{"id":6503,"date":"2023-06-12T15:34:04","date_gmt":"2023-06-12T06:34:04","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=6503"},"modified":"2024-05-12T20:27:27","modified_gmt":"2024-05-12T11:27:27","slug":"%e4%bb%bb%e6%84%8f%e5%91%a8%e6%9c%9f%e3%81%ae%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","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\/%e3%83%95%e3%83%bc%e3%83%aa%e3%82%a8%e8%a7%a3%e6%9e%90\/%e4%bb%bb%e6%84%8f%e3%81%ae%e5%91%a8%e6%9c%9f%e3%82%92%e3%82%82%e3%81%a4%e9%96%a2%e6%95%b0%e3%81%ae%e3%83%95%e3%83%bc%e3%83%aa%e3%82%a8%e7%b4%9a%e6%95%b0%e5%b1%95%e9%96%8b\/%e4%bb%bb%e6%84%8f%e5%91%a8%e6%9c%9f%e3%81%ae%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\/","title":{"rendered":"\u4efb\u610f\u5468\u671f\u306e\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u5c55\u958b\u306e\u4f8b"},"content":{"rendered":"

\u533a\u9593 \\([-1 : 1]\\) \u3067\u5b9a\u7fa9\u3055\u308c\u305f\u95a2\u6570 \\(f(x) = x\\) \u304c\uff0c<\/p>\n

\"\"<\/p>\n

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\u533a\u9593\u5916\u3067\u306f\u5468\u671f \\(2\\) \u306e\u5468\u671f\u95a2\u6570\u3067\u3042\u308b\u3068\u3057\u3066\uff0c<\/p>\n

\"\"<\/p>\n

\\(n = 3\\) \u307e\u3067\u306e\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u5c55\u958b\u3092\u6c42\u3081\u308b\u3002<\/p>\n

\n
\n
\n

\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

$$ f(x) = \\frac{a_0}{2} + \\sum_{n=1}^{\\infty} \\bigl( a_n \\cos \\left( \\frac{n\\pi x}{L}\\right) + b_n \\sin \\left(\\frac{n\\pi x}{L} \\right) \\bigr) $$
\n$$a_n = \\frac{1}{L} \\int_{-L}^{L} f(x) \\cos \\left(\\frac{n\\pi x}{L} \\right) \\,\u00a0 d{x} $$
\n$$b_n = \\frac{1}{L} \\int_{-L}^{L} f(x) \\sin \\left(\\frac{n\\pi x}{L} \\right)\\,\u00a0 d{x} $$<\/p>\n

\u3053\u306e\u4f8b\u984c\u306e\u5834\u5408\u306f\uff0c\\(f(x) = x, \\ L = 1\\) \u3068\u3057\u3066<\/p>\n

$$a_n = \\int_{-1}^{1} x \\cos \\left(n \\pi\u00a0 x\\right) \\,\u00a0 d{x} = 0$$<\/p>\n

\\begin{eqnarray}
\nb_n &=& \\int_{-1}^{1} x \\sin \\left(n \\pi x \\right)\\,\u00a0 d{x} \\\\
\n&=& \\left[ – x \\frac{\\cos n \\pi x}{n \\pi} \\right]_{-1}^{1} + \\int_{-1}^{1} \\frac{\\cos n \\pi x}{n \\pi} dx \\\\
\n&=& – \\frac{2 \\cos n\\pi}{n\\pi} \\\\
\n&=& \\frac{2}{n\\pi} (-1)^{n+1}
\n\\end{eqnarray}<\/p>\n

\\begin{eqnarray}
\n\\therefore\\ \\ f(x) &=& b_1 \\sin \\pi x + b_2 \\sin 2 \\pi x + b_3 \\sin 3 \\pi x + \\cdots\\\\
\n&=& \\frac{2}{\\pi} \\sin \\pi x – \\frac{1}{\\pi} \\sin 2 \\pi x + \\frac{2}{3 \\pi} \\sin 3 \\pi x + \\cdots
\n\\end{eqnarray}<\/p>\n

 <\/p>\n<\/div>\n<\/div>\n<\/div>\n

\"\"<\/p>\n

 <\/p>\n

$n=10$ \u307e\u3067\u306e\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u5c55\u958b<\/h3>\n

\"\"<\/p>\n

\u30a2\u30cb\u30e1\u30fc\u30b7\u30e7\u30f3<\/h3>\n