{"id":2401,"date":"2022-02-26T20:05:58","date_gmt":"2022-02-26T11:05:58","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2401"},"modified":"2023-04-13T15:15:04","modified_gmt":"2023-04-13T06:15:04","slug":"maxima-jupyter-%e3%81%a6%e3%82%99%e5%a4%9a%e9%87%8d%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\/maxima-%e3%81%a7%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6c\/maxima-jupyter-%e3%81%a6%e3%82%99%e5%a4%9a%e9%87%8d%e7%a9%8d%e5%88%86\/","title":{"rendered":"Maxima \u3066\u3099\u591a\u91cd\u7a4d\u5206"},"content":{"rendered":"

<\/p>\n

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\u591a\u5909\u6570\u95a2\u6570\u306e\u7a4d\u5206\u3067\u3042\u308b\u591a\u91cd\u7a4d\u5206\u306e\u3046\u3061\uff0c\u3082\u3063\u3068\u3082\u7c21\u5358\u306a2\u5909\u6570\u95a2\u6570\u306e\u7a4d\u5206\u3064\u307e\u308a2\u91cd\u7a4d\u5206\u306b\u3064\u3044\u3066\u307e\u3068\u3081\u308b\u3002<\/p>\n

2\u91cd\u7a4d\u5206\u306f\u7d2f\u6b21\u7a4d\u5206\u3067<\/h3>\n

Maxima \u306b\u304a\u3051\u308b2\u91cd\u7a4d\u5206\u306e\u8a08\u7b97\u306f\uff0c\u57fa\u672c\u7684\u306b\u7d2f\u6b21\u7a4d\u5206\u3067\u3084\u308a\u307e\u3059\u3002<\/p>\n

$\\displaystyle \\iint_D f(x, y)\\, dx dy$ \u306e\u9818\u57df $D$ \u304c\u4f55\u3068\u4f55\u3067\u56f2\u307e\u308c\u3066\u3044\u308b\u304b\u3092\u660e\u3089\u304b\u306b\u3057\u3066\u7a4d\u5206\u3059\u308b\u3002<\/p>\n

\u7d2f\u6b21\u7a4d\u5206 1<\/h4>\n

\u9818\u57df $D$ \u304c $x_1 \\leq x \\leq x_2,\\ y_1 \\leq y \\leq y_2$ \u3067\u3042\u308b\u5834\u5408\uff1a<\/p>\n

$$\\iint_D f(x, y) \\,dx\\,dy = \\int_{x_1}^{x_2} \\left\\{\\int_{y_1}^{y_2} f(x, y)\\,dy \\right\\} dx$$<\/p>\n

\u307e\u305f\u306f<\/p>\n

$$\\iint_D f(x, y) \\,dx\\,dy = \\int_{y_1}^{y_2} \\left\\{\\int_{x_1}^{x_2} f(x, y)\\,dx \\right\\} dy$$<\/p>\n

\u7d2f\u6b21\u7a4d\u5206 2<\/h4>\n

\u9818\u57df $D$ \u304c $x_1 \\leq x \\leq x_1,\\ \\phi_1(x) \\leq y \\leq \\phi_2(x)$ \u3067\u3042\u308b\u5834\u5408\uff1a<\/p>\n

$$ \\iint_D f(x, y) \\,dx\\,dy = \\int_{x_1}^{x_2} \\left\\{\\int_{\\phi_1(x)}^{\\phi_2(x)} f(x, y)\\,dy \\right\\} dx$$<\/p>\n

\u7d2f\u6b21\u7a4d\u5206 3<\/h4>\n

\u9818\u57df $D$ \u304c $\\psi_1(y)\\leq x \\leq \\psi_2(y),\\ y_1 \\leq y \\leq y_2$ \u3067\u3042\u308b\u5834\u5408\uff1a<\/p>\n

$$ \\iint_D f(x, y) \\,dx\\,dy = \\int_{y_1}^{y_2} \\left\\{\\int_{\\psi_1(y)}^{\\psi_2(y)} f(x, y)\\,dx \\right\\} dy$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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\u4f8b\u984c<\/h4>\n

$$I = \\iint_D y \\,dx dy, \\quad D: 0\\leq x\\leq 2, \\ 0 \\leq y \\leq 2$$<\/p>\n

\u3053\u308c\u306f\uff0c\u30b1\u30fc\u30b9 1\u3002\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u7d2f\u6b21\u7a4d\u5206\u306b\u3057\u3066…
\n\\begin{eqnarray}
\nf(x, y) &=& y \\\\
\nx_1 &=& 0\\\\
\nx_2 &=& 2\\\\
\ny_1 &=& 0\\\\
\ny_2 &=& 2
\n\\end{eqnarray}<\/p>\n

