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",[573,1103,1105,1127],{"className":1104},[576],[573,1106,1108],{"className":1107},[580],[582,1109,1110],{"xmlns":584},[586,1111,1112,1124],{},[589,1113,1114],{},[731,1115,1116,1118],{},[592,1117,924],{},[589,1119,1120,1122],{},[592,1121,594],{},[592,1123,597],{},[599,1125,1126],{"encoding":601},"e^{i\\varphi}",[573,1128,1130],{"className":1129,"ariaHidden":607},[606],[573,1131,1133,1136],{"className":1132},[611],[573,1134],{"className":1135,"style":1027},[615],[573,1137,1139,1142],{"className":1138},[620],[573,1140,924],{"className":1141},[620,621],[573,1143,1145],{"className":1144},[788],[573,1146,1148],{"className":1147},[792],[573,1149,1151],{"className":1150},[796],[573,1152,1154],{"className":1153,"style":1027},[800],[573,1155,1156,1159],{"style":804},[573,1157],{"className":1158,"style":809},[808],[573,1160,1162],{"className":1161},[813,814,815,816],[573,1163,1165,1168],{"className":1164},[620,816],[573,1166,594],{"className":1167},[620,621,816],[573,1169,597],{"className":1170},[620,621,816]," liegt auf dem ",[569,1173,1174],{},"Einheitskreis",", und zwar beim Winkel ",[573,1177,1179,1193],{"className":1178},[576],[573,1180,1182],{"className":1181},[580],[582,1183,1184],{"xmlns":584},[586,1185,1186,1190],{},[589,1187,1188],{},[592,1189,597],{},[599,1191,1192],{"encoding":601},"\\varphi",[573,1194,1196],{"className":1195,"ariaHidden":607},[606],[573,1197,1199,1202],{"className":1198},[611],[573,1200],{"className":1201,"style":659},[615],[573,1203,597],{"className":1204},[620,621]," zur positiven reellen Achse. Daraus folgt die Polarform ",[573,1207,1209,1243],{"className":1208},[576],[573,1210,1212],{"className":1211},[580],[582,1213,1214],{"xmlns":584},[586,1215,1216,1240],{},[589,1217,1218,1221,1223,1226,1230],{},[592,1219,1220],{},"z",[642,1222,762],{},[592,1224,1225],{},"r",[1227,1228,1229],"mtext",{}," ",[731,1231,1232,1234],{},[592,1233,924],{},[589,1235,1236,1238],{},[592,1237,594],{},[592,1239,597],{},[599,1241,1242],{"encoding":601},"z = r\\,e^{i\\varphi}",[573,1244,1246,1266],{"className":1245,"ariaHidden":607},[606],[573,1247,1249,1253,1257,1260,1263],{"className":1248},[611],[573,1250],{"className":1251,"style":1252},[615],"height:0.4306em;",[573,1254,1220],{"className":1255,"style":1256},[620,621],"margin-right:0.044em;",[573,1258],{"className":1259,"style":884},[667],[573,1261,762],{"className":1262},[888],[573,1264],{"className":1265,"style":884},[667],[573,1267,1269,1272,1276,1279],{"className":1268},[611],[573,1270],{"className":1271,"style":1027},[615],[573,1273,1225],{"className":1274,"style":1275},[620,621],"margin-right:0.0278em;",[573,1277],{"className":1278,"style":668},[667],[573,1280,1282,1285],{"className":1281},[620],[573,1283,924],{"className":1284},[620,621],[573,1286,1288],{"className":1287},[788],[573,1289,1291],{"className":1290},[792],[573,1292,1294],{"className":1293},[796],[573,1295,1297],{"className":1296,"style":1027},[800],[573,1298,1299,1302],{"style":804},[573,1300],{"className":1301,"style":809},[808],[573,1303,1305],{"className":1304},[813,814,815,816],[573,1306,1308,1311],{"className":1307},[620,816],[573,1309,594],{"className":1310},[620,621,816],[573,1312,597],{"className":1313},[620,621,816],[565,1315,1316],{},"Wichtige Folgerungen:",[1318,1319,1320,1619,1736,2216],"ul",{},[1321,1322,1323,1324,1427,1428,1427,1523,714],"li",{},"Besondere Werte: ",[573,1325,1327,1360],{"className":1326},[576],[573,1328,1330],{"className":1329},[580],[582,1331,1332],{"xmlns":584},[586,1333,1334,1357],{},[589,1335,1336,1353,1355],{},[731,1337,1338,1340],{},[592,1339,924],{},[589,1341,1342,1344,1347,1351],{},[592,1343,594],{},[592,1345,1346],{},"π",[592,1348,1350],{"mathvariant":1349},"normal","\u002F",[740,1352,742],{},[642,1354,762],{},[592,1356,594],{},[599,1358,1359],{"encoding":601},"e^{i\\pi\u002F2} = i",[573,1361,1363,1417],{"className":1362,"ariaHidden":607},[606],[573,1364,1366,1370,1408,1411,1414],{"className":1365},[611],[573,1367],{"className":1368,"style":1369},[615],"height:0.888em;",[573,1371,1373,1376],{"className":1372},[620],[573,1374,924],{"className":1375},[620,621],[573,1377,1379],{"className":1378},[788],[573,1380,1382],{"className":1381},[792],[573,1383,1385],{"className":1384},[796],[573,1386,1388],{"className":1387,"style":1369},[800],[573,1389,1390,1393],{"style":804},[573,1391],{"className":1392,"style":809},[808],[573,1394,1396],{"className":1395},[813,814,815,816],[573,1397,1399,1404],{"className":1398},[620,816],[573,1400,1403],{"className":1401,"style":1402},[620,621,816],"margin-right:0.0359em;","iπ",[573,1405,1407],{"className":1406},[620,816],"\u002F2",[573,1409],{"className":1410,"style":884},[667],[573,1412,762],{"className":1413},[888],[573,1415],{"className":1416,"style":884},[667],[573,1418,1420,1424],{"className":1419},[611],[573,1421],{"className":1422,"style":1423},[615],"height:0.6595em;",[573,1425,594],{"className":1426},[620,621],", ",[573,1429,1431,1460],{"className":1430},[576],[573,1432,1434],{"className":1433},[580],[582,1435,1436],{"xmlns":584},[586,1437,1438,1457],{},[589,1439,1440,1450,1452,1455],{},[731,1441,1442,1444],{},[592,1443,924],{},[589,1445,1446,1448],{},[592,1447,594],{},[592,1449,1346],{},[642,1451,762],{},[642,1453,1454],{},"−",[740,1456,765],{},[599,1458,1459],{"encoding":601},"e^{i\\pi} = -1",[573,1461,1463,1510],{"className":1462,"ariaHidden":607},[606],[573,1464,1466,1469,1501,1504,1507],{"className":1465},[611],[573,1467],{"className":1468,"style":1027},[615],[573,1470,1472,1475],{"className":1471},[620],[573,1473,924],{"className":1474},[620,621],[573,1476,1478],{"className":1477},[788],[573,1479,1481],{"className":1480},[792],[573,1482,1484],{"className":1483},[796],[573,1485,1487],{"className":1486,"style":1027},[800],[573,1488,1489,1492],{"style":804},[573,1490],{"className":1491,"style":809},[808],[573,1493,1495],{"className":1494},[813,814,815,816],[573,1496,1498],{"className":1497},[620,816],[573,1499,1403],{"className":1500,"style":1402},[620,621,816],[573,1502],{"className":1503,"style":884},[667],[573,1505,762],{"className":1506},[888],[573,1508],{"className":1509,"style":884},[667],[573,1511,1513,1517,1520],{"className":1512},[611],[573,1514],{"className":1515,"style":1516},[615],"height:0.7278em;vertical-align:-0.0833em;",[573,1518,1454],{"className":1519},[620],[573,1521,765],{"className":1522},[620],[573,1524,1526,1554],{"className":1525},[576],[573,1527,1529],{"className":1528},[580],[582,1530,1531],{"xmlns":584},[586,1532,1533,1551],{},[589,1534,1535,1547,1549],{},[731,1536,1537,1539],{},[592,1538,924],{},[589,1540,1541,1543,1545],{},[740,1542,742],{},[592,1544,1346],{},[592,1546,594],{},[642,1548,762],{},[740,1550,765],{},[599,1552,1553],{"encoding":601},"e^{2\\pi i} = 1",[573,1555,1557,1610],{"className":1556,"ariaHidden":607},[606],[573,1558,1560,1563,1601,1604,1607],{"className":1559},[611],[573,1561],{"className":1562,"style":1027},[615],[573,1564,1566,1569],{"className":1565},[620],[573,1567,924],{"className":1568},[620,621],[573,1570,1572],{"className":1571},[788],[573,1573,1575],{"className":1574},[792],[573,1576,1578],{"className":1577},[796],[573,1579,1581],{"className":1580,"style":1027},[800],[573,1582,1583,1586],{"style":804},[573,1584],{"className":1585,"style":809},[808],[573,1587,1589],{"className":1588},[813,814,815,816],[573,1590,1592,1595,1598],{"className":1591},[620,816],[573,1593,742],{"className":1594},[620,816],[573,1596,1346],{"className":1597,"style":1402},[620,621,816],[573,1599,594],{"className":1600},[620,621,816],[573,1602],{"className":1603,"style":884},[667],[573,1605,762],{"className":1606},[888],[573,1608],{"className":1609,"style":884},[667],[573,1611,1613,1616],{"className":1612},[611],[573,1614],{"className":1615,"style":898},[615],[573,1617,765],{"className":1618},[620],[1321,1620,1621,1623,1624,714],{},[569,1622,553],{},": ",[573,1625,1627,1658],{"className":1626},[576],[573,1628,1630],{"className":1629},[580],[582,1631,1632],{"xmlns":584},[586,1633,1634,1655],{},[589,1635,1636,1646,1648,1650,1652],{},[731,1637,1638,1640],{},[592,1639,924],{},[589,1641,1642,1644],{},[592,1643,594],{},[592,1645,1346],{},[642,1647,747],{},[740,1649,765],{},[642,1651,762],{},[740,1653,1654],{},"0",[599,1656,1657],{"encoding":601},"e^{i\\pi} + 