{"id":2129,"date":"2024-04-16T16:45:15","date_gmt":"2024-04-16T07:45:15","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2129"},"modified":"2024-08-07T11:20:58","modified_gmt":"2024-08-07T02:20:58","slug":"%e9%80%86%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\/%e9%80%86%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":"\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u5b9a\u7fa9\u3068\u305d\u306e\u5fae\u5206"},"content":{"rendered":"

\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u8868\u8a18\u306f\uff0c
\n$$ \\cosh^{-1} x, \\quad \\sinh^{-1} x, \\quad \\tanh^{-1} x$$
\n\u307e\u305f\u306f
\n$$ \\mbox{arcosh } x, \\quad \\mbox{arsinh } x, \\quad \\mbox{artanh } x$$\u5fae\u5206\u306f\uff0c<\/p>\n

$$(\\cosh^{-1} x)’ = \\frac{1}{\\sqrt{x^2-1}}, \\ \\ (\\sinh^{-1} x)’ =\u00a0 \\frac{1}{\\sqrt{x^2 + 1}}, \\ \\ (\\tanh^{-1} x)’ =\u00a0 \\frac{1}{1-x^2}$$<\/p>\n

\u9006\u4e09\u89d2\u95a2\u6570\u306e\u3068\u304d\u306b\u306f \\(\\arccos x\\) \u306e\u3088\u3046\u306b \\(\\bf\\mbox{arc}\\)\u300c\u30a2\u30fc\u30af<\/strong><\/span>\u300d\u3060\u3063\u305f\u304c\uff0c\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306f\\(\\bf\\mbox{area}\\) \u306e\u7565\u3067\u3042\u308b \\(\\bf\\mbox{ar}\\) \u3068\u66f8\u304f\u3079\u304d\u3067\u3042\u308a\uff0c\\(\\mbox{arc}\\)\u300c\u30a2\u30fc\u30af<\/strong><\/span>\u300d\u3068\u66f8\u304f\u3079\u304d\u3067\u306f\u306a\u3044\u3002<\/p>\n

\n

\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u9006\u95a2\u6570\u3092\u4e00\u822c\u306b\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u3068\u3044\u3046\u3002\u9006\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\u3057\u3066 2nd \u3092\u30af\u30ea\u30c3\u30af\u3059\u308b\u3002\uff09<\/p>\n

\"\"<\/p>\n

\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u5b9a\u7fa9<\/h3>\n

$\\cosh^{-1} x$ \u307e\u305f\u306f $\\operatorname{arcosh} x$<\/h4>\n

\\(y = \\cosh^{-1} x= \\mbox{arcosh } x\\) \u3092 \\(y = \\cosh x\\) \u306e\u9006\u95a2\u6570\uff08\u3064\u307e\u308a\uff0c\u65b9\u7a0b\u5f0f \\(x = \\cosh y\\) \u3092 \\(y\\) \u306b\u3064\u3044\u3066\u89e3\u3044\u305f\u3082\u306e\uff09\u3068\u3057\u3066\u5b9a\u7fa9\u3059\u308b\u3002\u5b9a\u7fa9\u57df\u3068\u5024\u57df\u306f<\/p>\n

$$ 1 \\leq x < \\infty\uff0c\\quad 0 \\leq y < \\infty$$<\/p>\n

\"\"<\/p>\n

\u3061\u306a\u307f\u306b\uff0c\u3053\u306e\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306f\u5bfe\u6570\u3092\u4f7f\u3063\u3066\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u7e70\u308a\u8fd4\u3059\u304c\uff0c\\(y = \\cosh^{-1} x\\) \u306f\u65b9\u7a0b\u5f0f \\(x = \\cosh y\\) \u3092 \\(y\\) \u306b\u3064\u3044\u3066\u89e3\u3044\u305f\u3082\u306e\u3060\u3063\u305f\u306e\u3060\u304b\u3089<\/p>\n

