{"id":2355,"date":"2022-02-26T10:23:41","date_gmt":"2022-02-26T01:23:41","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2355"},"modified":"2024-07-28T11:37:02","modified_gmt":"2024-07-28T02:37:02","slug":"%e3%82%ac%e3%82%a6%e3%82%b9%e7%a9%8d%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%a6c\/%e5%a4%9a%e9%87%8d%e7%a9%8d%e5%88%86%ef%bc%9a%e5%a4%9a%e5%a4%89%e6%95%b0%e9%96%a2%e6%95%b0%e3%81%ae%e7%a9%8d%e5%88%86\/%e3%82%ac%e3%82%a6%e3%82%b9%e7%a9%8d%e5%88%86\/","title":{"rendered":"\u30ac\u30a6\u30b9\u7a4d\u5206"},"content":{"rendered":"

\u4e0d\u5b9a\u7a4d\u5206 $\\displaystyle \\int e^{-x^2 } dx$ \u306f\u7a4d\u5206\u3067\u304d\u306a\u3044\u306e\u306b\uff0c\u306a\u305c\u304b
\n$$\\displaystyle \\int_{-\\infty}^{\\infty}\u00a0 e^{-x^2 } dx = \\sqrt{\\pi}$$ \u3068\u306a\u308b\u3053\u3068\u3002
\n<\/p>\n

\u30ac\u30a6\u30b9\u7a4d\u5206\u306e\u6e96\u5099\u3068\u3057\u3066\u306e2\u91cd\u7a4d\u5206<\/a><\/h3>\n

\"\"<\/p>\n

$$ I = \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} e^{-x^2 – y^2} dx\\,dy$$<\/p>\n

\u6975\u5ea7\u6a19\u306b\u5909\u63db\u3057\u3066\u8a08\u7b97\u3059\u308b\u3002<\/p>\n

$$I = \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} e^{-x^2 – y^2} dx\\,dy = \\int_0^{2\\pi} d\\theta \\int_0^{\\infty} e^{-r^2} r dr$$<\/p>\n

\u5909\u6570\u5909\u63db $$ r^2 = t, \\quad r dr = \\frac{1}{2} dt $$
\n$$\\therefore I = 2\\pi \\times \\frac{1}{2} \\int_0^{\\infty} e^{-t} dt = \\pi \\Bigl[ -e^{-t} \\Bigr]_0^{\\infty} = \\pi$$<\/p>\n

\u30ac\u30a6\u30b9\u7a4d\u5206<\/h3>\n

\"\"$$ I = \\int_{-\\infty}^{\\infty} e^{-x^2 } dx$$<\/p>\n

\u3053\u306e\u7a4d\u5206\u306f\uff0c\u7a4d\u5206\u5909\u6570 \\( x \\) \u3092 \\( y \\) \u306b\u3057\u3066
$$ I_y = \\int_{-\\infty}^{\\infty}\u00a0 e^{-y^2 } dy$$ \u3068\u3057\u3066\u3082\uff0c\u7a4d\u5206\u7bc4\u56f2\u304c\u540c\u3058\u306a\u3089\u5168\u304f\u540c\u3058\u7b54\u3048\u3092\u4e0e\u3048\u308b\u306f\u3059\u3067\u3059\u3088\u306d\u3002\u3064\u307e\u308a \\( I = I_y \\) \u3067\u3059\u304b\u3089\uff0c\u4ee5\u4e0b\u306e\u8a08\u7b97\u3067\uff0c\\(I^2 = I \\times I_y \\) \u3068\u3057\u3066\u3088\u3044\u3002<\/p>\n

\\(\\displaystyle\u00a0 \\int\u00a0 e^{-x^2 } dx\\)\u306f\u4e0d\u5b9a\u7a4d\u5206\u3067\u304d\u306a\u3044\uff01\u306e\u3067\uff0c\u5b9a\u7a4d\u5206\u3060\u3063\u3066\u3067\u304d\u306a\u3044\u306f\u305a… \u306a\u3093\u3060\u3051\u3069<\/p>\n

