{"id":2746,"date":"2022-03-29T17:26:41","date_gmt":"2022-03-29T08:26:41","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2746"},"modified":"2022-04-22T16:43:58","modified_gmt":"2022-04-22T07:43:58","slug":"%e5%8f%82%e8%80%83%ef%bc%9a%e9%9d%99%e9%9b%bb%e5%a0%b4%e3%82%92%e6%b1%82%e3%82%81%e3%82%8b%e9%9a%9b%e3%81%ab%e4%bd%bf%e3%81%a3%e3%81%9f%e7%a9%8d%e5%88%86%e3%82%92-maxima-jupyter-%e3%81%a7%e7%a2%ba","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e9%9b%bb%e7%a3%81%e6%b0%97%e5%ad%a6-i\/%e9%9d%99%e9%9b%bb%e5%a0%b4%ef%bc%9a%e9%9b%bb%e8%8d%b7%e5%af%86%e5%ba%a6%e3%81%8b%e3%82%89%e7%9b%b4%e6%8e%a5%e9%9b%bb%e5%a0%b4%e3%82%92%e6%b1%82%e3%82%81%e3%82%8b\/%e5%8f%82%e8%80%83%ef%bc%9a%e9%9d%99%e9%9b%bb%e5%a0%b4%e3%82%92%e6%b1%82%e3%82%81%e3%82%8b%e9%9a%9b%e3%81%ab%e4%bd%bf%e3%81%a3%e3%81%9f%e7%a9%8d%e5%88%86%e3%82%92-maxima-jupyter-%e3%81%a7%e7%a2%ba\/","title":{"rendered":"\u53c2\u8003\uff1a\u9759\u96fb\u5834\u3092\u6c42\u3081\u308b\u969b\u306b\u4f7f\u3063\u305f\u7a4d\u5206\u3092 Maxima-Jupyter \u3067\u78ba\u8a8d\u3059\u308b"},"content":{"rendered":"

<\/p>\n

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\u9759\u96fb\u5834\u3092\u6c42\u3081\u308b\u969b\u306b\u4f7f\u3063\u305f\u7a4d\u5206\u3092 Maxima-Jupyter \u3067\u78ba\u8a8d\u3059\u308b<\/h3>\n

\u4e00\u69d8\u306a\u7dda\u96fb\u8377\u306b\u3088\u308b\u96fb\u5834\u3067\u4f7f\u3063\u305f\u7a4d\u5206<\/h4>\n<\/div>\n<\/div>\n<\/div>\n
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<\/div>\n
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$$\\int_{-\\infty}^{\\infty}
\n\\frac{1}{\\left(x^2 + y^2 + (z-z’)^2\\right)^{3\/2}} dz’ =\u00a0 \\frac{2}{x^2 + y^2}$$\u304a\u3088\u3073<\/p>\n

$$\\int_{-\\infty}^{\\infty}
\n\\frac{z – z’}{\\left(x^2 + y^2 + (z-z’)^2\\right)^{3\/2}} dz’ =0$$<\/p>\n

\u306e\u78ba\u8a8d\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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In\u00a0[1]:<\/div>\n
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\/* \u5206\u6bcd\u304c\u30bc\u30ed\u306b\u306a\u3089\u306a\u3044\u3088\u3046\u306b\u4eee\u5b9a\u3057\u307e\u3059\u3002*\/<\/span>\r\nassume<\/span>(<\/span>x<\/span> ><\/span> 0<\/span>)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[2]:<\/div>\n
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'<\/span>integrate<\/span>(<\/span>1\/<\/span>(<\/span>x<\/span>**<\/span>2<\/span> +<\/span> y<\/span>**<\/span>2<\/span> +<\/span> (<\/span>z<\/span>-<\/span>z1<\/span>)<\/span>**<\/span>2<\/span>)<\/span>**<\/span>(<\/span>3\/<\/span>2<\/span>)<\/span>, z1<\/span>, -<\/span>inf<\/span>, inf<\/span>)<\/span> =<\/span> \r\n integrate<\/span>(<\/span>1\/<\/span>(<\/span>x<\/span>**<\/span>2<\/span> +<\/span> y<\/span>**<\/span>2<\/span> +<\/span> (<\/span>z<\/span>-<\/span>z1<\/span>)<\/span>**<\/span>2<\/span>)<\/span>**<\/span>(<\/span>3\/<\/span>2<\/span>)<\/span>, z1<\/span>, -<\/span>inf<\/span>, inf<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[2]:<\/div>\n
