{"id":2378,"date":"2022-02-26T13:40:19","date_gmt":"2022-02-26T04:40:19","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2378"},"modified":"2023-06-12T15:48:44","modified_gmt":"2023-06-12T06:48:44","slug":"maxima-%e3%81%a6%e3%82%99%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\/maxima-%e3%81%a7%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6c\/maxima-%e3%81%a6%e3%82%99%e3%83%95%e3%83%bc%e3%83%aa%e3%82%a8%e8%a7%a3%e6%9e%90\/","title":{"rendered":"Maxima \u3066\u3099\u30d5\u30fc\u30ea\u30a8\u89e3\u6790"},"content":{"rendered":"<p><!--more--><\/p>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h3 id=\"\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\">\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570<\/h3>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<ol>\n<li>\u533a\u9593 $-\\pi &lt; x &lt; \\pi$ \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&#8230;<\/li>\n<li>\u533a\u9593\u5916\u3067\u306f\uff0c\u5468\u671f $ 2\\pi $ \u306e\u5468\u671f\u95a2\u6570\u3068\u307f\u306a\u3057\u3066<\/li>\n<li>\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<p>\u3064\u307e\u308a\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3068\u3044\u3046\u3053\u3068\u3002<\/p>\n<p>$$ f(x) = \\frac{a_0}{2} + \\sum_{n=1}^{\\infty} \\bigl( a_n \\cos n x + b_n \\sin nx \\bigr) $$<\/p>\n<p>\u3053\u3053\u3067\uff0c\u30d5\u30fc\u30ea\u30a8\u4fc2\u6570 $a_n, b_n$ \u306f<\/p>\n<p>$$a_n = \\frac{1}{\\pi} \\int_{-\\pi}^{\\pi} f(x) \\cos nx \\, dx $$$$b_n = \\frac{1}{\\pi} \\int_{-\\pi}^{\\pi} f(x) \\sin nx \\, dx $$<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h4 id=\"\u4f8b\u984c\">\u4f8b\u984c<\/h4>\n<p>\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<p>\u5468\u671f $2 \\pi$ \u306e\u5468\u671f\u95a2\u6570\u3068\u3057\u3066\uff0c<\/p>\n<ul>\n<li>$[-3\\pi:-\\pi]$ \u3067\u306f $f(x) = (x+2 \\pi)^2$&#8230;<\/li>\n<li>$[-\\pi:\\pi]$ \u3067\u306f $f(x) = x^2$,<\/li>\n<li>$[\\pi:3\\pi]$ \u3067\u306f $f(x) = (x-2 \\pi)^2$&#8230;<\/li>\n<\/ul>\n<p>\u306e\u3088\u3046\u306b\u3059\u308c\u3070\u3044\u3044\u3002<\/p>\n<p>\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<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[1]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"nf\">f0<\/span><span class=\"p\">(<\/span><span class=\"nv\">x<\/span><span class=\"p\">)<\/span><span class=\"o\">:=<\/span> <span class=\"nv\">x<\/span><span class=\"o\">**<\/span><span class=\"mi\">2<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[1]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{1}$}f_{0}\\left(x\\right):=x^2\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[2]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"nf\">plot2d<\/span><span class=\"p\">(<\/span><span class=\"nf\">f0<\/span><span class=\"p\">(<\/span><span class=\"nv\">x<\/span><span class=\"p\">)<\/span>, <span class=\"p\">[<\/span><span class=\"nv\">x<\/span>, <span class=\"o\">-<\/span><span class=\"nv\">%pi<\/span>, <span class=\"nv\">%pi<\/span><span class=\"p\">])<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_svg output_subarea \">\n<p><!--?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?--><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-large wp-image-6146\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/mmathc401-1.svg\" alt=\"\" width=\"640\" height=\"481\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>$[-\\pi:\\pi]$ \u306e\u533a\u9593\u5916\u3067\u306f\uff0c\u5468\u671f $2 \\pi$ \u306e\u5468\u671f\u95a2\u6570\u3068\u3057\u3066\uff0c<\/p>\n<ul>\n<li>$[-3\\pi:-\\pi]$ \u3067\u306f $f(x) = (x+2 \\pi)^2$&#8230;<\/li>\n<li>$[-\\pi:\\pi]$ \u3067\u306f $f(x) = x^2$,<\/li>\n<li>$[\\pi:3\\pi]$ \u3067\u306f $f(x) = (x-2 \\pi)^2$&#8230;<\/li>\n<\/ul>\n<p>\u3092\u30b0\u30e9\u30d5\u306b\u63cf\u304f\u305f\u3081\u306b\uff0c\u3061\u3087\u3063\u3068\u5de5\u592b\u3092\u3057\u307e\u3059\u3002\uff08\u8a73\u7d30\u306f\u7701\u304d\u307e\u3059\u304c\uff0c$x$ \u306e\u7570\u306a\u308b\u7bc4\u56f2\u3054\u3068\u306b\u7570\u306a\u308b\u95a2\u6570\u306e\u30b0\u30e9\u30d5\u3092\u91cd\u306d\u3066\u63cf\u304f\u305f\u3081\uff0c\u5a92\u4ecb\u5909\u6570\u8868\u793a <code>parametric<\/code> \u3067\u30d7\u30ed\u30c3\u30c8\u3057\u3066\u3044\u307e\u3059\u3002\uff09<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[3]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"nf\">plot2d<\/span><span class=\"p\">([<\/span><span class=\"nv\">parametric<\/span>, <span class=\"nv\">t<\/span>, <span class=\"nf\">f0<\/span><span class=\"p\">(<\/span><span class=\"nv\">t<\/span><span class=\"p\">)<\/span>, <span class=\"p\">[<\/span><span class=\"nv\">t<\/span>, <span class=\"o\">-<\/span><span class=\"nv\">%pi<\/span>, <span class=\"nv\">%pi<\/span><span class=\"p\">]]<\/span>, \r\n       <span class=\"p\">[<\/span><span class=\"nv\">x<\/span>, <span class=\"o\">-<\/span>3<span class=\"o\">*<\/span><span class=\"nv\">%pi<\/span>, 3<span class=\"o\">*<\/span><span class=\"nv\">%pi<\/span><span class=\"p\">]<\/span>\r\n<span class=\"p\">)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_svg output_subarea \">\n<p><!