{"id":2127,"date":"2024-04-16T16:45:52","date_gmt":"2024-04-16T07:45:52","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2127"},"modified":"2024-04-18T10:55:31","modified_gmt":"2024-04-18T01:55:31","slug":"%e5%8f%8c%e6%9b%b2%e7%b7%9a%e9%96%a2%e6%95%b0%e3%81%ae%e5%be%ae%e5%88%86","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\/%e5%8f%8c%e6%9b%b2%e7%b7%9a%e9%96%a2%e6%95%b0%e3%81%ae%e5%be%ae%e5%88%86\/","title":{"rendered":"\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u5b9a\u7fa9\u3068\u305d\u306e\u5fae\u5206"},"content":{"rendered":"<p>\u4e09\u89d2\u95a2\u6570\u3068\u7d1b\u3089\u308f\u3057\u3044\u8868\u8a18\u306e\u300c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u53cc\u66f2\u7dda\u95a2\u6570<\/strong><\/span>\u300d\u306e\u5b9a\u7fa9\u3068\u305d\u306e\u5fae\u5206\u3002<br \/>\n$$\\cosh x \\equiv \\frac{e^x + e^{-x}}{2}, \\quad \\sinh x \\equiv \\frac{e^x -e^{-x}}{2}, \\quad \\tanh x \\equiv \\frac{\\sinh x}{\\cosh x}$$$$(\\cosh x)&#8217; = \\sinh x, \\quad (\\sinh x)&#8217; = \\cosh x, \\quad (\\tanh x)&#8217; = \\frac{1}{\\cosh^2 x}$$<br \/>\n<!--more--><\/p>\n<hr \/>\n<p id=\"yui_3_17_2_1_1645506795464_1412\" dir=\"ltr\">\u6307\u6570\u95a2\u6570\u304a\u3088\u3073\u4e09\u89d2\u95a2\u6570\u306b\u95a2\u9023\u3057\u3066\uff0c\u300c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong id=\"yui_3_17_2_1_1645506795464_1518\">\u53cc\u66f2\u7dda\u95a2\u6570<\/strong><\/span>\u300d\u306e\u5b9a\u7fa9\u3068\u305d\u306e\u5fae\u5206\u306b\u3064\u3044\u3066\u307e\u3068\u3081\u308b\u3002\u53cc\u66f2\u7dda\u95a2\u6570\u306f\u4e09\u89d2\u95a2\u6570\u3068\u7d1b\u3089\u308f\u3057\u3044\u8868\u8a18\u3067\u3042\u308a\uff0c\u305d\u306e\u6027\u8cea\u3082\u306a\u3093\u3068\u306a\u304f\u985e\u4f3c\u6027\u304c\u3042\u308b\u3002\u5f8c\u306b\u300c<a 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\/\"><span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u4eba\u985e\u306e\u81f3\u5b9d\uff1a\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f<\/strong><\/span><\/a>\u300d\u306e\u6bb5\u3067\uff0c\u53cc\u66f2\u7dda\u95a2\u6570\u3068\u4e09\u89d2\u95a2\u6570\u306f\u5bc6\u63a5\u306a\u95a2\u4fc2\u304c\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u306e\u3067\uff0c\u305d\u3053\u307e\u3067\u306f\u3057\u3070\u3089\u304f\u8f9b\u62b1\u3057\u3066\u304f\u3060\u3055\u3044\u3002\u53cc\u66f2\u7dda\u95a2\u6570\u306f\u30b9\u30de\u30db\u30a2\u30d7\u30ea\u306e\u300c\u8a08\u7b97\u6a5f\u300d\u3067\u3082\u4f7f\u3048\u307e\u3059\u3002\uff08\u4ee5\u4e0b\u306f