{"id":6200,"date":"2023-04-20T12:06:16","date_gmt":"2023-04-20T03:06:16","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=6200"},"modified":"2024-03-15T14:51:50","modified_gmt":"2024-03-15T05:51:50","slug":"sympy-%e3%81%a7%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\/sympy-%e3%81%a7%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6c\/sympy-%e3%81%a7%e5%a4%9a%e9%87%8d%e7%a9%8d%e5%88%86\/","title":{"rendered":"SymPy \u3067\u591a\u91cd\u7a4d\u5206"},"content":{"rendered":"

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\u5fc5\u8981\u306a\u30d1\u30c3\u30b1\u30fc\u30b8\u3092 import \u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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In\u00a0[1]:<\/div>\n
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from sympy.abc<\/span> import *<\/span> \r\nfrom<\/span> sympy<\/span> import<\/span> *<\/span>\r\n\r\n# SymPy Plotting Backends (SPB): \u30b0\u30e9\u30d5\u3092\u63cf\u304f\u969b\u306b\u5229\u7528<\/span>\r\nfrom<\/span> spb<\/span> import<\/span> *<\/span>\r\n\r\n# \u30b0\u30e9\u30d5\u3092 SVG \u3067 Notebook \u306b\u30a4\u30f3\u30e9\u30a4\u30f3\u8868\u793a<\/span>\r\n%<\/span>config<\/span> InlineBackend.figure_formats = ['svg']\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\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
2\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
SymPy \u3067\u306f2\u91cd\u7a4d\u5206\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3002<\/p>\n
$\\displaystyle \\int_{y_1}^{y_2} \\left\\{\\int_{x_1}^{x_2} f(x, y) dx \\right\\} dy =
\n$<\/p>\n
integrate(f(x, y), (x, x1, x2), (y, y1, y2))<\/code><\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[2]:<\/div>\n
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f<\/span> =<\/span> y<\/span>\r\nx1<\/span> =<\/span> 0<\/span>\r\nx2<\/span> =<\/span> 2<\/span>\r\ny1<\/span> =<\/span> 0<\/span>\r\ny2<\/span> =<\/span> 2<\/span>\r\n\r\n# \u624b\u3063\u53d6\u308a\u65e9\u304f\u7b54\u3048\u3060\u3051\u3092\u8868\u793a\u3055\u305b\u305f\u3044\u306a\u3089<\/span>\r\nintegrate<\/span>(<\/span>f<\/span>,<\/span> (<\/span>x<\/span>,<\/span> x1<\/span>,<\/span> x2<\/span>),<\/span> (<\/span>y<\/span>,<\/span> y1<\/span>,<\/span> y2<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle 4$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[3]:<\/div>\n
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# \u4f55\u3092\u8a08\u7b97\u3059\u308b\u304b\u8868\u793a\u3055\u305b\u3066\u304b\u3089\u7b54\u3048\u3092\u8868\u793a\u3055\u305b\u308b\u306a\u3089<\/span>\r\n\r\nEq<\/span>(<\/span>Integral<\/span>(<\/span>y<\/span>,<\/span> (<\/span>x<\/span>,<\/span> x1<\/span>,<\/span> x2<\/span>),<\/span> (<\/span>y<\/span>,<\/span> y1<\/span>,<\/span> y2<\/span>)),<\/span> \r\n   Integral<\/span>(<\/span>y<\/span>,<\/span> (<\/span>x<\/span>,<\/span> x1<\/span>,<\/span> x2<\/span>),<\/span> (<\/span>y<\/span>,<\/span> y1<\/span>,<\/span> y2<\/span>))<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle \\int\\limits_{0}^{2}\\int\\limits_{0}^{2} y\\, dx\\, dy = 4$<\/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
$x_1 \\leq x \\leq x_2$ \u306e\u7bc4\u56f2\u3067 $y = y1$ \u3068 $y = y2$ \u306b\u56f2\u307e\u308c\u305f\u9818\u57df\u3092\u5857\u308a\u3064\u3076\u3059\u65b9\u6cd5\u306f\uff0cSymPy Plotting Backends \u3067\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[4]:<\/div>\n