\\begin{eqnarray}
\ny &=& \\cosh^{-1} x \\\\
\n\\therefore\\ \\ x &=& \\cosh y = \\frac{e^y + e^{-y}}{2} \\\\
\n\\mbox{\u4e21\u8fba\u306b $2 e^y$ \u3092\u304b\u3051\u3066} \\quad 2 x e^y &=& \\left(e^{y}\\right)^2 + 1 \\\\ \\ \\\\
\n\\therefore\\ \\ (e^y)^2 -2 x e^y + 1 &=& 0
\n\\end{eqnarray}<\/p>\n

\u3053\u308c\u3092 \\(e^y\\) \u306b\u3064\u3044\u3066\u306e2\u6b21\u65b9\u7a0b\u5f0f\u3068\u898b\u505a\u3057\u3066\u89e3\u304f\u3068\uff0c
$$ e^y = x \\pm \\sqrt{x^2-1}, \\quad\\therefore\\ \\ y = \\log\\left(x \\pm \\sqrt{x^2-1}\\right)$$
\u5143\u306e \\( x = \\cosh y\\) \u304c \\(x\\) \u8ef8\u306b\u3064\u3044\u3066\u5bfe\u79f0\u306a\u5076\u95a2\u6570\u3067\u3042\u3063\u305f\u305f\u3081\u306b\uff0c\u305d\u306e\u9006\u95a2\u6570\u3067\u3042\u308b \\( y = \\cosh^{-1} x\\) \u304c2\u4fa1\u306e\u95a2\u6570\u306b\u306a\u308b\u306e\u306f\u4ed5\u65b9\u306a\u3044\u3068\u3053\u308d\u3002\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3059\u308b\u3068\uff0c
\\begin{eqnarray}
\u00a0\\log\\left(x -\\sqrt{x^2-1}\\right)
&=& \\log\\left(\\frac{x^2 -(x^2-1)}{x + \\sqrt{x^2-1}} \\right)\\\\
&=& \\log\\left(\\frac{1}{x + \\sqrt{x^2-1}} \\right)\\\\
&=& -\\log\\left(x + \\sqrt{x^2-1}\\right)
\\end{eqnarray} \u306a\u306e\u3067\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3082\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u3002
$$ y = \\cosh^{-1} x = \\pm \\log\\left(x + \\sqrt{x^2-1}\\right)$$\u4e3b\u5024\u3092 \\(y>0\\)\u00a0 \u3068\u3059\u308c\u3070\uff0c\\( y = \\cosh^{-1} x =\u00a0 \\log\\left(x + \\sqrt{x^2-1}\\right)\\) \u306e\u307f\u3068\u306a\u308b\u3002<\/p>\n

$\\sinh^{-1} x$ \u307e\u305f\u306f $\\operatorname{arsinh} x$<\/h4>\n

\\(y = \\sinh^{-1} x = \\mbox{arsinh } x\\) \u3092 \\(y = \\sinh x\\) \u306e\u9006\u95a2\u6570\uff08\u3064\u307e\u308a\uff0c\u65b9\u7a0b\u5f0f \\(x = \\sinh y\\) \u3092 \\(y\\) \u306b\u3064\u3044\u3066\u89e3\u3044\u305f\u3082\u306e\uff09\u3068\u3057\u3066\u5b9a\u7fa9\u3059\u308b\u3002\u5b9a\u7fa9\u57df\u3068\u5024\u57df\u306f<\/p>\n

$$-\\infty < x < \\infty\uff0c\\quad -\\infty < y < \\infty$$<\/p>\n

\"\"<\/p>\n

\u3061\u306a\u307f\u306b\uff0c\u3053\u306e\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306f\u5bfe\u6570\u3092\u4f7f\u3063\u3066\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u7e70\u308a\u8fd4\u3059\u304c\uff0c\\(y = \\sinh^{-1} x\\) \u306f\u65b9\u7a0b\u5f0f \\(x = \\sinh y\\) \u3092 \\(y\\) \u306b\u3064\u3044\u3066\u89e3\u3044\u305f\u3082\u306e\u3060\u3063\u305f\u306e\u3060\u304b\u3089<\/p>\n

