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",[932,1127,1118],{"class":946},[932,1129,1130],{"class":984}," 1",[932,1132,956],{"class":946},[932,1134,962],{"class":977},[932,1136,1137],{"class":946},");\n",[932,1139,1140],{"class":934,"line":991},[932,1141,1061],{"class":946},[918,1143,1144],{},"倘若除此之外，我们还需要实现 $\\displaystyle\\sum_^bi^2$ 和 $\\displaystyle\\sum_^b\\frac1{i^2}$。当然可以类似地写出：",[922,1146,1148],{"className":924,"code":1147,"language":926,"meta":927,"style":927},"int sum_sq(int a, int b) {\n    return a > b ? 0 : a * a + sum_sq(a + 1, b);\n}\ndouble sum_sq_rv(int a, int b) {\n    return a > b \n        ? 0.0 \n        : 1.0 \u002F (a * a) + sum_sq_rv(a + 1, b);\n}\n",[929,1149,1150,1173,1214,1218,1242,1253,1264,1304],{"__ignoreMap":927},[932,1151,1152,1154,1157,1159,1161,1163,1165,1167,1169,1171],{"class":934,"line":935},[932,1153,939],{"class":938},[932,1155,1156],{"class":942}," sum_sq",[932,1158,947],{"class":946},[932,1160,939],{"class":938},[932,1162,953],{"class":952},[932,1164,956],{"class":946},[932,1166,959],{"class":938},[932,1168,962],{"class":952},[932,1170,965],{"class":946},[932,1172,968],{"class":946},[932,1174,1175,1177,1179,1181,1183,1185,1187,1189,1191,1194,1196,1198,1200,1202,1204,1206,1208,1210,1212],{"class":934,"line":971},[932,1176,1051],{"class":994},[932,1178,1098],{"class":977},[932,1180,1101],{"class":946},[932,1182,1104],{"class":977},[932,1184,1107],{"class":946},[932,1186,1110],{"class":984},[932,1188,1113],{"class":946},[932,1190,1098],{"class":977},[932,1192,1193],{"class":946},"*",[932,1195,1098],{"class":977},[932,1197,1118],{"class":946},[932,1199,1156],{"class":942},[932,1201,947],{"class":946},[932,1203,1125],{"class":977},[932,1205,1118],{"class":946},[932,1207,1130],{"class":984},[932,1209,956],{"class":946},[932,1211,962],{"class":977},[932,1213,1137],{"class":946},[932,1215,1216],{"class":934,"line":991},[932,1217,1061],{"class":946},[932,1219,1220,1223,1226,1228,1230,1232,1234,1236,1238,1240],{"class":934,"line":1027},[932,1221,1222],{"class":938},"double",[932,1224,1225],{"class":942}," sum_sq_rv",[932,1227,947],{"class":946},[932,1229,939],{"class":938},[932,1231,953],{"class":952},[932,1233,956],{"class":946},[932,1235,959],{"class":938},[932,1237,962],{"class":952},[932,1239,965],{"class":946},[932,1241,968],{"class":946},[932,1243,1244,1246,1248,1250],{"class":934,"line":1042},[932,1245,1051],{"class":994},[932,1247,1098],{"class":977},[932,1249,1101],{"class":946},[932,1251,1252],{"class":977}," b \n",[932,1254,1255,1258,1261],{"class":934,"line":1048},[932,1256,1257],{"class":946},"        ?",[932,1259,1260],{"class":984}," 0.0",[932,1262,1263],{"class":977}," \n",[932,1265,1266,1269,1272,1275,1277,1279,1281,1283,1285,1288,1290,1292,1294,1296,1298,1300,1302],{"class":934,"line":1058},[932,1267,1268],{"class":946},"        :",[932,1270,1271],{"class":984}," 1.0",[932,1273,1274],{"class":946}," \u002F",[932,1276,998],{"class":946},[932,1278,1125],{"class":977},[932,1280,1193],{"class":946},[932,1282,953],{"class":977},[932,1284,965],{"class":946},[932,1286,1287],{"class":946}," +",[932,1289,1225],{"class":942},[932,1291,947],{"class":946},[932,1293,1125],{"class":977},[932,1295,1118],{"class":946},[932,1297,1130],{"class":984},[932,1299,956],{"class":946},[932,1301,962],{"class":977},[932,1303,1137],{"class":946},[932,1305,1307],{"class":934,"line":1306},8,[932,1308,1061],{"class":946},[918,1310,1311],{},"观察这三个函数的实现方法，你会发现它们是类似的。基本上，都长成这个模样：",[922,1313,1315],{"className":924,"code":1314,"language":926,"meta":927,"style":927},"double sth(int a, int b) {\n    return a > b ? 0.0 : 啥啥啥 + sth(a + 1, b);\n}\n",[929,1316,1317,1340,1377],{"__ignoreMap":927},[932,1318,1319,1321,1324,1326,1328,1330,1332,1334,1336,1338],{"class":934,"line":935},[932,1320,1222],{"class":938},[932,1322,1323],{"class":942}," sth",[932,1325,947],{"class":946},[932,1327,939],{"class":938},[932,1329,953],{"class":952},[932,1331,956],{"class":946},[932,1333,959],{"class":938},[932,1335,962],{"class":952},[932,1337,965],{"class":946},[932,1339,968],{"class":946},[932,1341,1342,1344,1346,1348,1350,1352,1354,1356,1359,1361,1363,1365,1367,1369,1371,1373,1375],{"class":934,"line":971},[932,1343,1051],{"class":994},[932,1345,1098],{"class":977},[932,1347,1101],{"class":946},[932,1349,1104],{"class":977},[932,1351,1107],{"class":946},[932,1353,1260],{"class":984},[932,1355,1113],{"class":946},[932,1357,1358],{"class":977}," 啥啥啥 ",[932,1360,1118],{"class":946},[932,1362,1323],{"class":942},[932,1364,947],{"class":946},[932,1366,1125],{"class":977},[932,1368,1118],{"class":946},[932,1370,1130],{"class":984},[932,1372,956],{"class":946},[932,1374,962],{"class":977},[932,1376,1137],{"class":946},[932,1378,1379],{"class":934,"line":991},[932,1380,1061],{"class":946},[918,1382,1383,1384,1387,1388,1391,1392,1395],{},"而这里的“啥啥啥”则是一个关于 $a$ 的函数。在 ",[929,1385,1386],{},"sum"," 里，“啥啥啥”是恒等函数 $f_1(a)=a$，在 ",[929,1389,1390],{},"sum_sq"," 里是 $f_2(a)=a^2$，在 ",[929,1393,1394],{},"sum_eq_rv"," 里，则是 $f_3(a)=\\dfrac1{a^2}$。其余的部分，则完全相同。对于这种大量重复代码的情形，我们希望用一个通用的函数来处理；而不同的部分则作为函数的参数。换句话说，我们希望“啥啥啥”是一个参数：",[922,1397,1399],{"className":924,"code":1398,"language":926,"meta":927,"style":927},"double sum(int a, int b, double term(int)) {\n    return a > b ? 