\\begin{eqnarray}
\n\\iint_D y \\,dx dy &=& \\int_{x_1}^{x_2} \\left\\{\\int_{y_1}^{y_2} f(x, y) dy \\right\\} dx\\\\
\n&=& \\cdots
\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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f<\/span>(<\/span>x<\/span>, y<\/span>)<\/span>:=<\/span> y<\/span>$\r\nx1<\/span>:<\/span> 0$\r\nx2<\/span>:<\/span> 2$\r\ny1<\/span>:<\/span> 0$\r\ny2<\/span>:<\/span> 2$\r\n\r\nI<\/span> =<\/span> integrate<\/span>(<\/span>\r\n      integrate<\/span>(<\/span>f<\/span>(<\/span>x<\/span>,y<\/span>)<\/span>, y<\/span>, y1<\/span>, y2<\/span>)<\/span>, \r\n      x<\/span>, x1<\/span>, x2<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{5}$}I=4\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u53c2\u8003\uff1a\u9818\u57df $D$ \u306e\u5857\u308a\u3064\u3076\u3057<\/h4>\n
\u3064\u3044\u3067\u306b\uff0c\u9818\u57df $D$ \u306e\u90e8\u5206\u3092\u5857\u308a\u3064\u3076\u3057\u3066\u63cf\u3044\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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f<\/span>(<\/span>x<\/span>, y<\/span>)<\/span>:=<\/span> y<\/span>$\r\nx1<\/span>:<\/span> 0$\r\nx2<\/span>:<\/span> 2$\r\ny1<\/span>:<\/span> 0$\r\ny2<\/span>:<\/span> 2$\r\n\r\ndraw2d<\/span>(<\/span>\r\n  title<\/span> =<\/span> \"\u9818\u57df D\"<\/span>,\r\n  \/* x \u8ef8 y \u8ef8\u306e\u8868\u793a\u7bc4\u56f2\u306e\u8a2d\u5b9a\u3002*\/<\/span> \r\n  xrange<\/span> =<\/span> [<\/span>-<\/span>0.5<\/span>, 2.<\/span>5]<\/span>, xaxis<\/span> =<\/span> true<\/span>, xlabel<\/span>=<\/span>\"x\"<\/span>, \r\n  yrange<\/span> =<\/span> [<\/span>-<\/span>0.5<\/span>, 2.<\/span>5]<\/span>, yaxis<\/span> =<\/span> true<\/span>, ylabel<\/span>=<\/span>\"y\"<\/span>, \r\n  user_preamble<\/span>=<\/span>\"set grid front\"<\/span>,\r\n  \/* \u7e26\u6a2a\u6bd4 *\/<\/span>\r\n  proportional_axes<\/span> =<\/span> xy<\/span>,\r\n  \r\n  \/* \u5857\u308a\u3064\u3076\u3059\u8272\u306e\u6307\u5b9a\u3002*\/<\/span>\r\n  fill_color<\/span> =<\/span> yellow<\/span>, \r\n  \r\n  \/* \u4e0a\u306e\u7dda\u3002y = y2 \u3092 filled_func \u306b\u4ee3\u5165\u3002*\/<\/span>\r\n  filled_func<\/span> =<\/span> y2<\/span>, \r\n  \r\n  \/* \u4e0b\u306e\u7dda\u3002y = y1 \u3092 x1 < x < x2 \u306e\u7bc4\u56f2\u3067\u63cf\u304f\u3002*\/<\/span>\r\n  explicit<\/span>(<\/span>y1<\/span>, x<\/span>, x1<\/span>, x2<\/span>)<\/span>  \r\n)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u3064\u3044\u3067\u306b\uff0c3\u6b21\u5143\u7684\u306a\u30b0\u30e9\u30d5\u3092\u63cf\u3044\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[3]:<\/div>\n
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f<\/span>(<\/span>x<\/span>, y<\/span>)<\/span>:=<\/span> y<\/span>$\r\nx1<\/span>:<\/span> 0$\r\nx2<\/span>:<\/span> 2$\r\ny1<\/span>:<\/span> 0$\r\ny2<\/span>:<\/span> 2$\r\n\r\ndraw3d<\/span>(<\/span>\r\n  \/* \u8868\u793a\u7bc4\u56f2 *\/<\/span>\r\n  xrange<\/span> =<\/span> [<\/span>-<\/span>0.5<\/span>, 2.<\/span>5]<\/span>,\r\n  yrange<\/span> =<\/span> [<\/span>-<\/span>0.5<\/span>, 2.<\/span>5]<\/span>, \r\n  zrange<\/span> =<\/span> [<\/span>0<\/span>, 2.