{"id":15660,"date":"2010-11-24T13:26:24","date_gmt":"2010-11-24T05:26:24","guid":{"rendered":"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/?p=15660"},"modified":"2021-10-06T16:30:41","modified_gmt":"2021-10-06T08:30:41","slug":"%e8%a4%87%e6%95%b8%e7%9a%84%ef%bd%8e%e6%ac%a1%e6%96%b9%e6%a0%b9%ef%bc%88nth-root-of-complex-number%ef%bc%89","status":"publish","type":"post","link":"http:\/\/localhost\/%e8%a4%87%e6%95%b8%e7%9a%84%ef%bd%8e%e6%ac%a1%e6%96%b9%e6%a0%b9%ef%bc%88nth-root-of-complex-number%ef%bc%89\/","title":{"rendered":"\u8907\u6578\u7684\uff4e\u6b21\u65b9\u6839\uff08nth root of complex number\uff09"},"content":{"rendered":"

\u8907\u6578\u7684\uff4e\u6b21\u65b9\u6839\uff08nth root of complex number\uff09<\/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\u6559\u6388\u8cac\u4efb\u7de8\u8f2f<\/span><\/strong><\/p>\n

\u6839\u64da\u4ee3\u6578\u57fa\u672c\u5b9a\u7406\u8207\u56e0\u5f0f\u5b9a\u7406\u5f97\u77e5\uff0c$$n$$ \u6b21\u65b9\u7a0b\u5f0f $$x^n={a}$$ ($${a}$$\u662f\u8907\u6578)\u6070\u6709 $$n$$ \u500b\u8907\u6578\u6839\uff0c\u9019 $$n$$ \u500b\u6839\u7a31\u70ba $${a}$$ \u7684 $$n$$ \u6b21\u65b9\u6839\u3002\u73fe\u5728\uff0c\u6211\u5011\u61c9\u7528\u68e3\u7f8e\u5f17\u5b9a\u7406\u6c42\u89e3\u6b64\u65b9\u7a0b\u5f0f\u3002<\/p>\n

\u5047\u8a2d\u8907\u6578 $$z$$ \u662f\u65b9\u7a0b\u5f0f $$x^n={a}$$ \u7684\u6839\uff0c\u5373 $$z^n={a}$$\u3002\u4ee5\u6975\u5f0f\u8868\u793a\uff1a<\/p>\n

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

\u96d6\u7136\u6574\u6578 $$k$$ \u6709\u7121\u9650\u591a\u500b\uff0c\u4e4d\u770b\u4e4b\u4e0b\u597d\u50cf\u4e5f\u6709\u7121\u9650\u591a\u500b $$\\theta$$\uff0c\u4f46 $$k=0,1,2,\\mbox{……},n-1$$ \u5f97\u51fa\u7684 $$n$$ \u7684\u8f3b\u89d2 $$\\theta$$\uff0c\u548c $$k=n,n+1,n+2,\\mbox{……},2n-1$$ \u5f97\u51fa\u7684 $$n$$ \u500b\u8f3b\u89d2 $$\\theta$$ \u662f\u540c\u754c\u89d2\uff0c\u800c\u4e14 $$\\sin\\theta$$\u3001$$\\cos\\theta$$ \u7684\u9031\u671f\u70ba $$2\\pi$$\uff0c\u6545\u53d6 $$k=0,1,2,\\mbox{……},n-1$$\uff0c\u5373\u8907\u6578 $$z$$ \u7684\u4e3b\u8f3b\u89d2 $$\\theta$$ \u70ba<\/p>\n

$$\\displaystyle \\frac{\\phi}{n},~~~\\frac{\\phi+2\\pi}{n},~~~\\frac{\\phi+4\\pi}{n},\\mbox{……},\\frac{\\phi+2(n-1)\\pi}{n}$$<\/p>\n