\\begin{eqnarray}
\ny &=& \\sinh^{-1} x \\\\
\n\\therefore\\ \\ x &=& \\sin y = \\frac{e^y -e^{-y}}{2} \\\\
\n\\mbox{\u4e21\u8fba\u306b $2 e^y$ \u3092\u304b\u3051\u3066} \\quad 2 x e^y &=& \\left(e^{y}\\right)^2 -1 \\\\ \\ \\\\
\n\\therefore\\ \\ (e^y)^2 -2 x e^y -1 &=& 0
\n\\end{eqnarray}<\/p>\n

\u3053\u308c\u3092 \\(e^y\\) \u306b\u3064\u3044\u3066\u306e2\u6b21\u65b9\u7a0b\u5f0f\u3068\u898b\u505a\u3057\u3066\u89e3\u304f\u3068\uff0c\\( e^y > 0\\) \u3067\u3042\u308b\u3053\u3068\u304b\u3089
$$ e^y = x + \\sqrt{x^2+1}, \\quad\\therefore\\ \\ y = \\sinh^{-1} x = \\log\\left(x + \\sqrt{x^2+1}\\right)$$<\/p>\n

$\\tanh^{-1} x$ \u307e\u305f\u306f $\\operatorname{artanh} x$<\/h4>\n

\\(y = \\tanh^{-1} x = \\mbox{artanh } x\\) \u3092 \\(y = \\tanh x\\) \u306e\u9006\u95a2\u6570\uff08\u3064\u307e\u308a\uff0c\u65b9\u7a0b\u5f0f \\(x = \\tanh y\\) \u3092 \\(y\\) \u306b\u3064\u3044\u3066\u89e3\u3044\u305f\u3082\u306e\uff09\u3068\u3057\u3066\u5b9a\u7fa9\u3059\u308b\u3002\u5b9a\u7fa9\u57df\u3068\u5024\u57df\u306f<\/p>\n

$$ -1 < x < 1\uff0c\\quad -\\infty < y < \\infty$$<\/p>\n

\"\"<\/p>\n

\u3061\u306a\u307f\u306b\uff0c\u3053\u306e\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306f\u5bfe\u6570\u3092\u4f7f\u3063\u3066\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u7e70\u308a\u8fd4\u3059\u304c\uff0c\\(y = \\tanh^{-1} x\\) \u306f\u65b9\u7a0b\u5f0f \\(x = \\tanh y\\) \u3092 \\(y\\) \u306b\u3064\u3044\u3066\u89e3\u3044\u305f\u3082\u306e\u3060\u3063\u305f\u306e\u3060\u304b\u3089<\/p>\n

\\begin{eqnarray}
\ny &=& \\tanh^{-1} x \\\\
\n\\therefore\\ \\ x &=& \\tanh y = \\frac{e^y -e^{-y}}{e^y + e^{-y}} \\\\
\n\\mbox{\u4e21\u8fba\u306b $e^y\\,(e^y + e^{-y})$ \u3092\u304b\u3051\u308b\u3068} \\quad && \\\\
\nx \\left( \\left(e^y\\right)^2 + 1\\right) &=& \\left(e^y\\right)^2 -1 \\\\
\n\\left(e^y\\right)^2 &=& \\frac{1 -x}{1 +x} \\\\
\ne^y &=& \\sqrt{\\frac{1+x}{1-x}} \\\\
\n\\therefore\\ \\ y
\n&=& \\log \\sqrt{\\frac{1+x}{1-x}} \\\\
\n&=& \\frac{1}{2} \\log\\left(\\frac{1+x}{1-x}\\right) \\\\
\n\\mbox{\u3064\u307e\u308a}\\quad y = \\tanh^{-1} x &=& \\frac{1}{2} \\log\\left(\\frac{1+x}{1-x}\\right)
\n\\end{eqnarray}<\/p>\n