iPhone \u306e\u4f8b\u3002\u6a2a\u5411\u304d\u306b\u3059\u308b\u3002\uff09<\/p>\n<p dir=\"ltr\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-large wp-image-8441\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/IMG_7599-640x360.png\" alt=\"\" width=\"640\" height=\"360\" srcset=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/\/IMG_7599-640x360.png 640w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/\/IMG_7599-300x169.png 300w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/\/IMG_7599-750x422.png 750w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/\/IMG_7599.png 1334w\" sizes=\"auto, (max-width: 640px) 100vw, 640px\" \/><\/p>\n<p dir=\"ltr\">\u307e\u305f\uff0c\u53cc\u66f2\u7dda\u95a2\u6570\u306a\u3093\u3066\u6240\u8a6e\uff0c\u6307\u6570\u95a2\u6570\u304b\u3089\u4f5c\u3089\u308c\u308b\u3060\u3051\u3060\u304b\u3089\uff0c\u6307\u6570\u95a2\u6570\u3055\u3048\u899a\u3048\u3066\u304a\u3051\u3070\u308f\u3056\u308f\u3056\u53cc\u66f2\u7dda\u95a2\u6570\u306a\u3069\u3068\u3042\u3089\u305f\u3081\u3066\u899a\u3048\u3066\u304a\u304f\u5fc5\u8981\u306f\u306a\u3044\u3060\u308d\u3046&#8230; \u3068\u3044\u3046\u4eba\u3082\u3044\u308b\u304b\u3082\u3057\u308c\u306a\u3044\u3002\u3057\u304b\u3057\uff0c\u529b\u5b66\u306e\u6e1b\u8870\u632f\u52d5\u554f\u984c\u3084\uff0c\u5b87\u5b99\u8ad6\u306b\u304a\u3044\u3066\u30d5\u30ea\u30fc\u30c9\u30de\u30f3\uff08FLRW\uff09\u30e2\u30c7\u30eb\u306b\u95a2\u308f\u308b\u554f\u984c\u3067\u306f\uff0c\u53cc\u66f2\u7dda\u95a2\u6570\u3092\u4f7f\u3063\u305f\u7d71\u4e00\u7684\u7406\u89e3\u304c\u4e0d\u53ef\u6b20\u3067\u3042\u308b\u3002<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u3060\u304b\u3089\uff0c\u5b87\u5b99\u8ad6\u3092\u52c9\u5f37\u3057\u305f\u3044\u4eba\uff08\u3084\uff0c\u5c11\u306a\u304f\u3068\u3082\u529b\u5b66\u306e\u6e1b\u8870\u632f\u52d5\u554f\u984c\u3092\u3059\u3063\u304d\u308a\u7406\u89e3\u3057\u305f\u3044\u4eba\uff09\u306f\uff0c\u3057\u3063\u304b\u308a\u3068\u53cc\u66f2\u7dda\u95a2\u6570\u3092\u7406\u89e3\u3057\u3066\u304a\u3044\u305f\u307b\u3046\u304c\u7d76\u5bfe\u304a\u5f97\uff01\u3067\u3059\uff08\u305f\u3076\u3093\uff09<\/strong><\/span>\u3002<\/p>\n<p dir=\"ltr\">\u305f\u3068\u3048\u3070\uff0c\u5b87\u5b99\u81a8\u5f35\u3092\u3042\u3089\u308f\u3059\u30b9\u30b1\u30fc\u30eb\u56e0\u5b50\u306e\u8868\u8a18\u306b\u306f\u4e09\u89d2\u95a2\u6570\u3084\u53cc\u66f2\u7dda\u95a2\u6570\u304c\u51fa\u3066\u304d\u307e\u3059\u3088\u3002\u4ee5\u4e0b\u306e\u30da\u30fc\u30b8\u306a\u3069\u3092\u53c2\u7167\uff1a<\/p>\n<ul>\n<li