{"id":6206,"date":"2023-04-20T15:31:27","date_gmt":"2023-04-20T06:31:27","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=6206"},"modified":"2025-08-05T15:45:03","modified_gmt":"2025-08-05T06:45:03","slug":"sympy-%e3%81%a7%e3%83%95%e3%83%bc%e3%83%aa%e3%82%a8%e8%a7%a3%e6%9e%90","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\/","title":{"rendered":"SymPy \u3067\u30d5\u30fc\u30ea\u30a8\u89e3\u6790"},"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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<\/div>\n
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\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570<\/h3>\n
    \n
  1. \u533a\u9593 $-\\pi < x < \\pi$ \u3067\u5b9a\u7fa9\u3055\u308c\u305f\u95a2\u6570 $f(x)$ \u306f\uff0c\u305d\u308c\u304c\u3069\u3093\u306a\u95a2\u6570\u3067\u3042\u3063\u3066\u3082…<\/li>\n
  2. \u533a\u9593\u5916\u3067\u306f\uff0c\u5468\u671f $ 2\\pi $ \u306e\u5468\u671f\u95a2\u6570\u3068\u307f\u306a\u3057\u3066<\/li>\n
  3. \u4e09\u89d2\u95a2\u6570 $\\cos, \\ \\sin$ \u306e\u91cd\u306d\u5408\u308f\u305b\u3066\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u308b\uff01<\/li>\n<\/ol>\n
    \u3064\u307e\u308a\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3068\u3044\u3046\u3053\u3068\u3002<\/p>\n
    $$ f(x) = \\frac{a_0}{2} + \\sum_{n=1}^{\\infty} \\bigl( a_n \\cos n x + b_n \\sin nx \\bigr) $$<\/p>\n
    \u3053\u3053\u3067\uff0c\u30d5\u30fc\u30ea\u30a8\u4fc2\u6570 $a_n, b_n$ \u306f<\/p>\n
    $$a_n = \\frac{1}{\\pi} \\int_{-\\pi}^{\\pi} f(x) \\cos nx \\, dx $$$$b_n = \\frac{1}{\\pi} \\int_{-\\pi}^{\\pi} f(x) \\sin nx \\, dx $$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    <\/div>\n
    \n
    \n
    \u4f8b\u984c<\/h4>\n
    \u533a\u9593 $[-\\pi: \\pi]$ \u3067\u5b9a\u7fa9\u3055\u308c\u305f\u95a2\u6570 $f(x) = x^2$ \u306e\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u5c55\u958b\u3002<\/p>\n
    \u5468\u671f $2 \\pi$ \u306e\u5468\u671f\u95a2\u6570\u3068\u3057\u3066\uff0c<\/p>\n
      \n
    • $[-3\\pi:-\\pi]$ \u3067\u306f $f(x) = (x+2 \\pi)^2$…<\/li>\n
    • $[-\\pi:\\pi]$ \u3067\u306f $f(x) = x^2$,<\/li>\n
    • $[\\pi:3\\pi]$ \u3067\u306f $f(x) = (x-2 \\pi)^2$…<\/li>\n<\/ul>\n
      \u306e\u3088\u3046\u306b\u3059\u308c\u3070\u3044\u3044\u3002<\/p>\n
      \u307e\u305a\uff0c1\u5468\u671f\u5206\u306e $f_0(x) \\equiv x^2$ \u3092\u5b9a\u7fa9\u3057\u3066\uff0c$[-\\pi:\\pi]$ \u306e\u533a\u9593\u3067\u30b0\u30e9\u30d5\u3092\u63cf\u304f\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
      \n
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      In\u00a0[2]:<\/div>\n
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      \n
      def<\/span> f0<\/span>(<\/span>x<\/span>):<\/span>\r\n    return<\/span> x<\/span>**<\/span>2<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
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      In\u00a0[3]:<\/div>\n
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      \n
      plot<\/span>(<\/span>f0<\/span>(<\/span>x<\/span>),<\/span> (<\/span>x<\/span>,<\/span> -<\/span>pi<\/span>,<\/span> pi<\/span>))<\/span>;<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      <\/div>\n
      \n
      <\/p>\n
      \"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      <\/div>\n
      \n
      \n
      $[-\\pi:\\pi]$ \u306e\u533a\u9593\u5916\u3067\u306f\uff0c\u5468\u671f $2 \\pi$ \u306e\u5468\u671f\u95a2\u6570\u3068\u3057\u3066\uff0c<\/p>\n
        \n
      • $[-3\\pi:-\\pi]$ \u3067\u306f $f(x) = f_0(x+2 \\pi)$…<\/li>\n
      • $[-\\pi:\\pi]$ \u3067\u306f $f(x) = f_0(x)$,<\/li>\n
      • $[\\pi:3\\pi]$ \u3067\u306f $f(x) = f_0(x-2 \\pi)$…<\/li>\n<\/ul>\n
        \u3092\u30b0\u30e9\u30d5\u306b\u63cf\u304f\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
        \n
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        In\u00a0[4]:<\/div>\n
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        \n
        plot<\/span>((<\/span>f0<\/span>(<\/span>x<\/span>+<\/span>2<\/span>*<\/span>pi<\/span>),<\/span> (<\/span>x<\/span>,<\/span> -<\/span>3<\/span>*<\/span>pi<\/span>,<\/span> -<\/span>pi<\/span>)),<\/span> \r\n     (<\/span>f0<\/span>(<\/span>x<\/span>),<\/span> (<\/span>x<\/span>,<\/span> -<\/span>pi<\/span>,<\/span> pi<\/span>)),<\/span> \r\n     (<\/span>f0<\/span>(<\/span>x<\/span>-<\/span>2<\/span>*<\/span>pi<\/span>),<\/span> (<\/span>x<\/span>,<\/span> pi<\/span>,<\/span> 3<\/span>*<\/span>pi<\/span>)),<\/span> legend<\/span> =<\/span> False<\/span>)<\/span>;<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