\\begin{eqnarray}
I^2 &=& I \\times I_y \\\\
&=& \\int_{-\\infty}^{\\infty}\u00a0 e^{-x^2 } dx \\int_{-\\infty}^{\\infty}\u00a0 e^{-y^2 } dy \\\\
&=& \\int_{-\\infty}^{\\infty} dx \\int_{-\\infty}^{\\infty} dy \\,e^{-x^2 -y^2}\\\\
&=& \\pi
\\end{eqnarray} \uff08\u4f55\u3067 \\(\\pi\\) \u306b\u306a\u308b\u306e\u304b\u306f\uff0c\u4e0a\u7bc0\u300c\u30ac\u30a6\u30b9\u7a4d\u5206\u306e\u6e96\u5099\u3068\u3057\u3066\u306e2\u91cd\u7a4d\u5206<\/span><\/strong><\/span><\/a>\u300d\u3092\u53c2\u7167\u3002\uff09
\n$$\\therefore \\ I = \\int_{-\\infty}^{\\infty}\u00a0 e^{-x^2 } dx = \\sqrt{\\pi} $$<\/p>\n

\u4e0d\u5b9a\u7a4d\u5206\u306f\u3067\u304d\u306a\u3044\u306e\u306b\uff0c\uff08\u7121\u9650\u533a\u9593\u3067\u306e\uff09\u5b9a\u7a4d\u5206\u306f\u8a08\u7b97\u3067\u304d\u308b\uff01\u3068\u3044\u3046\uff0c\u306a\u3093\u3068\u3082\u4e0d\u601d\u8b70\u306a\u7a4d\u5206<\/strong><\/span>\u3002\u3053\u306e\u5f62\u306e\u7a4d\u5206\u3092\u300c\u30ac\u30a6\u30b9\u7a4d\u5206<\/b><\/span>\u300d\u3068\u547c\u3093\u3067\u3044\u308b\u3002<\/p>\n

\u88ab\u7a4d\u5206\u95a2\u6570 \\(e^{-x^2}\\) \u306f\u5076\u95a2\u6570\u306a\u306e\u3067\uff0c
\n$$ \\int_{0}^{\\infty}\u00a0 e^{-x^2 } dx = \\frac{1}{2} \\int_{-\\infty}^{\\infty}\u00a0 e^{-x^2 } dx = \\frac{\\sqrt{\\pi}}{2}$$<\/p>\n

\u30ac\u30a6\u30b9\u7a4d\u5206\u306b\u95a2\u9023\u3057\u305f\u7a4d\u5206<\/h3>\n

$$ \\int_{0}^{\\infty} x^m e^{-x^2 } dx$$<\/p>\n

\\(m = 0\\) \u306e\u5834\u5408<\/h4>\n

$$\\int_{0}^{\\infty}\u00a0e^{-x^2 } dx = \\frac{\\sqrt{\\pi}}{2}$$<\/p>\n

 <\/p>\n

\\(m = 1\\) \u306e\u5834\u5408<\/h4>\n

\u5909\u6570\u5909\u63db \\( x^2 = t, \\ x dx = \\frac{1}{2} dt \\) \u3092\u4f7f\u3063\u3066
\n$$\\int_{0}^{\\infty} x e^{-x^2 } dx =\\frac{1}{2} \\int_{0}^{\\infty} e^{-t} dt = \\frac{1}{2} \\Bigl[ – e^{-t} \\Bigr]_0^{\\infty} = \\frac{1}{2}$$<\/p>\n

 <\/p>\n

\\(m\\) \u304c\u5076\u6570\u306e\u5834\u5408\u306e\u7a4d\u5206\u3092\u6c42\u3081\u308b\u6e96\u5099<\/h4>\n

\u307e\u305a\uff0c\u30ac\u30a6\u30b9\u7a4d\u5206\u3092\u3061\u3087\u3063\u3068\u4e00\u822c\u5316\u3057\u3066
\n$$ I_e = \\int_0^{\\infty} e^{-a x^2} dx = \\frac{1}{\\sqrt{a}} \\int_0^{\\infty} e^{-(\\sqrt{a} x)^2} \\sqrt{a} dx = a^{-\\frac{1}{2}} \\frac{\\sqrt{\\pi}}{2}$$<\/p>\n

 <\/p>\n

\\( m = 2 \\) \u306e\u5834\u5408<\/h4>\n

\\begin{eqnarray}
\n-\\frac{d}{da} I_e &=& -\\int_0^{\\infty} \\frac{\\partial}{\\partial a} e^{-a x^2} dx =\u00a0 \\int_0^{\\infty} x^2 e^{-a x^2} dx \\\\
\n&=& -\\frac{d}{da}a^{-\\frac{1}{2}} \\frac{\\sqrt{\\pi}}{2} = \\frac{1}{2} a^{-\\frac{3}{2}}\\frac{\\sqrt{\\pi}}{2}
\n\\end{eqnarray}<\/p>\n