1 = 0",[573,1659,1661,1709,1727],{"className":1660,"ariaHidden":607},[606],[573,1662,1664,1668,1700,1703,1706],{"className":1663},[611],[573,1665],{"className":1666,"style":1667},[615],"height:0.908em;vertical-align:-0.0833em;",[573,1669,1671,1674],{"className":1670},[620],[573,1672,924],{"className":1673},[620,621],[573,1675,1677],{"className":1676},[788],[573,1678,1680],{"className":1679},[792],[573,1681,1683],{"className":1682},[796],[573,1684,1686],{"className":1685,"style":1027},[800],[573,1687,1688,1691],{"style":804},[573,1689],{"className":1690,"style":809},[808],[573,1692,1694],{"className":1693},[813,814,815,816],[573,1695,1697],{"className":1696},[620,816],[573,1698,1403],{"className":1699,"style":1402},[620,621,816],[573,1701],{"className":1702,"style":829},[667],[573,1704,747],{"className":1705},[833],[573,1707],{"className":1708,"style":829},[667],[573,1710,1712,1715,1718,1721,1724],{"className":1711},[611],[573,1713],{"className":1714,"style":898},[615],[573,1716,765],{"className":1717},[620],[573,1719],{"className":1720,"style":884},[667],[573,1722,762],{"className":1723},[888],[573,1725],{"className":1726,"style":884},[667],[573,1728,1730,1733],{"className":1729},[611],[573,1731],{"className":1732,"style":898},[615],[573,1734,1654],{"className":1735},[620],[1321,1737,1738,1739,1427,1980,714],{},"Kosinus und Sinus durch die e-Funktion: ",[573,1740,1742,1796],{"className":1741},[576],[573,1743,1745],{"className":1744},[580],[582,1746,1747],{"xmlns":584},[586,1748,1749,1793],{},[589,1750,1751,1753,1755,1757,1759],{},[592,1752,640],{},[642,1754,644],{},[592,1756,597],{},[642,1758,762],{},[1760,1761,1762],"mstyle",{"scriptlevel":1654,"displaystyle":607},[1763,1764,1765,1791],"mfrac",{},[589,1766,1767,1777,1779],{},[731,1768,1769,1771],{},[592,1770,924],{},[589,1772,1773,1775],{},[592,1774,594],{},[592,1776,597],{},[642,1778,747],{},[731,1780,1781,1783],{},[592,1782,924],{},[589,1784,1785,1787,1789],{},[642,1786,1454],{},[592,1788,594],{},[592,1790,597],{},[740,1792,742],{},[599,1794,1795],{"encoding":601},"\\cos\\varphi = \\dfrac{e^{i\\varphi} + e^{-i\\varphi}}{2}",[573,1797,1799,1823],{"className":1798,"ariaHidden":607},[606],[573,1800,1802,1805,1808,1811,1814,1817,1820],{"className":1801},[611],[573,1803],{"className":1804,"style":659},[615],[573,1806,640],{"className":1807},[663],[573,1809],{"className":1810,"style":668},[667],[573,1812,597],{"className":1813},[620,621],[573,1815],{"className":1816,"style":884},[667],[573,1818,762],{"className":1819},[888],[573,1821],{"className":1822,"style":884},[667],[573,1824,1826,1830],{"className":1825},[611],[573,1827],{"className":1828,"style":1829},[615],"height:2.1877em;vertical-align:-0.686em;",[573,1831,1833,1837,1977],{"className":1832},[620],[573,1834],{"className":1835},[957,1836],"nulldelimiter",[573,1838,1840],{"className":1839},[1763],[573,1841,1843,1968],{"className":1842},[792,966],[573,1844,1846,1965],{"className":1845},[796],[573,1847,1850,1863,1874],{"className":1848,"style":1849},[800],"height:1.5017em;",[573,1851,1853,1857],{"style":1852},"top:-2.314em;",[573,1854],{"className":1855,"style":1856},[808],"height:3em;",[573,1858,1860],{"className":1859},[620],[573,1861,742],{"className":1862},[620],[573,1864,1866,1869],{"style":1865},"top:-3.23em;",[573,1867],{"className":1868,"style":1856},[808],[573,1870],{"className":1871,"style":1873},[1872],"frac-line","border-bottom-width:0.04em;",[573,1875,1877,1880],{"style":1876},"top:-3.677em;",[573,1878],{"className":1879,"style":1856},[808],[573,1881,1883,1918,1921,1924,1927],{"className":1882},[620],[573,1884,1886,1889],{"className":1885},[620],[573,1887,924],{"className":1888},[620,621],[573,1890,1892],{"className":1891},[788],[573,1893,1895],{"className":1894},[792],[573,1896,1898],{"className":1897},[796],[573,1899,1901],{"className":1900,"style":1027},[800],[573,1902,1903,1906],{"style":804},[573,1904],{"className":1905,"style":809},[808],[573,1907,1909],{"className":1908},[813,814,815,816],[573,1910,1912,1915],{"className":1911},[620,816],[573,1913,594],{"className":1914},[620,621,816],[573,1916,597],{"className":1917},[620,621,816],[573,1919],{"className":1920,"style":829},[667],[573,1922,747],{"className":1923},[833],[573,1925],{"className":1926,"style":829},[667],[573,1928,1930,1933],{"className":1929},[620],[573,1931,924],{"className":1932},[620,621],[573,1934,1936],{"className":1935},[788],[573,1937,1939],{"className":1938},[792],[573,1940,1942],{"className":1941},[796],[573,1943,1945],{"className":1944,"style":1027},[800],[573,1946,1947,1950],{"style":804},[573,1948],{"className":1949,"style":809},[808],[573,1951,1953],{"className":1952},[813,814,815,816],[573,1954,1956,1959,1962],{"className":1955},[620,816],[573,1957,1454],{"className":1958},[620,816],[573,1960,594],{"className":1961},[620,621,816],[573,1963,597],{"className":1964},[620,621,816],[573,1966,999],{"className":1967},[998],[573,1969,1971],{"className":1970},[796],[573,1972,1975],{"className":1973,"style":1974},[800],"height:0.686em;",[573,1976],{},[573,1978],{"className":1979},[1048,1836],[573,1981,1983,2039],{"className":1982},[576],[573,1984,1986],{"className":1985},[580],[582,1987,1988],{"xmlns":584},[586,1989,1990,2036],{},[589,1991,1992,1994,1996,1998,2000],{},[592,1993,687],{},[642,1995,644],{},[592,1997,597],{},[642,1999,762],{},[1760,2001,2002],{"scriptlevel":1654,"displaystyle":607},[1763,2003,2004,2030],{},[589,2005,2006,2016,2018],{},[731,2007,2008,2010],{},[592,2009,924],{},[589,2011,2012,2014],{},[592,2013,594],{},[592,2015,597],{},[642,2017,1454],{},[731,2019,2020,2022],{},[592,2021,924],{},[589,2023,2024,2026,2028],{},[642,2025,1454],{},[592,2027,594],{},[592,2029,597],{},[589,2031,2032,2034],{},[740,2033,742],{},[592,2035,594],{},[599,2037,2038],{"encoding":601},"\\sin\\varphi = \\dfrac{e^{i\\varphi} - e^{-i\\varphi}}{2i}",[573,2040,2042,2066],{"className":2041,"ariaHidden":607},[606],[573,2043,2045,2048,2051,2054,2057,2060,2063],{"className":2044},[611],[573,2046],{"className":2047,"style":704},[615],[573,2049,687],{"className":2050},[663],[573,2052],{"className":2053,"style":668},[667],[573,2055,597],{"className":2056},[620,621],[573,2058],{"className":2059,"style":884},[667],[573,2061,762],{"className":2062},[888],[573,2064],{"className":2065,"style":884},[667],[573,2067,2069,2072],{"className":2068},[611],[573,2070],{"className":2071,"style":1829},[615],[573,2073,2075,2078,2213],{"className":2074},[620],[573,2076],{"className":2077},[957,1836],[573,2079,2081],{"className":2080},[1763],[573,2082,2084,2205],{"className":2083},[792,966],[573,2085,2087,2202],{"className":2086},[796],[573,2088,2090,2104,2112],{"className":2089,"style":1849},[800],[573,2091,2092,2095],{"style":1852},[573,2093],{"className":2094,"style":1856},[808],[573,2096,2098,2101],{"className":2097},[620],[573,2099,742],{"className":2100},[620],[573,2102,594],{"className":2103},[620,621],[573,2105,2106,2109],{"style":1865},[573,2107],{"className":2108,"style":1856},[808],[573,2110],{"className":2111,"style":1873},[1872],[573,2113,2114,2117],{"style":1876},[573,2115],{"className":2116,"style":1856},[808],[573,2118,2120,2155,2158,2161,2164],{"className":2119},[620],[573,2121,2123,2126],{"className":2122},[620],[573,2124,924],{"className":2125},[620,621],[573,2127,2129],{"className":2128},[788],[573,2130,2132],{"className":2131},[792],[573,2133,2135],{"className":2134},[796],[573,2136,2138],{"className":2137,"style":1027},[800],[573,2139,2140,2143],{"style":804},[573,2141],{"className":2142,"style":809},[808],[573,2144,2146],{"className":2145},[813,814,815,816],[573,2147,2149,2152],{"className":2148},[620,816],[573,2150,594],{"className":2151},[620,621,816],[573,2153,597],{"className":2154},[620,621,816],[573,2156],{"className":2157,"style":829},[667],[573,2159,1454],{"className":2160},[833],[573,2162],{"className":2163,"style":829},[667],[573,2165,2167,2170],{"className":2166},[620],[573,2168,924],{"className":2169},[620,621],[573,2171,2173],{"className":2172},[788],[573,2174,2176],{"className":2175},[792],[573,2177,2179],{"className":2178},[796],[573,2180,2182],{"className":2181,"style":1027},[800],[573,2183,2184,2187],{"style":804},[573,2185],{"className":2186,"style":809},[808],[573,2188,2190],{"className":2189},[813,814,815,816],[573,2191,2193,2196,2199],{"className":2192},[620,816],[573,2194,1454],{"className":2195},[620,816],[573,2197,594],{"className":2198},[620,621,816],[573,2200,597],{"className":2201},[620,621,816],[573,2203,999],{"className":2204},[998],[573,2206,2208],{"className":2207},[796],[573,2209,2211],{"className":2210,"style":1974},[800],[573,2212],{},[573,2214],{"className":2215},[1048,1836],[1321,2217,2218,2219,2441],{},"Aus ",[573,2220,2222,2282],{"className":2221},[576],[573,2223,2225],{"className":2224},[580],[582,2226,2227],{"xmlns":584},[586,2228,2229,2279],{},[589,2230,2231,2254,2256,2266,2269],{},[731,2232,2233,2235],{},[592,2234,924],{},[589,2236,2237,2239,2243,2246,2248,2251],{},[592,2238,594],{},[642,2240,2242],{"stretchy":2241},"false","(",[592,2244,2245],{},"α",[642,2247,747],{},[592,2249,2250],{},"β",[642,2252,2253],{"stretchy":2241},")",[642,2255,762],{},[731,2257,2258,2260],{},[592,2259,924],{},[589,2261,2262,2264],{},[592,2263,594],{},[592,2265,2245],{},[642,2267,2268],{},"⋅",[731,2270,2271,2273],{},[592,2272,924],{},[589,2274,2275,2277],{},[592,2276,594],{},[592,2278,2250],{},[599,2280,2281],{"encoding":601},"e^{i(\\alpha + \\beta)} = e^{i\\alpha} \\cdot e^{i\\beta}",[573,2283,2285,2349,2399],{"className":2284,"ariaHidden":607},[606],[573,2286,2288,2291,2340,2343,2346],{"className":2287},[611],[573,2289],{"className":2290,"style":1369},[615],[573,2292,2294,2297],{"className":2293},[620],[573,2295,924],{"className":2296},[620,621],[573,2298,2300],{"className":2299},[788],[573,2301,2303],{"className":2302},[792],[573,2304,2306],{"className":2305},[796],[573,2307,2309],{"className":2308,"style":1369},[800],[573,2310,2311,2314],{"style":804},[573,2312],{"className":2313,"style":809},[808],[573,2315,2317],{"className":2316},[813,814,815,816],[573,2318,2320,2323,2326,2330,2333,2337],{"className":2319},[620,816],[573,2321,594],{"className":2322},[620,621,816],[573,2324,2242],{"className":2325},[957,816],[573,2327,2245],{"className":2328,"style":2329},[620,621,816],"margin-right:0.0037em;",[573,2331,747],{"className":2332},[833,816],[573,2334,2250],{"className":2335,"style":2336},[620,621,816],"margin-right:0.0528em;",[573,2338,2253],{"className":2339},[1048,816],[573,2341],{"className":2342,"style":884},[667],[573,2344,762],{"className":2345},[888],[573,2347],{"className":2348,"style":884},[667],[573,2350,2352,2355,2390,2393,2396],{"className":2351},[611],[573,2353],{"className":2354,"style":1027},[615],[573,2356,2358,2361],{"className":2357},[620],[573,2359,924],{"className":2360},[620,621],[573,2362,2364],{"className":2363},[788],[573,2365,2367],{"className":2366},[792],[573,2368,2370],{"className":2369},[796],[573,2371,2373],{"className":2372,"style":1027},[800],[573,2374,2375,2378],{"style":804},[573,2376],{"className":2377,"style":809},[808],[573,2379,2381],{"className":2380},[813,814,815,816],[573,2382,2384,2387],{"className":2383},[620,816],[573,2385,594],{"className":2386},[620,621,816],[573,2388,2245],{"className":2389,"style":2329},[620,621,816],[573,2391],{"className":2392,"style":829},[667],[573,2394,2268],{"className":2395},[833],[573,2397],{"className":2398,"style":829},[667],[573,2400,2402,2406],{"className":2401},[611],[573,2403],{"className":2404,"style":2405},[615],"height:0.8491em;",[573,2407,2409,2412],{"className":2408},[620],[573,2410,924],{"className":2411},[620,621],[573,2413,2415],{"className":2414},[788],[573,2416,2418],{"className":2417},[792],[573,2419,2421],{"className":2420},[796],[573,2422,2424],{"className":2423,"style":2405},[800],[573,2425,2426,2429],{"style":804},[573,2427],{"className":2428,"style":809},[808],[573,2430,2432],{"className":2431},[813,814,815,816],[573,2433,2435,2438],{"className":2434},[620,816],[573,2436,594],{"className":2437},[620,621,816],[573,2439,2250],{"className":2440,"style":2336},[620,621,816]," folgen durch Vergleich von Real- und Imaginärteil die Additionstheoreme von Sinus und Kosinus.",[560,2443,2445],{"id":2444},"herleitung","Herleitung",[565,2447,2448,2449,2482,2483,2775,2776,1427,2862,1427,2950,3032,3033,3062,3063,3091],{},"Man setzt ",[573,2450,2452,2467],{"className":2451},[576],[573,2453,2455],{"className":2454},[580],[582,2456,2457],{"xmlns":584},[586,2458,2459,2465],{},[589,2460,2461,2463],{},[592,2462,594],{},[592,2464,597],{},[599,2466,602],{"encoding":601},[573,2468,2470],{"className":2469,"ariaHidden":607},[606],[573,2471,2473,2476,2479],{"className":2472},[611],[573,2474],{"className":2475,"style":616},[615],[573,2477,594],{"className":2478},[620,621],[573,2480,597],{"className":2481},[620,621]," in die Potenzreihe ",[573,2484,2486,2540],{"className":2485},[576],[573,2487,2489],{"className":2488},[580],[582,2490,2491],{"xmlns":584},[586,2492,2493,2537],{},[589,2494,2495,2502,2504,2522],{},[731,2496,2497,2499],{},[592,2498,924],{},[592,2500,2501],{},"x",[642,2503,762],{},[2505,2506,2507,2510,2519],"msubsup",{},[642,2508,2509],{},"∑",[589,2511,2512,2515,2517],{},[592,2513,2514],{},"k",[642,2516,762],{},[740,2518,1654],{},[592,2520,2521],{"mathvariant":1349},"∞",[1763,2523,2524,2530],{},[731,2525,2526,2528],{},[592,2527,2501],{},[592,2529,2514],{},[589,2531,2532,2534],{},[592,2533,2514],{},[642,2535,2536],{"stretchy":2241},"!",[599,2538,2539],{"encoding":601},"e^x = \\sum_{k=0}^{\\infty} \\frac{x^k}{k!}",[573,2541,2543,2588],{"className":2542,"ariaHidden":607},[606],[573,2544,2546,2550,2579,2582,2585],{"className":2545},[611],[573,2547],{"className":2548,"style":2549},[615],"height:0.6644em;",[573,2551,2553,2556],{"className":2552},[620],[573,2554,924],{"className":2555},[620,621],[573,2557,2559],{"className":2558},[788],[573,2560,2562],{"className":2561},[792],[573,2563,2565],{"className":2564},[796],[573,2566,2568],{"className":2567,"style":2549},[800],[573,2569,2570,2573],{"style":804},[573,2571],{"className":2572,"style":809},[808],[573,2574,2576],{"className":2575},[813,814,815,816],[573,2577,2501],{"className":2578},[620,621,816],[573,2580],{"className":2581,"style":884},[667],[573,2583,762],{"className":2584},[888],[573,2586],{"className":2587,"style":884},[667],[573,2589,2591,2595,2666,2669],{"className":2590},[611],[573,2592],{"className":2593,"style":2594},[615],"height:1.3879em;vertical-align:-0.345em;",[573,2596,2598,2604],{"className":2597},[663],[573,2599,2509],{"className":2600,"style":2603},[663,2601,2602],"op-symbol","small-op","position:relative;top:0em;",[573,2605,2607],{"className":2606},[788],[573,2608,2610,2657],{"className":2609},[792,966],[573,2611,2613,2654],{"className":2612},[796],[573,2614,2617,2639],{"className":2615,"style":2616},[800],"height:0.8043em;",[573,2618,2620,2623],{"style":2619},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[573,2621],{"className":2622,"style":809},[808],[573,2624,2626],{"className":2625},[813,814,815,816],[573,2627,2629,2633,2636],{"className":2628},[620,816],[573,2630,2514],{"className":2631,"style":2632},[620,621,816],"margin-right:0.0315em;",[573,2634,762],{"className":2635},[888,816],[573,2637,1654],{"className":2638},[620,816],[573,2640,2642,2645],{"style":2641},"top:-3.2029em;margin-right:0.05em;",[573,2643],{"className":2644,"style":809},[808],[573,2646,2648],{"className":2647},[813,814,815,816],[573,2649,2651],{"className":2650},[620,816],[573,2652,2521],{"className":2653},[620,816],[573,2655,999],{"className":2656},[998],[573,2658,2660],{"className":2659},[796],[573,2661,2664],{"className":2662,"style":2663},[800],"height:0.2997em;",[573,2665],{},[573,2667],{"className":2668,"style":668},[667],[573,2670,2672,2675,2772],{"className":2671},[620],[573,2673],{"className":2674},[957,1836],[573,2676,2678],{"className":2677},[1763],[573,2679,2681,2763],{"className":2680},[792,966],[573,2682,2684,2760],{"className":2683},[796],[573,2685,2688,2706,2714],{"className":2686,"