\\[\\tag{${\\it \\%o}_{2}$}\\int_{-\\infty }^{\\infty }{\\frac{1}{\\left(\\left(z-z_{1}\\right)^2+y^2+x^2\\right)^{\\frac{3}{2}}}\\;dz_{1}}=\\frac{2}{y^2+x^2}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[3]:<\/div>\n
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'<\/span>integrate<\/span>(<\/span> (<\/span>z<\/span>-<\/span>z1<\/span>)<\/span>\/<\/span>(<\/span>x<\/span>**<\/span>2<\/span> +<\/span> y<\/span>**<\/span>2<\/span> +<\/span> (<\/span>z<\/span>-<\/span>z1<\/span>)<\/span>**<\/span>2<\/span>)<\/span>**<\/span>(<\/span>3\/<\/span>2<\/span>)<\/span> , z1<\/span>, -<\/span>inf<\/span>, inf<\/span>)<\/span> =<\/span> \r\n integrate<\/span>(<\/span> (<\/span>z<\/span>-<\/span>z1<\/span>)<\/span>\/<\/span>(<\/span>x<\/span>**<\/span>2<\/span> +<\/span> y<\/span>**<\/span>2<\/span> +<\/span> (<\/span>z<\/span>-<\/span>z1<\/span>)<\/span>**<\/span>2<\/span>)<\/span>**<\/span>(<\/span>3\/<\/span>2<\/span>)<\/span> , z1<\/span>, -<\/span>inf<\/span>, inf<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[3]:<\/div>\n
\\[\\tag{${\\it \\%o}_{3}$}\\int_{-\\infty }^{\\infty }{\\frac{z-z_{1}}{\\left(\\left(z-z_{1}\\right)^2+y^2+x^2\\right)^{\\frac{3}{2}}}\\;dz_{1}}=0\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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<\/div>\n
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\u8ef8\u5bfe\u79f0\u306a\u96fb\u8377\u5206\u5e03\u306b\u3088\u308b\u96fb\u5834\u3067\u4f7f\u3063\u305f\u7a4d\u5206\u306e\u78ba\u8a8d<\/h4>\n
$$\\int_0^{2\\pi} d\\phi’ \\frac{x-r’ \\cos\\phi’}{(x-r’ \\cos \\phi’)^2 + (y-r’ \\sin \\phi’)^2} = \\frac{2\\pi x}{x^2 + y^2} H(\\sqrt{x^2 + y^2} – r’)$$<\/p>\n
\u304a\u3088\u3073<\/p>\n
$$\\int_0^{2\\pi} d\\phi’ \\frac{y-r’ \\sin\\phi’}{(x-r’ \\cos \\phi’)^2 + (y-r’ \\sin \\phi’)^2} = \\frac{2\\pi y}{x^2 + y^2} H(\\sqrt{x^2 + y^2} – r’)$$<\/p>\n
\u306e\u78ba\u8a8d\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[4]:<\/div>\n
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\/* r > r' \u306e\u5834\u5408 *\/<\/span>\r\n\r\nfacts<\/span>(<\/span>r1<\/span>)<\/span>$\r\nforget<\/span>(<\/span>%<\/span>)<\/span>$\r\n\r\nassume<\/span>(<\/span>0<\/span> <<\/span> r1<\/span>, r1<\/span> <<\/span> sqrt<\/span>(<\/span>x<\/span>**<\/span>2+<\/span>y<\/span>**<\/span>2<\/span>))<\/span>$\r\n\r\n'<\/span>integrate<\/span>((<\/span>x<\/span>-<\/span>r1<\/span>*<\/span>cos<\/span>(<\/span>phi1<\/span>))<\/span>\/<\/span>\r\n    (<\/span>x<\/span>**<\/span>2+<\/span>y<\/span>**<\/span>2+<\/span>r1<\/span>**<\/span>2-<\/span>2*<\/span>r1<\/span>*<\/span>(<\/span>x<\/span>*<\/span>cos<\/span>(<\/span>phi1<\/span>)<\/span>+<\/span>y<\/span>*<\/span>sin<\/span>(<\/span>phi1<\/span>)))<\/span>, phi1<\/span>, 