\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u5fae\u5206<\/h3>\n

\\begin{eqnarray}
\n(\\cosh^{-1} x)’ &=& \\frac{d}{dx} \\log\\left(x + \\sqrt{x^2-1}\\right)\\\\
\n&=& \\frac{1}{\\sqrt{x^2-1}}\\\\ \\ \\\\
\n(\\sinh^{-1} x)’ &=& \\frac{d}{dx}\\log\\left(x + \\sqrt{x^2+1}\\right) \\\\
\n&=& \\frac{1}{\\sqrt{x^2 + 1}}\\\\ \\ \\\\
\n(\\tanh^{-1} x)’ &=& \\frac{d}{dx} \\frac{1}{2} \\log\\left(\\frac{1+x}{1-x}\\right)\\\\
\n&=& \\frac{1}{1-x^2}
\n\\end{eqnarray}<\/p>\n

\u8a3c\u660e\u306f\uff0c\u5bfe\u6570\u95a2\u6570\u8868\u8a18\u3092\u76f4\u63a5\u5fae\u5206\u3057\u3066\u3082\u3044\u3044\u3057\uff0c\u307e\u305f\u9006\u95a2\u6570\u306e\u5fae\u5206\u6cd5\u3092\u4f7f\u3063\u3066\u3082\u3088\u3044\u3002<\/p>\n

\u4f8b\u3068\u3057\u3066\uff0c\\((\\cosh^{-1} x)’\\) \u306b\u3064\u3044\u3066\u9006\u95a2\u6570\u306e\u5fae\u5206\u6cd5<\/a>\u3092\u4f7f\u3063\u3066\u8a3c\u660e\u3057\u3066\u307f\u308b\u3002\\(\\cosh^2 y -\\sinh^2 y = 1\\) \u3082\u4f7f\u3044\u307e\u3059\u3002\u307e\u305a\uff0c\\( y = \\cosh^{-1} x \\) \u306f \\( x = \\cosh y\\) \u306e\u9006\u95a2\u6570\u3067\u3042\u3063\u305f\u306e\u3067\uff0c\u3053\u306e\u4e21\u8fba\u3092 \\(y\\)\u00a0 \u3067\u5fae\u5206\u3057\u3066\uff0c
$$ \\frac{dx}{dy} = \\frac{d}{dy} \\cosh y = \\sinh y = \\sqrt{\\cosh^2 y -1} = \\sqrt{x^2-1}$$
$$ \\therefore \\ \\ \\frac{d}{dx} \\cosh^{-1} x = \\frac{dy}{dx} = \\frac{1}{\\frac{dx}{dy}} = \\frac{1}{\\sqrt{x^2-1}}$$<\/p>\n

\u5bfe\u6570\u95a2\u6570\u8868\u8a18\u3092\u76f4\u63a5\u5fae\u5206\u3059\u308b\u5834\u5408\u306f\uff0c<\/p>\n

\\begin{eqnarray}
\n\\frac{d}{dx} \\cosh^{-1} x &=& \\frac{d}{dx} \\log\\left(x + \\sqrt{x^2-1}\\right) \\\\
\n&=& \\frac{1 + \\frac{x}{\\sqrt{x^2-1}}}{x + \\sqrt{x^2-1}} \\\\
\n&=& \\frac{1}{\\sqrt{x^2\u00a0 -1}}
\n\\end{eqnarray}<\/p>\n

$(\\sinh^{-1} x)’$ \u306b\u3064\u3044\u3066\u3082\uff0c\u4e0a\u306e\u5bfe\u6570\u95a2\u6570\u8868\u8a18\u3092\u76f4\u63a5\u5fae\u5206\u3057\u305f\u7d50\u679c\u3067 $\\sqrt{x^2 -1} \\rightarrow \\sqrt{x^2 +1}$ \u306e\u7f6e\u304d\u63db\u3048\u3092\u3059\u308c\u3070\u3044\u3044\u306e\u3067\u7c21\u5358\u3002<\/p>\n