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# NumPy \u306b\u983c\u3089\u306a\u3044\u4f8b<\/span>\r\n# x1 ~ x2 \u3092 100 \u7b49\u5206\u3057\u305f\u6570\u5024\u30ea\u30b9\u30c8\u3092\u4f5c\u6210<\/span>\r\nx_list<\/span> =<\/span> [(<\/span>x1<\/span> +<\/span> (<\/span>x2<\/span>-<\/span>x1<\/span>)<\/span>*<\/span>i<\/span>\/<\/span>100<\/span>)<\/span> for<\/span> i<\/span> in<\/span> range<\/span>(<\/span>101<\/span>)]<\/span>\r\n# x_array \u306b\u5bfe\u5fdc\u3057\u305f y1 \u306e\u6570\u5024\u30ea\u30b9\u30c8\u3092\u4f5c\u6210<\/span>\r\ny1_list<\/span> =<\/span> lambdify<\/span>(<\/span>x<\/span>,<\/span> y1<\/span>)(<\/span>x_list<\/span>)<\/span>\r\n# x_array \u306b\u5bfe\u5fdc\u3057\u305f y2 \u306e\u6570\u5024\u30ea\u30b9\u30c8\u3092\u4f5c\u6210<\/span>\r\ny2_list<\/span> =<\/span> lambdify<\/span>(<\/span>x<\/span>,<\/span> y2<\/span>)(<\/span>x_list<\/span>)<\/span>\r\n\r\nplot<\/span>(<\/span>xlim<\/span> =<\/span> (<\/span>-<\/span>0.5<\/span>,<\/span> 2.5<\/span>),<\/span> \r\n     ylim<\/span> =<\/span> (<\/span>-<\/span>0.5<\/span>,<\/span> 2.5<\/span>),<\/span> \r\n     aspect<\/span> =<\/span> \"equal\"<\/span>,<\/span>\r\n     title<\/span> =<\/span> \"\u9818\u57df $D$\"<\/span>,<\/span>\r\n     xlabel<\/span> =<\/span> \"x\"<\/span>,<\/span> ylabel<\/span> =<\/span> \"y\"<\/span>,<\/span>\r\n     fill<\/span>=<\/span>{<\/span>'x'<\/span>:<\/span> x_list<\/span>,<\/span>\r\n           'y1'<\/span>:<\/span>y1_list<\/span>,<\/span>\r\n           'y2'<\/span>:<\/span>y2_list<\/span>,<\/span>\r\n           'color'<\/span>:<\/span>'yellow'<\/span>})<\/span>;<\/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[5]:<\/div>\n
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plot3d<\/span>((<\/span>0<\/span>,<\/span> (<\/span>x<\/span>,<\/span> x1<\/span>,<\/span> x2<\/span>),<\/span> (<\/span>y<\/span>,<\/span> y1<\/span>,<\/span> y2<\/span>)),<\/span> \r\n       (<\/span>f<\/span>,<\/span> (<\/span>x<\/span>,<\/span> x1<\/span>,<\/span> x2<\/span>),<\/span> (<\/span>y<\/span>,<\/span> y1<\/span>,<\/span> y2<\/span>)),<\/span>\r\n       xlim<\/span> =<\/span> (<\/span>-<\/span>0.5<\/span>,<\/span> 2.5<\/span>),<\/span> ylim<\/span> =<\/span> (<\/span>-<\/span>0.5<\/span>,<\/span> 2.5<\/span>),<\/span> \r\n       zlim<\/span> =<\/span> (<\/span>0<\/span>,<\/span> 3<\/span>),<\/span> n<\/span> =<\/span> 10<\/span>,<\/span>\r\n       rendering_kw<\/span>=<\/span>[{<\/span>\"color\"<\/span>:<\/span>\"yellow\"<\/span>},<\/span>\r\n                     {<\/span>\"color\"<\/span>:<\/span>\"lightblue\"<\/span>}])<\/span>;<\/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\u8584\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> 1<\/span>\r\nx1<\/span> =<\/span> -<\/span>1<\/span>\r\nx2<\/span> =<\/span> 1<\/span>\r\nphi1<\/span> =<\/span> -<\/span>sqrt<\/span>(<\/span>1<\/span>-<\/span>x<\/span>**<\/span>2<\/span>)<\/span>\r\nphi2<\/span> =<\/span> sqrt<\/span>(<\/span>1<\/span>-<\/span>x<\/span>**<\/span>2<\/span>)<\/span>\r\n\r\nEq<\/span>(<\/span>Integral<\/span>(<\/span>f<\/span>,<\/span> (<\/span>y<\/span>,<\/span> phi1<\/span>,<\/span> phi2<\/span>),<\/span> (<\/span>x<\/span>,<\/span> x1<\/span>,<\/span> x2<\/span>)),<\/span> \r\n   Integral<\/span>(<\/span>f<\/span>,<\/span> (<\/span>y<\/span>,<\/span> phi1<\/span>,<\/span> phi2<\/span>),<\/span> (<\/span>x<\/span>,<\/span> x1<\/span>,<\/span> x2<\/span>))<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle \\int\\limits_{-1}^{1}\\int\\limits_{- \\sqrt{1 – x^{2}}}^{\\sqrt{1 – x^{2}}} 1\\, dy\\, dx = \\pi$<\/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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In\u00a0[7]:<\/div>\n
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# NumPy \u3092 import \u3057\u3066\u4f7f\u3046\u5834\u5408<\/span>\r\nimport<\/span> numpy<\/span> as<\/span> np<\/span>\r\nx_array<\/span> =<\/span> np<\/span>.<\/span>linspace<\/span>(<\/span>x1<\/span>,<\/span> x2<\/span>,<\/span> 100<\/span>)<\/span>\r\ndef<\/span> y1<\/span>(<\/span>x<\/span>):<\/span>\r\n    return<\/span> -<\/span>np<\/span>.<\/span>sqrt<\/span>(<\/span>1<\/span>-<\/span>x<\/span>**<\/span>2<\/span>)<\/span>\r\ndef<\/span> y2<\/span>(<\/span>x<\/span>):<\/span>\r\n    return<\/span> np<\/span>.