0<\/span>, 2*<\/span>%pi<\/span>)<\/span>=<\/span>\r\n    integrate<\/span>((<\/span>x<\/span>-<\/span>r1<\/span>*<\/span>cos<\/span>(<\/span>phi1<\/span>))<\/span>\/<\/span>\r\n    (<\/span>x<\/span>**<\/span>2+<\/span>y<\/span>**<\/span>2+<\/span>r1<\/span>**<\/span>2-<\/span>2*<\/span>r1<\/span>*<\/span>(<\/span>x<\/span>*<\/span>cos<\/span>(<\/span>phi1<\/span>)<\/span>+<\/span>y<\/span>*<\/span>sin<\/span>(<\/span>phi1<\/span>)))<\/span>, phi1<\/span>, 0<\/span>, 2*<\/span>%pi<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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<\/div>\n
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\u001bX           2    2\r\n          y  + x\r\nIs --------------------- - 1 positive, negative or zero?\r\n          2  2     2  2\r\n   sqrt(r1  y  + r1  x )\r\n\u001b\\           2    2\r\n          y  + x\r\nIs --------------------- - 1 positive, negative or zero?\r\n          2  2     2  2\r\n   sqrt(r1  y  + r1  x )\r\npositive;\r\n\r\n<\/pre>\n<\/div>\n<\/div>\n
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Out[4]:<\/div>\n
\\[\\tag{${\\it \\%o}_{7}$}\\int_{0}^{2\\,\\pi}{\\frac{x-\\cos \\varphi_{1}\\,r_{1}}{y^2-2\\,r_{1}\\,\\left(\\sin \\varphi_{1}\\,y+\\cos \\varphi_{1}\\,x\\right)+x^2+r_{1}^2}\\;d\\varphi_{1}}=\\frac{2\\,\\pi\\,x}{y^2+x^2}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[5]:<\/div>\n
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\/* r < r' \u306e\u5834\u5408 *\/<\/span>\r\n\r\nfacts<\/span>(<\/span>r1<\/span>)<\/span>$\r\nforget<\/span>(<\/span>%<\/span>)<\/span>$\r\n\r\nassume<\/span>(<\/span>0<\/span> <<\/span> r1<\/span>, r1<\/span> ><\/span> sqrt<\/span>(<\/span>x<\/span>**<\/span>2+<\/span>y<\/span>**<\/span>2<\/span>))<\/span>$\r\n\r\n'<\/span>integrate<\/span>((<\/span>x<\/span>-<\/span>r1<\/span>*<\/span>cos<\/span>(<\/span>phi1<\/span>))<\/span>\/<\/span>\r\n    (<\/span>x<\/span>**<\/span>2+<\/span>y<\/span>**<\/span>2+<\/span>r1<\/span>**<\/span>2-<\/span>2*<\/span>r1<\/span>*<\/span>(<\/span>x<\/span>*<\/span>cos<\/span>(<\/span>phi1<\/span>)<\/span>+<\/span>y<\/span>*<\/span>sin<\/span>(<\/span>phi1<\/span>)))<\/span>, phi1<\/span>, 0<\/span>, 2*<\/span>%pi<\/span>)<\/span>=<\/span>\r\n    integrate<\/span>((<\/span>x<\/span>-<\/span>r1<\/span>*<\/span>cos<\/span>(<\/span>phi1<\/span>))<\/span>\/<\/span>\r\n    (<\/span>x<\/span>**<\/span>2+<\/span>y<\/span>**<\/span>2+<\/span>r1<\/span>**<\/span>2-<\/span>2*<\/span>r1<\/span>*<\/span>(<\/span>x<\/span>*<\/span>cos<\/span>(<\/span>phi1<\/span>)<\/span>+<\/span>y<\/span>*<\/span>sin<\/span>(<\/span>phi1<\/span>)))<\/span>, phi1<\/span>, 0<\/span>, 2*<\/span>%pi<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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<\/div>\n