<\/span>5]<\/span>, \r\n  \/* \u7e26\u6a2a\u6bd4 *\/<\/span>\r\n  proportional_axes<\/span> =<\/span> xyz<\/span>, xyplane<\/span>=<\/span>0<\/span>,\r\n\r\n  \/* \u9818\u57df D*\/<\/span>\r\n  color<\/span>=<\/span>yellow<\/span>,\r\n  parametric_surface<\/span>(<\/span>x<\/span>, y<\/span>, 0<\/span>, \r\n                     x<\/span>, x1<\/span>, x2<\/span>, y<\/span>, y1<\/span>, y2<\/span>)<\/span>,\r\n  \/* z = f(x, y) *\/<\/span>\r\n  color<\/span>=<\/span>blue<\/span>,\r\n  explicit<\/span>(<\/span>f<\/span>(<\/span>x<\/span>, y<\/span>)<\/span>, x<\/span>, x1<\/span>, x2<\/span>, y<\/span>, y1<\/span>, y2<\/span>)<\/span>\r\n)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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2\u91cd\u7a4d\u5206\u306e\u7d50\u679c\u306f\uff0c\u9ec4\u8272\u3067\u5857\u308a\u3064\u3076\u3055\u308c\u305f\u9818\u57df $D$ \u3068\u9752\u8272\u3067\u5857\u3089\u308c\u305f\u5e73\u9762\u3067\u631f\u307e\u308c\u305f\u90e8\u5206\u306e\u4f53\u7a4d\u3092\u8868\u3059\u3002<\/p>\n
\u7e26\u6a2a\u9ad8\u3055\u3068\u3082\u306b $2$ \u306e\u7acb\u65b9\u4f53\u3092\u659c\u3081\u306b2\u3064\u306b\u5207\u3063\u305f\u7acb\u4f53\u3067\u3042\u308b\u306e\u3067\u4f53\u7a4d\u306f<\/p>\n
$$\\frac{2\\cdot 2\\cdot 2}{2} = 4$$<\/p>\n
\u306b\u306a\u308b\u306f\u305a\u3067\uff0c\u7b54\u3048\u3082\u78ba\u304b\u306b\u305d\u3046\u306a\u3063\u3066\u3044\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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\u4f8b\u984c<\/h4>\n
$$I = \\iint_D dx dy, \\quad D: x^2 + y^2 \\leq 1$$<\/p>\n
\u7d2f\u6b21\u7a4d\u5206\u306e\u5f62\u306b\u3059\u308b\u3002\u307e\u305a\u5148\u306b $-\\sqrt{1-x^2} < y < \\sqrt{1-x^2}$ \u306e\u7bc4\u56f2\u3067 $y$ \u3067\u5b9a\u7a4d\u5206\u3057\uff0c\u305d\u306e\u3042\u3068\u306b $ -1 < x < 1$ \u306e\u7bc4\u56f2\u3067 $x$ \u3067\u5b9a\u7a4d\u5206\u3059\u308b\u3002
\n\u88ab\u7a4d\u5206\u95a2\u6570\u306f $1$\u3002<\/p>\n
\u30b1\u30fc\u30b9 2 \u306e\u7d2f\u6b21\u7a4d\u5206\u306e\u5f62\uff1a
\n\\begin{eqnarray}
\nf(x, y) &=& 1 \\\\
\nx_1 &=& -1\\\\
\nx_2 &=& 1 \\\\
\n\\phi_1(x) &=& -\\sqrt{1-x^2}\\\\
\n\\phi_2(x) &=& \\sqrt{1-x^2}
\n\\end{eqnarray}<\/p>\n
\\begin{eqnarray}
\n\\iint_D dx dy &=& \\int_{x_1}^{x_2} \\left\\{\\int_{\\phi_1(x)}^{\\phi_2(x) } f(x, y)\\,dy \\right\\} dx \\\\
\n&=& \\cdots
\n\\end{eqnarray}<\/p>\n
$I$ \u306f\u534a\u5f84 $1$ \u306e\u5186\u306e\u9762\u7a4d\u3067\u3042\u308b\u304b\u3089\uff0c$\\pi r^2 = \\pi \\ (\\because r=1)$ \u3068\u306a\u308b\u306f\u305a\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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f<\/span>(<\/span>x<\/span>, y<\/span>)<\/span>:=<\/span> 1$\r\nphi1<\/span>(<\/span>x<\/span>)<\/span>:=<\/span> -<\/span>sqrt<\/span>(<\/span>1-<\/span>x<\/span>**<\/span>2<\/span>)<\/span>$\r\nphi2<\/span>(<\/span>x<\/span>)<\/span>:=<\/span> sqrt<\/span>(<\/span>1-<\/span>x<\/span>**<\/span>2<\/span>)<\/span>$\r\nx1<\/span>:<\/span> -<\/span>1$\r\nx2<\/span>:<\/span> 1$\r\n\r\nI<\/span> =<\/span> integrate<\/span>(<\/span> \r\n      integrate<\/span>(<\/span>f<\/span>(<\/span>x<\/span>, y<\/span>)<\/span>, y<\/span>, phi1<\/span>(<\/span>x<\/span>)<\/span>, phi2<\/span>(<\/span>x<\/span>))<\/span>, \r\n      x<\/span>, x1<\/span>, x2<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{25}$}I=\\pi\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u53c2\u8003\uff1a\u9818\u57df $D$ \u306e\u5857\u308a\u3064\u3076\u3057<\/h4>\n
\u3064\u3044\u3067\u306b\uff0c\u9818\u57df $D$ \u306e\u90e8\u5206\u3092\u5857\u308a\u3064\u3076\u3057\u3066\u63cf\u3044\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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phi1<\/span>(<\/span>x<\/span>)<\/span>:=<\/span> -<\/span>sqrt<\/span>(<\/span>1-<\/span>x<\/span>**<\/span>2<\/span>)<\/span>$\r\nphi2<\/span>(<\/span>x<\/span>)<\/span>:=<\/span> sqrt<\/span>(<\/span>1-<\/span>x<\/span>**<\/span>2<\/span>)<\/span>$\r\nx1<\/span>:<\/span> -<\/span>1$\r\nx2<\/span>:<\/span> 1$\r\n\r\ndraw2d<\/span>(<\/span>\r\n  title<\/span>=<\/span>\"\u9818\u57df D\"<\/span>, \r\n  \/* x \u8ef8 y \u8ef8\u306e\u8868\u793a\u7bc4\u56f2\u306e\u8a2d\u5b9a\u3002*\/<\/span> \r\n  xrange<\/span> =<\/span> [<\/span>-<\/span>1.5<\/span>, 1.