{"id":7958,"date":"2024-03-08T13:01:35","date_gmt":"2024-03-08T04:01:35","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=7958"},"modified":"2024-03-08T13:01:35","modified_gmt":"2024-03-08T04:01:35","slug":"sympy-%e3%81%a7%e3%83%95%e3%83%bc%e3%83%aa%e3%82%a8%e7%b4%9a%e6%95%b0%e5%b1%95%e9%96%8b%e3%81%ae%e4%be%8b%e9%a1%8c%e3%82%92%e8%a7%a3%e3%81%8f","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%e3%83%95%e3%83%bc%e3%83%aa%e3%82%a8%e8%a7%a3%e6%9e%90\/sympy-%e3%81%a7%e3%83%95%e3%83%bc%e3%83%aa%e3%82%a8%e7%b4%9a%e6%95%b0%e5%b1%95%e9%96%8b%e3%81%ae%e4%be%8b%e9%a1%8c%e3%82%92%e8%a7%a3%e3%81%8f\/","title":{"rendered":"SymPy \u3067\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u5c55\u958b\u306e\u4f8b\u984c\u3092\u89e3\u304f"},"content":{"rendered":"

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\u30e2\u30b8\u30e5\u30fc\u30eb\u306e import<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[1]:<\/div>\n
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from<\/span> sympy.abc<\/span> import<\/span> *<\/span>\r\nfrom<\/span> sympy<\/span> import<\/span> *<\/span>\r\n# SymPy Plotting Backends (SPB) <\/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\r\ninit_printing<\/span>();<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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mathtext.fontset \u306e\u8a2d\u5b9a<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[2]:<\/div>\n
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# \u30b0\u30e9\u30d5\u3092\u63cf\u304f\u305f\u3081\u3067\u306f\u306a\u304f\u30c7\u30d5\u30a9\u30eb\u30c8\u8a2d\u5b9a\u306e\u305f\u3081<\/span>\r\nimport<\/span> matplotlib.pyplot<\/span> as<\/span> plt<\/span>\r\n\r\nplt<\/span>.<\/span>rcParams<\/span>[<\/span>'mathtext.fontset'<\/span>]<\/span> =<\/span> 'cm'<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u95a2\u6570\u306e\u5b9a\u7fa9<\/h3>\n
\u533a\u9593 $[-1: 1]$ \u3067\u5b9a\u7fa9\u3055\u308c\u305f\u95a2\u6570 $f(x) = 1 – |x|$ \u304c\u533a\u9593\u5916\u3067\u306f\u5468\u671f $2$ \u306e\u5468\u671f\u95a2\u6570\u3060\u3068\u3057\u3066\uff0c\u305d\u306e\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u5c55\u958b\u3092\u6c42\u3081\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[3]:<\/div>\n
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# -1 <= x <= 1 \u3067\u5b9a\u7fa9\u3055\u308c\u305f\u95a2\u6570 f0(x)<\/span>\r\ndef<\/span> f0<\/span>(<\/span>x<\/span>):<\/span>\r\n    return<\/span> 1<\/span> -<\/span> abs<\/span>(<\/span>x<\/span>)<\/span>\r\n\r\n# f0(x) \u3092\u4efb\u610f\u5468\u671f 2 \u306e\u5468\u671f\u95a2\u6570\u306b\u3059\u308b\u5c0f\u9053\u5177<\/span>\r\ndef<\/span> xcyc<\/span>(<\/span>x<\/span>):<\/span>\r\n    return<\/span> x<\/span> -<\/span> 2<\/span>*<\/span>floor<\/span>((<\/span>x<\/span>+<\/span>1<\/span>)<\/span>\/<\/span>2<\/span>)<\/span>\r\n\r\n# \u5468\u671f 2 \u306e\u5468\u671f\u95a2\u6570 f(x)<\/span>\r\ndef<\/span> f<\/span>(<\/span>x<\/span>):<\/span>\r\n    return<\/span> f0<\/span>(<\/span>xcyc<\/span>(<\/span>x<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u95a2\u6570\u306e\u30b0\u30e9\u30d5<\/h3>\n
\u533a\u9593 $-1 \\le x \\le 1$ \u3067\u5b9a\u7fa9\u3055\u308c\u305f\u95a2\u6570 $f_0(x)$ \u3068\uff0c\u5468\u671f $2$ \u306e\u5468\u671f\u95a2\u6570 $f(x)$ \u3092\u305d\u308c\u305e\u308c\u30b0\u30e9\u30d5\u306b\u3057\u3066\u307f\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[4]:<\/div>\n