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\u001bX           2    2\r\n          y  + x\r\nIs --------------------- - 1 positive, negative or zero?\r\n          2  2     2  2\r\n   sqrt(r1  y  + r1  x )\r\n\u001b\\           2    2\r\n          y  + x\r\nIs --------------------- - 1 positive, negative or zero?\r\n          2  2     2  2\r\n   sqrt(r1  y  + r1  x )\r\nnegative;\r\n\r\n<\/pre>\n<\/div>\n<\/div>\n
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Out[5]:<\/div>\n
\\[\\tag{${\\it \\%o}_{11}$}\\int_{0}^{2\\,\\pi}{\\frac{x-\\cos \\varphi_{1}\\,r_{1}}{y^2-2\\,r_{1}\\,\\left(\\sin \\varphi_{1}\\,y+\\cos \\varphi_{1}\\,x\\right)+x^2+r_{1}^2}\\;d\\varphi_{1}}=0\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[6]:<\/div>\n
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\/* r > r' \u306e\u5834\u5408 *\/<\/span>\r\n\r\nfacts<\/span>(<\/span>r1<\/span>)<\/span>$\r\nforget<\/span>(<\/span>%<\/span>)<\/span>$\r\n\r\nassume<\/span>(<\/span>0<\/span> <<\/span> r1<\/span>, r1<\/span> <<\/span> sqrt<\/span>(<\/span>x<\/span>**<\/span>2+<\/span>y<\/span>**<\/span>2<\/span>))<\/span>$\r\n\r\n'<\/span>integrate<\/span>((<\/span>y<\/span>-<\/span>r1<\/span>*<\/span>sin<\/span>(<\/span>phi1<\/span>))<\/span>\/<\/span>\r\n    (<\/span>x<\/span>**<\/span>2+<\/span>y<\/span>**<\/span>2+<\/span>r1<\/span>**<\/span>2-<\/span>2*<\/span>r1<\/span>*<\/span>(<\/span>x<\/span>*<\/span>cos<\/span>(<\/span>phi1<\/span>)<\/span>+<\/span>y<\/span>*<\/span>sin<\/span>(<\/span>phi1<\/span>)))<\/span>, phi1<\/span>, 0<\/span>, 2*<\/span>%pi<\/span>)<\/span>=<\/span>\r\n    integrate<\/span>((<\/span>y<\/span>-<\/span>r1<\/span>*<\/span>sin<\/span>(<\/span>phi1<\/span>))<\/span>\/<\/span>\r\n    (<\/span>x<\/span>**<\/span>2+<\/span>y<\/span>**<\/span>2+<\/span>r1<\/span>**<\/span>2-<\/span>2*<\/span>r1<\/span>*<\/span>(<\/span>x<\/span>*<\/span>cos<\/span>(<\/span>phi1<\/span>)<\/span>+<\/span>y<\/span>*<\/span>sin<\/span>(<\/span>phi1<\/span>)))<\/span>, phi1<\/span>, 0<\/span>, 2*<\/span>%pi<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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<\/div>\n
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\u001bX           2    2\r\n          y  + x\r\nIs --------------------- - 1 positive, negative or zero?\r\n          2  2     2  2\r\n   sqrt(r1  y  + r1  x )\r\n\u001b\\           2    2\r\n          y  + x\r\nIs --------------------- - 1 positive, negative or zero?\r\n          2  2     2  2\r\n   sqrt(r1  y  + r1  x )\r\npositive;\r\n\r\n<\/pre>\n<\/div>\n<\/div>\n
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Out[6]:<\/div>\n