        \n
        \n
        \n
        <\/div>\n
        \n
        <\/p>\n
        \"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
        \n
        <\/div>\n
        \n
        \n
        \u5468\u671f\u6bce\u306b\u5225\u306e\u95a2\u6570\u3092\u63cf\u304f\u306e\u306f\u9762\u5012\u306a\u306e\u3067\uff0c\u5207\u308a\u6368\u3066\u308b\u95a2\u6570 floor()<\/code> \u3092\u4f7f\u3063\u3066\u5468\u671f $2\\pi$ \u306e\u95a2\u6570 $f(x)$ \u3092\u5b9a\u7fa9\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
        \n
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        In\u00a0[5]:<\/div>\n
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        \n
        def<\/span> f<\/span>(<\/span>x<\/span>):<\/span>\r\n    return<\/span> f0<\/span>(<\/span>x<\/span>-<\/span>2<\/span>*<\/span>pi<\/span>*<\/span>floor<\/span>((<\/span>x<\/span>+<\/span>pi<\/span>)<\/span>\/<\/span>(<\/span>2<\/span>*<\/span>pi<\/span>)))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
        \n
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        In\u00a0[6]:<\/div>\n
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        plot<\/span>(<\/span>f<\/span>(<\/span>x<\/span>),<\/span> (<\/span>x<\/span>,<\/span> -<\/span>3<\/span>*<\/span>pi<\/span>,<\/span> 3<\/span>*<\/span>pi<\/span>))<\/span>;<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
        \n
        \n
        \n
        <\/div>\n
        \n
        \/usr\/local\/lib\/python3.8\/dist-packages\/spb\/series.py:587: UserWarning: NumPy is unable to evaluate with complex numbers some of the functions included in this symbolic expression: [floor]. Hence, the evaluation will use real numbers. If you believe the resulting plot is incorrect, change the evaluation module by setting the `modules` keyword argument.\r\n  warnings.warn(\"NumPy is unable to evaluate with complex \"\r\n<\/pre>\n<\/div>\n<\/div>\n
        \n
        <\/div>\n
        \n
        <\/p>\n
        \"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
        \n
        <\/div>\n
        \n
        \n
        \u5c11\u3057\u30aa\u30d7\u30b7\u30e7\u30f3\u3092\u8a2d\u5b9a\u3057\u3066\u63cf\u304f\u4f8b\uff1a<\/p>\n
        \u4ee5\u4e0b\u306e\u4f8b\u3067\u306f\uff0c<\/p>\n
          \n
        1. \u51e1\u4f8b\u3092\u8a2d\u5b9a\u3057\uff0c<\/li>\n
        2. \u6a2a\u8ef8\u306e\u8868\u793a\u7bc4\u56f2\u3092\u8a2d\u5b9a\u3057\uff0c<\/li>\n
        3. \u7e26\u8ef8\u306e\u8868\u793a\u7bc4\u56f2\u3092\u8a2d\u5b9a\u3057\uff0c<\/li>\n
        4. $x$ \u8ef8\u306e\u30d5\u30a9\u30fc\u30de\u30c3\u30c8\u3092 $\\pi$ \u306e\u500d\u6570\u306b\u306a\u308b\u3088\u3046\u306b\u8a2d\u5b9a\u3057\u3066\u3044\u307e\u3059\u3002<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n
          \n
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          In\u00a0[7]:<\/div>\n
          \n
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          \n
          fpi<\/span> =<\/span> float<\/span>(<\/span>pi<\/span>)<\/span>\r\np<\/span> =<\/span> plot<\/span>(<\/span>f<\/span>(<\/span>x<\/span>),<\/span> (<\/span>x<\/span>,<\/span> -<\/span>3<\/span>*<\/span>pi<\/span>,<\/span> 3<\/span>*<\/span>pi<\/span>),<\/span> \r\n         \"$f(x)$\"<\/span>,<\/span> legend<\/span>=<\/span>True<\/span>,<\/span> \r\n         xlim<\/span> =<\/span> (<\/span>-<\/span>3<\/span>*<\/span>pi<\/span>,<\/span> 3<\/span>*<\/span>pi<\/span>),<\/span> \r\n         ylim<\/span> =<\/span> (<\/span>-<\/span>2<\/span>,<\/span> 12<\/span>),<\/span> show<\/span> =<\/span> False<\/span>)<\/span>\r\n\r\nax<\/span> =<\/span> p<\/span>.<\/span>ax<\/span>\r\nax<\/span>.<\/span>set_xticks<\/span>([(<\/span>i<\/span>*<\/span>fpi<\/span>)<\/span> for<\/span> i<\/span> in<\/span> range<\/span>(<\/span>-<\/span>3<\/span>,<\/span>4<\/span>)])<\/span>\r\nxlab<\/span> =<\/span> [<\/span>\"-3 $\\pi$\"<\/span>,<\/span> \"-2 $\\pi$\"<\/span>,<\/span> \"-$\\pi$\"<\/span>,<\/span> \"0\"<\/span>,<\/span> \r\n        \"$\\pi$\"<\/span>,<\/span> \"$2 \\pi$\"<\/span>,<\/span> \"3 $\\pi$\"<\/span>]<\/span>\r\nax<\/span>.<\/span>set_xticklabels<\/span>(<\/span>xlab<\/span>);<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          \n
          <\/div>\n
          \n
          <\/p>\n
          \"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          <\/div>\n
          \n
          \n