<\/span>sqrt<\/span>(<\/span>1<\/span>-<\/span>x<\/span>**<\/span>2<\/span>)<\/span>\r\n\r\ny1_array<\/span> =<\/span> y1<\/span>(<\/span>x_array<\/span>)<\/span>\r\ny2_array<\/span> =<\/span> y2<\/span>(<\/span>x_array<\/span>)<\/span>\r\n\r\nplot<\/span>(<\/span>xlim<\/span> =<\/span> (<\/span>-<\/span>1.5<\/span>,<\/span> 1.5<\/span>),<\/span> \r\n     ylim<\/span> =<\/span> (<\/span>-<\/span>1.5<\/span>,<\/span> 1.5<\/span>),<\/span> \r\n     aspect<\/span> =<\/span> \"equal\"<\/span>,<\/span>\r\n     title<\/span> =<\/span> \"\u9818\u57df $D$\"<\/span>,<\/span>\r\n     xlabel<\/span> =<\/span> \"x\"<\/span>,<\/span> ylabel<\/span> =<\/span> \"y\"<\/span>,<\/span>\r\n     fill<\/span>=<\/span>{<\/span>'x'<\/span>:<\/span> x_array<\/span>,<\/span>\r\n           'y1'<\/span>:<\/span>y1_array<\/span>,<\/span>\r\n           'y2'<\/span>:<\/span>y2_array<\/span>,<\/span>\r\n           'color'<\/span>:<\/span>'yellow'<\/span>})<\/span>;<\/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
\u30e4\u30b3\u30d3\u884c\u4f8b<\/h5>\n
SymPy \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[8]:<\/div>\n
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X<\/span> =<\/span> Matrix<\/span>([<\/span>r<\/span>*<\/span>cos<\/span>(<\/span>theta<\/span>),<\/span> r<\/span>*<\/span>sin<\/span>(<\/span>theta<\/span>)])<\/span>\r\nY<\/span> =<\/span> Matrix<\/span>([<\/span>r<\/span>,<\/span> theta<\/span>])<\/span>\r\n\r\njaco<\/span> =<\/span> X<\/span>.<\/span>jacobian<\/span>(<\/span>Y<\/span>)<\/span>\r\njaco<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[8]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}\\cos{\\left(\\theta \\right)} & – r \\sin{\\left(\\theta \\right)}\\\\\\sin{\\left(\\theta \\right)} & r \\cos{\\left(\\theta \\right)}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
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\n
\u30e4\u30b3\u30d3\u30a2\u30f3\uff08\u30e4\u30b3\u30d3\u884c\u5217\u5f0f\uff09<\/h5>\n
jacobian()<\/code> \u306f\u30e4\u30b3\u30d3\u884c\u5217\u3002\u3053\u306e\u884c\u5217\u306e\u884c\u5217\u5f0f\u304c\uff0c\u6211\u3005\u304c\u6c42\u3081\u305f\u3044\u30e4\u30b3\u30d3\u30a2\u30f3\u306f\u30e4\u30b3\u30d3\u884c\u5217\u306e\u884c\u5217\u5f0f\u3002det()<\/code> \u3067\u884c\u5217\u5f0f\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[9]:<\/div>\n
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det<\/span>(<\/span>jaco<\/span>)<\/span>.<\/span>simplify<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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Out[9]:<\/div>\n
$\\displaystyle r$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\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 = \\cdots
\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
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In\u00a0[10]:<\/div>\n
\n
\n
\n
Eq<\/span>(<\/span>Integral<\/span>(<\/span>exp<\/span>(<\/span>-<\/span>r<\/span>**<\/span>2<\/span>)<\/span>*<\/span>r<\/span>,<\/span> (<\/span>theta<\/span>,<\/span> 0<\/span>,<\/span> 2<\/span>*<\/span>pi<\/span>),<\/span> (<\/span>r<\/span>,<\/span> 0<\/span>,<\/span> oo<\/span>)),<\/span>\r\n   Integral<\/span>(<\/span>exp<\/span>(<\/span>-<\/span>r<\/span>**<\/span>2<\/span>)<\/span>*<\/span>r<\/span>,<\/span> (<\/span>theta<\/span>,<\/span> 0<\/span>,<\/span> 2<\/span>*<\/span>pi<\/span>),<\/span> (<\/span>r<\/span>,<\/span> 0<\/span>,<\/span> oo<\/span>))<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[10]:<\/div>\n
$\\displaystyle \\int\\limits_{0}^{\\infty}\\int\\limits_{0}^{2 \\pi} r e^{- r^{2}}\\, d\\theta\\, dr = \\pi$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
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\n