<\/span>5]<\/span>, xaxis<\/span> =<\/span> true<\/span>, xlabel<\/span>=<\/span>\"x\"<\/span>, \r\n  yrange<\/span> =<\/span> [<\/span>-<\/span>1.5<\/span>, 1.<\/span>5]<\/span>, yaxis<\/span> =<\/span> true<\/span>, ylabel<\/span>=<\/span>\"y\"<\/span>, \r\n  proportional_axes<\/span> =<\/span> xy<\/span>,   \/* \u5186\u3092\u307e\u3093\u307e\u308b\u304f\u8868\u793a\u3057\u305f\u3044\u3068\u304d\u306b *\/<\/span>\r\n  user_preamble<\/span>=<\/span>\"set grid front\"<\/span>,\r\n\r\n  fill_color<\/span> =<\/span> yellow<\/span>, \r\n  \/* \u7e01\u3092\u306a\u3081\u3089\u304b\u306b\u3059\u308b\u306b\u306f\u9069\u5b9c\u5927\u304d\u3044\u5024\u3092 *\/<\/span>\r\n  x_voxel<\/span> =<\/span> 50<\/span>, y_voxel<\/span> =<\/span> 50<\/span>, \r\n  \/* \u4e0d\u7b49\u5f0f\u3067\u793a\u3055\u308c\u305f\u9818\u57df\u3092\u5857\u308a\u3064\u3076\u3059\u4f8b *\/<\/span>\r\n  region<\/span>(<\/span>x<\/span>**<\/span>2<\/span> +<\/span> y<\/span>**<\/span>2<\/span> <<\/span> 1<\/span>, x<\/span>, -<\/span>1.5<\/span>, 1.5<\/span>, y<\/span>, -<\/span>1.5<\/span>, 1.5<\/span>)<\/span>\r\n)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u30ac\u30a6\u30b9\u7a4d\u5206\u306e\u6e96\u5099\u3068\u3057\u3066\u306e2\u91cd\u7a4d\u5206<\/h3>\n
$$ S = \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} e^{-x^2 – y^2} dx\\,dy$$<\/p>\n
\u6975\u5ea7\u6a19\u306b\u5909\u63db\u3057\u3066\u7a4d\u5206<\/h4>\n
\u6975\u5ea7\u6a19\u306b\u5909\u63db\u3057\u3066\u8a08\u7b97\u3059\u308b\u3002<\/p>\n
\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19 $x, y$ \u30682\u6b21\u5143\u6975\u5ea7\u6a19 $r, \\theta$ \u3068\u306e\u9593\u306e\u95a2\u4fc2\u306f
\n\\begin{eqnarray}
\nx &=& r\\cos\\theta\\\\
\ny &=& r\\sin\\theta
\n\\end{eqnarray}
\n\u3067\u3042\u308a\uff0c\u5fae\u5c0f\u9762\u7a4d\u8981\u7d20 $dx\\,dy$ \u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5909\u63db\u3055\u308c\u308b\u3002<\/p>\n
\\begin{eqnarray}
\ndx\\,dy &=& \\frac{\\partial(x, y)}{\\partial(r,\\theta)} \\,dr\\, d\\theta
\n\\end{eqnarray}<\/p>\n
\u3053\u3053\u3067\uff0c\u3053\u3053\u3067\uff0c$$ \\frac{\\partial(x,y)}{\\partial(r,\\theta)} \\equiv
\n\\begin{vmatrix}
\n\\frac{\\partial x}{\\partial r} & \\frac{\\partial x}{\\partial \\theta}\\\\
\n\\frac{\\partial y}{\\partial r} & \\frac{\\partial y}{\\partial \\theta}\\\\
\n\\end{vmatrix}
\n= \\frac{\\partial x}{\\partial r} \\frac{\\partial y}{\\partial \\theta} – \\frac{\\partial y}{\\partial r}\\frac{\\partial x}{\\partial \\theta}$$ \u3092\u30e4\u30b3\u30d3\u30a2\u30f3\u3068\u3044\u3046\u3002<\/p>\n
Maxima \u3067\u30e4\u30b3\u30d3\u30a2\u30f3\u3092\u8a08\u7b97\u3059\u308b\u4f8b\uff1a<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[6]:<\/div>\n
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x<\/span>:<\/span> r<\/span> *<\/span> cos<\/span>(<\/span>theta<\/span>)<\/span>;\r\ny<\/span>:<\/span> r<\/span> *<\/span> sin<\/span>(<\/span>theta<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{32}$}r\\,\\cos \\vartheta\\]<\/div>\n<\/div>\n
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Out[6]:<\/div>\n
\\[\\tag{${\\it \\%o}_{33}$}r\\,\\sin \\vartheta\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[7]:<\/div>\n
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jacobian<\/span>([<\/span>x<\/span>, y<\/span>]<\/span>, [<\/span>r<\/span>, theta<\/span>])<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[7]:<\/div>\n