          \u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u307e\u3057\u305f\u3002<\/p>\n
          $$ f(x) = \\frac{a_0}{2} + \\sum_{n=1}^{\\infty} \\bigl( a_n \\cos n x + b_n \\sin nx \\bigr) $$<\/p>\n
          \u3053\u3053\u3067\uff0c\u30d5\u30fc\u30ea\u30a8\u4fc2\u6570 $a_n, b_n$ \u306f<\/p>\n
          $$a_n = \\frac{1}{\\pi} \\int_{-\\pi}^{\\pi} f(x) \\cos nx \\, dx $$$$b_n = \\frac{1}{\\pi} \\int_{-\\pi}^{\\pi} f(x) \\sin nx \\, dx $$<\/p>\n
          SymPy \u306b\u306f\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u3092\u6271\u3046\u95a2\u6570\u304c\u3042\u308a\u307e\u3059\u304c\uff0c\u7c21\u5358\u306a\u306e\u3067\uff0c\u7a4d\u5206\u3068\u548c\u3092\u3068\u308b\u6f14\u7b97\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3057\u3066\u3057\u307e\u3044\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[8]:<\/div>\n
          \n
          \n
          \n
          # \u30d5\u30fc\u30ea\u30a8\u4fc2\u6570\u306e\u5b9a\u7fa9\uff0c\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u306e\u5b9a\u7fa9\u3002<\/span>\r\n\r\nn<\/span> =<\/span> Symbol<\/span>(<\/span>'n'<\/span>,<\/span> integer<\/span> =<\/span> True<\/span>,<\/span> positive<\/span> =<\/span> True<\/span>)<\/span>\r\n\r\ndef<\/span> a<\/span>(<\/span>n<\/span>):<\/span>\r\n    return<\/span> 1<\/span>\/<\/span>pi<\/span>*<\/span>integrate<\/span>(<\/span>f0<\/span>(<\/span>x<\/span>)<\/span>*<\/span>cos<\/span>(<\/span>n<\/span>*<\/span>x<\/span>),<\/span> (<\/span>x<\/span>,<\/span> -<\/span>pi<\/span>,<\/span> pi<\/span>))<\/span>\r\ndef<\/span> b<\/span>(<\/span>n<\/span>):<\/span>\r\n    return<\/span> 1<\/span>\/<\/span>pi<\/span>*<\/span>integrate<\/span>(<\/span>f0<\/span>(<\/span>x<\/span>)<\/span>*<\/span>sin<\/span>(<\/span>n<\/span>*<\/span>x<\/span>),<\/span> (<\/span>x<\/span>,<\/span> -<\/span>pi<\/span>,<\/span> pi<\/span>))<\/span>\r\n\r\ndef<\/span> Fourier<\/span>(<\/span>n<\/span>,<\/span> x<\/span>):<\/span>\r\n    var<\/span>(<\/span>'i'<\/span>)<\/span>\r\n    return<\/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>x<\/span>)<\/span> +<\/span> b<\/span>(<\/span>i<\/span>)<\/span>*<\/span>sin<\/span>(<\/span>i<\/span>*<\/span>x<\/span>),<\/span> (<\/span>i<\/span>,<\/span> 1<\/span>,<\/span> n<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          <\/div>\n
          \n
          \n
          \u3061\u306a\u307f\u306b\uff0c\u4e0a\u306e\u30bb\u30eb\u306e\u5b9a\u7fa9\u3067\u4f7f\u308f\u308c\u3066\u3044\u308b\u95a2\u6570 integrate()<\/code> \u306f\u7a4d\u5206\u3092\u5b9f\u884c\u3057\uff0c summation()<\/code> \u306f\u548c\u3092\u3068\u308a\uff0c\u8aac\u660e\u306f\u4ee5\u4e0b\u306e\u901a\u308a\u3067\u3059\u3002<\/p>\n
          $\\displaystyle \\int_a^b f(x)\\, dx = $ integrate(f(x), x, a, b);<\/code><\/p>\n
          $\\displaystyle \\sum_{i = 1}^n a_i = $ summation(a(i), (i, 1, n));<\/code><\/p>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          <\/div>\n
          \n
          \n
          $f(x) = x^2$ \u306f\u5076\u95a2\u6570\u306a\u306e\u3067\uff0c\u5947\u95a2\u6570\u90e8\u5206\u306e\u30d5\u30fc\u30ea\u30a8\u4fc2\u6570\u3067\u3042\u308b $b_n$ \u306f\u30bc\u30ed\u306e\u306f\u305a\u3002\u5b9f\u969b\u306b\u8abf\u3079\u3066\u307f\u308b\u3068…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[9]:<\/div>\n
          \n
          \n
          \n
          for<\/span> i<\/span> in<\/span> range<\/span>(<\/span>1<\/span>,<\/span> 5<\/span>):<\/span>\r\n    print<\/span>(<\/span>\"b(<\/span>%d<\/span>) = \"<\/span> %<\/span> i<\/span>,<\/span>b<\/span>(<\/span>i<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          \n
          <\/div>\n
          \n
          b(1) =  0\r\nb(2) =  0\r\nb(3) =  0\r\nb(4) =  0\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[10]:<\/div>\n
          \n
          \n
          \n
          # \u4e00\u822c\u306e n \u306b\u3064\u3044\u3066\u306f...<\/span>\r\nb<\/span>(<\/span>n<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          \n
          Out[10]:<\/div>\n
          $\\displaystyle 0$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[11]:<\/div>\n
          \n
          \n
          \n
          a<\/span>(<\/span>0<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          \n
          Out[11]:<\/div>\n
          $\\displaystyle \\frac{2 \\pi^{2}}{3}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[12]:<\/div>\n
          \n
          \n
          \n