0.0 : term(a) + sum(a + 1, b, term);\n}\n",[929,1400,1401,1436,1483],{"__ignoreMap":927},[932,1402,1403,1405,1407,1409,1411,1413,1415,1417,1419,1421,1424,1427,1429,1431,1434],{"class":934,"line":935},[932,1404,1222],{"class":938},[932,1406,943],{"class":942},[932,1408,947],{"class":946},[932,1410,939],{"class":938},[932,1412,953],{"class":952},[932,1414,956],{"class":946},[932,1416,959],{"class":938},[932,1418,962],{"class":952},[932,1420,956],{"class":946},[932,1422,1423],{"class":938}," double",[932,1425,1426],{"class":942}," term",[932,1428,947],{"class":946},[932,1430,939],{"class":938},[932,1432,1433],{"class":946},"))",[932,1435,968],{"class":946},[932,1437,1438,1440,1442,1444,1446,1448,1450,1452,1454,1456,1459,1461,1463,1465,1467,1469,1471,1473,1475,1477,1479,1481],{"class":934,"line":971},[932,1439,1051],{"class":994},[932,1441,1098],{"class":977},[932,1443,1101],{"class":946},[932,1445,1104],{"class":977},[932,1447,1107],{"class":946},[932,1449,1260],{"class":984},[932,1451,1113],{"class":946},[932,1453,1426],{"class":942},[932,1455,947],{"class":946},[932,1457,1458],{"class":977},"a",[932,1460,965],{"class":946},[932,1462,1287],{"class":946},[932,1464,943],{"class":942},[932,1466,947],{"class":946},[932,1468,1125],{"class":977},[932,1470,1118],{"class":946},[932,1472,1130],{"class":984},[932,1474,956],{"class":946},[932,1476,962],{"class":977},[932,1478,956],{"class":946},[932,1480,1426],{"class":977},[932,1482,1137],{"class":946},[932,1484,1485],{"class":934,"line":991},[932,1486,1061],{"class":946},[918,1488,1489,1490,1493,1494,1496,1497,1499,1500,1502],{},"这里，我就使用了 ",[929,1491,1492],{},"term"," 参数。根据刚才的分析，它是一个关于 ",[929,1495,1458],{}," 的“函数”（传入一个 ",[929,1498,939],{},"，返回 ",[929,1501,1222],{}," 类型）。使用的时候，需要这样做：",[922,1504,1506],{"className":924,"code":1505,"language":926,"meta":927,"style":927},"#include \u003Ciostream>\n\n\u002F\u002F f1(a) = a\ndouble identity(int a) {\n    return a;\n}\n\u002F\u002F f2(a) = a^2\ndouble square(int a) {\n    return a * a;\n}\n\u002F\u002F f3(a) = 1 \u002F (a^2)\ndouble square_rev(int a) {\n    return 1.0 \u002F (a * a);\n}\n\ndouble sum(int a, int b, double term(int)) {\n    return a > b ? 0.0 : term(a) + sum(a + 1, b, term);\n}\n\nint main() {\n    \u002F\u002F 1 + 2 + 3 + 4 + 5 = 15\n    std::cout \u003C\u003C sum(1, 5, identity) \u003C\u003C std::endl;\n\n    \u002F\u002F 3^2 + 4^2 + 5^2 = 50\n    std::cout \u003C\u003C sum(3, 5, square) \u003C\u003C std::endl;\n\n    \u002F\u002F ~ 1.63 ~= π^2\u002F6\n    std::cout \u003C\u003C sum(1, 100, square_rev) \u003C\u003C std::endl;\n}\n",[929,1507,1508,1523,1529,1535,1552,1560,1564,1569,1586,1599,1604,1610,1628,1647,1652,1657,1690,1737,1742,1747,1760,1766,1813,1818,1824,1862,1867,1873,1911],{"__ignoreMap":927},[932,1509,1510,1513,1516,1520],{"class":934,"line":935},[932,1511,1512],{"class":994},"#include",[932,1514,1515],{"class":946}," \u003C",[932,1517,1519],{"class":1518},"sfazB","iostream",[932,1521,1522],{"class":946},">\n",[932,1524,1525],{"class":934,"line":971},[932,1526,1528],{"emptyLinePlaceholder":1527},true,"\n",[932,1530,1531],{"class":934,"line":991},[932,1532,1534],{"class":1533},"sHwdD","\u002F\u002F