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plot<\/span>(<\/span>f0<\/span>(<\/span>x<\/span>),<\/span> (<\/span>x<\/span>,<\/span> -<\/span>1<\/span>,<\/span> 1<\/span>),<\/span> xlim<\/span>=<\/span>(<\/span>-<\/span>5<\/span>,<\/span> 5<\/span>),<\/span> ylabel<\/span>=<\/span>\"$f_0(x)$\"<\/span>)<\/span>;<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\"\"<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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# plot(f(x), (x, -5, 5)) \u3068\u3059\u308b\u3068\u3053\u308d\u3060\u304c<\/span>\r\n# plot(f, (x, -5, 5)) \u3068\u3059\u308b\u3068\u8b66\u544a\u304c\u51fa\u306a\u3044\u3002<\/span>\r\n\r\nplot<\/span>(<\/span>f<\/span>,<\/span> (<\/span>x<\/span>,<\/span> -<\/span>5<\/span>,<\/span> 5<\/span>),<\/span> xlim<\/span>=<\/span>(<\/span>-<\/span>5<\/span>,<\/span> 5<\/span>),<\/span> ylabel<\/span>=<\/span>\"$f(x)$\"<\/span>)<\/span>;<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\"\"<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u306e\u5b9a\u7fa9<\/h3>\n
\u5b9a\u7fa9\u3069\u304a\u308a\u306b\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u5c55\u958b\u3092\u5b9a\u7fa9\u3059\u308b\u3068\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[6]:<\/div>\n
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var<\/span>(<\/span>'n'<\/span>,<\/span> integer<\/span> =<\/span> True<\/span>)<\/span>\r\n\r\ndef<\/span> a<\/span>(<\/span>n<\/span>):<\/span>\r\n    return<\/span> integrate<\/span>(<\/span>f0<\/span>(<\/span>x<\/span>)<\/span>*<\/span>cos<\/span>(<\/span>n<\/span>*<\/span>pi<\/span>*<\/span>x<\/span>),<\/span> (<\/span>x<\/span>,<\/span> -<\/span>1<\/span>,<\/span> 1<\/span>))<\/span>\r\n\r\ndef<\/span> b<\/span>(<\/span>n<\/span>):<\/span>\r\n    return<\/span> integrate<\/span>(<\/span>f0<\/span>(<\/span>x<\/span>)<\/span>*<\/span>sin<\/span>(<\/span>n<\/span>*<\/span>pi<\/span>*<\/span>x<\/span>),<\/span> (<\/span>x<\/span>,<\/span> -<\/span>1<\/span>,<\/span> 1<\/span>))<\/span>\r\n\r\ndef<\/span> Fourier<\/span>(<\/span>n<\/span>):<\/span>\r\n    sums<\/span> =<\/span> a<\/span>(<\/span>0<\/span>)<\/span>\/<\/span>2<\/span> +<\/span> summation<\/span>(<\/span>\r\n                        a<\/span>(<\/span>i<\/span>)<\/span>*<\/span>cos<\/span>(<\/span>i<\/span>*<\/span>pi<\/span>*<\/span>x<\/span>)<\/span> +<\/span> \r\n                        b<\/span>(<\/span>i<\/span>)<\/span>*<\/span>sin<\/span>(<\/span>i<\/span>*<\/span>pi<\/span>*<\/span>x<\/span>),<\/span> (<\/span>i<\/span>,<\/span> 1<\/span>,<\/span> n<\/span>))<\/span>\r\n    return<\/span> sums<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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SymPy \u306f\u512a\u79c0\u3067\uff0c\u7d76\u5bfe\u5024 abs(x)<\/code> \u3092\u542b\u3080\u7a4d\u5206\u3067\u3082\u7279\u306b\u554f\u984c\u306a\u304f\u7a4d\u5206\u5024\u3092\u8fd4\u3057\u3066\u304f\u308c\u308b\u3002\u305f\u3068\u3048\u3070\uff0ca(3)<\/code> \u3084 Fourier(3)<\/code> \u306f…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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a<\/span>(<\/span>3<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[7]:<\/div>\n
$\\displaystyle \\frac{4}{9 \\pi^{2}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[8]:<\/div>\n
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Fourier<\/span>(<\/span>3<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[8]:<\/div>\n
$\\displaystyle \\frac{4 \\cos{\\left(\\pi x \\right)}}{\\pi^{2}} + \\frac{4 \\cos{\\left(3 \\pi x \\right)}}{9 \\pi^{2}} + \\frac{1}{2}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u30d5\u30fc\u30ea\u30a8\u4fc2\u6570<\/h3>\n
\u30d5\u30fc\u30ea\u30a8\u4fc2\u6570 $a_n, b_n$ \u306f…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[9]:<\/div>\n