style":2687},[800],"height:1.0429em;",[573,2689,2691,2694],{"style":2690},"top:-2.655em;",[573,2692],{"className":2693,"style":1856},[808],[573,2695,2697],{"className":2696},[813,814,815,816],[573,2698,2700,2703],{"className":2699},[620,816],[573,2701,2514],{"className":2702,"style":2632},[620,621,816],[573,2704,2536],{"className":2705},[1048,816],[573,2707,2708,2711],{"style":1865},[573,2709],{"className":2710,"style":1856},[808],[573,2712],{"className":2713,"style":1873},[1872],[573,2715,2717,2720],{"style":2716},"top:-3.394em;",[573,2718],{"className":2719,"style":1856},[808],[573,2721,2723],{"className":2722},[813,814,815,816],[573,2724,2726],{"className":2725},[620,816],[573,2727,2729,2732],{"className":2728},[620,816],[573,2730,2501],{"className":2731},[620,621,816],[573,2733,2735],{"className":2734},[788],[573,2736,2738],{"className":2737},[792],[573,2739,2741],{"className":2740},[796],[573,2742,2745],{"className":2743,"style":2744},[800],"height:0.927em;",[573,2746,2748,2752],{"style":2747},"top:-2.931em;margin-right:0.0714em;",[573,2749],{"className":2750,"style":2751},[808],"height:2.5em;",[573,2753,2757],{"className":2754},[813,2755,2756,816],"reset-size3","size1",[573,2758,2514],{"className":2759,"style":2632},[620,621,816],[573,2761,999],{"className":2762},[998],[573,2764,2766],{"className":2765},[796],[573,2767,2770],{"className":2768,"style":2769},[800],"height:0.345em;",[573,2771],{},[573,2773],{"className":2774},[1048,1836]," ein. Wegen ",[573,2777,2779,2803],{"className":2778},[576],[573,2780,2782],{"className":2781},[580],[582,2783,2784],{"xmlns":584},[586,2785,2786,2800],{},[589,2787,2788,2794,2796,2798],{},[731,2789,2790,2792],{},[592,2791,594],{},[740,2793,742],{},[642,2795,762],{},[642,2797,1454],{},[740,2799,765],{},[599,2801,2802],{"encoding":601},"i^2 = -1",[573,2804,2806,2850],{"className":2805,"ariaHidden":607},[606],[573,2807,2809,2812,2841,2844,2847],{"className":2808},[611],[573,2810],{"className":2811,"style":801},[615],[573,2813,2815,2818],{"className":2814},[620],[573,2816,594],{"className":2817},[620,621],[573,2819,2821],{"className":2820},[788],[573,2822,2824],{"className":2823},[792],[573,2825,2827],{"className":2826},[796],[573,2828,2830],{"className":2829,"style":801},[800],[573,2831,2832,2835],{"style":804},[573,2833],{"className":2834,"style":809},[808],[573,2836,2838],{"className":2837},[813,814,815,816],[573,2839,742],{"className":2840},[620,816],[573,2842],{"className":2843,"style":884},[667],[573,2845,762],{"className":2846},[888],[573,2848],{"className":2849,"style":884},[667],[573,2851,2853,2856,2859],{"className":2852},[611],[573,2854],{"className":2855,"style":1516},[615],[573,2857,1454],{"className":2858},[620],[573,2860,765],{"className":2861},[620],[573,2863,2865,2890],{"className":2864},[576],[573,2866,2868],{"className":2867},[580],[582,2869,2870],{"xmlns":584},[586,2871,2872,2887],{},[589,2873,2874,2881,2883,2885],{},[731,2875,2876,2878],{},[592,2877,594],{},[740,2879,2880],{},"3",[642,2882,762],{},[642,2884,1454],{},[592,2886,594],{},[599,2888,2889],{"encoding":601},"i^3 = -i",[573,2891,2893,2937],{"className":2892,"ariaHidden":607},[606],[573,2894,2896,2899,2928,2931,2934],{"className":2895},[611],[573,2897],{"className":2898,"style":801},[615],[573,2900,2902,2905],{"className":2901},[620],[573,2903,594],{"className":2904},[620,621],[573,2906,2908],{"className":2907},[788],[573,2909,2911],{"className":2910},[792],[573,2912,2914],{"className":2913},[796],[573,2915,2917],{"className":2916,"style":801},[800],[573,2918,2919,2922],{"style":804},[573,2920],{"className":2921,"style":809},[808],[573,2923,2925],{"className":2924},[813,814,815,816],[573,2926,2880],{"className":2927},[620,816],[573,2929],{"className":2930,"style":884},[667],[573,2932,762],{"className":2933},[888],[573,2935],{"className":2936,"style":884},[667],[573,2938,2940,2944,2947],{"className":2939},[611],[573,2941],{"className":2942,"style":2943},[615],"height:0.7429em;vertical-align:-0.0833em;",[573,2945,1454],{"className":2946},[620],[573,2948,594],{"className":2949},[620,621],[573,2951,2953,2976],{"className":2952},[576],[573,2954,2956],{"className":2955},[580],[582,2957,2958],{"xmlns":584},[586,2959,2960,2973],{},[589,2961,2962,2969,2971],{},[731,2963,2964,2966],{},[592,2965,594],{},[740,2967,2968],{},"4",[642,2970,762],{},[740,2972,765],{},[599,2974,2975],{"encoding":601},"i^4 = 1",[573,2977,2979,3023],{"className":2978,"ariaHidden":607},[606],[573,2980,2982,2985,3014,3017,3020],{"className":2981},[611],[573,2983],{"className":2984,"style":801},[615],[573,2986,2988,2991],{"className":2987},[620],[573,2989,594],{"className":2990},[620,621],[573,2992,2994],{"className":2993},[788],[573,2995,2997],{"className":2996},[792],[573,2998,3000],{"className":2999},[796],[573,3001,3003],{"className":3002,"style":801},[800],[573,3004,3005,3008],{"style":804},[573,3006],{"className":3007,"style":809},[808],[573,3009,3011],{"className":3010},[813,814,815,816],[573,3012,2968],{"className":3013},[620,816],[573,3015],{"className":3016,"style":884},[667],[573,3018,762],{"className":3019},[888],[573,3021],{"className":3022,"style":884},[667],[573,3024,3026,3029],{"className":3025},[611],[573,3027],{"className":3028,"style":898},[615],[573,3030,765],{"className":3031},[620]," werden die Glieder mit geradem ",[573,3034,3036,3049],{"className":3035},[576],[573,3037,3039],{"className":3038},[580],[582,3040,3041],{"xmlns":584},[586,3042,3043,3047],{},[589,3044,3045],{},[592,3046,2514],{},[599,3048,2514],{"encoding":601},[573,3050,3052],{"className":3051,"ariaHidden":607},[606],[573,3053,3055,3059],{"className":3054},[611],[573,3056],{"className":3057,"style":3058},[615],"height:0.6944em;",[573,3060,2514],{"className":3061,"style":2632},[620,621]," reell, die mit ungeradem ",[573,3064,3066,3079],{"className":3065},[576],[573,3067,3069],{"className":3068},[580],[582,3070,3071],{"xmlns":584},[586,3072,3073,3077],{},[589,3074,3075],{},[592,3076,2514],{},[599,3078,2514],{"encoding":601},[573,3080,3082],{"className":3081,"ariaHidden":607},[606],[573,3083,3085,3088],{"className":3084},[611],[573,3086],{"className":3087,"style":3058},[615],[573,3089,2514],{"className":3090,"style":2632},[620,621]," imaginär:",[573,3093,3096],{"className":3094},[3095],"katex-display",[573,3097,3099,3218],{"className":3098},[576],[573,3100,3102],{"className":3101},[580],[582,3103,3105],{"xmlns":584,"display":3104},"block",[586,3106,3107,3215],{},[589,3108,3109,3119,3121,3166,3168,3170],{},[731,3110,3111,3113],{},[592,3112,924],{},[589,3114,3115,3117],{},[592,3116,594],{},[592,3118,597],{},[642,3120,762],{},[589,3122,3123,3125,3127,3129,3143,3145,3159,3161,3164],{},[642,3124,2242],{"fence":607},[740,3126,765],{},[642,3128,1454],{},[1763,3130,3131,3137],{},[731,3132,3133,3135],{},[592,3134,597],{},[740,3136,742],{},[589,3138,3139,3141],{},[740,3140,742],{},[642,3142,2536],{"stretchy":2241},[642,3144,747],{},[1763,3146,3147,3153],{},[731,3148,3149,3151],{},[592,3150,597],{},[740,3152,2968],{},[589,3154,3155,3157],{},[740,3156,2968],{},[642,3158,2536],{"stretchy":2241},[642,3160,1454],{},[642,3162,3163],{},"…",[642,3165,2253],{"fence":607},[642,3167,747],{},[592,3169,594],{},[589,3171,3172,3174,3176,3178,3192,3194,3209,3211,3213],{},[642,3173,2242],{"fence":607},[592,3175,597],{},[642,3177,1454],{},[1763,3179,3180,3186],{},[731,3181,3182,3184],{},[592,3183,597],{},[740,3185,2880],{},[589,3187,3188,3190],{},[740,3189,2880],{},[642,3191,2536],{"stretchy":2241},[642,3193,747],{},[1763,3195,3196,3203],{},[731,3197,3198,3200],{},[592,3199,597],{},[740,3201,3202],{},"5",[589,3204,3205,3207],{},[740,3206,3202],{},[642,3208,2536],{"stretchy":2241},[642,3210,1454],{},[642,3212,3163],{},[642,3214,2253],{"fence":607},[599,3216,3217],{"encoding":601},"e^{i\\varphi}\n= \\left(1 - \\frac{\\varphi^2}{2!} + \\frac{\\varphi^4}{4!} - \\ldots\\right)\n+ i\\left(\\varphi - \\frac{\\varphi^3}{3!} + \\frac{\\varphi^5}{5!