\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\u3002SymPy \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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Eq<\/span>(<\/span>Integral<\/span>(<\/span>exp<\/span>(<\/span>-<\/span>x<\/span>**<\/span>2<\/span>),<\/span> (<\/span>x<\/span>,<\/span> -<\/span>oo<\/span>,<\/span> oo<\/span>)),<\/span>\r\n   Integral<\/span>(<\/span>exp<\/span>(<\/span>-<\/span>x<\/span>**<\/span>2<\/span>),<\/span> (<\/span>x<\/span>,<\/span> -<\/span>oo<\/span>,<\/span> oo<\/span>))<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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Out[11]:<\/div>\n
$\\displaystyle \\int\\limits_{-\\infty}^{\\infty} e^{- x^{2}}\\, dx = \\sqrt{\\pi}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
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\n
\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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Eq<\/span>(<\/span>Integral<\/span>(<\/span>exp<\/span>(<\/span>-<\/span>x<\/span>**<\/span>2<\/span>),<\/span> (<\/span>x<\/span>,<\/span> 0<\/span>,<\/span> oo<\/span>)),<\/span>\r\n   Integral<\/span>(<\/span>exp<\/span>(<\/span>-<\/span>x<\/span>**<\/span>2<\/span>),<\/span> (<\/span>x<\/span>,<\/span> 0<\/span>,<\/span> oo<\/span>))<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[12]:<\/div>\n
$\\displaystyle \\int\\limits_{0}^{\\infty} e^{- x^{2}}\\, dx = \\frac{\\sqrt{\\pi}}{2}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
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\n
\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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Eq<\/span>(<\/span>Integral<\/span>(<\/span>exp<\/span>(<\/span>-<\/span>t<\/span>**<\/span>2<\/span>),<\/span> (<\/span>t<\/span>,<\/span> 0<\/span>,<\/span> x<\/span>)),<\/span>\r\n   Integral<\/span>(<\/span>exp<\/span>(<\/span>-<\/span>t<\/span>**<\/span>2<\/span>),<\/span> (<\/span>t<\/span>,<\/span> 0<\/span>,<\/span> x<\/span>))<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[13]:<\/div>\n
$\\displaystyle \\int\\limits_{0}^{x} e^{- t^{2}}\\, dt = \\frac{\\sqrt{\\pi} \\operatorname{erf}{\\left(x \\right)}}{2}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\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
\n
<\/div>\n
\n
\n
\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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def<\/span> I<\/span>(<\/span>m<\/span>):<\/span>\r\n    return<\/span> integrate<\/span>(<\/span>x<\/span>**<\/span>m<\/span> *<\/span> exp<\/span>(<\/span>-<\/span>x<\/span>**<\/span>2<\/span>),<\/span> (<\/span>x<\/span>,<\/span> 0<\/span>,<\/span> oo<\/span>))<\/span>\r\n<\/pre>\n<\/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
$\\displaystyle \\frac{\\sqrt{\\pi}}{2}$<\/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\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[16]:<\/div>\n
$\\displaystyle \\frac{\\sqrt{\\pi}}{4}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[17]:<\/div>\n
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I<\/span>(<\/span>4<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[17]:<\/div>\n
$\\displaystyle \\frac{3 \\sqrt{\\pi}}{8}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[18]:<\/div>\n
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I<\/span>(<\/span>6<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[18]:<\/div>\n
$\\displaystyle \\frac{15 \\sqrt{\\pi}}{16}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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<\/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\u307f\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[19]:<\/div>\n
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I<\/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[19]:<\/div>\n
$\\displaystyle \\frac{1}{2}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[20]:<\/div>\n
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I<\/span>(<\/span>4<\/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[20]:<\/div>\n
$\\displaystyle \\frac{3}{4}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[21]:<\/div>\n