          for<\/span> i<\/span> in<\/span> range<\/span>(<\/span>1<\/span>,<\/span> 5<\/span>):<\/span>\r\n    print<\/span>(<\/span>\"a(<\/span>%d<\/span>) = \"<\/span> %<\/span> i<\/span>,<\/span>a<\/span>(<\/span>i<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          \n
          <\/div>\n
          \n
          a(1) =  -4\r\na(2) =  1\r\na(3) =  -4\/9\r\na(4) =  1\/4\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[13]:<\/div>\n
          \n
          \n
          \n
          # \u4e00\u822c\u306e n \u306b\u3064\u3044\u3066\u306f...<\/span>\r\na<\/span>(<\/span>n<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          \n
          Out[13]:<\/div>\n
          $\\displaystyle \\frac{4 \\left(-1\\right)^{n}}{n^{2}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[14]:<\/div>\n
          \n
          \n
          \n
          f1<\/span> =<\/span> Fourier<\/span>(<\/span>1<\/span>,<\/span> x<\/span>)<\/span>\r\nf1<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          \n
          Out[14]:<\/div>\n
          $\\displaystyle – 4 \\cos{\\left(x \\right)} + \\frac{\\pi^{2}}{3}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[15]:<\/div>\n
          \n
          \n
          \n
          f2<\/span> =<\/span> Fourier<\/span>(<\/span>2<\/span>,<\/span> x<\/span>)<\/span>\r\nf2<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          \n
          Out[15]:<\/div>\n
          $\\displaystyle – 4 \\cos{\\left(x \\right)} + \\cos{\\left(2 x \\right)} + \\frac{\\pi^{2}}{3}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[16]:<\/div>\n
          \n
          \n
          \n
          f3<\/span> =<\/span> Fourier<\/span>(<\/span>3<\/span>,<\/span> x<\/span>)<\/span>\r\nf3<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          \n
          Out[16]:<\/div>\n
          $\\displaystyle – 4 \\cos{\\left(x \\right)} + \\cos{\\left(2 x \\right)} – \\frac{4 \\cos{\\left(3 x \\right)}}{9} + \\frac{\\pi^{2}}{3}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[17]:<\/div>\n
          \n
          \n
          \n
          f4<\/span> =<\/span> Fourier<\/span>(<\/span>4<\/span>,<\/span> x<\/span>)<\/span>\r\nf4<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          \n
          Out[17]:<\/div>\n
          $\\displaystyle – 4 \\cos{\\left(x \\right)} + \\cos{\\left(2 x \\right)} – \\frac{4 \\cos{\\left(3 x \\right)}}{9} + \\frac{\\cos{\\left(4 x \\right)}}{4} + \\frac{\\pi^{2}}{3}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          <\/div>\n
          \n
          \n
          \u305d\u308c\u305e\u308c\u306e\u6b21\u6570\u307e\u3067\u306e\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u5c55\u958b\u306e\u5f0f\u3068\uff0c\u3082\u3068\u306e\u5468\u671f\u95a2\u6570 $f(x)$ \u3092\u91cd\u306d\u3066\u30b0\u30e9\u30d5\u306b\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[18]:<\/div>\n
          \n
          \n
          \n
          p<\/span>=<\/span>plot<\/span>(<\/span>\r\n     (<\/span>f<\/span>(<\/span>x<\/span>),<\/span> (<\/span>x<\/span>,<\/span> -<\/span>3<\/span>*<\/span>pi<\/span>,<\/span> 3<\/span>*<\/span>pi<\/span>),<\/span> \"f(x)\"<\/span>),<\/span> \r\n     (<\/span>f4<\/span>,<\/span> (<\/span>x<\/span>,<\/span> -<\/span>3<\/span>*<\/span>pi<\/span>,<\/span> 3<\/span>*<\/span>pi<\/span>),<\/span> \"n=4\"<\/span>,<\/span> {<\/span>\"linewidth\"<\/span>:<\/span>0.7<\/span>}),<\/span>\r\n     (<\/span>f3<\/span>,<\/span> (<\/span>x<\/span>,<\/span> -<\/span>3<\/span>*<\/span>pi<\/span>,<\/span> 3<\/span>*<\/span>pi<\/span>),<\/span> \"n=3\"<\/span>,<\/span> {<\/span>\"linewidth\"<\/span>:<\/span>0.7<\/span>}),<\/span>\r\n     (<\/span>f2<\/span>,<\/span> (<\/span>x<\/span>,<\/span> -<\/span>3<\/span>*<\/span>pi<\/span>,<\/span> 3<\/span>*<\/span>pi<\/span>),<\/span> \"n=2\"<\/span>,<\/span> {<\/span>\"linewidth\"<\/span>:<\/span>0.7<\/span>}),<\/span>\r\n     (<\/span>f1<\/span>,<\/span> (<\/span>x<\/span>,<\/span> -<\/span>3<\/span>*<\/span>pi<\/span>,<\/span> 3<\/span>*<\/span>pi<\/span>),<\/span> \"n=1\"<\/span>,<\/span> {<\/span>\"linewidth\"<\/span>:<\/span>0.7<\/span>}),<\/span>\r\n     xlim<\/span> =<\/span> (<\/span>-<\/span>3<\/span>*<\/span>pi<\/span>,<\/span> 3<\/span>*<\/span>pi<\/span>),<\/span> ylim<\/span> =<\/span> (<\/span>-<\/span>2<\/span>,<\/span> 12<\/span>),<\/span> \r\n     size<\/span>=<\/span>(<\/span>6<\/span>,<\/span>6<\/span>),<\/span> show<\/span> =<\/span> False<\/span>\r\n)<\/span>\r\n\r\nax<\/span> =<\/span> p<\/span>.<\/span>ax<\/span>\r\nax<\/span>.<\/span>set_xticks<\/span>([(<\/span>i<\/span>*<\/span>fpi<\/span>)<\/span> for<\/span> i<\/span> in<\/span> range<\/span>(<\/span>-<\/span>3<\/span>,<\/span>4<\/span>)])<\/span>\r\nxlab<\/span> =<\/span> [<\/span>\"-3 $\\pi$\"<\/span>,<\/span> \"-2 $\\pi$\"<\/span>,<\/span> \"-$\\pi$\"<\/span>,<\/span> \"0\"<\/span>,<\/span> \r\n        \"$\\pi$\"<\/span>,<\/span> \"$2 \\pi$\"<\/span>,<\/span> \"3 $\\pi$\"<\/span>]<\/span>\r\nax<\/span>.<\/span>set_xticklabels<\/span>(<\/span>xlab<\/span>);<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          \n
