{"id":15618,"date":"2010-11-24T11:35:38","date_gmt":"2010-11-24T03:35:38","guid":{"rendered":"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/?p=15618"},"modified":"2021-10-06T16:30:41","modified_gmt":"2021-10-06T08:30:41","slug":"%e6%a3%a3%e7%be%8e%e5%bc%97%e5%ae%9a%e7%90%86-de-moivres-theorem","status":"publish","type":"post","link":"http:\/\/localhost\/%e6%a3%a3%e7%be%8e%e5%bc%97%e5%ae%9a%e7%90%86-de-moivres-theorem\/","title":{"rendered":"\u68e3\u7f8e\u5f17\u5b9a\u7406"},"content":{"rendered":"

\u68e3\u7f8e\u5f17\u5b9a\u7406 (De Moivre’s Theorem)<\/span><\/strong>
\n\u570b\u7acb\u5c4f\u6771\u9ad8\u7d1a\u4e2d\u5b78 \u6578\u5b78\u79d1\u694a\u74ca\u8339\u8001\u5e2b\/\u570b\u7acb\u81fa\u7063\u5e2b\u7bc4\u5927\u5b78\u6578\u5b78\u7cfb\u6d2a\u842c\u751f\u9000\u4f11\u6559\u6388\u8cac\u4efb\u7de8\u8f2f<\/span><\/strong><\/p>\n

\u662f\u8ab0\u8b93\u8718\u86db\u5982\u68e3\u7f8e\u5f17\u90a3\u822c\u7cbe\u78ba\uff0c\u4e0d\u9760\u91cf\u5c3a\u6e96\u7e69\u5c31\u8a2d\u8a08\u51fa\u5716\u6a23\uff1f<\/p>\n

\u9019\u6bb5\u8a71\u662f\u82f1\u570b\u8a69\u4eba\u6ce2\u666e (Alexander Pope) \u5728\u4ed6\u7684\u8457\u4f5c\u300a\u4eba\u7684\u8b9a\u79ae\u300b\u4e2d\uff0c\u5c0d\u68e3\u7f8e\u5f17\u7684\u6578\u5b78\u80fd\u529b\u8868\u793a\u656c\u610f\u3002\u68e3\u7f8e\u5f17 (Abraham De Moivre, 1667-1754 ) \u51fa\u751f\u65bc\u6cd5\u570b\u9999\u6ab3\u7701\u7dad\u5d14\u93ae\u7684\u65b0\u6559\u5f92\u5bb6\u5ead\uff0c\u56e0\u70ba\u65b0\u820a\u6559\u6d3e\u7684\u9b25\u722d\u800c\u906d\u5230\u62d8\u7981\u5169\u5e74\uff0c\u96a8\u5f8c\u9077\u5c45\u82f1\u570b\uff0c\u6210\u70ba\u725b\u9813\u548c\u54c8\u96f7\u7684\u646f\u53cb\uff0c\u9084\u7372\u9078\u70ba\u82f1\u570b\u7687\u5bb6\u5b78\u6703\u6703\u54e1\u3001\u67cf\u6797\u79d1\u5b78\u9662\u58eb\u548c\u6cd5\u570b\u79d1\u5b78\u9662\u58eb\u3002\u4ed6\u65bc1718 \u5e74\u767c\u8868\u300a\u6a5f\u9047\u8ad6\u300b(The Doctrine of Chances) \u4e00\u66f8\uff0c\u6210\u70ba\u6a5f\u7387\u8ad6\u7684\u5148\u9a45\u3002<\/p>\n

\u8457\u540d\u7684\u68e3\u7f8e\u5f17\u5b9a\u7406\uff1a<\/p>\n

$$({\\cos\\theta+i\\sin\\theta})^n=\\cos{n\\theta}+i\\sin{n\\theta}~~~,n\\in{N}$$<\/p>\n