\\[\\tag{${\\it \\%o}_{34}$}\\begin{pmatrix}\\cos \\vartheta & -r\\,\\sin \\vartheta \\\\ \\sin \\vartheta & r\\,\\cos \\vartheta \\\\ \\end{pmatrix}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Maxima \u306e jacobian()<\/code> \u95a2\u6570\u306f\u300c\u95a2\u6570\u884c\u5217\u300d\u3092\u4e0e\u3048\u308b\u306e\u3067\uff0c\u300c\u95a2\u6570\u884c\u5217\u5f0f\u300d\u306b\u3059\u308b\u306b\u306f\uff0c\u3055\u3089\u306b determinant()<\/code> \u3092\u3068\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[8]:<\/div>\n
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determinant<\/span>(<\/span>jacobian<\/span>([<\/span>x<\/span>, y<\/span>]<\/span>, [<\/span>r<\/span>, theta<\/span>]))<\/span>;\r\ntrigsimp<\/span>(<\/span>%<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[8]:<\/div>\n
\\[\\tag{${\\it \\%o}_{35}$}r\\,\\sin ^2\\vartheta+r\\,\\cos ^2\\vartheta\\]<\/div>\n<\/div>\n
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Out[8]:<\/div>\n
\\[\\tag{${\\it \\%o}_{36}$}r\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u30e4\u30b3\u30d3\u30a2\u30f3\u304c $r$ \u3068\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u3063\u305f\u3002<\/p>\n
\\begin{eqnarray}
\n\\therefore\\ \\ S &=& \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} e^{-x^2 – y^2} dx\\,dy \\\\
\n&=& \\int_0^{2\\pi} \\int_0^{\\infty} e^{-r^2} r \\,dr \\,d\\theta\\\\
\n&=& 2 \\pi \\int_0^{\\infty} e^{-r^2} r \\,dr
\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[9]:<\/div>\n
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2*<\/span>%pi<\/span>*<\/span> '<\/span>integrate<\/span>(<\/span>exp<\/span>(<\/span>-<\/span>r<\/span>**<\/span>2<\/span>)<\/span> *<\/span> r<\/span>, r<\/span>, 0<\/span>, inf<\/span>)<\/span> =<\/span>\r\n 2*<\/span>%pi<\/span>*<\/span> integrate<\/span>(<\/span>exp<\/span>(<\/span>-<\/span>r<\/span>**<\/span>2<\/span>)<\/span> *<\/span> r<\/span>, r<\/span>, 0<\/span>, inf<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[9]:<\/div>\n
\\[\\tag{${\\it \\%o}_{37}$}2\\,\\pi\\,\\int_{0}^{\\infty }{r\\,e^ {- r^2 }\\;dr}=\\pi\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[10]:<\/div>\n
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\/* \u6b21\u306b\u9032\u3080\u524d\u306b *\/<\/span>\r\n\/* x \u3084 y \u306e\u5b9a\u7fa9\u3092\u306a\u3057\u306b\u3057\u3066\u304a\u304f\u3002*\/<\/span>\r\nkill<\/span>(<\/span>x<\/span>, y<\/span>)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u30ac\u30a6\u30b9\u7a4d\u5206<\/h3>\n
$$ I = \\int_{-\\infty}^{\\infty} e^{-x^2 } dx$$<\/p>\n
\u3068\u3059\u308b\u3068\uff0c\u4e0a\u3067<\/p>\n
\\begin{eqnarray}
\nS &=& \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} e^{-x^2 – y^2} dx\\,dy \\\\
\n&=& I^2 \\\\
\n&=& \\pi
\n\\end{eqnarray}<\/p>\n
\u3068\u3044\u3046\u3053\u3068\u304c\u308f\u304b\u3063\u305f\u306e\u3067\uff0c<\/p>\n
$$ I = \\int_{-\\infty}^{\\infty} e^{-x^2 } dx = \\sqrt{S} = \\sqrt{\\pi}$$<\/p>\n
\u3068\u3044\u3046\u3053\u3068\u306b\u306a\u308b\u3002Maxima \u304c\u30ac\u30a6\u30b9\u7a4d\u5206 $I$ \u3092\u77e5\u3063\u3066\u3044\u308b\u3053\u3068\u3092\u78ba\u8a8d\u3059\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[11]:<\/div>\n
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'<\/span>integrate<\/span>(<\/span>exp<\/span>(<\/span>-<\/span>x<\/span>**<\/span>2<\/span>)<\/span>, x<\/span>, -<\/span>inf<\/span>, inf<\/span>)<\/span> =<\/span> \r\n integrate<\/span>(<\/span>exp<\/span>(<\/span>-<\/span>x<\/span>**<\/span>2<\/span>)<\/span>, x<\/span>, -<\/span>inf<\/span>, inf<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[11]:<\/div>\n
