{"id":4518,"date":"2013-12-26T06:00:35","date_gmt":"2013-12-25T21:00:35","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=4518"},"modified":"2014-09-10T16:47:14","modified_gmt":"2014-09-10T07:47:14","slug":"post-4518","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=4518","title":{"rendered":"\u8907\u7d20\u6570"},"content":{"rendered":"

\u3000\u8907\u7d20\u6570\u306f x^2 + 1 = 0 \u3084 x^2 + x + 1 = 0 \u3068\u3044\u3063\u305f\u5b9f\u6570\u89e3\u3092\u6301\u305f\u306a\u3044\u6574\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u969b\u306b\u51fa\u73fe\u3057\u307e\u3059\uff0e\u8907\u7d20\u6570\u306f a + bi<\/span> \u3068\u3044\u3046\u5f62\u3092\u3057\u3066\u304a\u308a\uff0ca, b<\/span> \u306f\u5b9f\u6570\u90e8\u3067\u3042\u308a\uff0ci<\/span> \u306f\u865a\u6570\u90e8\u3068\u547c\u3070\u308c\uff0ci2<\/sup><\/span> = -1 \u3068\u3044\u3046\u6027\u8cea\u3092\u6301\u3061\u307e\u3059\uff0e\u8907\u7d20\u6570\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u307e\u3059\uff0e<\/p>\n

    \n
  1. \u52a0\u7b97
    \n\\displaystyle (a + bi) + (c + di) = (a + c) + (b + d)i<\/li>\n
  2. \u6e1b\u7b97
    \n\\displaystyle (a + bi) - (c + di) = (a - c) + (b - d)i<\/li>\n
  3. \u4e57\u7b97
    \n\\displaystyle (a + bi)\\times(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i<\/li>\n
  4. \u9664\u7b97
    \n\\displaystyle \\frac{a + bi}{c + di} = \\frac{a + bi}{c + di}\\times\\frac{c - di}{c - di} = \\frac{ac + bd}{c^2 + d^2} + \\frac{bc - ad}{c^2 + d^2}i<\/li>\n<\/ol>\n

    \u3000\u3044\u304b\u306a\u308b\u72b6\u6cc1\u3067\u3082 i2<\/sup><\/span> \u3092 -1 \u3067\u7f6e\u63db\u3059\u308b\u3053\u3068\u3092\u9664\u3051\u3070\uff0c\u4ee3\u6570\u306e\u901a\u5e38\u306e\u5b9a\u7fa9\u304c\u7528\u3044\u3089\u308c\u307e\u3059\uff0e\u4ea4\u63db\u6cd5\u5247\uff0c\u7d50\u5408\u6cd5\u5247\uff0c\u5206\u914d\u6cd5\u5247\u3082\u307e\u305f\u8907\u7d20\u6570\u306b\u9069\u7528\u3055\u308c\u307e\u3059\uff0ea + bi<\/span> \u306e a<\/span> \u304a\u3088\u3073 b<\/span> \u306f\u305d\u308c\u305e\u308c \u5b9f\u6570\u90e8<\/span> \u304a\u3088\u3073 \u865a\u6570\u90e8<\/span> \u3068\u547c\u3070\u308c\u307e\u3059\uff0e\u4e8c\u3064\u306e\u8907\u7d20\u6570\u304c\u7b49\u3057\u3044\u3068\u306f\uff0c\u5b9f\u6570\u90e8\u3068\u865a\u6570\u90e8\u3068\u304c\u305d\u308c\u305e\u308c\u7b49\u3057\u3044\u6642\u306b\u9650\u3089\u308c\u307e\u3059\uff0e<\/p>\n

    \u3000\u3042\u308b\u8907\u7d20\u6570 z = x + iy<\/span> \u306f xy<\/span> \u76f4\u4ea4\u5e73\u9762\u4e0a\u306e (x, y<\/span>) \u5ea7\u6a19\u306e\u70b9 P \u3068\u307f\u306a\u3059\u3053\u3068\u304c\u3067\u304d\uff0c\u3053\u306e\u5834\u5408\u3053\u306e\u5e73\u9762\u3092 \u8907\u7d20\u5e73\u9762<\/span> \u307e\u305f\u306f \u30a2\u30eb\u30ac\u30f3\u56f3<\/span> \u3068\u547c\u3073\u307e\u3059\uff0e\u4eee\u306b\u539f\u70b9 O<\/span> \u304b\u3089\u70b9 P<\/span> \u3078\u306e\u76f4\u7dda\u3092\u5f15\u304d\uff0cOP<\/span> \u9593\u306e\u8ddd\u96e2\u3092 ρ \u3068\u3057\uff0cOP<\/span> \u3068 x<\/span> \u8ef8\u306e\u306a\u3059\u89d2\u5ea6\u3092 φ<\/span> \u3068\u3059\u308b\u3068\uff0c\u6b21\u306e\u56f3\u3092\u5f97\u307e\u3059\uff0e<\/p>\n

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

    \\displaystyle x = \\rho \\cos\\phi,\\ y = \\rho\\sin\\phi,\\ \\rho = \\sqrt{x^2 + y^2}<\/p>\n