\\[\\tag{${\\it \\%o}_{15}$}\\int_{0}^{2\\,\\pi}{\\frac{y-\\sin \\varphi_{1}\\,r_{1}}{y^2-2\\,r_{1}\\,\\left(\\sin \\varphi_{1}\\,y+\\cos \\varphi_{1}\\,x\\right)+x^2+r_{1}^2}\\;d\\varphi_{1}}=\\frac{2\\,\\pi\\,y}{y^2+x^2}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[7]:<\/div>\n
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\/* r < r' \u306e\u5834\u5408 *\/<\/span>\r\n\r\nfacts<\/span>(<\/span>r1<\/span>)<\/span>$\r\nforget<\/span>(<\/span>%<\/span>)<\/span>$\r\n\r\nassume<\/span>(<\/span>0<\/span> <<\/span> r1<\/span>, r1<\/span> ><\/span> sqrt<\/span>(<\/span>x<\/span>**<\/span>2+<\/span>y<\/span>**<\/span>2<\/span>))<\/span>$\r\n\r\n'<\/span>integrate<\/span>((<\/span>y<\/span>-<\/span>r1<\/span>*<\/span>sin<\/span>(<\/span>phi1<\/span>))<\/span>\/<\/span>\r\n    (<\/span>x<\/span>**<\/span>2+<\/span>y<\/span>**<\/span>2+<\/span>r1<\/span>**<\/span>2-<\/span>2*<\/span>r1<\/span>*<\/span>(<\/span>x<\/span>*<\/span>cos<\/span>(<\/span>phi1<\/span>)<\/span>+<\/span>y<\/span>*<\/span>sin<\/span>(<\/span>phi1<\/span>)))<\/span>, phi1<\/span>, 0<\/span>, 2*<\/span>%pi<\/span>)<\/span>=<\/span>\r\n    integrate<\/span>((<\/span>y<\/span>-<\/span>r1<\/span>*<\/span>sin<\/span>(<\/span>phi1<\/span>))<\/span>\/<\/span>\r\n    (<\/span>x<\/span>**<\/span>2+<\/span>y<\/span>**<\/span>2+<\/span>r1<\/span>**<\/span>2-<\/span>2*<\/span>r1<\/span>*<\/span>(<\/span>x<\/span>*<\/span>cos<\/span>(<\/span>phi1<\/span>)<\/span>+<\/span>y<\/span>*<\/span>sin<\/span>(<\/span>phi1<\/span>)))<\/span>, phi1<\/span>, 0<\/span>, 2*<\/span>%pi<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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<\/div>\n
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\u001bX           2    2\r\n          y  + x\r\nIs --------------------- - 1 positive, negative or zero?\r\n          2  2     2  2\r\n   sqrt(r1  y  + r1  x )\r\n\u001b\\           2    2\r\n          y  + x\r\nIs --------------------- - 1 positive, negative or zero?\r\n          2  2     2  2\r\n   sqrt(r1  y  + r1  x )\r\nnegative;\r\n\r\n<\/pre>\n<\/div>\n<\/div>\n
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Out[7]:<\/div>\n
\\[\\tag{${\\it \\%o}_{19}$}\\int_{0}^{2\\,\\pi}{\\frac{y-\\sin \\varphi_{1}\\,r_{1}}{y^2-2\\,r_{1}\\,\\left(\\sin \\varphi_{1}\\,y+\\cos \\varphi_{1}\\,x\\right)+x^2+r_{1}^2}\\;d\\varphi_{1}}=0\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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<\/div>\n
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\u4e00\u69d8\u306a\u9762\u96fb\u8377\u306b\u3088\u308b\u96fb\u5834\u3067\u4f7f\u3063\u305f\u7a4d\u5206<\/h4>\n
$$\\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty}\\frac{x }{\\left(x^2 + (y-y’)^2 + (z-z’)^2\\right)^{3\/2}}dy’ dz’ = 2 \\pi \\frac{x}{|x|}$$<\/p>\n