 <\/p>\n

\\(a = 1\\) \u3068\u304a\u304f\u3068<\/p>\n

$$ \\int_0^{\\infty} x^2 e^{- x^2} dx = \\frac{\\sqrt{\\pi}}{4}$$<\/p>\n

 <\/p>\n

\\( m = 4 \\) \u306e\u5834\u5408<\/h4>\n

\\begin{eqnarray}
\n(-1)^2 \\frac{d^2}{da^2} \u00a0I_e &=& \\int_0^{\\infty} \\frac{\\partial^2}{\\partial a^2} e^{-a x^2} dx =\u00a0 \\int_0^{\\infty} x^4 e^{-a x^2} dx \\\\
\n&=& \\frac{d^2}{da^2}a^{-\\frac{1}{2}} \\frac{\\sqrt{\\pi}}{2} = \\frac{3}{4} a^{-\\frac{5}{2}}\\frac{\\sqrt{\\pi}}{2}
\n\\end{eqnarray}<\/p>\n

 <\/p>\n

\\(a = 1\\) \u3068\u304a\u304f\u3068<\/p>\n

$$ \\int_0^{\\infty} x^4 e^{- x^2} dx = \\frac{3\\sqrt{\\pi}}{8}$$<\/p>\n

\u4e00\u822c\u306b \\( m = 2 n\\) \uff08\\(n\\) \u306f\u6b63\u306e\u6574\u6570\uff09\u306e\u5834\u5408<\/h4>\n

\\begin{eqnarray}
\n\\int_0^{\\infty} x^{2n} e^{-a x^2} dx &=& (-1)^n \\frac{d^n}{da^n} I_e \\\\
\n&=& (-1)^n \\frac{d^n}{da^n}a^{-\\frac{1}{2}} \\frac{\\sqrt{\\pi}}{2} \\\\
\n&=& \\frac{(2n-1)!!}{2^n} a^{-\\frac{2n+1}{2}} \\frac{\\sqrt{\\pi}}{2}
\n\\end{eqnarray}<\/p>\n

\\(a = 1\\) \u3068\u304a\u304f\u3068<\/p>\n

$$ \\int_0^{\\infty} x^{2n} e^{- x^2} dx = \\frac{(2n-1)!!}{2^{n+1}}\\sqrt{\\pi}$$<\/p>\n

\u3053\u3053\u3067\uff0c$$ (2n-1)!! = (2n-1)\\cdot (2n-3)\\cdots 3\\cdot 1 $$ \u4f8b\u3048\u3070\uff0c$$ 5!! = 5\\cdot 3 \\cdot 1 = 15$$<\/p>\n

<\/h4>\n

\\(m\\) \u304c\u5947\u6570\u306e\u5834\u5408\u306e\u7a4d\u5206\u3092\u6c42\u3081\u308b\u6e96\u5099<\/h4>\n

\\begin{eqnarray}I_o &=& \\int_0^{\\infty} x e^{-a x^2} dx \\\\
\n&=& \\frac{1}{a} \\int_0^{\\infty} (\\sqrt{a} x) e^{-(\\sqrt{a} x)^2} \\sqrt{a} dx \\\\
\n&=& \\frac{1}{a}\\int_0^{\\infty} t e^{-t^2} dt = a^{-1} \\frac{1}{2}\\end{eqnarray}<\/p>\n

 <\/p>\n

\u4e00\u822c\u306b \\( m = 2 n+1\\) \uff08\\(n\\) \u306f\u6b63\u306e\u6574\u6570\uff09\u306e\u5834\u5408<\/h4>\n

\\begin{eqnarray}
\n\\int_0^{\\infty} x^{2n+1} e^{-a x^2} dx &=& (-1)^n \\frac{d^n}{da^n} I_o \\\\
\n&=& (-1)^n \\frac{d^n}{da^n}a^{-1} \\frac{1}{2} \\\\
\n&=& \\dots
\n\\end{eqnarray}<\/p>\n","protected":false},"excerpt":{"rendered":"

\u4e0d\u5b9a\u7a4d\u5206 $\\displaystyle \\int e^{-x^2 } dx$ \u306f\u7a4d\u5206\u3067\u304d\u306a\u3044\u306e\u306b\uff0c\u306a\u305c\u304b $$\\displaystyle \\int_{-\\infty}^{\\infty}\u00a0 e^{-x^2 } dx = \\sqrt{\\pi}$$ \u3068\u306a\u308b\u3053\u3068\u3002<\/p>

\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"parent":2228,"menu_order":4,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-2355","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\/2355","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=2355"}],"version-history":[{"count":10,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2355\/revisions"}],"predecessor-version":[{"id":9267,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2355\/revisions\/9267"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2228"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=2355"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}