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a<\/span>(<\/span>n<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[9]:<\/div>\n
$\\displaystyle \\begin{cases} – \\frac{2 \\left(-1\\right)^{n}}{\\pi^{2} n^{2}} + \\frac{2}{\\pi^{2} n^{2}} & \\text{for}\\: n \\neq 0 \\\\1 & \\text{otherwise} \\end{cases}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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b<\/span>(<\/span>n<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[10]:<\/div>\n
$\\displaystyle 0$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$a_n$ \u306f $n$ \u306e\u5076\u5947\u306b\u3088\u3063\u3066\u30bc\u30ed\u306b\u306a\u308b\u5834\u5408\u304c\u3042\u308b\u306e\u3067\uff0c$n=10$ \u307e\u3067\u8868\u793a\u3057\u3066\u307f\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[11]:<\/div>\n
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for<\/span> n<\/span> in<\/span> range<\/span>(<\/span>11<\/span>):<\/span>\r\n    print<\/span>(<\/span>'a(<\/span>%2d<\/span>) = '<\/span> %<\/span> n<\/span>,<\/span> a<\/span>(<\/span>n<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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a( 0) =  1\r\na( 1) =  4\/pi**2\r\na( 2) =  0\r\na( 3) =  4\/(9*pi**2)\r\na( 4) =  0\r\na( 5) =  4\/(25*pi**2)\r\na( 6) =  0\r\na( 7) =  4\/(49*pi**2)\r\na( 8) =  0\r\na( 9) =  4\/(81*pi**2)\r\na(10) =  0\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u306e\u30b0\u30e9\u30d5<\/h3>\n
$n = 3, 5, 7$ \u306e\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u3092\u6c42\u3081\uff0c\u30b0\u30e9\u30d5\u306b\u3057\u3066\u307f\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[12]:<\/div>\n
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f3<\/span> =<\/span> Fourier<\/span>(<\/span>3<\/span>)<\/span>\r\nf5<\/span> =<\/span> Fourier<\/span>(<\/span>5<\/span>)<\/span>\r\nf7<\/span> =<\/span> Fourier<\/span>(<\/span>7<\/span>)<\/span>\r\n\r\nplot<\/span>(<\/span>\r\n     (<\/span>f<\/span>,<\/span> \"$f(x)$\"<\/span>),<\/span> \r\n     (<\/span>f3<\/span>,<\/span> \"$n=3$\"<\/span>),<\/span> \r\n     (<\/span>f5<\/span>,<\/span> \"$n=5$\"<\/span>),<\/span> \r\n     (<\/span>f7<\/span>,<\/span> \"$n=7$\"<\/span>),<\/span> (<\/span>x<\/span>,<\/span> -<\/span>5<\/span>,<\/span> 5<\/span>),<\/span> {<\/span>\"lw\"<\/span>:<\/span>1<\/span>},<\/span> \r\n     xlim<\/span> =<\/span> (<\/span>-<\/span>5<\/span>,<\/span> 6.5<\/span>),<\/span> ylabel<\/span> =<\/span> False<\/span>)<\/span>;<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$n=15$ \u304f\u3089\u3044\u3060\u3068\uff0c\u304b\u306a\u308a $f(x)$ \u306b\u8fd1\u3044\u611f\u3058\u304c\u3042\u3089\u308f\u308c\u3066\u3044\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[13]:<\/div>\n
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f15<\/span> =<\/span> Fourier<\/span>(<\/span>15<\/span>)<\/span>\r\n\r\nplot<\/span>(<\/span>\r\n     (<\/span>f<\/span>,<\/span> \"$f(x)$\"<\/span>),<\/span> \r\n     (<\/span>f15<\/span>,<\/span> \"$n=15$\"<\/span>),<\/span> (<\/span>x<\/span>,<\/span> -<\/span>5<\/span>,<\/span> 5<\/span>),<\/span> {<\/span>\"lw\"<\/span>:<\/span>1<\/span>},<\/span>\r\n     xlim<\/span> =<\/span> (<\/span>-<\/span>5<\/span>,<\/span> 6.5<\/span>),<\/span> ylabel<\/span> =<\/span> False<\/span>)<\/span>;<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\"\"<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u53c2\u8003\uff1afourier_series()<\/code> \u3092\u4f7f\u3046\u5834\u5408<\/h3>\n
SymPy \u306b\u306f\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u5c55\u958b\u3059\u308b\u95a2\u6570\u304c\u3042\u308a\u307e\u3059\u3002\u3053\u308c\u3092\u4f7f\u3046\u4f8b\u3002<\/p>\n