\u662f\u4ed6\u57281707 \u5e74\u767c\u73fe\uff0c1722 \u5e74\u6b63\u5f0f\u767c\u8868\uff0c\u9996\u5ea6\u628a\u8907\u6578\u7d0d\u5165\u4e09\u89d2\u51fd\u6578\u4e2d\uff0c\u6c42\u89e3\u4e00\u5143 $$n$$ \u6b21\u65b9\u7a0b\u5f0f\u3002<\/p>\n

\"\"<\/a><\/p>\n

\u8aa0\u5982\u9ea5\u51f1 (Herbert Mc Kay) \u5728\u4ed6\u7684\u8457\u4f5c\u300a\u6578\u7684\u4e16\u754c\u300b\u4e2d\u8aaa\u5230\uff1a\u300c\u68e3\u7f8e\u5f17\u5b9a\u7406\u638c\u63e1\u4e86\u9032\u5165\u8907\u4e09\u89d2\u5b78\u65b0\u4e16\u754c\u4e4b\u9470\u3002\u300d\u8332\u5c07\u6b64\u5b9a\u7406\u6539\u5beb\u5982\u4e0b\uff1a<\/p>\n

\u82e5 $$z=\\cos\\theta+i\\sin\\theta$$\uff0c\u5247 $$z^n=\\cos{n}\\theta+i\\sin{n}\\theta$$\uff0c$$n\\in{N}$$\u3002<\/p>\n

\u9019\u500b\u5b9a\u7406\u7531\u8907\u6578\u6975\u5f0f\u7684\u4e58\u6cd5\u5ef6\u4f38\u4e0d\u96e3\u7406\u89e3\uff0c\u5176\u9053\u7406\u4e5f\u4e0d\u8a00\u53ef\u55bb\u3002\u5728\u6b64\uff0c\u6211\u5011\u904b\u7528\u7528\u6578\u5b78\u6b78\u7d0d\u6cd5\u52a0\u4ee5\u8b49\u660e\uff1a<\/p>\n

(\uff11)\u7576 $$n=1$$ \u6642\uff0c
\n$$z^1=\\cos\\theta+i\\sin\\theta=\\cos 1\\cdot(\\theta)+i\\sin 1 \\cdot(\\theta)$$ \u7b49\u5f0f\u6210\u7acb\u3002<\/p>\n

(\uff12)\u8a2d $$n=k$$ \u6642\uff0c\u7b49\u5f0f\u6210\u7acb\u3002\u5373 $$z^k=\\cos {k\\theta}+i\\sin {k\\theta}$$\u3002
\n\u5247\u7576 $$n=k+1$$ \u6642\uff0c
\n$$\\begin{array}{ll}z^{k+1} &=z^k\\cdot z^1\\\\&=(\\cos {k\\theta}+i\\sin{k\\theta})\\cdot(\\cos\\theta+i\\sin\\theta)\\\\&=\\cos(k\\theta+\\theta)+i\\sin(k\\theta+\\theta)\\\\&=\\cos[(k+1)\\theta]+i\\sin[(k+1)\\theta]\\end{array}$$
\n\u7b49\u5f0f\u4ea6\u6210\u7acb\u3002<\/p>\n

\u6545\u7531\u6578\u5b78\u6b78\u7d0d\u6cd5\u5f97\u77e5\uff0c\u5c0d\u6240\u6709\u7684\u6b63\u6574\u6578 $$n$$\uff0c$$z^n=\\cos{n\\theta}+i\\sin{n\\theta}$$\u90fd\u6210\u7acb\u3002<\/p>\n

\u4e0a\u9762\u6558\u8ff0\u7684\u8907\u6578\u6975\u5f0f\u662f $$r=1$$ \u548c $$n$$ \u70ba\u6b63\u6574\u6578\u7684\u60c5\u6cc1\uff0c\u9084\u53ef\u4ee5\u63a8\u5ee3\u5230\u4efb\u610f\u7684\u6b63\u5be6\u6578 $$r$$ \u548c\u6574\u6578 $$n$$\u3002<\/p>\n