dir=\"ltr\"><a href=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e4%b8%80%e8%88%ac%e7%9b%b8%e5%af%be%e8%ab%96%e7%9a%84%e5%ae%87%e5%ae%99%e8%ab%96\/%e5%ae%87%e5%ae%99%e8%ab%96%e3%83%91%e3%83%a9%e3%83%a1%e3%83%bc%e3%82%bf%e3%81%a8%e5%ae%87%e5%ae%99%e5%b9%b4%e9%bd%a2\/%e8%a3%9c%e8%b6%b3%ef%bc%9a%e3%82%b9%e3%82%b1%e3%83%bc%e3%83%ab%e5%9b%a0%e5%ad%90%e3%81%ae%e8%a7%a3\/\" target=\"_blank\" rel=\"noopener\">\u88dc\u8db3\uff1a\u30b9\u30b1\u30fc\u30eb\u56e0\u5b50\u306e\u89e3<\/a><\/li>\n<\/ul>\n<h3 id=\"yui_3_17_2_1_1645506795464_1519\">\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u5b9a\u7fa9<\/h3>\n<h4 id=\"yui_3_17_2_1_1645506795464_1521\">\u30cf\u30a4\u30d1\u30dc\u30ea\u30c3\u30af\u30b3\u30b5\u30a4\u30f3 $\\cosh x$<\/h4>\n<p id=\"yui_3_17_2_1_1645506795464_1522\" dir=\"ltr\">$$y = \\cosh x \\equiv \\frac{e^x + e^{-x}}{2}$$ \\(\\cosh x\\) \u306e\u8aad\u307f\u65b9\u306f\u300c\u30cf\u30a4\u30d1\u30dc\u30ea\u30c3\u30af\u30b3\u30b5\u30a4\u30f3\u30fb\u30a8\u30c3\u30af\u30b9\u300d\u3002\u5b9a\u7fa9\u57df\u3068\u5024\u57df\u306f\uff0c<\/p>\n<p dir=\"ltr\">$$ -\\infty &lt; x &lt; \\infty, \\qquad 1 \\leq y &lt; \\infty$$<\/p>\n<p dir=\"ltr\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-large wp-image-8392\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/spbmathBcosh.svg\" alt=\"\" width=\"640\" height=\"481\" \/><\/p>\n<h4 id=\"yui_3_17_2_1_1645506795464_1524\">\u30cf\u30a4\u30d1\u30dc\u30ea\u30c3\u30af\u30b5\u30a4\u30f3 $\\sinh x$<\/h4>\n<p dir=\"ltr\">$$y = \\sinh x \\equiv \\frac{e^x -e^{-x}}{2}$$ \\(\\sinh x\\) \u306e\u8aad\u307f\u65b9\u306f\u300c\u30cf\u30a4\u30d1\u30dc\u30ea\u30c3\u30af\u30b5\u30a4\u30f3\u30fb\u30a8\u30c3\u30af\u30b9\u300d\u3002<\/p>\n<p id=\"yui_3_17_2_1_1645506795464_1522\" dir=\"ltr\">\u5b9a\u7fa9\u57df\u3068\u5024\u57df\u306f\uff0c<\/p>\n<p dir=\"ltr\">$$ -\\infty &lt; x &lt; \\infty, \\qquad -\\infty &lt; y &lt; \\infty$$<\/p>\n<p dir=\"ltr\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-large wp-image-8393\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/spbmathBsinh.svg\" alt=\"\" width=\"640\" height=\"481\" \/><\/p>\n<h4 id=\"yui_3_17_2_1_1645506795464_1527\">\u30cf\u30a4\u30d1\u30dc\u30ea\u30c3\u30af\u30bf\u30f3\u30b8\u30a7\u30f3\u30c8 $\\tanh x$<\/h4>\n<p>$$y = \\tanh x \\equiv \\frac{\\sinh x}{\\cosh x} = \\frac{e^x -e^{-x}}{e^x + e^{-x}}$$ \\(\\tanh x\\) \u306e\u8aad\u307f\u65b9\u306f\u300c\u30cf\u30a4\u30d1\u30dc\u30ea\u30c3\u30af\u30bf\u30f3\u30b8\u30a7\u30f3\u30c8\u30fb\u30a8\u30c3\u30af\u30b9\u300d\u3002<\/p>\n<p