\u306e\u78ba\u8a8d\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[8]:<\/div>\n
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\/* x > 0 \u306e\u5834\u5408 *\/<\/span>\r\nfacts<\/span>(<\/span>x<\/span>)<\/span>$\r\nforget<\/span>(<\/span>%<\/span>)<\/span>$\r\n\/* \u3042\u3089\u305f\u3081\u3066 x > 0 \u3068\u4eee\u5b9a\u3057\u307e\u3059\u3002*\/<\/span>\r\nassume<\/span>(<\/span>x<\/span> ><\/span> 0<\/span>)<\/span>$\r\n\r\n'<\/span>integrate<\/span>(<\/span>\r\n    '<\/span>integrate<\/span>(<\/span>\r\n      x<\/span>\/<\/span>(<\/span>x<\/span>**<\/span>2<\/span> +<\/span> (<\/span>y<\/span>-<\/span>y1<\/span>)<\/span>**<\/span>2<\/span> +<\/span> (<\/span>z<\/span>-<\/span>z1<\/span>)<\/span>**<\/span>2<\/span>)<\/span>**<\/span>(<\/span>3\/<\/span>2<\/span>)<\/span>, \r\n      y1<\/span>, -<\/span>inf<\/span>, inf<\/span>)<\/span>, \r\n  z1<\/span>, -<\/span>inf<\/span>, inf<\/span>)<\/span> =<\/span>\r\n integrate<\/span>(<\/span>\r\n    integrate<\/span>(<\/span>\r\n      x<\/span>\/<\/span>(<\/span>x<\/span>**<\/span>2<\/span> +<\/span> (<\/span>y<\/span>-<\/span>y1<\/span>)<\/span>**<\/span>2<\/span> +<\/span> (<\/span>z<\/span>-<\/span>z1<\/span>)<\/span>**<\/span>2<\/span>)<\/span>**<\/span>(<\/span>3\/<\/span>2<\/span>)<\/span>, \r\n      y1<\/span>, -<\/span>inf<\/span>, inf<\/span>)<\/span>, \r\n  z1<\/span>, -<\/span>inf<\/span>, inf<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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<\/div>\n
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\u001bX     2             2    2\r\nIs z1  - 2 z z1 + z  + x  zero or nonzero?\r\n\u001b\\     2             2    2\r\nIs z1  - 2 z z1 + z  + x  zero or nonzero?\r\nnonzero;\r\n\r\n<\/pre>\n<\/div>\n<\/div>\n
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Out[8]:<\/div>\n
\\[\\tag{${\\it \\%o}_{23}$}x\\,\\int_{-\\infty }^{\\infty }{\\int_{-\\infty }^{\\infty }{\\frac{1}{\\left(\\left(z-z_{1}\\right)^2+\\left(y-y_{1}\\right)^2+x^2\\right)^{\\frac{3}{2}}}\\;dy_{1}}\\;dz_{1}}=2\\,\\pi\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[9]:<\/div>\n
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\/* x < 0 \u306e\u5834\u5408 *\/<\/span>\r\nfacts<\/span>(<\/span>x<\/span>)<\/span>$\r\nforget<\/span>(<\/span>%<\/span>)<\/span>$\r\n\r\n\/* \u3042\u3089\u305f\u3081\u3066 x < 0 \u3068\u4eee\u5b9a\u3057\u307e\u3059\u3002*\/<\/span>\r\nassume<\/span>(<\/span>x<\/span> <<\/span> 0<\/span>)<\/span>$\r\n\r\n'<\/span>integrate<\/span>(<\/span>\r\n    '<\/span>integrate<\/span>(<\/span>\r\n      x<\/span>\/<\/span>(<\/span>x<\/span>**<\/span>2<\/span> +<\/span> (<\/span>y<\/span>-<\/span>y1<\/span>)<\/span>**<\/span>2<\/span> +<\/span> (<\/span>z<\/span>-<\/span>z1<\/span>)<\/span>**<\/span>2<\/span>)<\/span>**<\/span>(<\/span>3\/<\/span>2<\/span>)<\/span>, \r\n      