          <\/div>\n
          \n
          <\/p>\n
          \"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          <\/div>\n
          \n
          \n
          \u4efb\u610f\u306e\u5468\u671f\u3092\u3082\u3064\u95a2\u6570\u306e\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u5c55\u958b<\/h3>\n
          \u5468\u671f $2\\pi$ \u306e\u6c7a\u3081\u6253\u3061\u306e\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u3067\u306f\u306a\u304f\uff0c\u4efb\u610f\u306e\u5468\u671f\u3092\u3082\u3064\u5834\u5408\u306f\uff0c\u4e00\u822c\u306b\u5468\u671f\u3092 $2L$ \u3068\u3057\u3066…<\/p>\n
          $$ f(x) = \\frac{a_0}{2} + \\sum_{n=1}^{\\infty} \\bigl( a_n \\cos \\frac{n\\pi x}{L} + b_n \\sin \\frac{n\\pi x}{L} \\bigr) $$$$a_n = \\frac{1}{L} \\int_{-L}^{L} f(x) \\cos \\frac{n\\pi x}{L} \\, d{x} $$$$b_n = \\frac{1}{L} \\int_{-L}^{L} f(x) \\sin \\frac{n\\pi x}{L} \\, d{x} $$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          <\/div>\n
          \n
          \n
          \u4f8b\u984c<\/h4>\n
          \u533a\u9593 $[-1:1]$ \u3067\u5b9a\u7fa9\u3055\u308c\u305f\u95a2\u6570 $f(x) = x$ \u304c\uff0c\u533a\u9593\u5916\u3067\u306f\u5468\u671f $2$ \u306e\u5468\u671f\u95a2\u6570\u3067\u3042\u308b\u3068\u3057\u3066\uff0c$n=5$ \u307e\u3067\u306e\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u5c55\u958b\u3092\u6c42\u3081\u308b\u3002<\/p>\n
          \u4e0a\u8a18\u306e\u5f0f\u3067 $L = 1$ \u3068\u3059\u308c\u3070\u3088\u3044\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          <\/div>\n
          \n
          \n
          \u4e00\u822c\u306b\uff0c1\u533a\u95931\u5468\u671f\u3067 $f_0(x)$ \u3068\u3057\u3066\u5b9a\u7fa9\u3055\u308c\u305f\u95a2\u6570\u3092\uff0c\u305d\u306e\u533a\u9593\u5916\u3067\u5468\u671f $2L$ \u306e\u5468\u671f\u95a2\u6570\u306b\u3059\u308b\u306b\u306f\uff0c\u4e0a\u8a18\u306e\u3088\u3046\u306a $x_{\\rm cyc}$ \u3092\u4f7f\u3048\u3070\u3088\u3044\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[19]:<\/div>\n
          \n
          \n
          \n
          def<\/span> xcyc<\/span>(<\/span>x<\/span>,<\/span> L<\/span>):<\/span>\r\n    return<\/span> x<\/span> -<\/span> 2<\/span>*<\/span>L<\/span>*<\/span>floor<\/span>((<\/span>x<\/span>+<\/span>L<\/span>)<\/span>\/<\/span>(<\/span>2<\/span>*<\/span>L<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[20]:<\/div>\n
          \n
          \n
          \n
          def<\/span> f0<\/span>(<\/span>x<\/span>):<\/span>\r\n    return<\/span> x<\/span>\r\ndef<\/span> f<\/span>(<\/span>x<\/span>):<\/span>\r\n    return<\/span> f0<\/span>(<\/span>xcyc<\/span>(<\/span>x<\/span>,<\/span> 1<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[21]:<\/div>\n
          \n
          \n
          \n
          plot<\/span>(<\/span>f<\/span>(<\/span>x<\/span>),<\/span> (<\/span>x<\/span>,<\/span> -<\/span>5<\/span>,<\/span> 5<\/span>))<\/span>;<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          \n
          <\/div>\n
          \n
          <\/p>\n
          \"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[22]:<\/div>\n
          \n
          \n
          \n
          # L = 1 \u6c7a\u3081\u3046\u3061\u3067<\/span>\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\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> x<\/span>):<\/span>\r\n    var<\/span>(<\/span>'i'<\/span>)<\/span>\r\n    return<\/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> 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<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          <\/div>\n
          \n
          \n
          $f_0(x) = x$ \u306f\u5947\u95a2\u6570\u306a\u306e\u3067\uff0c\u5076\u95a2\u6570\u90e8\u5206\u306e\u30d5\u30fc\u30ea\u30a8\u4fc2\u6570\u3067\u3042\u308b $a_n$ \u306f\u30bc\u30ed\u306e\u306f\u305a\u3002\u5b9f\u969b\u306b\u8abf\u3079\u3066\u307f\u308b\u3068…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[23]:<\/div>\n
