{"id":2142,"date":"2024-05-17T21:00:24","date_gmt":"2024-05-17T12:00:24","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2142"},"modified":"2024-05-23T14:07:09","modified_gmt":"2024-05-23T05:07:09","slug":"%e4%ba%ba%e9%a1%9e%e3%81%ae%e8%87%b3%e5%ae%9d%ef%bc%9a%e3%82%aa%e3%82%a4%e3%83%a9%e3%83%bc%e3%81%ae%e5%85%ac%e5%bc%8f","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6b\/%e4%ba%ba%e9%a1%9e%e3%81%ae%e8%87%b3%e5%ae%9d%ef%bc%9a%e3%82%aa%e3%82%a4%e3%83%a9%e3%83%bc%e3%81%ae%e5%85%ac%e5%bc%8f\/","title":{"rendered":"\u4eba\u985e\u306e\u81f3\u5b9d : \u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f"},"content":{"rendered":"<p>\uff08\u3042\u3093\u305f\u306e\u516c\u5f0f\u3067\u306f\u306a\u304f\uff09<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f<\/strong><\/span>\u3068\u306f\uff0c<\/p>\n<p>$$e^{i \\theta} = \\cos \\theta + i \\sin \\theta$$<\/p>\n<p>\uff08\u5927\u4eba\u306b\u306a\u3063\u3066\uff09<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f<\/strong><\/span>\u304c\u308f\u304b\u308b\u3088\u3046\u306b\u306a\u308b\u3068\uff0c<span style=\"font-family: helvetica, arial, sans-serif; color: #ff0000;\"><strong>\u30a2\u30a4\u3067\u7d50\u3070\u308c\u305f\u5bc6\u63a5\u306a\u95a2\u4fc2<\/strong><\/span>\u3082\u308f\u304b\u308b\u3088\u3046\u306b\u306a\u308a\u307e\u3059\uff01\uff1f<!--more--><\/p>\n<hr \/>\n<h3 id=\"yui_3_17_2_1_1645509517222_1431\">\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\uff08\u30de\u30af\u30ed\u30fc\u30ea\u30f3\u5c55\u958b\uff09\u306b\u3088\u308b\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f\u306e\u8a3c\u660e<\/h3>\n<p>\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<p>$$f(x)\u00a0 = f(0)+ f'(x) x + \\frac{1}{2} f^{&#8221;}(0) x^2 + \\cdots + \\frac{1}{n!} f^{(n)}(0) x^n + \\cdots$$<\/p>\n<p>\u306e\u516c\u5f0f\u306b\u5f93\u3063\u3066\uff0c<\/p>\n<p>$$ e^{x} = 1 + x + \\frac{1}{2} x^2 + \\frac{1}{3!} x^3 + \\frac{1}{4!} x^4 + \\cdots$$<\/p>\n<p>\\( x = i \\theta \\) \u3092\u5165\u308c\u308b\u3068<\/p>\n<p>\\begin{eqnarray}<br \/>\ne^{i \\theta} &amp;=&amp;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\\\\<br \/>\n&amp;=&amp;\u00a0 {\\color{red}{1 -\\frac{1}{2!} \\theta^2 + \\frac{1}{4!} \\theta^4 -\\cdots}} \\\\<br \/>\n&amp;&amp; + i\\,\\left\\{ {\\color{blue}{ \\theta -\\frac{1}{3!} \\theta^3 + \\frac{1}{5!} \\theta^5 -\\cdots}}\\right\\}<br \/>\n\\end{eqnarray}<\/p>\n<p>\u4e00\u65b9\uff0c\\(\\cos \\theta, \\sin \\theta\\) \u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u304c<\/p>\n<p>$${\\color{red}{\\cos \\theta}} = \\color{red}{1 -\\frac{1}{2!} \\theta^2 + \\frac{1}{4!} \\theta^4 -\\cdots}$$<\/p>\n<p>$${\\color{blue}{\\sin \\theta}} = \\color{blue}{\\theta -\\frac{1}{3!} \\theta^3 + \\frac{1}{5!