    \u307e\u305f\u8907\u7d20\u6570\u3092\u3044\u308f\u3086\u308b \u6975\u5ea7\u6a19\u7cfb\u5f0f<\/span> \u3068\u3057\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8a18\u8ff0\u3067\u304d\u307e\u3059\uff0e<\/p>\n

    \\displaystyle z = x + iy = \\rho(\\cos\\phi + i\\sin\\phi) = \\rho cis\\phi<\/p>\n

    \u3000\u3057\u3070\u3057\u3070 ρ<\/span> \u306f z<\/span> \u306e \u30e2\u30b8\u30e5\u30e9\u30b9<\/span> \u307e\u305f\u306f \u7d76\u5bfe\u5024<\/span> \u3068\u547c\u3070\u308c\uff0c|z<\/span>| \u3068\u8a18\u8ff0\u3057\u307e\u3059\uff0e\u89d2 φ<\/span> \u306f z<\/span> \u306e amplitude<\/span> \u307e\u305f\u306f argument<\/span> \u3068\u547c\u3070\u308c\uff0carg z<\/span> \u3068\u7565\u8a18\u3057\u307e\u3059\uff0e\\rho = \\sqrt{z\\bar{z}} \u3068\u3082\u8a18\u8ff0\u3067\u304d\u307e\u3059\uff0e\u305f\u3060\u3057 \\bar{z} = x - iy \u306f z = x + iy \u306e \u5171\u5f79\u8907\u7d20\u6570<\/span> \u3068\u547c\u3070\u308c\u307e\u3059\uff0e<\/p>\n

    \u3000\u6975\u5ea7\u6a19\u3067\u8907\u7d20\u6570\u3092\u8a18\u8ff0\u3059\u308b\u3068\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\uff0e<\/p>\n

    \\displaystyle z_1 = \\rho_1(\\cos\\phi_1 + i\\sin\\phi_1),\\ z_2 = \\rho_2(\\cos\\phi_2 + i\\sin\\phi_2)<\/p>\n

    \u3059\u308b\u3068<\/p>\n

    \\displaystyle z_1z_2 = \\rho_1\\rho_2[\\cos(\\phi_1 + \\phi_2) + i\\sin(\\phi_1 + \\phi_2)]\\\\\\vspace{0.2 in} \\frac{z_1}{z_2} = \\frac{\\rho_1}{\\rho_2}[\\cos(\\phi_1 - \\phi_2) + i\\sin(\\phi_1 - \\phi_2)]<\/p>\n

    \u3000\u4eee\u306b n<\/span> \u3092\u4efb\u610f\u306e\u5b9f\u6570\u3068\u3059\u308b\u3068\u4ee5\u4e0b\u3092\u5f97\u307e\u3059\uff0e<\/p>\n

    \\displaystyle z^n = [\\rho(\\cos\\phi + i\\sin\\phi)]^n = \\rho^n(\\cos n\\phi + i\\sin n\\phi)<\/p>\n

    \u3053\u308c\u306f \u30c9\u30fb\u30e2\u30a2\u30d6\u30eb\u306e\u5b9a\u7406<\/span> \u3068\u3057\u3066\u77e5\u3089\u308c\u3066\u3044\u307e\u3059\uff0e\u3053\u308c\u3092\u7528\u3044\u3066\u8907\u7d20\u6570\u306e\u6839\u3092\u5b9a\u7fa9\u3067\u304d\u307e\u3059\uff0e\u4f8b\u3048\u3070\u4eee\u306b n<\/span> \u3092\u6b63\u306e\u6574\u6570\u3068\u4eee\u5b9a\u3059\u308b\u3068\uff0c<\/p>\n

    \\displaystyle z^{\\frac{1}{n}} = [\\rho(\\cos\\phi + i\\sin\\phi)]^\\frac{1}{n} = \\rho^{\\frac{1}{n}}\\left\\{\\cos\\left(\\frac{\\phi + 2k\\pi}{n}\\right) + i\\sin\\left(\\frac{\\phi + 2k\\pi}{n}\\right)\\right\\}\\\\\\vspace{0.2 in} \\ \\ k = 0,\\ 1,\\ 2,\\ \\cdots,\\ n-1<\/p>\n

    ex<\/sup><\/span>, sin x<\/span>, cos x<\/span> \u306e\u305f\u3081\u306e\u7d1a\u6570\u3092\u7528\u3044\u308b\u3068\u4ee5\u4e0b\u306e\u5b9a\u7fa9\u306b\u5c0e\u304b\u308c\u307e\u3059\uff0e<\/p>\n

    \\displaystyle e^{i\\phi} = \\cos\\phi + i\\sin\\phi,\\ e^{-i\\phi} = \\cos\\phi - i\\sin\\phi<\/p>\n

    \u3053\u308c\u3089\u306f \u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f<\/span> \u3068\u547c\u3070\u308c\uff0c\u65b9\u7a0b\u5f0f\u3092\u6307\u6570\u95a2\u6570\u306e\u89b3\u70b9\u304b\u3089\u66f8\u304d\u76f4\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\uff0e<\/p>\n