          \n
          \n
          \n
          # \u4efb\u610f\u306e n \u306b\u3064\u3044\u3066...<\/span>\r\na<\/span>(<\/span>n<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          \n
          Out[23]:<\/div>\n
          $\\displaystyle 0$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[24]:<\/div>\n
          \n
          \n
          \n
          b<\/span>(<\/span>1<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          \n
          Out[24]:<\/div>\n
          $\\displaystyle \\frac{2}{\\pi}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[25]:<\/div>\n
          \n
          \n
          \n
          b<\/span>(<\/span>2<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          \n
          Out[25]:<\/div>\n
          $\\displaystyle – \\frac{1}{\\pi}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[26]:<\/div>\n
          \n
          \n
          \n
          b<\/span>(<\/span>3<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          \n
          Out[26]:<\/div>\n
          $\\displaystyle \\frac{2}{3 \\pi}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[27]:<\/div>\n
          \n
          \n
          \n
          b<\/span>(<\/span>n<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          \n
          Out[27]:<\/div>\n
          $\\displaystyle – \\frac{2 \\left(-1\\right)^{n}}{\\pi n}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[28]:<\/div>\n
          \n
          \n
          \n
          f2<\/span> =<\/span> Fourier<\/span>(<\/span>2<\/span>,<\/span> x<\/span>)<\/span>\r\nf2<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          \n
          Out[28]:<\/div>\n
          $\\displaystyle – \\frac{2 \\left(- \\sin{\\left(\\pi x \\right)} + \\frac{\\sin{\\left(2 \\pi x \\right)}}{2}\\right)}{\\pi}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[29]:<\/div>\n
          \n
          \n
          \n
          f3<\/span> =<\/span> Fourier<\/span>(<\/span>3<\/span>,<\/span> x<\/span>)<\/span>\r\nf3<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          \n
          Out[29]:<\/div>\n
          $\\displaystyle – \\frac{2 \\left(- \\sin{\\left(\\pi x \\right)} + \\frac{\\sin{\\left(2 \\pi x \\right)}}{2} – \\frac{\\sin{\\left(3 \\pi x \\right)}}{3}\\right)}{\\pi}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[30]:<\/div>\n
          \n
          \n
          \n
          f4<\/span> =<\/span> Fourier<\/span>(<\/span>4<\/span>,<\/span> x<\/span>)<\/span>\r\nf4<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          \n
          Out[30]:<\/div>\n
          $\\displaystyle – \\frac{2 \\left(- \\sin{\\left(\\pi x \\right)} + \\frac{\\sin{\\left(2 \\pi x \\right)}}{2} – \\frac{\\sin{\\left(3 \\pi x \\right)}}{3} + \\frac{\\sin{\\left(4 \\pi x \\right)}}{4}\\right)}{\\pi}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[31]:<\/div>\n
          \n
          \n
          \n
          plot<\/span>(<\/span>\r\n     (<\/span>f<\/span>(<\/span>x<\/span>),<\/span> (<\/span>x<\/span>,<\/span> -<\/span>5<\/span>,<\/span> 5<\/span>),<\/span> \"f(x)\"<\/span>),<\/span> \r\n     (<\/span>f4<\/span>,<\/span> (<\/span>x<\/span>,<\/span> -<\/span>5<\/span>,<\/span> 5<\/span>),<\/span> \"n=4\"<\/span>,<\/span> {<\/span>\"linewidth\"<\/span>:<\/span>0.7<\/span>}),<\/span>\r\n     (<\/span>f3<\/span>,<\/span> (<\/span>x<\/span>,<\/span> -<\/span>5<\/span>,<\/span> 5<\/span>),<\/span> \"n=3\"<\/span>,<\/span> {<\/span>\"linewidth\"<\/span>:<\/span>0.7<\/span>}),<\/span>\r\n     (<\/span>f2<\/span>,<\/span> (<\/span>x<\/span>,<\/span> -<\/span>5<\/span>,<\/span> 5<\/span>),<\/span> \"n=2\"<\/span>,<\/span> {<\/span>\"linewidth\"<\/span>:<\/span>0.7<\/span>}),<\/span>\r\n     xlim<\/span> =<\/span> (<\/span>-<\/span>5<\/span>,<\/span> 5<\/span>),<\/span> ylim<\/span> =<\/span> (<\/span>-<\/span>1.3<\/span>,<\/span> 1.8<\/span>),<\/span> \r\n     size<\/span> =<\/span> (<\/span>6<\/span>,<\/span> 5<\/span>)<\/span>\r\n)<\/span>;<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          \n
          <\/div>\n
          \n
          <\/p>\n
          \"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          <\/div>\n
          \n
          \n
          \u30d5\u30fc\u30ea\u30a8\u7a4d\u5206\u30fb\u30d5\u30fc\u30ea\u30a8\u5909\u63db<\/h3>\n