id=\"yui_3_17_2_1_1645506795464_1522\" dir=\"ltr\">\u5b9a\u7fa9\u57df\u3068\u5024\u57df\u306f\uff0c<\/p>\n<p dir=\"ltr\">$$ -\\infty &lt; x &lt; \\infty, \\qquad -1 &lt; y &lt; 1$$<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-large wp-image-8394\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/spbmathBtanh.svg\" alt=\"\" width=\"640\" height=\"481\" \/><\/p>\n<h3 id=\"yui_3_17_2_1_1645506795464_1529\">\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u6027\u8cea<\/h3>\n<p id=\"yui_3_17_2_1_1645506795464_1530\">\\( \\cosh x\\) \u306f\u5076\u95a2\u6570\u3067\u3042\u308a\uff0c\\(x = 0\\) \u3067\u306e\u5024\u306f \\(1\\)\u3002\uff08\u305d\u3046\u3044\u3048\u3070 \\(\\cos x\\) \u3082\u5076\u95a2\u6570\u3067\u3042\u308a\uff0c\\(x = 0\\) \u3067\u306e\u5024\u306f\\(1\\) \u3060\u3063\u305f\u306a\u3041\u3002\uff09<\/p>\n<p id=\"yui_3_17_2_1_1645506795464_1531\">$$ \\cosh (-x) = \\frac{e^{-x} + e^{x}}{2} = \\frac{e^x + e^{-x}}{2} = \\cosh x$$<br id=\"yui_3_17_2_1_1645506795464_1532\" \/>$$ \\cosh\u00a0 0 = \\frac{e^0 + e^{-0}}{2} = \\frac{1 + 1}{2} = 1$$<\/p>\n<p id=\"yui_3_17_2_1_1645506795464_1534\">\\( \\sinh x\\) \u306f\u5947\u95a2\u6570\u3067\u3042\u308a\uff0c\\(x = 0\\) \u3067\u306e\u5024\u306f \\(0\\)\u3002\uff08\u305d\u3046\u3044\u3048\u3070 \\(\\sin x\\) \u3082\u5947\u95a2\u6570\u3067\u3042\u308a\uff0c\\(x = 0\\) \u3067\u306e\u5024\u306f\\(0\\) \u3060\u3063\u305f\u306a\u3041\u3002\uff09<\/p>\n<p>$$ \\sinh (-x) = \\frac{e^{-x} -e^{x}}{2} = -\\frac{e^x -e^{-x}}{2} = -\\sinh x$$<\/p>\n<p id=\"yui_3_17_2_1_1645506795464_1537\">$$ \\sinh\u00a0 0 = \\frac{e^0 -e^{-0}}{2} = \\frac{ 1 -1}{2} = 0$$<\/p>\n<p id=\"yui_3_17_2_1_1645506795464_1538\">\\( (\\cosh x)^2\\) \u3068 \\( (\\sinh x)^2\\) \u3068\u306e\u9593\u306e\u95a2\u4fc2\u3002<\/p>\n<p id=\"yui_3_17_2_1_1645506795464_1540\">$$\\cosh^2 x -\\sinh^2 x = \\frac{e^{2x} + 2 + e^{-2x}}{4} -\\frac{e^{2x} -2 + e^{-2x}}{4} = 1$$\uff08\u305d\u3046\u3044\u3048\u3070\uff0c\\(\\cos^2 x + \\sin^2 x = 1\\) \u3068\u7b26\u53f7\u304c\u3061\u3087\u3063\u3068\u9055\u3046\u3051\u3069\u4f3c\u3066\u308b\u306a\u3041\u3002\uff09<\/p>\n<p id=\"yui_3_17_2_1_1645506795464_1541\">\u3055\u3089\u306b\u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u300c\u52a0\u6cd5\u5b9a\u7406\u300d\u3082\u6210\u308a\u7acb\u3064\u3002<\/p>\n<p id=\"yui_3_17_2_1_1645506795464_1542\">$$\\cosh(x + y) = \\cosh x \\cosh y + \\sinh x \\sinh y$$<br id=\"yui_3_17_2_1_1645506795464_1543\" \/>$$\\sinh(x + y) = \\sinh x \\cosh y + \\cosh x \\sinh y$$<br id=\"yui_3_17_2_1_1645506795464_1544\" \/>\uff08\u305d\u3046\u3044\u3048\u3070\uff0c<br id=\"yui_3_17_2_1_1645506795464_1545\" \/>$$ \\cos(x + y) = \\cos x \\cos y -\\sin x \\sin y, \\ \\ \\sin(x+y) = \\sin x \\cos y + \\cos x \\sin y$$\u3068\u7b26\u53f7\u304c\u3061\u3087\u3063\u3068\u9055\u3046\u3051\u3069\u4f3c\u3066\u308b\u306a\u3041\u3002\uff09<\/p>\n<p id=\"yui_3_17_2_1_1645506795464_1546\">\u76f4\u63a5\u8a08\u7b97\u3057\u3066\u8a3c\u660e\u3057\u3066\u307f\u308b\u3002<br id=\"yui_3_17_2_1_1645506795464_1547\" \/>\\begin{eqnarray}<br id=\"yui_3_17_2_1_1645506795464_1548\" \/>\\cosh x \\cosh y + \\sinh x \\sinh y &amp;=&amp; \\frac{e^x + e^{-x}}{2} \\frac{e^y + e^{-y}}{2} + \\frac{e^x -e^{-x}}{2}\\frac{e^y -e^{-y}}{2}\\\\<br id=\"yui_3_17_2_1_1645506795464_1549\" \/>&amp;=&amp; \\frac{e^{x+y} + e^{x-y} +e^{-x+y} +e^{-x-y} }{4} \\\\<br id=\"yui_3_17_2_1_1645506795464_1550\" \/>&amp;&amp; \\quad\u00a0 + \\frac{e^{x+y} -e^{x-y} -e^{-x+y} +e^{-x-y}}{4} \\\\<br id=\"yui_3_17_2_1_1645506795464_1551\" \/>&amp;=&amp; \\frac{2e^{x+y} + 2e^{-(x+y)}}{4} = \\cosh (x + y)<br id=\"yui_3_17_2_1_1645506795464_1552\" \/>\\end{eqnarray}\\(\\sinh(x + y)\\) \u306b\u3064\u3044\u3066\u3082\u540c\u69d8\u3002<\/p>\n<h3 id=\"yui_3_17_2_1_1645506795464_1555\">\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u5fae\u5206<\/h3>\n<p id=\"yui_3_17_2_1_1645506795464_1556\">\u5148\u306b\u7b54\u3048\u3092\u307e\u3068\u3081\u3066\u304a\u304f\u3002<\/p>\n<p id=\"yui_3_17_2_1_1645506795464_1557\">$$(\\cosh x)&#8217; = \\sinh x, \\quad (\\sinh x)&#8217; = \\cosh x, \\quad (\\tanh x)&#8217; = \\frac{1}{\\cosh^2 x}$$<\/p>\n<p id=\"yui_3_17_2_1_1645506795464_1558\">\uff08\u305d\u3046\u3044\u3048\u3070\uff0c$$ (\\cos x)&#8217; = -\\sin x, \\quad (\\sin x)&#8217; = \\cos x, \\quad (\\tan x)&#8217; = \\frac{1}{\\cos^2 x}$$\u3068\u7b26\u53f7\u304c\u3061\u3087\u3063\u3068\u9055\u3046\u3051\u3069\u306b\u3066\u308b\u306a\u3041\u3002\uff09<\/p>\n<p id=\"yui_3_17_2_1_1645506795464_1559\">\\((\\cosh x)&#8217;, (\\sinh x)&#8217; \\) \u306b\u3064\u3044\u3066\u306f\u5b9a\u7fa9\u304b\u3089\u660e\u3089\u304b\u3002\u5ff5\u306e\u305f\u3081\u306b \\( (\\cosh x)&#8217; \\) \u306b\u3064\u3044\u3066\u5fae\u5206\u3057\u3066\u307f\u308b\u3068\uff0c<br id=\"yui_3_17_2_1_1645506795464_1560\" \/>$$(\\cosh x)&#8217; = \\left(\\frac{e^x + e^{-x}}{2}\\right)&#8217; = \\frac{e^x -e^{-x}}{2} = \\sinh x$$ \\( (\\sinh