$(\\tanh^{-1} x)’ $ \u306b\u3064\u3044\u3066\u3082\u5bfe\u6570\u95a2\u6570\u8868\u8a18\u3092\u76f4\u63a5\u5fae\u5206\u3057\u3066\u3084\u3063\u3066\u307f\u308b\u3068\uff0c<\/p>\n

\\begin{eqnarray}
\n\\frac{d}{dx} \\tanh^{-1} x &=& \\frac{d}{dx} \\frac{1}{2} \\log\\left(\\frac{1+x}{1-x}\\right) \\\\
\n&=& \\frac{1}{2} \\frac{d}{dx}\u00a0 \\bigl\\{ \\log(1+x) -\\log(1-x)\\bigr\\} \\\\
\n&=& \\frac{1}{2} \\left( \\frac{1}{1+x} + \\frac{1}{1-x}\\right) \\\\
\n&=& \\frac{1}{1-x^2}
\n\\end{eqnarray}<\/p>\n

\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u8868\u8a18<\/h3>\n

\\(\\displaystyle\u00a0 (\\cosh x)^{-1} = \\frac{1}{\\cosh x} \\) \u306a\u306e\u3067\uff0c\u3064\u3044\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u9593\u9055\u3044\u3092\u3057\u3066\u3057\u307e\u3046\u3053\u3068\u3082\u3042\u308b\u304b\u3082\u3057\u308c\u306a\u3044\u3002<\/span><\/p>\n

\\(\\displaystyle\u00a0 \\cosh^{-1} x = \\frac{x}{\\cosh } \\)\uff08\u5927\u9593\u9055\u3044!!<\/strong><\/span>\uff09\u3068\u304b\uff0c\\(\\displaystyle\u00a0 \\cosh^{-1} x = \\frac{1}{\\cosh x} \\)\uff08\u9593\u9055\u3044!!<\/strong><\/span>\uff09\u3068\u304b…
<\/span><\/span><\/span><\/p>\n

\u8aa4\u89e3\u3057\u3066\u3057\u307e\u3046\u6050\u308c\u304c\u3042\u308b\u306e\u3067\uff0c\u3053\u306e\u3088\u3046\u306a\u304a\u8336\u76ee\u306a\u9593\u9055\u3044\u3092\u8a98\u767a\u3057\u306a\u3044\u3088\u3046\u306b\uff0c\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306b\u5bfe\u3057\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u8868\u8a18\u6cd5\u3092\u7528\u3044\u308b\u3053\u3068\u3082\u3042\u308b\u3002<\/span><\/span><\/span><\/p>\n

$$ \\cosh^{-1} x = \\mbox{arcosh } x, \\quad \\sinh^{-1} x = \\mbox{arsinh } x, \\quad \\tanh^{-1} x = \\mbox{artanh } x$$
<\/span>\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306f \\(\\bf\\mbox{area}\\) \u306e\u7565\u3067\u3042\u308b \\(\\bf\\mbox{ar}\\) \u3068\u66f8\u304f\u3079\u304d\u3067\u3042\u308a\uff0c\\(\\mbox{arc}\\)\u300c\u30a2\u30fc\u30af<\/strong><\/span>\u300d\u3068\u66f8\u3044\u305f\u308a\u8aad\u3093\u3060\u308a\u3059\u3079\u304d\u3067\u306f\u306a\u3044\u3002\u306a\u305c\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u304c\u9762\u7a4d \\(\\bf\\mbox{area}\\) \u306a\u306e\u304b<\/strong><\/span>\u306b\u3064\u3044\u3066\u306f\uff0c\u4ee5\u4e0b\u3092\u53c2\u7167\uff1a<\/p>\n