          \u4efb\u610f\u306e\u5468\u671f $2L$ \u3092\u3082\u3064\u95a2\u6570\u306e\u8907\u7d20\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u5c55\u958b\u3092\uff0c\u975e\u5468\u671f\u7684\u73fe\u8c61\u306b\u307e\u3067\u62e1\u5f35\u3057\u305f\u3082\u306e\u304c\u300c\u30d5\u30fc\u30ea\u30a8\u7a4d\u5206\u300d\u3067\u3042\u308a\uff0c\u30d5\u30fc\u30ea\u30a8\u4fc2\u6570\u306e\u62e1\u5f35\u304c\u300c\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u300d\u3002<\/p>\n
          \u5468\u671f\u6027\u306e\u306a\u3044\u95a2\u6570\u306b\u5bfe\u3059\u308b\uff08\u9023\u7d9a\u6975\u9650\u3068\u3057\u3066\u306e\uff09\u30d5\u30fc\u30ea\u30a8\u7a4d\u5206<\/p>\n
          $$ f(x) = \\frac{1}{2\\pi} \\int_{-\\infty}^{\\infty}\\ F(k)\\ e^{i k x}\\ dk, \\quad F(k) \\equiv \\int_{-\\infty}^{\\infty} f(x) \\ e^{-i k x} \\ dx $$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          <\/div>\n
          \n
          \n
          \u4f8b\u984c<\/h4>\n
          \u4ee5\u4e0b\u306e\u5468\u671f\u6027\u306e\u306a\u3044\u95a2\u6570 $f(x)$ \u306e\u30d5\u30fc\u30ea\u30a8\u5909\u63db $F(k)$ \u3092\u6c42\u3081\u3088\u3002<\/p>\n
          $$ f(x) = \\begin{cases}
          \n\\frac{1}{a} & (|x| \\leq \\frac{a}{2}) \\\\
          \n0 & (|x|> \\frac{a}{2})
          \n\\end{cases}$$<\/p>\n
          \u6b21\u306b\uff0c\u4ee5\u4e0b\u306e\u6975\u9650\u3092\u6c42\u3081\u3088\u3002$$ \\lim_{a \\rightarrow 0} F(k) =?$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          <\/div>\n
          \n
          \n
          \u3082\u3057 $\\displaystyle |x| \\leq \\frac{a}{2}$ \u306a\u3089 $\\displaystyle \\frac{1}{a}$ \u3092\u8fd4\u3057\uff0c\u305d\u308c\u4ee5\u5916\u306a\u3089 $0$ \u3092\u8fd4\u3059\u95a2\u6570 $f(x)$ \u306e\u5b9a\u7fa9\u3002<\/p>\n
          if \u6587\u3092\u4f7f\u3044\u305d\u3046\u306b\u306a\u308b\u304c\uff0c\u3053\u306e\u5834\u5408\u306f Piecewise()<\/code> \u3092\u4f7f\u3046\u3089\u3057\u3044\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
          \n
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          In\u00a0[32]:<\/div>\n
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          \n
          def<\/span> f<\/span>(<\/span>a<\/span>,<\/span> x<\/span>):<\/span>\r\n    return<\/span> Piecewise<\/span>((<\/span>1<\/span>\/<\/span>a<\/span>,<\/span> abs<\/span>(<\/span>x<\/span>)<\/span> <=<\/span> a<\/span>\/<\/span>2<\/span>),<\/span> \r\n                     (<\/span>0<\/span>,<\/span> abs<\/span>(<\/span>x<\/span>)<\/span> ><\/span> a<\/span>\/<\/span>2<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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          In\u00a0[33]:<\/div>\n
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          plot<\/span>(<\/span>f<\/span>(<\/span>2<\/span>,<\/span> x<\/span>),<\/span> (<\/span>x<\/span>,<\/span> -<\/span>5<\/span>,<\/span> 5<\/span>),<\/span> ylim<\/span> =<\/span> (<\/span>-<\/span>0.5<\/span>,<\/span> 1<\/span>))<\/span>;<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
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          <\/div>\n
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          <\/p>\n
          \"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          <\/div>\n
          \n
          \n
          $$ F(k) = \\int_{-\\infty}^{\\infty} f(x) \\ e^{-i k x} \\ dx = \\int_{-\\frac{a}{2}}^{\\frac{a}{2}} \\frac{1}{a}\\ e^{-i k x} \\ dx$$\u3068\u3044\u3046\u3053\u3068\u3092\u4f7f\u3063\u3066\uff0c\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u306e\u7121\u9650\u7a4d\u5206\u3092\u6709\u9650\u533a\u9593\u306e\u7a4d\u5206\u306b\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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          In\u00a0[34]:<\/div>\n
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          var<\/span>(<\/span>'a k'<\/span>,<\/span> nonzero<\/span> =<\/span> True<\/span>)<\/span>\r\n\r\nF<\/span> =<\/span> integrate<\/span>(<\/span>1<\/span>\/<\/span>a<\/span>*<\/span>exp<\/span>(<\/span>-<\/span>I<\/span>*<\/span>k<\/span>*<\/span>x<\/span>),<\/span> (<\/span>x<\/span>,<\/span> -<\/span>a<\/span>\/<\/span>2<\/span>,<\/span> a<\/span>\/<\/span>2<\/span>))<\/span>\r\nF<\/span>.<\/span>simplify<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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          Out[34]:<\/div>\n
          $\\displaystyle \\frac{2 \\sin{\\left(\\frac{a k}{2} \\right)}}{a k}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
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          In\u00a0[35]:<\/div>\n