\u56e0\u6b64\uff0c$$a$$ \u7684 $$n$$ \u500b $$n$$ \u6b21\u65b9\u6839\u70ba<\/p>\n

$$\\displaystyle z_k=\\sqrt[n]{|a|}(\\cos\\frac{\\phi+2k\\pi}{n}+i\\sin\\frac{\\phi+2k\\pi}{n})$$<\/p>\n

\u5176\u4e2d$$k=0,1,2,\\mbox{……},n-1$$\u3002<\/p>\n

\u5728\u8907\u6578\u5e73\u9762\u4e0a\uff0c\u9019 $$n$$ \u500b $$n$$ \u6b21\u65b9\u6839\u843d\u5728\u4ee5\u539f\u9ede\u70ba\u5713\u5fc3\u3001$$\\sqrt[n]{|a|}$$\u00a0\u70ba\u534a\u5f91\u7684\u5713\u4e0a\uff0c\u4e26\u4e14\u5e73\u5747\u5206\u5e03\u5728\u5713\u5167\u63a5\u6b63 $$n$$ \u908a\u5f62\u7684\u9802\u9ede\u4e0a\u3002\u7576\u7136\uff0c\u65b9\u7a0b\u5f0f $$x^n={a}$$ \u7684 $$n$$ \u6b21\u65b9\u6839\u6db5\u84cb\u4e86\u4e4b\u524d\u8a0e\u8ad6\u904e\u7684 $$1$$ \u7684 $$n$$ \u6b21\u65b9\u6839\u3002\u5e95\u4e0b\uff0c\u6211\u5011\u8209\u500b\u4f8b\u5b50\u4f86\u719f\u6089\u8907\u6578\u7684 $$n$$ \u6b21\u65b9\u6839\u3002<\/p>\n

\u4f8b\u984c\uff1a\u6c42\u89e3 $$-8+8\\sqrt{3}i$$ \u7684\u56db\u6b21\u65b9\u6839\u3002<\/p>\n

\u89e3\uff1a\u5047\u8a2d\u8907\u6578 $$z$$ \u662f $$-8+8\\sqrt{3}i$$ \u7684\u56db\u6b21\u65b9\u6839\uff0c<\/p>\n

\u4ee5\u6975\u5f0f\u8868\u793a\uff0c$$z=r(\\cos\\theta+i\\sin\\theta)$$\u3001$$-8+8\\sqrt{3}i=16(\\cos{\\frac{2\\pi}{3}}+i\\sin\\frac{2\\pi}{3})$$\uff0c<\/p>\n

\u6240\u4ee5 $$z^4=-8+8\\sqrt{3}i\\Rightarrow[r(\\cos{\\theta}+i\\sin\\theta)]^4=16(\\cos\\frac{2\\pi}{3}+i\\sin\\frac{2\\pi}{2})$$\uff0c<\/p>\n

\u7531\u68e3\u7f8e\u5f17\u5b9a\u7406 $$[r^4(\\cos 4\\theta+i\\sin 4\\theta)]=16(\\cos\\frac{2\\pi}{3}+i\\sin\\frac{2\\pi}{3})$$\uff0c<\/p>\n

\u7531\u8907\u6578\u7684\u76f8\u7b49\u6027\u8cea\uff08$$r$$ \u662f\u5927\u65bc $$0$$ \u7684\u5be6\u6578\uff0c$$4\\theta$$ \u548c $$\\frac{2\\pi}{3}$$ \u662f\u540c\u754c\u89d2\uff09\uff0c<\/p>\n

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

\u56e0\u6b64\uff0c$$z_k=2(\\cos(\\frac{\\pi}{6}+\\frac{2k\\pi}{4})+i\\sin(\\frac{\\pi}{6}+\\frac{2k\\pi}{4}))$$\uff0c\u5176\u4e2d$$k=0,1,2,3.$$\u3002
\n\u5373<\/p>\n