\\[\\tag{${\\it \\%o}_{39}$}\\int_{-\\infty }^{\\infty }{e^ {- x^2 }\\;dx}=\\sqrt{\\pi}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u88ab\u7a4d\u5206\u95a2\u6570\u306f\u5076\u95a2\u6570\u3067\u3042\u308b\u304b\u3089\uff0c\u7a4d\u5206\u7bc4\u56f2\u3092 $0$ \u304b\u3089\u306b\u3059\u308b\u3068\u7b54\u3048\u306f\u534a\u5206\u306b\u306a\u308b\u306f\u305a\u3067\uff0c\u3053\u308c\u3082\u78ba\u8a8d\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[12]:<\/div>\n
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'<\/span>integrate<\/span>(<\/span>exp<\/span>(<\/span>-<\/span>x<\/span>**<\/span>2<\/span>)<\/span>, x<\/span>, 0<\/span>, inf<\/span>)<\/span> =<\/span> \r\n integrate<\/span>(<\/span>exp<\/span>(<\/span>-<\/span>x<\/span>**<\/span>2<\/span>)<\/span>, x<\/span>, 0<\/span>, inf<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[12]:<\/div>\n
\\[\\tag{${\\it \\%o}_{40}$}\\int_{0}^{\\infty }{e^ {- x^2 }\\;dx}=\\frac{\\sqrt{\\pi}}{2}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u30ac\u30a6\u30b9\u7a4d\u5206\u306f\uff0c\u7a4d\u5206\u306e\u4e0a\u9650\u304c $\\infty$ \u306e\u3068\u304d\u306b\u3060\u3051\u89e3\u6790\u7684\u306b\u7a4d\u5206\u3067\u304d\u308b\u4f8b\u3002<\/p>\n
\u7a4d\u5206\u7bc4\u56f2\u304c $0$ \u304b\u3089\u4e00\u822c\u306b $x$ \u307e\u3067\u3060\u3068\u7a4d\u5206\u3067\u304d\u306a\u3044\uff08\u7d50\u679c\u304c\u521d\u7b49\u95a2\u6570\u3067\u8868\u305b\u306a\u3044\uff09\u3002\u5b9f\u969b\u306b\u3084\u3063\u3066\u307f\u308b\u3068…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[13]:<\/div>\n
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assume<\/span>(<\/span>x<\/span> ><\/span> 0<\/span>)<\/span>$\r\n'<\/span>integrate<\/span>(<\/span>exp<\/span>(<\/span>-<\/span>t<\/span>**<\/span>2<\/span>)<\/span>, t<\/span>, 0<\/span>, x<\/span>)<\/span> =<\/span> \r\n integrate<\/span>(<\/span>exp<\/span>(<\/span>-<\/span>t<\/span>**<\/span>2<\/span>)<\/span>, t<\/span>, 0<\/span>, x<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[13]:<\/div>\n
\\[\\tag{${\\it \\%o}_{42}$}\\int_{0}^{x}{e^ {- t^2 }\\;dt}=\\frac{\\sqrt{\\pi}\\,\\mathrm{erf}\\left(x\\right)}{2}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u53c2\u8003\uff1a\u8aa4\u5dee\u95a2\u6570<\/h4>\n
$\\operatorname{erf} (x)$ \u306f\u300c\u8aa4\u5dee\u95a2\u6570\u300d\u3068\u547c\u3070\u308c\uff0c\u3053\u308c\u304c\u5b9a\u7fa9\uff1a<\/p>\n
\\begin{eqnarray}
\n\\operatorname{erf} (x) &\\equiv& \\frac{2}{\\sqrt{\\pi}} \\int_0^x e^{-t^2} \\, dt
\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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\u30ac\u30a6\u30b9\u7a4d\u5206\u306b\u95a2\u9023\u3057\u305f\u7a4d\u5206<\/h3>\n
$$I(m) \\equiv \\int_{0}^{\\infty} x^m e^{-x^2 } dx$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[14]:<\/div>\n
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I<\/span>(<\/span>m<\/span>)<\/span>:=<\/span> integrate<\/span>(<\/span> x<\/span>**<\/span>m<\/span> *<\/span> exp<\/span>(<\/span>-<\/span>x<\/span>**<\/span>2<\/span>)<\/span>, x<\/span>, 0<\/span>, inf<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[14]:<\/div>\n
\\[\\tag{${\\it \\%o}_{43}$}I\\left(m\\right):={\\it integrate}\\left(x^{m}\\,\\exp \\left(-x^2\\right) , x , 0 , \\infty \\right)\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[15]:<\/div>\n
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I<\/span>(<\/span>0<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[15]:<\/div>\n
