align:-0.35em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">i\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">sin\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">φ\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mclose\">\u003Cspan class=\"delimsizing size1\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","university",{},true,"\u002Fde\u002Fformulas\u002Fmath\u002Falgebra\u002Fformel-von-moivre",[],[4892,4893,4894,4895],"polar-form","euler-formula","complex-roots","sine-double-angle",{"title":548,"description":4828},"de\u002Fformulas\u002Fmath\u002Falgebra\u002Fformel-von-moivre","Eine komplexe Zahl in Polarform potenziert man, indem man den Betrag potenziert und den Winkel mit n multipliziert.",[4900,4901,4902,4903,4904],"komplexe zahlen","potenz","polarform","winkelfunktionen","moivre",[4906,4909,4913,4916],{"symbol":683,"meaning":4907,"symbolHtml":4908},"Betrag der komplexen Zahl","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>r\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">r\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">r\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":4910,"meaning":4911,"symbolHtml":4912},"\\varphi","Argument (Winkel) der komplexen Zahl im Bogenmaß","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>φ\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\varphi\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">φ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":597,"meaning":4914,"symbolHtml":4915},"Exponent, eine ganze Zahl","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">n\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":697,"meaning":4917,"symbolHtml":4918},"imaginäre Einheit mit i² = −1","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">i\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6595em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","8Fm8dTtppFnk7e72dtMopBMmiMhNK5cu4B81tbwXxps",[4921,4924,4927,4930],{"stem":4922,"title":4923,"category":5},"de\u002Fformulas\u002Fmath\u002Falgebra\u002Feulersche-formel","Eulersche Formel",{"stem":4925,"title":4926,"category":5},"de\u002Fformulas\u002Fmath\u002Falgebra\u002Fn-te-wurzel-komplexe-zahl","n-te Wurzeln einer komplexen Zahl",{"stem":4928,"title":4929,"category":5},"de\u002Fformulas\u002Fmath\u002Falgebra\u002Fpolarform-komplexe-zahl","Polarform einer komplexen Zahl",{"stem":4931,"title":4932,"category":518},"de\u002Fformulas\u002Fmath\u002Ftrigonometry\u002Fdoppelwinkelformel-sinus","Doppelwinkelformel für den Sinus",1793824271984]