x)&#8217; \\) \u306b\u3064\u3044\u3066\u3082\u7c21\u5358\u306b\u5c0e\u3051\u308b\u3002<\/p>\n<p id=\"yui_3_17_2_1_1645506795464_1419\">\u307e\u305f\uff0c\\((\\tanh x)&#8217;\\) \u306b\u3064\u3044\u3066\u306f\uff0c<br id=\"yui_3_17_2_1_1645506795464_1562\" \/>\\begin{eqnarray}<br id=\"yui_3_17_2_1_1645506795464_1563\" \/>(\\tanh x)&#8217; &amp;=&amp; \\left(\\frac{\\sinh x}{\\cosh x}\\right)&#8217; \\\\<br id=\"yui_3_17_2_1_1645506795464_1564\" \/>&amp;=&amp; \\frac{(\\sinh x)&#8217; \\cosh x -\\sinh x (\\cosh x)&#8217;}{\\cosh^2 x} \\\\<br id=\"yui_3_17_2_1_1645506795464_1565\" \/>&amp;=&amp; \\frac{\\cosh x\\, \\cosh x -\\sinh x \\, \\sinh x}{\\cosh^2 x} = \\frac{1}{\\cosh^2 x}<br id=\"yui_3_17_2_1_1645506795464_1566\" \/>\\end{eqnarray}<\/p>\n<h3>\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u30b0\u30e9\u30d5<\/h3>\n<p>3\u3064\u307e\u3068\u3081\u3066\u30b0\u30e9\u30d5\u306b\u3059\u308b\u3068&#8230;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-large wp-image-8108\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/pmathB09-2.svg\" alt=\"\" width=\"640\" height=\"640\" \/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u4e09\u89d2\u95a2\u6570\u3068\u7d1b\u3089\u308f\u3057\u3044\u8868\u8a18\u306e\u300c\u53cc\u66f2\u7dda\u95a2\u6570\u300d\u306e\u5b9a\u7fa9\u3068\u305d\u306e\u5fae\u5206\u3002 $$\\cosh x \\equiv \\frac{e^x + e^{-x}}{2}, \\quad \\sinh x \\equiv \\frac{e^x -e^{-x}}{2}, \\quad \\tanh x \\equiv \\frac{\\sinh x}{\\cosh x}$$$$(\\cosh x)&#8217; = \\sinh x, \\quad (\\sinh x)&#8217; = \\cosh x, \\quad (\\tanh x)&#8217; = \\frac{1}{\\cosh^2 x}$$<\/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\/%e5%8f%8c%e6%9b%b2%e7%b7%9a%e9%96%a2%e6%95%b0%e3%81%ae%e5%be%ae%e5%88%86\/\">\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"parent":2068,"menu_order":8,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-2127","page","type-page","status-publish","hentry","nodate","item-wrap"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2127","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/users\/33"}],"replies":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/comments?post=2127"}],"version-history":[{"count":14,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2127\/revisions"}],"predecessor-version":[{"id":8444,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2127\/revisions\/8444"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2068"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=2127"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}