} \\theta^5 -\\cdots}$$<\/p>\n<p>\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<p>$$e^{i \\theta} = {\\color{red}{\\cos \\theta}} + i\\,\u00a0 {\\color{blue}{\\sin\\theta}}$$<\/p>\n<p>\u3053\u308c\u3053\u305d\u304c\uff0c\u4eba\u985e\u306e\u81f3\u5b9d\uff01<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f<\/strong><\/span>\u3067\u3042\u308b\uff01<\/p>\n<ul>\n<li>\u30d5\u30a1\u30a4\u30f3\u30de\u30f3\u7269\u7406\u5b66 \u2160 \u529b\u5b66\uff0cp.307\uff0c\u300c\u3053\u308c\u306f\u6211\u3005\u306e\u81f3\u5b9d\u3067\u3042\u308b\u3002\u300d<\/li>\n<\/ul>\n<h3>\u30aa\u30a4\u30e9\u30fc\u306e\u7b49\u5f0f<\/h3>\n<p>\\( \\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<p>$$ e^{i \\pi} = \\cos \\pi + i \\sin \\pi = -1$$$$\\therefore\\ \\\u00a0 e^{i \\pi} + 1 = 0$$<\/p>\n<p>\u7279\u306b\u3053\u306e\u5f0f\u3092<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\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<ul>\n<li>\u30bc\u30ed \\(0\\)\uff0c\u5358\u4f4d\u5143 \\(1\\) \u3068\u3044\u3046\u6574\u6570\u306e\u3082\u3063\u3068\u3082\u57fa\u672c\u3068\u306a\u308b\u6570<\/li>\n<li>\u7121\u7406\u6570\u306e\u4ee3\u8868\u9078\u624b\uff0c\u5186\u5468\u7387 \\(\\pi\\)\uff0c\u81ea\u7136\u5bfe\u6570\u306e\u5e95 \\(e\\)<\/li>\n<li>\u305d\u3057\u3066\u865a\u6570\u5358\u4f4d \\(i\\)<\/li>\n<\/ul>\n<p>\u3068\u3044\u3046\u3044\u305a\u308c\u52a3\u3089\u306c\u30b9\u30fc\u30d1\u30fc\u30b9\u30bf\u30fc\u9054\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<h3>\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<p>\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f$$e^{i\\theta} = \\cos\\theta + i \\sin\\theta$$ \u3068\uff0c \\(\\theta\\) \u3092\\( &#8211; \\theta\\) \u306b\u3057\u305f<\/p>\n<p>\\begin{eqnarray}<br \/>\ne^{ &#8211; i\\theta} &amp;=&amp; \\cos( &#8211; \\theta) + i\\sin( &#8211; \\theta) \\\\<br \/>\n&amp;=&amp; \\cos\\theta -i \\sin\\theta<br \/>\n\\end{eqnarray}<\/p>\n<p>\u306e\u8fba\u3005\uff08\u5de6\u8fba\u540c\u58eb\uff0c\u53f3\u8fba\u540c\u58eb\uff09\u3092\u8db3\u3059\u3068&#8230;<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\ \u00a0 &amp; e^{i\\theta} &amp;= \\cos\\theta + i \\sin\\theta \\\\<br \/>\n+)&amp; e^{ &#8211; i \\theta} &amp;= \\cos\\theta -i \\sin\\theta \\\\<br \/>\n\\hline<br \/>\n&amp;e^{i\\theta} + e^{ &#8211; i\\theta} &amp;= 2 \\cos\\theta \\\\ \\ \\\\<br \/>\n\\therefore\\ \\ &amp;\\frac{e^{i\\theta} + e^{ &#8211; i\\theta}}{2} &amp;= \\cos\\theta<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3042\u308b\u3044\u306f\u8fba\u3005\u5f15\u304f\u3068&#8230;<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\ &amp; e^{i\\theta} &amp;= \\cos\\theta + i \\sin\\theta \\\\<br \/>\n-)&amp;e^{ &#8211; i\\theta} &amp;= \\cos\\theta -i \\sin\\theta \\\\<br \/>\n\\hline<br \/>\n&amp;e^{i\\theta} -e^{ &#8211; i\\theta} &amp;= 2 i \\sin\\theta \\\\ \\ \\\\<br \/>\n\\therefore\\ \\ &amp;\\frac{e^{i\\theta} -e^{ &#8211; i\\theta}}{2 