\uff08\u3042\u3093\u305f\u306e\u516c\u5f0f\u3067\u306f\u306a\u304f\uff09\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f\u3068\u306f\uff0c<\/p>\n
$$e^{i \\theta} = \\cos \\theta + i \\sin \\theta$$<\/p>\n
<\/p>\n
\u6307\u6570\u95a2\u6570\u306e\u80a9\u304c\u5b9f\u6570\u3060\u308d\u3046\u304c\u865a\u6570\u3060\u308d\u3046\u304c\uff0c\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\uff08\\(x=0\\)\u00a0 \u306e\u307e\u308f\u308a\u306e\u5c55\u958b\u306a\u306e\u3067\u7279\u306b\u30de\u30af\u30ed\u30fc\u30ea\u30f3\u5c55\u958b\u3068\u3082\u3044\u3046\uff09\u306f\u3067\u304d\u308b\u306e\u3067<\/p>\n
$$f(x)\u00a0 = f(0)+ f'(x) x + \\frac{1}{2} f^{”}(0) x^2 + \\cdots + \\frac{1}{n!} f^{(n)}(0) x^n + \\cdots$$<\/p>\n
\u306e\u516c\u5f0f\u306b\u5f93\u3063\u3066\uff0c<\/p>\n
$$ e^{x} = 1 + x + \\frac{1}{2} x^2 + \\frac{1}{3!} x^3 + \\frac{1}{4!} x^4 + \\cdots$$<\/p>\n
\\( x = i \\theta \\) \u3092\u5165\u308c\u308b\u3068<\/p>\n
\\begin{eqnarray}
\ne^{i \\theta} &=&1 + i\\theta + \\frac{1}{2!}(i\\theta)^2 + \\frac{1}{3!} (i\\theta)^3 + \\frac{1}{4!}(i\\theta)^4 + \\frac{1}{5!}(i\\theta)^5 + \\cdots\\\\
\n&=&\u00a0 {\\color{red}{1 -\\frac{1}{2!} \\theta^2 + \\frac{1}{4!} \\theta^4 -\\cdots}} \\\\
\n&& + i\\left\\{ {\\color{blue}{ \\theta -\\frac{1}{3!} \\theta^3 + \\frac{1}{5!} \\theta^5 -\\cdots}}\\right\\}
\n\\end{eqnarray}<\/p>\n
\u4e00\u65b9\uff0c\\(\\cos \\theta, \\sin \\theta\\) \u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u304c<\/p>\n
$$\\cos \\theta = \\color{red}{1 -\\frac{1}{2!} \\theta^2 + \\frac{1}{4!} \\theta^4 -\\cdots}$$<\/p>\n
$$\\sin \\theta = \\color{blue}{\\theta -\\frac{1}{3!} \\theta^3 + \\frac{1}{5!} \\theta^5 -\\cdots}$$<\/p>\n
\u3067\u3042\u308b\u3053\u3068\u3092\u4f7f\u3046\u3068\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n
$$e^{i \\theta} = {\\color{red}{\\cos \\theta}} + i\\,\u00a0 {\\color{blue}{\\sin\\theta}}$$<\/p>\n
\u3053\u308c\u3053\u305d\u304c\uff0c\u4eba\u985e\u306e\u81f3\u5b9d\uff01\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f<\/strong><\/span>\u3067\u3042\u308b\uff01<\/p>\n \\( \\theta = \\pi \\) \u306e\u3068\u304d\u306e\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f\u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\uff0c<\/p>\n $$ e^{i \\pi} = \\cos \\pi + i \\sin \\pi = -1$$$$\\therefore e^{i \\pi} + 1 = 0$$<\/p>\n \u7279\u306b\u3053\u306e\u5f0f\u3092\u30aa\u30a4\u30e9\u30fc\u306e\u7b49\u5f0f<\/strong><\/span>\u3068\u547c\u3093\u3067\u3044\u308b\u3002\u30aa\u30a4\u30e9\u30fc\u306e\u7b49\u5f0f\u306e\u4f55\u304c\u3059\u3054\u3044\u304b\u3068\u3044\u3046\u3068<\/p>\n \u3068\u3044\u3046\u5f79\u8005\u304c\uff0c\u52a0\u6cd5\uff0c\u4e57\u6cd5\uff0c\u6307\u6570\u95a2\u6570\u306b\u3088\u3063\u3066\u898b\u4e8b\u306b\u7d50\u3073\u4ed8\u3051\u3089\u308c\u3066\u3044\u308b\u3068\u3044\u3046\u3053\u3068\uff01<\/p>\n \u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f$$e^{i\\theta} = \\cos\\theta + i \\sin\\theta$$ \u3068\uff0c \\(\\theta\\) \u3092\\(-\\theta\\) \u306b\u3057\u305f $$e^{-i\\theta} = \\cos(-\\theta) + i\\sin(-\\theta) = \\cos\\theta -i \\sin\\theta$$ \u3092\u8db3\u3057\u305f\u308a\u5f15\u3044\u305f\u308a\u3059\u308b\u3068\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n \\begin{eqnarray} \u4e00\u65b9\uff0c\u53cc\u66f2\u7dda\u95a2\u6570\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u3066\u3044\u305f\u3002<\/p>\n \\begin{eqnarray} \u4e09\u89d2\u95a2\u6570\u3084\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u5909\u6570\u304c\u865a\u6570\u3067\u3082\u3044\u3044\u306e\u3060\u3068\u62e1\u5f35\u3059\u308b\u3068\uff0c<\/p>\n \\begin{eqnarray} \u3053\u308c\u3089\u304c\uff0c\u4e09\u89d2\u95a2\u6570\u3068\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u30a2\u30a4\u3067\u7d50\u3070\u308c\u305f<\/strong><\/span>\u95a2\u4fc2\u3067\u3042\u308b\u3002<\/p>\n \u4e09\u89d2\u95a2\u6570\u3068\u53cc\u66f2\u7dda\u95a2\u6570\u306f\uff0c\u305f\u3060\u307e\u304e\u3089\u308f\u3057\u3044\u307b\u3069\u306b\u4f3c\u305f\u8868\u8a18\u306a\u3060\u3051\u3067\u306a\u304f\uff0c\u5bc6\u63a5\u306a\uff08\u30a2\u30a4\u3067\u7d50\u3070\u308c\u305f<\/strong><\/span>\uff09\u95a2\u4fc2\u306a\u306e\u3060\u3068\u3044\u3046\u3053\u3068\u304c\u308f\u304b\u308b\u3068\u601d\u3046\u3002<\/p>\n <\/p>\n \u3042\u308a\u304c\u3068\u3046! \u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f!!<\/strong><\/span><\/p>\n <\/p>\n","protected":false},"excerpt":{"rendered":" \uff08\u3042\u3093\u305f\u306e\u516c\u5f0f\u3067\u306f\u306a\u304f\uff09\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f\u3068\u306f\uff0c<\/p>\n $$e^{i \\theta} = \\cos \\theta + i \\sin \\theta$$<\/p>\u30aa\u30a4\u30e9\u30fc\u306e\u7b49\u5f0f<\/h3>\n
\n
\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f\u304b\u3089\u307f\u305f\u4e09\u89d2\u95a2\u6570\u3068\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u95a2\u4fc2<\/h3>\n
\n\\cos\\theta &=& \\frac{e^{i\\theta} + e^{-i\\theta}}{2} \\\\
\n\\sin\\theta &=& \\frac{e^{i\\theta} -e^{-i\\theta}}{2 i}
\n\\end{eqnarray}<\/p>\n
\n\\cosh x &=& \\frac{e^x + e^{-x}}{2}\\\\
\n\\sinh x &=& \\frac{e^x -e^{-x}}{2}
\n\\end{eqnarray}<\/p>\n
\n\\cosh (i\\theta) &=& \\frac{e^{i\\theta} + e^{-i\\theta}}{2} = \\cos\\theta\\\\
\n\\sinh (i\\theta) &=& \\frac{e^{i\\theta} -e^{-i\\theta}}{2} = i \\sin\\theta\\\\
\n\\cos(i x) &=& \\frac{e^{i\\cdot i x} + e^{-i\\cdot i x}}{2} = \\frac{e^{x} + e^{-x}}{2} = \\cosh x\\\\
\n\\sin(i x) &=&\\frac{e^{i\\cdot i x} -e^{-i\\cdot i x}}{2 i} = i \\frac{e^x -e^{-x}}{2} = i \\sinh x
\n\\end{eqnarray}<\/p>\n