f1(a) = a\n",[932,1536,1537,1539,1542,1544,1546,1548,1550],{"class":934,"line":1027},[932,1538,1222],{"class":938},[932,1540,1541],{"class":942}," identity",[932,1543,947],{"class":946},[932,1545,939],{"class":938},[932,1547,953],{"class":952},[932,1549,965],{"class":946},[932,1551,968],{"class":946},[932,1553,1554,1556,1558],{"class":934,"line":1042},[932,1555,1051],{"class":994},[932,1557,953],{"class":977},[932,1559,1039],{"class":946},[932,1561,1562],{"class":934,"line":1048},[932,1563,1061],{"class":946},[932,1565,1566],{"class":934,"line":1058},[932,1567,1568],{"class":1533},"\u002F\u002F f2(a) = a^2\n",[932,1570,1571,1573,1576,1578,1580,1582,1584],{"class":934,"line":1306},[932,1572,1222],{"class":938},[932,1574,1575],{"class":942}," square",[932,1577,947],{"class":946},[932,1579,939],{"class":938},[932,1581,953],{"class":952},[932,1583,965],{"class":946},[932,1585,968],{"class":946},[932,1587,1589,1591,1593,1595,1597],{"class":934,"line":1588},9,[932,1590,1051],{"class":994},[932,1592,1098],{"class":977},[932,1594,1193],{"class":946},[932,1596,953],{"class":977},[932,1598,1039],{"class":946},[932,1600,1602],{"class":934,"line":1601},10,[932,1603,1061],{"class":946},[932,1605,1607],{"class":934,"line":1606},11,[932,1608,1609],{"class":1533},"\u002F\u002F f3(a) = 1 \u002F (a^2)\n",[932,1611,1613,1615,1618,1620,1622,1624,1626],{"class":934,"line":1612},12,[932,1614,1222],{"class":938},[932,1616,1617],{"class":942}," square_rev",[932,1619,947],{"class":946},[932,1621,939],{"class":938},[932,1623,953],{"class":952},[932,1625,965],{"class":946},[932,1627,968],{"class":946},[932,1629,1631,1633,1635,1637,1639,1641,1643,1645],{"class":934,"line":1630},13,[932,1632,1051],{"class":994},[932,1634,1271],{"class":984},[932,1636,1274],{"class":946},[932,1638,998],{"class":946},[932,1640,1125],{"class":977},[932,1642,1193],{"class":946},[932,1644,953],{"class":977},[932,1646,1137],{"class":946},[932,1648,1650],{"class":934,"line":1649},14,[932,1651,1061],{"class":946},[932,1653,1655],{"class":934,"line":1654},15,[932,1656,1528],{"emptyLinePlaceholder":1527},[932,1658,1660,1662,1664,1666,1668,1670,1672,1674,1676,1678,1680,1682,1684,1686,1688],{"class":934,"line":1659},16,[932,1661,1222],{"class":938},[932,1663,943],{"class":942},[932,1665,947],{"class":946},[932,1667,939],{"class":938},[932,1669,953],{"class":952},[932,1671,956],{"class":946},[932,1673,959],{"class":938},[932,1675,962],{"class":952},[932,1677,956],{"class":946},[932,1679,1423],{"class":938},[932,1681,1426],{"class":942},[932,1683,947],{"class":946},[932,1685,939],{"class":938},[932,1687,1433],{"class":946},[932,1689,968],{"class":946},[932,1691,1693,1695,1697,1699,1701,1703,1705,1707,1709,1711,1713,1715,1717,1719,1721,1723,1725,1727,1729,1731,1733,1735],{"class":934,"line":1692},17,[932,1694,1051],{"class":994},[932,1696,1098],{"class":977},[932,1698,1101