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          Eq<\/span>(<\/span>Limit<\/span>(<\/span>F<\/span>.<\/span>simplify<\/span>(),<\/span> a<\/span>,<\/span> 0<\/span>),<\/span> \r\n   Limit<\/span>(<\/span>F<\/span>.<\/span>simplify<\/span>(),<\/span> a<\/span>,<\/span> 0<\/span>)<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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          Out[35]:<\/div>\n
          $\\displaystyle \\lim_{a \\to 0^+}\\left(\\frac{2 \\sin{\\left(\\frac{a k}{2} \\right)}}{a k}\\right) = 1$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          <\/div>\n
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          \u7b97\u6570\u306e\u554f\u984c\u3068\u3057\u3066\uff0c<\/p>\n
          $$ \\lim_{a \\rightarrow 0} F(k) = \\lim_{a \\rightarrow 0} \\frac{\\sin\\frac{ak}{2}}{\\frac{ak}{2}} = 1$$<\/p>\n
          \u3068\u306a\u308b\u3053\u3068\u306f\u308f\u304b\u3063\u305f\u3002\u3067\u306f\uff0c\u30d5\u30fc\u30ea\u30a8\u5909\u63db $F(k)$ \u304c $1$ \u3068\u306a\u308b $f(x)$ \u3068\u306f\u3069\u306e\u3088\u3046\u306a\u95a2\u6570\u3067\u3042\u308d\u3046\u304b\u3002<\/p>\n
          \u4ee5\u4e0b\u306e\u30b0\u30e9\u30d5\u304b\u3089\u63a8\u5bdf\u3055\u308c\u308b\u3088\u3046\u306b\uff0c$a$ \u306e\u5024\u3092\u5c0f\u3055\u304f\u3057\u3066\u3044\u304f\u3068 $f(x)$ \u306f $x = 0$ \u306e\u8fd1\u304f\u3067\u306e\u307f\u5024\u3092\u3082\u3064\u95a2\u6570\u306b\u306a\u308b\u3002<\/p>\n
          $\\displaystyle \\int_{-\\infty}^{\\infty} f(x)\\, dx =\\int_{-\\frac{a}{2}}^{\\frac{a}{2}} \\frac{1}{a} \\,dx = 1$ \u3068\u3044\u3046\u9762\u7a4d\u3092\u4fdd\u3061\u306a\u304c\u3089\uff0c$a \\rightarrow 0$ \u3067\u5e45 $a$ \u304c\u3069\u3093\u3069\u3093\u72ed\u304f\u306a\u3063\u3066\u3044\u304f\u305f\u3081\uff0c\u9ad8\u3055 $\\displaystyle \\frac{1}{a}$ \u304c\u3069\u3093\u3069\u3093\u9ad8\u304f\u306a\u3063\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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          In\u00a0[36]:<\/div>\n
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          plot<\/span>((<\/span>f<\/span>(<\/span>0.1<\/span>,<\/span> x<\/span>),<\/span> (<\/span>x<\/span>,<\/span> -<\/span>5<\/span>,<\/span> 5<\/span>),<\/span> \"a=0.1\"<\/span>),<\/span> \r\n     (<\/span>f<\/span>(<\/span>0.2<\/span>,<\/span> x<\/span>),<\/span> (<\/span>x<\/span>,<\/span> -<\/span>5<\/span>,<\/span> 5<\/span>),<\/span> \"a=0.2\"<\/span>),<\/span> \r\n     (<\/span>f<\/span>(<\/span>0.5<\/span>,<\/span> x<\/span>),<\/span> (<\/span>x<\/span>,<\/span> -<\/span>5<\/span>,<\/span> 5<\/span>),<\/span> \"a=0.5\"<\/span>),<\/span> \r\n     (<\/span>f<\/span>(<\/span>2<\/span>,<\/span> x<\/span>),<\/span> (<\/span>x<\/span>,<\/span> -<\/span>5<\/span>,<\/span> 5<\/span>),<\/span> \"a=2\"<\/span>),<\/span> legend<\/span> =<\/span> True<\/span>)<\/span>;<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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          <\/div>\n
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          <\/p>\n
          \"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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          <\/div>\n
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          \n
          \u3053\u306e\u95a2\u6570 $f(x)$ \u306e $a \\rightarrow 0$ \u306e\u6975\u9650\u306f\uff0c\u300c\u30c7\u30eb\u30bf\u95a2\u6570\u300d$\\delta(x)$ \u3068\u547c\u3070\u308c\u307e\u3059\u3002\u5b9a\u7fa9\u306f\uff0c\uff08\u30d5\u30fc\u30ea\u30a8\u5909\u63db $F(k)$ \u304c $1$ \u3067\u3042\u308b\u30d5\u30fc\u30ea\u30a8\u7a4d\u5206\u3067\u3042\u308b\u304b\u3089\uff09<\/p>\n
          $$\\delta(x) \\equiv \\frac{1}{2\\pi} \\int_{-\\infty}^{\\infty} e^{i k x}\\ dk$$<\/p>\n
          \u30c7\u30eb\u30bf\u95a2\u6570 \\(\\delta(x) \\) \u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u6027\u8cea\u3092\u6301\u3061\u307e\u3059\u3002<\/p>\n
          $$\\delta(x) = 0\\\u00a0 \\ \\mbox{for $x \\neq 0$}$$$$\\int_{-\\infty}^{\\infty} g(x) \\delta(x -a)\\, dx = g(a)$$<\/p>\n
          \u7279\u306b\uff0c<\/p>\n
          $$\\int_{-\\infty}^{\\infty} \\delta(x)\\, dx = 1$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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          <\/div>\n
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          \u53c2\u8003\uff1afourier_series()<\/code> \u95a2\u6570\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