\\[\\tag{${\\it \\%o}_{44}$}\\frac{\\sqrt{\\pi}}{2}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[16]:<\/div>\n
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I<\/span>(<\/span>2<\/span>)<\/span>;\r\nI<\/span>(<\/span>4<\/span>)<\/span>;\r\nI<\/span>(<\/span>6<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[16]:<\/div>\n
\\[\\tag{${\\it \\%o}_{45}$}\\frac{\\sqrt{\\pi}}{4}\\]<\/div>\n<\/div>\n
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Out[16]:<\/div>\n
\\[\\tag{${\\it \\%o}_{46}$}\\frac{3\\,\\sqrt{\\pi}}{8}\\]<\/div>\n<\/div>\n
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Out[16]:<\/div>\n
\\[\\tag{${\\it \\%o}_{47}$}\\frac{15\\,\\sqrt{\\pi}}{16}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$m = 2 n$ ($n$ \u306f\u6574\u6570) \u306e\u5834\u5408\u3092\u63a8\u6e2c\u3059\u308b\u305f\u3081\u306b\uff0c$I(0)$ \u306e\u5024\u3067\u5272\u3063\u3066\uff0cfactor()<\/code> \u95a2\u6570\u3067\u5206\u6bcd\u5206\u5b50\u3092\u7d20\u56e0\u6570\u5206\u89e3\u3057\u3066\u307f\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[17]:<\/div>\n
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I<\/span>(<\/span>2<\/span>)<\/span>\/<\/span>I<\/span>(<\/span>0<\/span>)<\/span>, factor<\/span>;\r\nI<\/span>(<\/span>4<\/span>)<\/span>\/<\/span>I<\/span>(<\/span>0<\/span>)<\/span>, factor<\/span>;\r\nI<\/span>(<\/span>6<\/span>)<\/span>\/<\/span>I<\/span>(<\/span>0<\/span>)<\/span>, factor<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[17]:<\/div>\n
\\[\\tag{${\\it \\%o}_{48}$}\\frac{1}{2}\\]<\/div>\n<\/div>\n
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Out[17]:<\/div>\n
\\[\\tag{${\\it \\%o}_{49}$}\\frac{3}{2^2}\\]<\/div>\n<\/div>\n
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Out[17]:<\/div>\n
\\[\\tag{${\\it \\%o}_{50}$}\\frac{3\\,5}{2^3}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u3068\u3044\u3046\u3053\u3068\u306f\uff0c<\/p>\n
\\begin{eqnarray}
\nI(2) &=& 1\\times \\frac{1}{2} \\times I(0), \\\\
\nI(4) &=& \\frac{3\\cdot 1}{2^2} \\times I(0), \\\\
\nI(6) &=& \\frac{5\\cdot3\\cdot 1}{2^3} \\times I(0), \\dots
\n\\end{eqnarray}<\/p>\n
\u3060\u304b\u3089\uff0c<\/p>\n
$$\\displaystyle I(2 n) = \\frac{(2n-1)!!}{2^{n}} \\frac{\\sqrt{\\pi}}{2}$$<\/p>\n
\u3068\u63a8\u6e2c\u3055\u308c\u308b\u3002\u5b9f\u969b\u78ba\u304b\u3081\u3066\u307f\u308b\u3068…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[18]:<\/div>\n
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\/* m >= 2 \u306e\u5076\u6570\u306b\u5bfe\u3057\u3066\uff0c\u4e0a\u8a18\u306e 2 n \u3092 m \u3068\u304a\u3044\u3066 *\/<\/span>\r\n\r\nEvenI<\/span>(<\/span>m<\/span>)<\/span>:=<\/span> (<\/span>m<\/span>-<\/span>1<\/span>)<\/span>!!\/<\/span>2**<\/span>(<\/span>m<\/span>\/<\/span>2<\/span>)<\/span> *<\/span> I<\/span>(<\/span>0<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[18]:<\/div>\n
\\[\\tag{${\\it \\%o}_{51}$}{\\it EvenI}\\left(m\\right):=\\frac{{\\it genfact}\\left(m-1 , \\frac{m-1}{2} , 2\\right)\\,I\\left(0\\right)}{2^{\\frac{m}{2}}}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[19]:<\/div>\n
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\/* \u305f\u3068\u3048\u3070 m = 8 \u306e\u3068\u304d *\/<\/span>\r\nEvenI<\/span>(<\/span>8<\/span>)<\/span> -<\/span> I<\/span>(<\/span>8<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[19]:<\/div>\n
\\[\\tag{${\\it \\%o}_{52}$}0\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u3053\u308c\u3067 $m \\geq 2$ \u304c\u5076\u6570\u306e\u5834\u5408\u306e $I(m)$ \u306f\u6c42\u307e\u3063\u305f\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[20]:<\/div>\n