} - \\ldots\\right)",[573,3219,3221,3273,3522],{"className":3220,"ariaHidden":607},[606],[573,3222,3224,3228,3264,3267,3270],{"className":3223},[611],[573,3225],{"className":3226,"style":3227},[615],"height:0.8747em;",[573,3229,3231,3234],{"className":3230},[620],[573,3232,924],{"className":3233},[620,621],[573,3235,3237],{"className":3236},[788],[573,3238,3240],{"className":3239},[792],[573,3241,3243],{"className":3242},[796],[573,3244,3246],{"className":3245,"style":3227},[800],[573,3247,3249,3252],{"style":3248},"top:-3.113em;margin-right:0.05em;",[573,3250],{"className":3251,"style":809},[808],[573,3253,3255],{"className":3254},[813,814,815,816],[573,3256,3258,3261],{"className":3257},[620,816],[573,3259,594],{"className":3260},[620,621,816],[573,3262,597],{"className":3263},[620,621,816],[573,3265],{"className":3266,"style":884},[667],[573,3268,762],{"className":3269},[888],[573,3271],{"className":3272,"style":884},[667],[573,3274,3276,3280,3513,3516,3519],{"className":3275},[611],[573,3277],{"className":3278,"style":3279},[615],"height:2.4411em;vertical-align:-0.95em;",[573,3281,3283,3291,3294,3297,3300,3303,3395,3398,3401,3404,3495,3498,3501,3504,3507],{"className":3282},[953],[573,3284,3288],{"className":3285,"style":3287},[957,3286],"delimcenter","top:0em;",[573,3289,2242],{"className":3290},[961,815],[573,3292,765],{"className":3293},[620],[573,3295],{"className":3296,"style":829},[667],[573,3298,1454],{"className":3299},[833],[573,3301],{"className":3302,"style":829},[667],[573,3304,3306,3309,3392],{"className":3305},[620],[573,3307],{"className":3308},[957,1836],[573,3310,3312],{"className":3311},[1763],[573,3313,3315,3384],{"className":3314},[792,966],[573,3316,3318,3381],{"className":3317},[796],[573,3319,3322,3336,3344],{"className":3320,"style":3321},[800],"height:1.4911em;",[573,3323,3324,3327],{"style":1852},[573,3325],{"className":3326,"style":1856},[808],[573,3328,3330,3333],{"className":3329},[620],[573,3331,742],{"className":3332},[620],[573,3334,2536],{"className":3335},[1048],[573,3337,3338,3341],{"style":1865},[573,3339],{"className":3340,"style":1856},[808],[573,3342],{"className":3343,"style":1873},[1872],[573,3345,3346,3349],{"style":1876},[573,3347],{"className":3348,"style":1856},[808],[573,3350,3352],{"className":3351},[620],[573,3353,3355,3358],{"className":3354},[620],[573,3356,597],{"className":3357},[620,621],[573,3359,3361],{"className":3360},[788],[573,3362,3364],{"className":3363},[792],[573,3365,3367],{"className":3366},[796],[573,3368,3370],{"className":3369,"style":801},[800],[573,3371,3372,3375],{"style":804},[573,3373],{"className":3374,"style":809},[808],[573,3376,3378],{"className":3377},[813,814,815,816],[573,3379,742],{"className":3380},[620,816],[573,3382,999],{"className":3383},[998],[573,3385,3387],{"className":3386},[796],[573,3388,3390],{"className":3389,"style":1974},[800],[573,3391],{},[573,3393],{"className":3394},[1048,1836],[573,3396],{"className":3397,"style":829},[667],[573,3399,747],{"className":3400},[833],[573,3402],{"className":3403,"style":829},[667],[573,3405,3407,3410,3492],{"className":3406},[620],[573,3408],{"className":3409},[957,1836],[573,3411,3413],{"className":3412},[1763],[573,3414,3416,3484],{"className":3415},[792,966],[573,3417,3419,3481],{"className":3418},[796],[573,3420,3422,3436,3444],{"className":3421,"style":3321},[800],[573,3423,3424,3427],{"style":1852},[573,3425],{"className":3426,"style":1856},[808],[573,3428,3430,3433],{"className":3429},[620],[573,3431,2968],{"className":3432},[620],[573,3434,2536],{"className":3435},[1048],[573,3437,3438,3441],{"style":1865},[573,3439],{"className":3440,"style":1856},[808],[573,3442],{"className":3443,"style":1873},[1872],[573,3445,3446,3449],{"style":1876},[573,3447],{"className":3448,"style":1856},[808],[573,3450,3452],{"className":3451},[620],[573,3453,3455,3458],{"className":3454},[620],[573,3456,597],{"className":3457},[620,621],[573,3459,3461],{"className":3460},[788],[573,3462,3464],{"className":3463},[792],[573,3465,3467],{"className":3466},[796],[573,3468,3470],{"className":3469,"style":801},[800],[573,3471,3472,3475],{"style":804},[573,3473],{"className":3474,"style":809},[808],[573,3476,3478],{"className":3477},[813,814,815,816],[573,3479,2968],{"className":3480},[620,816],[573,3482,999],{"className":3483},[998],[573,3485,3487],{"className":3486},[796],[573,3488,3490],{"className":3489,"style":1974},[800],[573,3491],{},[573,3493],{"className":3494},[1048,1836],[573,3496],{"className":3497,"style":829},[667],[573,3499,1454],{"className":3500},[833],[573,3502],{"className":3503,"style":829},[667],[573,3505,3163],{"className":3506},[953],[573,3508,3510],{"className":3509,"style":3287},[1048,3286],[573,3511,2253],{"className":3512},[961,815],[573,3514],{"className":3515,"style":829},[667],[573,3517,747],{"className":3518},[833],[573,3520],{"className":3521,"style":829},[667],[573,3523,3525,3528,3531,3534],{"className":3524},[611],[573,3526],{"className":3527,"style":3279},[615],[573,3529,594],{"className":3530},[620,621],[573,3532],{"className":3533,"style":668},[667],[573,3535,3537,3543,3546,3549,3552,3555,3646,3649,3652,3655,3746,3749,3752,3755,3758],{"className":3536},[953],[573,3538,3540],{"className":3539,"style":3287},[957,3286],[573,3541,2242],{"className":3542},[961,815],[573,3544,597],{"className":3545},[620,621],[573,3547],{"className":3548,"style":829},[667],[573,3550,1454],{"className":3551},[833],[573,3553],{"className":3554,"style":829},[667],[573,3556,3558,3561,3643],{"className":3557},[620],[573,3559],{"className":3560},[957,1836],[573,3562,3564],{"className":3563},[1763],[573,3565,3567,3635],{"className":3566},[792,966],[573,3568,3570,3632],{"className":3569},[796],[573,3571,3573,3587,3595],{"className":3572,"style":3321},[800],[573,3574,3575,3578],{"style":1852},[573,3576],{"className":3577,"style":1856},[808],[573,3579,3581,3584],{"className":3580},[620],[573,3582,2880],{"className":3583},[620],[573,3585,2536],{"className":3586},[1048],[573,3588,3589,3592],{"style":1865},[573,3590],{"className":3591,"style":1856},[808],[573,3593],{"className":3594,"style":1873},[1872],[573,3596,3597,3600],{"style":1876},[573,3598],{"className":3599,"style":1856},[808],[573,3601,3603],{"className":3602},[620],[573,3604,3606,3609],{"className":3605},[620],[573,3607,597],{"className":3608},[620,621],[573,3610,3612],{"className":3611},[788],[573,3613,3615],{"className":3614},[792],[573,3616,3618],{"className":3617},[796],[573,3619,3621],{"className":3620,"style":801},[800],[573,3622,3623,3626],{"style":804},[573,3624],{"className":3625,"style":809},[808],[573,3627,3629],{"className":3628},[813,814,815,816],[573,3630,2880],{"className":3631},[620,816],[573,3633,999],{"className":3634},[998],[573,3636,3638],{"className":3637},[796],[573,3639,3641],{"className":3640,"style":1974},[800],[573,3642],{},[573,3644],{"className":3645},[1048,1836],[573,3647],{"className":3648,"style":829},[667],[573,3650,747],{"className":3651},[833],[573,3653],{"className":3654,"style":829},[667],[573,3656,3658,3661,3743],{"className":3657},[620],[573,3659],{"className":3660},[957,1836],[573,3662,3664],{"className":3663},[1763],[573,3665,3667,3735],{"className":3666},[792,966],[573,3668,3670,3732],{"className":3669},[796],[573,3671,3673,3687,3695],{"className":3672,"style":3321},[800],[573,3674,3675,3678],{"style":1852},[573,3676],{"className":3677,"style":1856},[808],[573,3679,3681,3684],{"className":3680},[620],[573,3682,3202],{"className":3683},[620],[573,3685,2536],{"className":3686},[1048],[573,3688,3689,3692],{"style":1865},[573,3690],{"className":3691,"style":1856},[808],[573,3693],{"className":3694,"style":1873},[1872],[573,3696,3697,3700],{"style":1876},[573,3698],{"className":3699,"style":1856},[808],[573,3701,3703],{"className":3702},[620],[573,3704,3706,3709],{"className":3705},[620],[573,3707,597],{"className":3708},[620,621],[573,3710,3712],{"className":3711},[788],[573,3713,3715],{"className":3714},[792],[573,3716,3718],{"className":3717},[796],[573,3719,3721],{"className":3720,"style":801},[800],[573,3722,3723,3726],{"style":804},[573,3724],{"className":3725,"style":809},[808],[573,3727,3729],{"className":3728},[813,814,815,816],[573,3730,3202],{"className":3731},[620,816],[573,3733,999],{"className":3734},[998],[573,3736,3738],{"className":3737},[796],[573,3739,3741],{"className":3740,"style":1974},[800],[573,3742],{},[573,3744],{"className":3745},[1048,1836],[573,3747],{"className":3748,"style":829},[667],[573,3750,1454],{"className":3751},[833],[573,3753],{"className":3754,"style":829},[667],[573,3756,3163],{"className":3757},[953],[573,3759,3761],{"className":3760,"style":3287},[1048,3286],[573,3762,2253],{"className":3763},[961,815],[565,3765,3766,3767,3805,3806,714],{},"Die