i} &amp;= \\sin\\theta<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3064\u307e\u308a\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\cos\\theta &amp;=&amp; \\frac{e^{i\\theta} + e^{ &#8211; i\\theta}}{2} \\\\<br \/>\n\\sin\\theta &amp;=&amp; \\frac{e^{i\\theta} -e^{ &#8211; i\\theta}}{2 i}<br \/>\n\\end{eqnarray}<\/p>\n<p>\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<p>\\begin{eqnarray}<br \/>\n\\cosh x &amp;=&amp; \\frac{e^x + e^{ &#8211; x}}{2}\\\\<br \/>\n\\sinh x &amp;=&amp; \\frac{e^x -e^{ &#8211; x}}{2}<br \/>\n\\end{eqnarray}<\/p>\n<p>\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<p>\\begin{eqnarray}<br \/>\n\\cosh (i\\theta) &amp;=&amp; \\frac{e^{i\\theta} + e^{ &#8211; i\\theta}}{2} = \\cos\\theta\\\\<br \/>\n\\sinh (i\\theta) &amp;=&amp; \\frac{e^{i\\theta} -e^{ &#8211; i\\theta}}{2} = i \\sin\\theta\\\\<br \/>\n\\cos(i x) &amp;=&amp; \\frac{e^{i\\cdot i x} + e^{ &#8211; i\\cdot i x}}{2} = \\frac{e^{x} + e^{ &#8211; x}}{2} = \\cosh x\\\\<br \/>\n\\sin(i x) &amp;=&amp;\\frac{e^{i\\cdot i x} -e^{ &#8211; i\\cdot i x}}{2 i} = i \\frac{e^x -e^{ &#8211; x}}{2} = i \\sinh x<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3053\u308c\u3089\u304c\uff0c\u4e09\u89d2\u95a2\u6570\u3068\u53cc\u66f2\u7dda\u95a2\u6570\u306e<span style=\"font-family: helvetica, arial, sans-serif; color: #ff0000;\"><strong>\u30a2\u30a4\u3067\u7d50\u3070\u308c\u305f<\/strong><\/span>\u95a2\u4fc2\u3067\u3042\u308b\u3002<\/p>\n<p>\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\u5b9f\u306b\u5bc6\u63a5\u306a\uff08<span style=\"font-family: helvetica, arial, sans-serif; color: #ff0000;\"><strong>\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 id=\"yui_3_17_2_1_1645509517222_1435\" style=\"text-align: center;\"><span style=\"font-family: helvetica, arial, sans-serif; color: #ff0000;\"><strong>\u3042\u308a\u304c\u3068\u3046! \u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f!!<\/strong><\/span><\/p>\n<h3>\u9006\u4e09\u89d2\u95a2\u6570\u3068\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u95a2\u4fc2<\/h3>\n<p>\u3064\u3044\u3067\u306b\uff0c\u5b87\u5b99\u8ad6\u306e\u3068\u3053\u308d\u3067\u5b9f\u969b\u306b\u4f7f\u3046\u306e\u3067\uff0c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u9006\u4e09\u89d2\u95a2\u6570\u3068\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306e<span style=\"color: #ff0000;\">\u30a2\u30a4\u3067\u7d50\u3070\u308c\u305f<\/span>\u95a2\u4fc2<\/strong><\/span>\u306b\u3064\u3044\u3066\u3082\u307e\u3068\u3081\u3066\u304a\u304f\u3002<\/p>\n<p>\u307e\u305a\uff0c$\\cos$ \u7cfb\u306b\u3064\u3044\u3066\u306f<\/p>\n<p>$$\\cosh (i\\theta) = \\cos\\theta \\equiv x$$<\/p>\n<p>\u3068\u304a\u304f\u3068\uff0c$x = \\cosh (i\\theta), \\ x = \\cos\\theta$ \u3088\u308a<\/p>\n<p>$$\\cosh^{ &#8211; 1} x = i\\theta, \\quad \\cos^{ &#8211; 1} x = \\theta$$<\/p>\n<p>$$\\therefore\\ \\ \\cos^{ &#8211; 1} x = -i \\cosh^{ &#8211; 1} x$$<\/p>\n<p>$\\sin$ \u7cfb\u306b\u3064\u3044\u3066\u306f<\/p>\n<p>$$\\sinh (i\\theta) = i \\sin\\theta \\equiv x$$<\/p>\n<p>\u3068\u304a\u304f\u3068\uff0c$x = \\sinh (i\\theta), \\ -i x = \\sin\\theta$ \u3088\u308a<\/p>\n<p>$$\\sinh^{ &#8211; 1} x = i\\theta, \\quad \\sin^{ &#8211; 1} (i x)\u00a0 = -\\theta$$<\/p>\n<p>$$\\therefore\\ \\ \\sin^{ &#8211; 1} (i x) = i \\sinh^{ &#8211; 1} x$$<\/p>\n<p>\u540c\u69d8\u306b\u3057\u3066\uff0c\u4ee5\u4e0b\u3082\u308f\u304b\u308b\u3060\u308d\u3046\u3002<\/p>\n<p>$$\\sinh^{ &#8211; 1} (i x) = i \\sin^{ &#8211; 1} x$$<\/p>\n<p>$\\tan$ \u7cfb\u306b\u3064\u3044\u3066\u306f<\/p>\n<p>$$\\tanh (i\\theta) = i \\tan\\theta \\equiv x$$<\/p>\n<p>\u3068\u304a\u304f\u3068\uff0c$x = \\tanh (i\\theta), \\ -i x = \\tan\\theta$ \u3088\u308a<\/p>\n<p>$$\\tanh^{ &#8211; 1} x = i\\theta, \\quad \\tan^{ &#8211; 1} (i x)\u00a0 = -\\theta$$<\/p>\n<p>$$\\therefore\\ \\ \\tan^{ &#8211; 1} (i x) = i \\tanh^{ &#8211; 1} x$$<\/p>\n<p>\u307e\u305f\uff0c\u4ee5\u4e0b\u3082\u308f\u304b\u308b\u3067\u3042\u308d\u3046\u3002<\/p>\n<p>$$\\tanh^{ &#8211; 1} (i x) = i \\tan^{ &#8211; 1} x$$<\/p>\n<p>&#8230; \u4ee5\u4e0a\uff0c\u6570\u5b66\u306e\u6388\u696d\u3067\u611b\uff08\u30a2\u30a4\uff09\u3092\u8a9e\u308b\u6559\u54e1\uff0c\u845b\u897f\u3067\u3057\u305f&#8230;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\uff08\u3042\u3093\u305f\u306e\u516c\u5f0f\u3067\u306f\u306a\u304f\uff09\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f\u3068\u306f\uff0c<\/p>\n<p>$$e^{i \\theta} = \\cos \\theta + i \\sin \\theta$$<\/p>\n<p>\uff08\u5927\u4eba\u306b\u306a\u3063\u3066\uff09\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f\u304c\u308f\u304b\u308b\u3088\u3046\u306b\u306a\u308b\u3068\uff0c\u30a2\u30a4\u3067\u7d50\u3070\u308c\u305f\u5bc6\u63a5\u306a\u95a2\u4fc2\u3082\u308f\u304b\u308b\u3088\u3046\u306b\u306a\u308a\u307e\u3059\uff01\uff1f<\/p><p><a class=\"more-link btn\" href=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6b\/%e4%ba%ba%e9%a1%9e%e3%81%ae%e8%87%b3%e5%ae%9d%ef%bc%9a%e3%82%aa%e3%82%a4%e3%83%a9%e3%83%bc%e3%81%ae%e5%85%ac%e5%bc%8f\/\">\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"parent":2068,"menu_order":13,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-2142","page","type-page","status-publish","hentry","nodate","item-wrap"],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO 5.0.0.1 - aioseo.com -->\n\t<meta name=\"description\" content=\"\uff08\u3042\u3093\u305f\u306e\u516c\u5f0f\u3067\u306f\u306a\u304f\uff09\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f\u3068\u306f\uff0c $$e^{i \\theta} = \\cos \\theta + i\" \/>\n\t<meta name=\"robots\" content=\"max-image-preview:large\" \/>\n\t<link rel=\"canonical\" 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