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I<\/span>(<\/span>6<\/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[21]:<\/div>\n
$\\displaystyle \\frac{15}{8}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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<\/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
\uff08SymPy \u3067\u306f $n! = $ factorial(n)<\/code>\uff0c$n!! = $ factorial2(n)<\/code>\uff09<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[22]:<\/div>\n
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# m >= 2 \u306e\u5076\u6570\u306b\u5bfe\u3057\u3066\uff0c\u4e0a\u8a18\u306e 2n \u3092 m \u3068\u304a\u3044\u3066<\/span>\r\n\r\ndef<\/span> EvenI<\/span>(<\/span>m<\/span>):<\/span>\r\n    return<\/span> factorial2<\/span>(<\/span>m<\/span>-<\/span>1<\/span>)<\/span>\/<\/span>2<\/span>**<\/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<\/div>\n
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In\u00a0[23]:<\/div>\n
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# \u4f8b\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[23]:<\/div>\n
$\\displaystyle 0$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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<\/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
\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[24]:<\/div>\n
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I<\/span>(<\/span>1<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[24]:<\/div>\n
$\\displaystyle \\frac{1}{2}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[25]:<\/div>\n
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I<\/span>(<\/span>3<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[25]:<\/div>\n
$\\displaystyle \\frac{1}{2}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[26]:<\/div>\n
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I<\/span>(<\/span>5<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[26]:<\/div>\n
$\\displaystyle 1$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[27]:<\/div>\n
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I<\/span>(<\/span>7<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[27]:<\/div>\n
$\\displaystyle 3$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[28]:<\/div>\n
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I<\/span>(<\/span>3<\/span>)<\/span>\/<\/span>I<\/span>(<\/span>1<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[28]:<\/div>\n
$\\displaystyle 1$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[29]:<\/div>\n
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I<\/span>(<\/span>5<\/span>)<\/span>\/<\/span>I<\/span>(<\/span>1<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[29]:<\/div>\n
$\\displaystyle 2$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[30]:<\/div>\n
\n
\n
\n
I<\/span>(<\/span>7<\/span>)<\/span>\/<\/span>I<\/span>(<\/span>1<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[30]:<\/div>\n
$\\displaystyle 6$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
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\u4ee5\u4e0a\u306e\u3053\u3068\u304b\u3089\u6b21\u306e\u3088\u3046\u306a\u63a8\u6e2c\u304c\u3067\u304d\u308b\u3002<\/p>\n
$$ I(2n+1) = n! \\times I(1) $$<\/p>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":6176,"menu_order":30,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-6200","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\/6200","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=6200"}],"version-history":[{"count":2,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6200\/revisions"}],"predecessor-version":[{"id":8093,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6200\/revisions\/8093"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6176"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=6200"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}