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\/* m = 2 \u304b\u3089 10 \u307e\u3067 MyEvenI(m) \u306e\u5024\u3092\u8868\u793a\u3055\u305b\u308b\u3002*\/<\/span>\r\nfor<\/span> m<\/span>:<\/span> 2<\/span> thru<\/span> 10<\/span> step<\/span> 2<\/span>\r\n    do<\/span>(<\/span>print<\/span>(<\/span>\"m = \"<\/span>, m<\/span>, \" \u306e\u3068\u304d: \"<\/span>, \r\n    '<\/span>integrate<\/span>(<\/span> x<\/span>**<\/span>m<\/span> *<\/span> exp<\/span>(<\/span>-<\/span>x<\/span>**<\/span>2<\/span>)<\/span>, x<\/span>, 0<\/span>, inf<\/span>)<\/span> =<\/span> EvenI<\/span>(<\/span>m<\/span>)))<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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m = \\(2\\) \u306e\u3068\u304d: \\(\\int_{0}^{\\infty }{x^2\\,e^ {- x^2 }\\;dx}=\\frac{\\sqrt{\\pi}}{4}\\)<\/div>\n<\/div>\n
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m = \\(4\\) \u306e\u3068\u304d: \\(\\int_{0}^{\\infty }{x^4\\,e^ {- x^2 }\\;dx}=\\frac{3\\,\\sqrt{\\pi}}{8}\\)<\/div>\n<\/div>\n
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m = \\(6\\) \u306e\u3068\u304d: \\(\\int_{0}^{\\infty }{x^6\\,e^ {- x^2 }\\;dx}=\\frac{15\\,\\sqrt{\\pi}}{16}\\)<\/div>\n<\/div>\n
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m = \\(8\\) \u306e\u3068\u304d: \\(\\int_{0}^{\\infty }{x^8\\,e^ {- x^2 }\\;dx}=\\frac{105\\,\\sqrt{\\pi}}{32}\\)<\/div>\n<\/div>\n
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m = \\(10\\) \u306e\u3068\u304d: \\(\\int_{0}^{\\infty }{x^{10}\\,e^ {- x^2 }\\;dx}=\\frac{945\\,\\sqrt{\\pi}}{64}\\)<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u6b21\u306b\uff0c$m$ \u304c\u5947\u6570\uff08$m = 2 n +1$\uff09\u306e\u5834\u5408\u306e $I(m)$ \u306e\u4e00\u822c\u5f62\u3092\u4e88\u60f3\u3057\u3066\u307f\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[21]:<\/div>\n
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I<\/span>(<\/span>1<\/span>)<\/span>;\r\nI<\/span>(<\/span>3<\/span>)<\/span>;\r\nI<\/span>(<\/span>5<\/span>)<\/span>;\r\nI<\/span>(<\/span>7<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[21]:<\/div>\n
\\[\\tag{${\\it \\%o}_{54}$}\\frac{1}{2}\\]<\/div>\n<\/div>\n
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Out[21]:<\/div>\n
\\[\\tag{${\\it \\%o}_{55}$}\\frac{1}{2}\\]<\/div>\n<\/div>\n
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Out[21]:<\/div>\n
\\[\\tag{${\\it \\%o}_{56}$}1\\]<\/div>\n<\/div>\n
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Out[21]:<\/div>\n
\\[\\tag{${\\it \\%o}_{57}$}3\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[22]:<\/div>\n
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I<\/span>(<\/span>3<\/span>)<\/span>\/<\/span>I<\/span>(<\/span>1<\/span>)<\/span>, factor<\/span>;\r\nI<\/span>(<\/span>5<\/span>)<\/span>\/<\/span>I<\/span>(<\/span>1<\/span>)<\/span>, factor<\/span>;\r\nI<\/span>(<\/span>7<\/span>)<\/span>\/<\/span>I<\/span>(<\/span>1<\/span>)<\/span>, factor<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[22]:<\/div>\n
\\[\\tag{${\\it \\%o}_{58}$}1\\]<\/div>\n<\/div>\n
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Out[22]:<\/div>\n
\\[\\tag{${\\it \\%o}_{59}$}2\\]<\/div>\n<\/div>\n
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Out[22]:<\/div>\n
\\[\\tag{${\\it \\%o}_{60}$}2\\,3\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\n
\u4ee5\u4e0a\u306e\u3053\u3068\u304b\u3089\u6b21\u306e\u3088\u3046\u306a\u63a8\u6e2c\u304c\u3067\u304d\u308b\u3002<\/p>\n
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