\u5373\uff1a\u82e5 $$z=r(\\cos\\theta+i\\sin\\theta)$$\uff0c\u5247 $$z^n=r^n(\\cos{n\\theta}+i\\sin{n\\theta})$$\uff0c$$n{\\in}Z$$\u3002<\/p>\n

\u5e95\u4e0b\uff0c\u6211\u5011\u7531\u4f8b\u984c\u4f86\u719f\u6089\u68e3\u7f8e\u5f17\u5b9a\u7406\u3002<\/p>\n

\u4f8b\u984c\uff1a\u8a2d $$z=-1+{\\sqrt{3}}i$$\uff0c\u6c42 $$z^{10}$$\u3002<\/p>\n

\u89e3\uff1a\u5148\u5c07 $$z$$ \u8f49\u63db\u6210\u6975\u5f0f\uff0c<\/p>\n

\"\"<\/a><\/p>\n

\u5f9e\u5728\u9762\u4f8b\u984c\u53ef\u4ee5\u770b\u5230\u68e3\u7f8e\u5f17\u5b9a\u7406\u5728\u6307\u6578\u554f\u984c\u4e0a\u7684\u6548\u7528\uff0c\u5b83\u8b93\u904b\u7b97\u8b8a\u5f97\u7c21\u55ae\uff01<\/p>\n

\n

\u53c3\u8003\u8cc7\u6599<\/strong><\/p>\n

    \n
  • \u6bdb\u723e(Eli Moar)(2000).\u300a\u6bdb\u8d77\u4f86\u8aaa\u4e09\u89d2\u300b\uff08\u80e1\u5b88\u4ec1\u8b6f\uff09\uff0c\u53f0\u5317\uff1a\u5929\u4e0b\u6587\u5316\u3002<\/li>\n
  • \u675c\u745e\u829d\u4e3b\u7de8(2000). \u300a\u6578\u5b78\u53f2\u8fad\u5178\u300b\uff0c\u6fdf\u5357\uff1a\u5c71\u6771\u6559\u80b2\u51fa\u7248\u793e\u3002<\/li>\n<\/ul>\n<\/div>","protected":false},"excerpt":{"rendered":"

    \u8457\u540d\u7684\u68e3\u7f8e\u5f17\u5b9a\u7406\u57281722\u5e74\u6b63\u5f0f\u767c\u8868\uff0c\u5176\u9053\u7406\u5f9e\u8907\u6578\u6975\u5f0f\u7684\u4e58\u6cd5\u5ef6\u4f38\u4e26\u4e0d\u96e3\u7406\u89e3\uff0c\u4f46\u537b\u5f71\u97ff\u6df1\u9060\uff0c\u8aa0\u5982\u9ea5\u51f1( Herbert Mc Kay )\u8aaa\u5230\uff1a\u300c\u68e3\u7f8e\u5f17\u5b9a\u7406\u638c\u63e1\u4e86\u9032\u5165\u8907\u4e09\u89d2\u5b78\u65b0\u4e16\u754c\u4e4b\u9470\u3002\u300d<\/p>\n","protected":false},"author":50,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[111,216],"tags":[7823,444,445,446],"class_list":["post-15618","post","type-post","status-publish","format-standard","hentry","category-mathematics00","category-math02","tag-de-moivre","tag-444","tag-445","tag-446","loop-entry","cat-111","cat-216","no-thumbnail"],"views":15977,"_links":{"self":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/15618","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/users\/50"}],"replies":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/comments?post=15618"}],"version-history":[{"count":1,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/15618\/revisions"}],"predecessor-version":[{"id":89305,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/15618\/revisions\/89305"}],"wp:attachment":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/media?parent=15618"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/categories?post=15618"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/tags?post=15618"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}