erste Klammer ist die Reihe von ",[573,3768,3770,3787],{"className":3769},[576],[573,3771,3773],{"className":3772},[580],[582,3774,3775],{"xmlns":584},[586,3776,3777,3785],{},[589,3778,3779,3781,3783],{},[592,3780,640],{},[642,3782,644],{},[592,3784,597],{},[599,3786,649],{"encoding":601},[573,3788,3790],{"className":3789,"ariaHidden":607},[606],[573,3791,3793,3796,3799,3802],{"className":3792},[611],[573,3794],{"className":3795,"style":659},[615],[573,3797,640],{"className":3798},[663],[573,3800],{"className":3801,"style":668},[667],[573,3803,597],{"className":3804},[620,621],", die zweite die von ",[573,3807,3809,3826],{"className":3808},[576],[573,3810,3812],{"className":3811},[580],[582,3813,3814],{"xmlns":584},[586,3815,3816,3824],{},[589,3817,3818,3820,3822],{},[592,3819,687],{},[642,3821,644],{},[592,3823,597],{},[599,3825,694],{"encoding":601},[573,3827,3829],{"className":3828,"ariaHidden":607},[606],[573,3830,3832,3835,3838,3841],{"className":3831},[611],[573,3833],{"className":3834,"style":704},[615],[573,3836,687],{"className":3837},[663],[573,3839],{"className":3840,"style":668},[667],[573,3842,597],{"className":3843},[620,621],[560,3845,3847],{"id":3846},"beispiele","Beispiele",[573,3849,3851],{"className":3850},[3095],[573,3852,3854,3912],{"className":3853},[576],[573,3855,3857],{"className":3856},[580],[582,3858,3859],{"xmlns":584,"display":3104},[586,3860,3861,3909],{},[589,3862,3863,3873,3875,3877,3879,3881,3883,3885,3887,3889,3891,3893,3895,3897,3899,3901,3903,3905,3907],{},[731,3864,3865,3867],{},[592,3866,924],{},[589,3868,3869,3871],{},[592,3870,594],{},[592,3872,1346],{},[642,3874,762],{},[592,3876,640],{},[642,3878,644],{},[592,3880,1346],{},[642,3882,747],{},[592,3884,594],{},[592,3886,687],{},[642,3888,644],{},[592,3890,1346],{},[642,3892,762],{},[642,3894,1454],{},[740,3896,765],{},[642,3898,747],{},[740,3900,1654],{},[592,3902,594],{},[642,3904,762],{},[642,3906,1454],{},[740,3908,765],{},[599,3910,3911],{"encoding":601},"e^{i\\pi} = \\cos\\pi + i\\sin\\pi = -1 + 0i = -1",[573,3913,3915,3962,3987,4018,4039,4060],{"className":3914,"ariaHidden":607},[606],[573,3916,3918,3921,3953,3956,3959],{"className":3917},[611],[573,3919],{"className":3920,"style":3227},[615],[573,3922,3924,3927],{"className":3923},[620],[573,3925,924],{"className":3926},[620,621],[573,3928,3930],{"className":3929},[788],[573,3931,3933],{"className":3932},[792],[573,3934,3936],{"className":3935},[796],[573,3937,3939],{"className":3938,"style":3227},[800],[573,3940,3941,3944],{"style":3248},[573,3942],{"className":3943,"style":809},[808],[573,3945,3947],{"className":3946},[813,814,815,816],[573,3948,3950],{"className":3949},[620,816],[573,3951,1403],{"className":3952,"style":1402},[620,621,816],[573,3954],{"className":3955,"style":884},[667],[573,3957,762],{"className":3958},[888],[573,3960],{"className":3961,"style":884},[667],[573,3963,3965,3969,3972,3975,3978,3981,3984],{"className":3964},[611],[573,3966],{"className":3967,"style":3968},[615],"height:0.6667em;vertical-align:-0.0833em;",[573,3970,640],{"className":3971},[663],[573,3973],{"className":3974,"style":668},[667],[573,3976,1346],{"className":3977,"style":1402},[620,621],[573,3979],{"className":3980,"style":829},[667],[573,3982,747],{"className":3983},[833],[573,3985],{"className":3986,"style":829},[667],[573,3988,3990,3994,3997,4000,4003,4006,4009,4012,4015],{"className":3989},[611],[573,3991],{"className":3992,"style":3993},[615],"height:0.6679em;",[573,3995,594],{"className":3996},[620,621],[573,3998],{"className":3999,"style":668},[667],[573,4001,687],{"className":4002},[663],[573,4004],{"className":4005,"style":668},[667],[573,4007,1346],{"className":4008,"style":1402},[620,621],[573,4010],{"className":4011,"style":884},[667],[573,4013,762],{"className":4014},[888],[573,4016],{"className":4017,"style":884},[667],[573,4019,4021,4024,4027,4030,4033,4036],{"className":4020},[611],[573,4022],{"className":4023,"style":1516},[615],[573,4025,1454],{"className":4026},[620],[573,4028,765],{"className":4029},[620],[573,4031],{"className":4032,"style":829},[667],[573,4034,747],{"className":4035},[833],[573,4037],{"className":4038,"style":829},[667],[573,4040,4042,4045,4048,4051,4054,4057],{"className":4041},[611],[573,4043],{"className":4044,"style":1423},[615],[573,4046,1654],{"className":4047},[620],[573,4049,594],{"className":4050},[620,621],[573,4052],{"className":4053,"style":884},[667],[573,4055,762],{"className":4056},[888],[573,4058],{"className":4059,"style":884},[667],[573,4061,4063,4066,4069],{"className":4062},[611],[573,4064],{"className":4065,"style":1516},[615],[573,4067,1454],{"className":4068},[620],[573,4070,765],{"className":4071},[620],[573,4073,4075],{"className":4074},[3095],[573,4076,4078,4153],{"className":4077},[576],[573,4079,4081],{"className":4080},[580],[582,4082,4083],{"xmlns":584,"display":3104},[586,4084,4085,4150],{},[589,4086,4087,4101,4103,4105,4107,4113,4115,4117,4119,4121,4127,4129,4135,4137,4146,4148],{},[731,4088,4089,4091],{},[592,4090,924],{},[589,4092,4093,4095,4097,4099],{},[592,4094,594],{},[592,4096,1346],{},[592,4098,1350],{"mathvariant":1349},[740,4100,2880],{},[642,4102,762],{},[592,4104,640],{},[642,4106,644],{},[1763,4108,4109,4111],{},[592,4110,1346],{},[740,4112,2880],{},[642,4114,747],{},[592,4116,594],{},[592,4118,687],{},[642,4120,644],{},[1763,4122,4123,4125],{},[592,4124,1346],{},[740,4126,2880],{},[642,4128,762],{},[1763,4130,4131,4133],{},[740,4132,765],{},[740,4134,742],{},[642,4136,747],{},[1763,4138,4139,4144],{},[4140,4141,4142],"msqrt",{},[740,4143,2880],{},[740,4145,742],{},[1227,4147,1229],{},[592,4149,594],{},[599,4151,4152],{"encoding":601},"e^{i\\pi\u002F3} = \\cos\\frac{\\pi}{3} + i\\sin\\frac{\\pi}{3} = \\frac{1}{2} + 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0 400000 1080","xMinYMin slice",[992,4561],{"d":4562},"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",[573,4564,999],{"className":4565},[998],[573,4567,4569],{"className":4568},[796],[573,4570,4573],{"className":4571,"style":4572},[800],"height:0.1328em;",[573,4574],{},[573,4576,999],{"className":4577},[998],[573,4579,4581],{"className":4580},[796],[573,4582,4584],{"className":4583,"style":1974},[800],[573,4585],{},[573,4587],{"className":4588},[1048,1836],[573,4590],{"className":4591,"style":668},[667],[573,4593,594],{"className":4594},[620,621],[573,4596,4598],{"className":4597},[3095],[573,4599,4601,4657],{"className":4600},[576],[573,4602,4604],{"className":4603},[580],[582,4605,4606],{"xmlns":584,"display":3104},[586,4607,4608,4654],{},[589,4609,4610,4624,4626,4628,4630,4636,4638,4640,4642,4644,4650,4652],{},[731,4611,4612,4614],{},[592,4613,924],{},[589,4615,4616,4618,4620,4622],{},[592,4617,594],{},[592,4619,1346],{},[592,4621,1350],{"mathvariant":1349},[740,4623,742],{},[642,4625,762],{},[592,4627,640],{},[642,4629,644],{},[1763,4631,4632,4634],{},[592,4633,1346],{},[740,4635,742],{},[642,4637,747],{},[592,4639,594],{},[592,4641,687],{},[642,4643,644],{},[1763,4645,4646,4648],{},[592,4647,1346],{},[740,4649,742],{},[642,4651,762],{},[592,4653,594],{},[599,4655,4656],{"encoding":601},"e^{i\\pi\u002F2} = \\cos\\frac{\\pi}{2} + i\\sin\\frac{\\pi}{2} = i",[573,4658,4660,4710,4793,4882],{"className":4659,"ariaHidden":607},[606],[573,4661,4663,4666,4701,4704,4707],{"className":4662},[611],[573,4664],{"className":4665,"style":4162},[615],[573,4667,4669,4672],{"className":4668},[620],[573,4670,924],{"className":4671},[620,621],[573,4673,4675],{"className":4674},[788],[573,4676,4678],{"className":4677},[792],[573,4679,4681],{"className":4680},[796],[573,4682,4684],{"className":4683,"style":4162