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

\u6240\u4ee5\uff0c$$-8+8\\sqrt{3}i$$ \u7684\u56db\u6b21\u65b9\u6839\u70ba $$\\sqrt{3}+i,-1+\\sqrt{3}i,-\\sqrt{3}+i,1-\\sqrt{3}i$$\u3002<\/p>\n

\u4e0a\u8ff0\u662f\u6a19\u6e96\u7684\u89e3\u6cd5\uff0c\u4f46\u662f\u5c0d\u591a\u6578\u7684\u5b78\u751f\u800c\u8a00\u662f\u76f8\u7576\u4e0d\u5bb9\u6613\u7684\u3002\u5e95\u4e0b\uff0c\u7528\u300c\u56de\u5230$$1$$\u300d\u7684\u89c0\u5ff5\u63d0\u4f9b\u53e6\u4e00\u500b\u7c21\u55ae\u7684\u89e3\u6cd5\u3002<\/p>\n

\u89e3\uff1a\u9996\u5148\uff0c\u627e\u51fa\u4e00\u500b$$-8+8\\sqrt{3}i$$\u7684\u56db\u6b21\u65b9\u6839\uff0c<\/p>\n

$$-8+8\\sqrt{3}i=16(\\cos\\frac{2\\pi}{3}+i{sin}\\frac{2\\pi}{3})=(2(\\cos\\frac{\\pi}{6}+i{sin}\\frac{\\pi}{6}))^4=(\\sqrt{3}+1)^4$$<\/p>\n

\u5c07\u65b9\u7a0b\u5f0f\u6574\u7406\uff0c\u300c\u56de\u5230$$1$$\u300d\uff0c<\/p>\n

$$z^4=-8+8\\sqrt{3}i\\Rightarrow\\frac{z^4}{-8+8\\sqrt{3}i}=1\\Rightarrow\\frac{z^4}{(\\sqrt{3}+i)^4}=1\\Rightarrow(\\frac{z}{\\sqrt{3}+i})^4=1$$\uff0c<\/p>\n

$$1$$\u7684\u56db\u6b21\u65b9\u6839\u986f\u7136\u5bb9\u6613\u591a\u4e86\uff0c\u6240\u4ee5$$\\frac{z}{\\sqrt{3}+i}=1,i,-1,-i$$\uff0c<\/p>\n

\u56e0\u6b64$$z=\\sqrt{3}+i, -1+\\sqrt{3}i, -\\sqrt{3}-i, 1-\\sqrt{3}i$$\u3002<\/p>\n<\/div>","protected":false},"excerpt":{"rendered":"

\u7531\u300c\u4ee3\u6578\u57fa\u672c\u5b9a\u7406\u300d\u8207\u300c\u56e0\u5f0f\u5b9a\u7406\u300d\u5f97\u77e5\uff0cn\u6b21\u65b9\u7a0b\u5f0f\u6070\u6709n\u500b\u8907\u6578\u6839\u3002\u4f46\u662f\u4f7f\u7528\u300c\u68e3\u7f8e\u5f17\u5b9a\u7406\u300d\u89e3\u6839\u7684\u6a19\u6e96\u7b97\u6cd5\uff0c\u5c0d\u591a\u6578\u7684\u5b78\u751f\u800c\u8a00\u662f\u76f8\u7576\u4e0d\u5bb9\u6613\u7684\u3002\u4e0d\u59a8\u904b\u7528\u201c\u56de\u52301\u201d\u7684\u89c0\u5ff5\u5c07\u89e3\u6cd5\u7c21\u55ae\u5316\u3002<\/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":[447],"class_list":["post-15660","post","type-post","status-publish","format-standard","hentry","category-mathematics00","category-math02","tag-n","loop-entry","cat-111","cat-216","no-thumbnail"],"views":17352,"_links":{"self":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/15660","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=15660"}],"version-history":[{"count":1,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/15660\/revisions"}],"predecessor-version":[{"id":89304,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/15660\/revisions\/89304"}],"wp:attachment":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/media?parent=15660"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/categories?post=15660"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/tags?post=15660"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}