y1<\/span>, -<\/span>inf<\/span>, inf<\/span>)<\/span>, \r\n  z1<\/span>, -<\/span>inf<\/span>, inf<\/span>)<\/span> =<\/span>\r\n integrate<\/span>(<\/span>\r\n    integrate<\/span>(<\/span>\r\n      x<\/span>\/<\/span>(<\/span>x<\/span>**<\/span>2<\/span> +<\/span> (<\/span>y<\/span>-<\/span>y1<\/span>)<\/span>**<\/span>2<\/span> +<\/span> (<\/span>z<\/span>-<\/span>z1<\/span>)<\/span>**<\/span>2<\/span>)<\/span>**<\/span>(<\/span>3\/<\/span>2<\/span>)<\/span>, \r\n      y1<\/span>, -<\/span>inf<\/span>, inf<\/span>)<\/span>, \r\n  z1<\/span>, -<\/span>inf<\/span>, inf<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u001bX     2             2    2\r\nIs z1  - 2 z z1 + z  + x  zero or nonzero?\r\n\u001b\\     2             2    2\r\nIs z1  - 2 z z1 + z  + x  zero or nonzero?\r\nnonzero;\r\n\r\n<\/pre>\n<\/div>\n<\/div>\n
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Out[9]:<\/div>\n
\\[\\tag{${\\it \\%o}_{27}$}x\\,\\int_{-\\infty }^{\\infty }{\\int_{-\\infty }^{\\infty }{\\frac{1}{\\left(\\left(z-z_{1}\\right)^2+\\left(y-y_{1}\\right)^2+x^2\\right)^{\\frac{3}{2}}}\\;dy_{1}}\\;dz_{1}}=-2\\,\\pi\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u7403\u5bfe\u79f0\u306a\u96fb\u8377\u5206\u5e03\u306b\u3088\u308b\u96fb\u5834\u3067\u4f7f\u3063\u305f\u7a4d\u5206<\/h4>\n
\\begin{eqnarray}
\n\\int_0^{\\pi} \\sin\\theta’ d\\theta’
\n\\frac{(r-r’\\cos\\theta’)}{\\left\\{r^2 + (r’)^2 -2 r r’ \\cos\\theta’\\right\\}^{3\/2}}
\n&=& \\frac{2\\pi}{r^2} \\left(\\frac{r+r’}{|r+r’|}+\\frac{r-r’}{|r-r’|} \\right)\\\\
\n&=& \\left\\{
\n\\begin{array}{ll}
\n\\frac{4\\pi}{r^2} & (r’ < r)\\\\ \\ \\\\
\n0 & (r’ > r)
\n\\end{array}
\n\\right.
\n\\end{eqnarray}<\/p>\n
\u306e\u78ba\u8a8d\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[10]:<\/div>\n
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integrate<\/span>(<\/span>\r\n  (<\/span>r<\/span>-<\/span>r1<\/span>*<\/span>cos<\/span>(<\/span>theta1<\/span>))<\/span>*<\/span>sin<\/span>(<\/span>theta1<\/span>)<\/span>\r\n   \/<\/span>(<\/span>r<\/span>**<\/span>2<\/span> +<\/span> r1<\/span>**<\/span>2<\/span> -<\/span> 2*<\/span>r<\/span>*<\/span>r1<\/span>*<\/span>cos<\/span>(<\/span>theta1<\/span>))<\/span>**<\/span>(<\/span>3\/<\/span>2<\/span>)<\/span>, theta1<\/span>, 0<\/span>, %pi<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[10]:<\/div>\n
\\[\\tag{${\\it \\%o}_{28}$}\\frac{r_{1}+r}{r^2\\,\\sqrt{r_{1}^2+2\\,r\\,r_{1}+r^2}}-\\frac{r_{1}-r}{r^2\\,\\sqrt{r_{1}^2-2\\,r\\,r_{1}+r^2}}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":2673,"menu_order":1,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-2746","page","type-page","status-publish","hentry","nodate","item-wrap"],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t