},[800],[573,4685,4686,4689],{"style":3248},[573,4687],{"className":4688,"style":809},[808],[573,4690,4692],{"className":4691},[813,814,815,816],[573,4693,4695,4698],{"className":4694},[620,816],[573,4696,1403],{"className":4697,"style":1402},[620,621,816],[573,4699,1407],{"className":4700},[620,816],[573,4702],{"className":4703,"style":884},[667],[573,4705,762],{"className":4706},[888],[573,4708],{"className":4709,"style":884},[667],[573,4711,4713,4716,4719,4722,4784,4787,4790],{"className":4712},[611],[573,4714],{"className":4715,"style":4214},[615],[573,4717,640],{"className":4718},[663],[573,4720],{"className":4721,"style":668},[667],[573,4723,4725,4728,4781],{"className":4724},[620],[573,4726],{"className":4727},[957,1836],[573,4729,4731],{"className":4730},[1763],[573,4732,4734,4773],{"className":4733},[792,966],[573,4735,4737,4770],{"className":4736},[796],[573,4738,4740,4751,4759],{"className":4739,"style":4239},[800],[573,4741,4742,4745],{"style":1852},[573,4743],{"className":4744,"style":1856},[808],[573,4746,4748],{"className":4747},[620],[573,4749,742],{"className":4750},[620],[573,4752,4753,4756],{"style":1865},[573,4754],{"className":4755,"style":1856},[808],[573,4757],{"className":4758,"style":1873},[1872],[573,4760,4761,4764],{"style":1876},[573,4762],{"className":4763,"style":1856},[808],[573,4765,4767],{"className":4766},[620],[573,4768,1346],{"className":4769,"style":1402},[620,621],[573,4771,999],{"className":4772},[998],[573,4774,4776],{"className":4775},[796],[573,4777,4779],{"className":4778,"style":1974},[800],[573,4780],{},[573,4782],{"className":4783},[1048,1836],[573,4785],{"className":4786,"style":829},[667],[573,4788,747],{"className":4789},[833],[573,4791],{"className":4792,"style":829},[667],[573,4794,4796,4799,4802,4805,4808,4811,4873,4876,4879],{"className":4795},[611],[573,4797],{"className":4798,"style":4214},[615],[573,4800,594],{"className":4801},[620,621],[573,4803],{"className":4804,"style":668},[667],[573,4806,687],{"className":4807},[663],[573,4809],{"className":4810,"style":668},[667],[573,4812,4814,4817,4870],{"className":4813},[620],[573,4815],{"className":4816},[957,1836],[573,4818,4820],{"className":4819},[1763],[573,4821,4823,4862],{"className":4822},[792,966],[573,4824,4826,4859],{"className":4825},[796],[573,4827,4829,4840,4848],{"className":4828,"style":4239},[800],[573,4830,4831,4834],{"style":1852},[573,4832],{"className":4833,"style":1856},[808],[573,4835,4837],{"className":4836},[620],[573,4838,742],{"className":4839},[620],[573,4841,4842,4845],{"style":1865},[573,4843],{"className":4844,"style":1856},[808],[573,4846],{"className":4847,"style":1873},[1872],[573,4849,4850,4853],{"style":1876},[573,4851],{"className":4852,"style":1856},[808],[573,4854,4856],{"className":4855},[620],[573,4857,1346],{"className":4858,"style":1402},[620,621],[573,4860,999],{"className":4861},[998],[573,4863,4865],{"className":4864},[796],[573,4866,4868],{"className":4867,"style":1974},[800],[573,4869],{},[573,4871],{"className":4872},[1048,1836],[573,4874],{"className":4875,"style":884},[667],[573,4877,762],{"className":4878},[888],[573,4880],{"className":4881,"style":884},[667],[573,4883,4885,4888],{"className":4884},[611],[573,4886],{"className":4887,"style":1423},[615],[573,4889,594],{"className":4890},[620,621],[560,4892,4894],{"id":4893},"typische-fehler","Typische Fehler",[1318,4896,4897,5132,5212],{},[1321,4898,4899,4900,4975,4976,5010,5011,5039,5040,714],{},"Den Winkel in Grad einsetzen: ",[573,4901,4903,4928],{"className":4902},[576],[573,4904,4906],{"className":4905},[580],[582,4907,4908],{"xmlns":584},[586,4909,4910,4925],{},[589,4911,4912],{},[731,4913,4914,4916],{},[592,4915,924],{},[589,4917,4918,4920,4922],{},[592,4919,594],{},[642,4921,2268],{},[740,4923,4924],{},"180",[599,4926,4927],{"encoding":601},"e^{i \\cdot 180}",[573,4929,4931],{"className":4930,"ariaHidden":607},[606],[573,4932,4934,4937],{"className":4933},[611],[573,4935],{"className":4936,"style":1027},[615],[573,4938,4940,4943],{"className":4939},[620],[573,4941,924],{"className":4942},[620,621],[573,4944,4946],{"className":4945},[788],[573,4947,4949],{"className":4948},[792],[573,4950,4952],{"className":4951},[796],[573,4953,4955],{"className":4954,"style":1027},[800],[573,4956,4957,4960],{"style":804},[573,4958],{"className":4959,"style":809},[808],[573,4961,4963],{"className":4962},[813,814,815,816],[573,4964,4966,4969,4972],{"className":4965},[620,816],[573,4967,594],{"className":4968},[620,621,816],[573,4970,2268],{"className":4971},[833,816],[573,4973,4924],{"className":4974},[620,816]," ist nicht ",[573,4977,4979,4995],{"className":4978},[576],[573,4980,4982],{"className":4981},[580],[582,4983,4984],{"xmlns":584},[586,4985,4986,4992],{},[589,4987,4988,4990],{},[642,4989,1454],{},[740,4991,765],{},[599,4993,4994],{"encoding":601},"-1",[573,4996,4998],{"className":4997,"ariaHidden":607},[606],[573,4999,5001,5004,5007],{"className":5000},[611],[573,5002],{"className":5003,"style":1516},[615],[573,5005,1454],{"className":5006},[620],[573,5008,765],{"className":5009},[620],", denn ",[573,5012,5014,5027],{"className":5013},[576],[573,5015,5017],{"className":5016},[580],[582,5018,5019],{"xmlns":584},[586,5020,5021,5025],{},[589,5022,5023],{},[740,5024,4924],{},[599,5026,4924],{"encoding":601},[573,5028,5030],{"className":5029,"ariaHidden":607},[606],[573,5031,5033,5036],{"className":5032},[611],[573,5034],{"className":5035,"style":898},[615],[573,5037,4924],{"className":5038},[620]," wird als Bogenmaß gelesen. Richtig ist ",[573,5041,5043,5070],{"className":5042},[576],[573,5044,5046],{"className":5045},[580],[582,5047,5048],{"xmlns":584},[586,5049,5050,5068],{},[589,5051,5052,5062,5064,5066],{},[731,5053,5054,5056],{},[592,5055,924],{},[589,5057,5058,5060],{},[592,5059,594],{},[592,5061,1346],{},[642,5063,762],{},[642,5065,1454],{},[740,5067,765],{},[599,5069,1459],{"encoding":601},[573,5071,5073,5120],{"className":5072,"ariaHidden":607},[606],[573,5074,5076,5079,5111,5114,5117],{"className":5075},[611],[573,5077],{"className":5078,"style":1027},[615],[573,5080,5082,5085],{"className":5081},[620],[573,5083,924],{"className":5084},[620,621],[573,5086,5088],{"className":5087},[788],[573,5089,5091],{"className":5090},[792],[573,5092,5094],{"className":5093},[796],[573,5095,5097],{"className":5096,"style":1027},[800],[573,5098,5099,5102],{"style":804},[573,5100],{"className":5101,"style":809},[808],[573,5103,5105],{"className":5104},[813,814,815,816],[573,5106,5108],{"className":5107},[620,816],[573,5109,1403],{"className":5110,"style":1402},[620,621,816],[573,5112],{"className":5113,"style":884},[667],[573,5115,762],{"className":5116},[888],[573,5118],{"className":5119,"style":884},[667],[573,5121,5123,5126,5129],{"className":5122},[611],[573,5124],{"className":5125,"style":1516},[615],[573,5127,1454],{"className":5128},[620],[573,5130,765],{"className":5131},[620],[1321,5133,5134,5135,5173,5174,714],{},"Sinus und Kosinus vertauschen. Der Realteil ist ",[573,5136,5138,5155],{"className":5137},[576],[573,5139,5141],{"className":5140},[580],[582,5142,5143],{"xmlns":584},[586,5144,5145,5153],{},[589,5146,5147,5149,5151],{},[592,5148,640],{},[642,5150,644],{},[592,5152,597],{},[599,5154,649],{"encoding":601},[573,5156,5158],{"className":5157,"ariaHidden":607},[606],[573,5159,5161,5164,5167,5170],{"className":5160},[611],[573,5162],{"className":5163,"style":659},[615],[573,5165,640],{"className":5166},[663],[573,5168],{"className":5169,"style":668},[667],[573,5171,597],{"className":5172},[620,621],", der Imaginärteil ",[573,5175,5177,5194],{"className":5176},[576],[573,5178,5180],{"className":5179},[580],[582,5181,5182],{"xmlns":584},[586,5183,5184,5192],{},[589,5185,5186,5188,5190],{},[592,5187,687],{},[642,5189,644],{},[592,5191,597],{},[599,5193,694],{"encoding":601},[573,5195,5197],{"className":5196,"ariaHidden":607},[606],[573,5198,5200,5203,5206,5209],{"className":5199},[611],[573,5201],{"className":5202,"style":704},[615],[573,5204,687],{"className":5205},[663],[573,5207],{"className":5208,"style":668},[667],[573,5210,597],{"className":5211},[620,621],[1321,5213,5214,5421,5422,5450,5451,714],{},[573,5215,5217,5253],{"className":5216},[576],[573,5218,5220],{"className":5219},[580],[582,5221,5222],{"xmlns":584},[586,5223,5224,5250],{},[589,5225,5226,5242,5244],{},[589,5227,5228,5230,5240],{},[642,5229,919],{"fence":607},[731,5231,5232,5234],{},[592,5233,924],{},[589,5235,5236,5238],{},[592,5237,594],{},[592,5239,597],{},[642,5241,919],{"fence":607},[642,5243,762],{},[731,5245,5246,5248],{},[592,5247,924],{},[592,5249,597],{},[599,5251,5252],{"encoding":601},"\\left|e^{i\\varphi}\\right| = e^{\\varphi}",[573,5254,5256,5383],{"className":5255,"ariaHidden":607},[606],[573,5257,5259,5262,5374,5377,5380],{"className":5258},[611],[573,5260],{"className":5261,"style":949},[615],[573,5263,5265,5302,5337],{"className":5264},[953],[573,5266,5268],{"className":5267},[957],[573,5269,5271],{"className":5270},[961,962],[573,5272,5274,5294],{"className":5273},[792,966],[573,5275,5277,5291],{"className":5276},[796],[573,5278,5280],{"className":5279,"style":973},[800],[573,5281,5282,5285],{"style":976},[573,5283],{"className":5284,"style":980},[808],[573,5286,5287],{"style":983},[985,5288,5289],{"xmlns":987,"width":988,"height":989,"viewBox":990},[992,5290],{"d":994},[573,5292,999],{"className":5293},[998],[573,5295,5297],{"className":5296},[796],[573,5298,5300],{"className":5299,"style":1006},[800],[573,5301],{},[573,5303,5305,5308],{"className":5304},[620],[573,5306,924],{"className":5307},[620,621],[573,5309,5311],{"className":5310},[788],[573,5312,5314],{"className":5313},[792],[573,5315,5317],{"className":5316},[796],[573,5318,5320],{"className":5319,"style":1027},[800],[573,5321,5322,5325],{"style":804},[573,5323],{"className":5324,"style":809},[808],[573,5326,5328],{"className":5327},[813,814,815,816],[573,5329,5331,5334],{"className":5330},[620,816],[573,5332,594],{"className":5333},[620,621,816],[573,5335,597],{"className":5336},[620,621,816],[573,5338,5340],{"className":5339},[1048],[573,5341,5343],{"className":5342},[961,962],[573,5344,5346,5366],{"className":5345},[792,966],[573,5347,5349,5363],{"className":5348},[796],[573,5350,5352],{"className":5351,"style":973},[800],[573,5353,5354,5357],{"style":976},[573,5355],{"className":5356,"style":980},[808],[573,5358,5359],{"style":983},[985,5360,5361],{"xmlns":987,"width":988,"height":989,"viewBox":990},[992,5362],{"d":994},[573,5364,999],{"className":5365},[998],[573,5367,5369],{"className":5368},[796],[573,5370,5372],{"className":5371,"style":1006},[800],[573,5373],{},[573,5375],{"className":5376,"style":884},[667],[573,5378,762],{"className":5379},[888],[573,5381],{"className":5382,"style":884},[667],[573,5384,5386,5389],{"className":5385},[611],[573,5387],{"className":5388,"style":2549},[615],[573,5390,5392,5395],{"className":5391},[620],[573,5393,924],{"className":5394},[620,621],[573,5396,5398],{"className":5397},[788],[573,5399,5401],{"className":5400},[792],[573,5402,5404],{"className":5403},[796],[573,5405,5407],{"className":5406,"style":2549},[800],[573,5408,5409,5412],{"style":804},[573,5410],{"className":5411,"style":809},[808],[573,5413,5415],{"className":5414},[813,814,815,816],[573,5416,5418],{"className":5417},[620,816],[573,5419,597],{"className":5420},[620,621,816]," annehmen. Für reelles ",[573,5423,5425,5438],{"className":5424},[576],[573,5426,5428],{"className":5427},[580],[582,5429,5430],{"xmlns":584},[586,5431,5432,5436],{},[589,5433,5434],{},[592,5435,597],{},[599,5437,1192],{"encoding":601},[573,5439,5441],{"className":5440,"ariaHidden":607},[606],[573,5442,5444,5447],{"className":5443},[611],[573,5445],{"className":5446,"style":659},[615],[573,5448,597],{"className":5449},[620,621]," ist der Betrag immer ",[573,5452,5454,5467],{"className":5453},[576],[573,5455,5457],{"className":5456},[580],[582,5458,5459],{"xmlns":584},[586,5460,5461,5465],{},[589,5462,5463],{},[740,5464,765],{},[599,5466,765],{"encoding":601},[573,5468,5470],{"className":5469,"ariaHidden":607},[606],[573,5471,5473,5476],{"className":5472},[611],[573,5474],{"className":5475,"style":898},[615],[573,5477,765],{"className":5478},[620],{"title":5480,"searchDepth":5481,"depth":5481,"links":5482},"",2,[5483,5484,5485,5486],{"id":562,"depth":5481,"text":563},{"id":2444,"depth":5481,"text":2445},{"id":3846,"depth":5481,"text":3847},{"id":4893,"depth":5481,"text":4894},[5488,5491,5494,5496,5498,5500,5503,5505,5507,5509,5511,5513,5515,5518],{"lhs":5489,"rhs":5490},"re(exp(i*x))","cos(x)",{"lhs":5492,"rhs":5493},"im(exp(i*x))","sin(x)",{"lhs":5495,"rhs":765},"abs(exp(i*x))",{"lhs":5497,"rhs":5490},"re((exp(i*x) + exp(-i*x)) \u002F 2)",{"lhs":5499,"rhs":5493},"re((exp(i*x) - exp(-i*x)) \u002F (2*i))",{"lhs":5501,"rhs":5502},"re(exp(i*x) * exp(i*y))","cos(x + y)",{"lhs":5504,"rhs":4994},"re(exp(i*pi))",{"lhs":5506,"rhs":1654},"im(exp(i*pi))",{"lhs":5508,"rhs":1654},"re(exp(i*pi) + 1)",{"lhs":5510,"rhs":1654},"re(exp(i*pi\u002F2))",{"lhs":5512,"rhs":765},"im(exp(i*pi\u002F2))",{"lhs":5514,"rhs":765},"re(exp(2*pi*i))",{"lhs":5516,"rhs":5517},"re(exp(i*pi\u002F3))","0.5",{"lhs":5519,"rhs":5520},"im(exp(i*pi\u002F3))","sqrt(3)\u002F2","\\varphi \\in \\mathbb{R}","\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>\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:0.6889em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathbb\">R\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",null,"md","euler-formula","e^{i\\varphi} = \\cos\\varphi + i\\sin\\varphi","\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>\u003Cmsup>\u003Cmi>e\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>φ\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsup>\u003Cmo>=\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>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">e^{i\\varphi} = \\cos\\varphi + i\\sin\\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:0.8747em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">e\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8747em;\">\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 mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">φ\u003C\u002Fspan>\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:0.7778em;vertical-align:-0.1944em;\">\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:0.8623em;vertical-align:-0.1944em;\">\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>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","university",{},true,"\u002Fde\u002Fformulas\u002Fmath\u002Falgebra\u002Feulersche-formel",[],[5534,5535,5536,5537],"polar-form","de-moivre","natural-exponential-function","cosine-double-angle",{"title":548,"description":5480},"de\u002Fformulas\u002Fmath\u002Falgebra\u002Feulersche-formel","Die Eulersche Formel verbindet die e-Funktion mit Kosinus und Sinus und beschreibt Punkte auf dem Einheitskreis.",[5542,5543,5544,5545,5546],"komplexe zahlen","e-funktion","sinus","kosinus","einheitskreis",[5548,5551,5554],{"symbol":924,"meaning":5549,"symbolHtml":5550},"Eulersche Zahl, e ≈ 2,71828","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>e\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">e\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\">e\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":594,"meaning":5552,"symbolHtml":5553},"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>",{"symbol":1192,"meaning":5555,"symbolHtml":5556},"Winkel 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>","RKKl9xZaJ1WzLTvF_-W6Di2DTsMtzdL7Kb116vz34M8",[5559,5562,5565,5568],{"stem":5560,"title":5561,"category":5},"de\u002Fformulas\u002Fmath\u002Falgebra\u002Fformel-von-moivre","Formel von Moivre",{"stem":5563,"title":5564,"category":5},"de\u002Fformulas\u002Fmath\u002Falgebra\u002Fpolarform-komplexe-zahl","Polarform einer komplexen Zahl",{"stem":5566,"title":5567,"category":208},"de\u002Fformulas\u002Fmath\u002Ffunctions\u002Fnatuerliche-exponentialfunktion","Natürliche Exponentialfunktion",{"stem":5569,"title":5570,"category":518},"de\u002Fformulas\u002Fmath\u002Ftrigonometry\u002Fdoppelwinkelformel-kosinus","Doppelwinkelformel für den Kosinus",1793824271944]