],{"class":946},[932,1700,1104],{"class":977},[932,1702,1107],{"class":946},[932,1704,1260],{"class":984},[932,1706,1113],{"class":946},[932,1708,1426],{"class":942},[932,1710,947],{"class":946},[932,1712,1458],{"class":977},[932,1714,965],{"class":946},[932,1716,1287],{"class":946},[932,1718,943],{"class":942},[932,1720,947],{"class":946},[932,1722,1125],{"class":977},[932,1724,1118],{"class":946},[932,1726,1130],{"class":984},[932,1728,956],{"class":946},[932,1730,962],{"class":977},[932,1732,956],{"class":946},[932,1734,1426],{"class":977},[932,1736,1137],{"class":946},[932,1738,1740],{"class":934,"line":1739},18,[932,1741,1061],{"class":946},[932,1743,1745],{"class":934,"line":1744},19,[932,1746,1528],{"emptyLinePlaceholder":1527},[932,1748,1750,1752,1755,1758],{"class":934,"line":1749},20,[932,1751,939],{"class":938},[932,1753,1754],{"class":942}," main",[932,1756,1757],{"class":946},"()",[932,1759,968],{"class":946},[932,1761,1763],{"class":934,"line":1762},21,[932,1764,1765],{"class":1533},"    \u002F\u002F 1 + 2 + 3 + 4 + 5 = 15\n",[932,1767,1769,1773,1776,1779,1782,1784,1786,1789,1791,1794,1796,1798,1800,1803,1806,1808,1811],{"class":934,"line":1768},22,[932,1770,1772],{"class":1771},"sBMFI","    std",[932,1774,1775],{"class":946},"::",[932,1777,1778],{"class":977},"cout ",[932,1780,1781],{"class":946},"\u003C\u003C",[932,1783,943],{"class":942},[932,1785,947],{"class":946},[932,1787,1788],{"class":984},"1",[932,1790,956],{"class":946},[932,1792,1793],{"class":984}," 5",[932,1795,956],{"class":946},[932,1797,1541],{"class":977},[932,1799,965],{"class":946},[932,1801,1802],{"class":946}," \u003C\u003C",[932,1804,1805],{"class":1771}," std",[932,1807,1775],{"class":946},[932,1809,1810],{"class":977},"endl",[932,1812,1039],{"class":946},[932,1814,1816],{"class":934,"line":1815},23,[932,1817,1528],{"emptyLinePlaceholder":1527},[932,1819,1821],{"class":934,"line":1820},24,[932,1822,1823],{"class":1533},"    \u002F\u002F 3^2 + 4^2 + 5^2 = 50\n",[932,1825,1827,1829,1831,1833,1835,1837,1839,1842,1844,1846,1848,1850,1852,1854,1856,1858,1860],{"class":934,"line":1826},25,[932,1828,1772],{"class":1771},[932,1830,1775],{"class":946},[932,1832,1778],{"class":977},[932,1834,1781],{"class":946},[932,1836,943],{"class":942},[932,1838,947],{"class":946},[932,1840,1841],{"class":984},"3",[932,1843,956],{"class":946},[932,1845,1793],{"class":984},[932,1847,956],{"class":946},[932,1849,1575],{"class":977},[932,1851,965],{"class":946},[932,1853,1802],{"class":946},[932,1855,1805],{"class":1771},[932,1857,1775],{"class":946},[932,1859,1810],{"class":977},[932,1861,1039],{"class":946},[932,1863,1865],{"class":934,"line":1864},26,[932,1866,1528],{"emptyLinePlaceholder":1527},[932,1868,1870],{"class":934,"line":1869},27,[932,1871,1872],{"class":1533},"    \u002F\u002F ~ 1.63 ~= π^2\u002F6\n",[932,1874,1876,1878,1880,1882,1884,1886,1888,1890,1892,1895,1897,1899,1901,1903,1905,1907,1909],{"class":934,"line":1875},28,[932,1877,1772],{"class":1771},[932,1879,1775],{"class":946},[932,1881,1778],{"class":977},[932,1883,1781],{"class":946},[932,1885,943],{"class":942},[932,1887,947],{"class":946},[932,1889,1788],{"class":984},[932,1891,956],{"class":946},[932,1893,1894],{"class":984}," 100",[932,1896,956],{"class":946},[932,1898,1617],{"class":977},[932,1900,965],{"class":946},[932,1902,1802],{"class":946},[932,1904,1805],{"class":1771},[932,1906,1775],{"class":946},[932,1908,1810],{"class":977},[932,1910,1039],{"class":946},[932,1912,1914],{"class":934,"line":1913},29,[932,1915,1061],{"class":946},[918,1917,1918,1919,1921,1922,1924,1925,1927],{},"观察 main 函数中调用 ",[929,1920,1386],{}," 的方法。我们“传入”对应的 ",[929,1923,1492],{}," 函数以决定求和的每一项，然后 ",[929,1926,1386],{}," 就会按照这样的规则去求值。",[918,1929,1930,1931,1935],{},"看上去很美好，函数作为实参，调用函数类型的形参……但实际上，背后的语法并没有那么简单；容我再稍微费一些口舌。实际上，C\u002FC++ 不允许函数作为参数，这里所有将函数传参的行为实际上传递的是",[1458,1932,1934],{"href":1933},"\u002Fch04\u002Fpointer\u002Fpointer_def_2#idx_%E5%87%BD%E6%95%B0%E6%8C%87%E9%92%88","指向函数的指针","。",[918,1937,1938,1939,1943,1944,1947,1948,1950,1951,1953,1954,1956,1957,1959,1960,1963,1964,1966,1967,1970,1971,1974],{},"和数组一样，如果在不允许使用函数的地方出现了函数，则发生 ",[1940,1941,1942],"em",{},"函数到指针转换","，函数 ",[929,1945,1946],{},"f"," 将转换为指向 ",[929,1949,1946],{}," 的指针。故尽管 ",[929,1952,1386],{}," 函数中声明了一个“函数类型的形参” ",[929,1955,1492],{},"，但实际上 ",[929,1958,1492],{}," 是指针类型的，也就是 ",[929,1961,1962],{},"double (*)(int)"," 类型的。类似地，在调用 ",[929,1965,1386],{}," 时，传入 ",[929,1968,1969],{},"identity"," 函数实际上传入的是 ",[929,1972,1973],{},"&identity"," 这个指针，隐式转换自动发生了。",[918,1976,1977,1978,1980,1981,1984,1985,1988,1989,1992,1993,1996],{},"所以，如果你在 ",[929,1979,1386],{}," 里对 ",[929,1982,1983],{},"sizeof(term)"," 求值，得到的是指针的大小。如果对普通的函数做 sizeof 运算则是一个编译错误。此外函数指针定义了 ",[929,1986,1987],{},"operator()","，所以 ",[929,1990,1991],{},"term(a)"," 就相当于 ",[929,1994,1995],{},"(*term)(a)","。在各种语言特性的加持下，好像函数就自然而然地成为了普通的参数。",[918,1998,1999],{},"这也就是我这一节标题所述的“函数也即对象”。在函数式编程里，很重要的一点就是把函数看做和普通变量没有差别的事物来看待。我们这里演示了函数作为参数，之后还会有函数作为返回值等等用法。许多编程语言如 JavaScript 天然地支持这些语法，人们习惯称这种特性为“函数是第一公民”。",[918,2001,2002],{},"不过在 C\u002FC++ 的语境下，函数是比“第二公民”还要弱的存在：函数不能赋值、传递，或者作为局部对象。所以我们刚才必须强调，在这一节代码中传递的始终是函数指针而非函数本身。为了解决这些麻烦，我们需要了解更多的语言特性和常用手段。",[2004,2005,2006],"style",{},"html pre.shiki code .spNyl, html code.shiki .spNyl{--shiki-light:#9C3EDA;--shiki-default:#C792EA;--shiki-dark:#C792EA}html pre.shiki code .s2Zo4, html code.shiki .s2Zo4{--shiki-light:#6182B8;--shiki-default:#82AAFF;--shiki-dark:#82AAFF}html pre.shiki code .sMK4o, html code.shiki .sMK4o{--shiki-light:#39ADB5;--shiki-default:#89DDFF;--shiki-dark:#89DDFF}html pre.shiki code .sHdIc, html code.shiki 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