--?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?--><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-large wp-image-6147\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/mmathc402-1.svg\" alt=\"\" width=\"640\" height=\"481\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[4]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"nf\">plot2d<\/span><span class=\"p\">([[<\/span><span class=\"nv\">parametric<\/span>, <span class=\"nv\">x<\/span>, <span class=\"nf\">f0<\/span><span class=\"p\">(<\/span><span class=\"nv\">x<\/span> <span class=\"o\">+<\/span> 2<span class=\"o\">*<\/span><span class=\"nv\">%pi<\/span><span class=\"p\">)<\/span>, <span class=\"p\">[<\/span><span class=\"nv\">x<\/span>, <span class=\"o\">-<\/span>3<span class=\"o\">*<\/span><span class=\"nv\">%pi<\/span>, <span class=\"o\">-<\/span><span class=\"nv\">%pi<\/span><span class=\"p\">]]<\/span>,\r\n        <span class=\"p\">[<\/span><span class=\"nv\">parametric<\/span>, <span class=\"nv\">x<\/span>, <span class=\"nf\">f0<\/span><span class=\"p\">(<\/span><span class=\"nv\">x<\/span><span class=\"p\">)<\/span>,         <span class=\"p\">[<\/span><span class=\"nv\">x<\/span>,   <span class=\"o\">-<\/span><span class=\"nv\">%pi<\/span>,  <span class=\"nv\">%pi<\/span><span class=\"p\">]]<\/span>,\r\n        <span class=\"p\">[<\/span><span class=\"nv\">parametric<\/span>, <span class=\"nv\">x<\/span>, <span class=\"nf\">f0<\/span><span class=\"p\">(<\/span><span class=\"nv\">x<\/span> <span class=\"o\">-<\/span> 2<span class=\"o\">*<\/span><span class=\"nv\">%pi<\/span><span class=\"p\">)<\/span>, <span class=\"p\">[<\/span><span class=\"nv\">x<\/span>,    <span class=\"nv\">%pi<\/span>, 3<span class=\"o\">*<\/span><span class=\"nv\">%pi<\/span><span class=\"p\">]]<\/span>\r\n       <span class=\"p\">]<\/span>, <span class=\"p\">[<\/span><span class=\"nv\">x<\/span>, <span class=\"o\">-<\/span>3<span class=\"o\">*<\/span><span class=\"nv\">%pi<\/span>, 3<span class=\"o\">*<\/span><span class=\"nv\">%pi<\/span><span class=\"p\">]<\/span>\r\n<span class=\"p\">)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_svg output_subarea \">\n<p><!--?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?--><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-large wp-image-6148\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/mmathc403-1.svg\" alt=\"\" width=\"640\" height=\"481\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\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 <code>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<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[5]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"nf\">f<\/span><span class=\"p\">(<\/span><span class=\"nv\">x<\/span><span class=\"p\">)<\/span><span class=\"o\">:=<\/span>  <span class=\"nf\">f0<\/span><span class=\"p\">(<\/span><span class=\"nv\">x<\/span> <span class=\"o\">-<\/span> 2<span class=\"o\">*<\/span><span class=\"nv\">%pi<\/span><span class=\"o\">*<\/span><span class=\"nf\">floor<\/span><span class=\"p\">((<\/span><span class=\"nv\">x<\/span><span class=\"o\">+<\/span><span class=\"nv\">%pi<\/span><span class=\"p\">)<\/span><span class=\"o\">\/<\/span><span class=\"p\">(<\/span>2<span class=\"o\">*<\/span><span class=\"nv\">%pi<\/span><span class=\"p\">)))<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[5]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{8}$}f\\left(x\\right):=f_{0}\\left(x-2\\,\\pi\\,\\left \\lfloor \\frac{x+\\pi}{2\\,\\pi} \\right \\rfloor\\right)\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u4f55\u3053\u308c\uff1f\u3068\u601d\u3063\u305f\u4eba\u3082\u3044\u308b\u304b\u3082\u3057\u308c\u306a\u3044\u304c\uff0c\u5b9f\u969b\u306b\u30b0\u30e9\u30d5\u3092\u63cf\u304f\u3068\u78ba\u304b\u306b\u5468\u671f\u95a2\u6570\u306b\u306a\u3063\u3066\u3044\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n<p>\u95a2\u6570 $f(x)$ \u3092 $a \\leq x \\leq b$ \u306e\u7bc4\u56f2\u3067\u30b0\u30e9\u30d5\u306b\u3059\u308b\u306e\u306f\uff0cMaxima \u3067\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304f\u3002<\/p>\n<p><code>plot2d(f(x), [x, a, b]);<\/code><\/p>\n<p>\u4e0a\u3067\u5b9a\u7fa9\u3057\u305f\u5468\u671f\u95a2\u6570 $f(x)$ \u3092 $-3\\pi \\leq x 3\\pi$ \u306e\u7bc4\u56f2\u3067\u30b0\u30e9\u30d5\u3092\u63cf\u304f\u306b\u306f&#8230;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[6]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"nf\">plot2d<\/span><span class=\"p\">(<\/span><span class=\"nf\">f<\/span><span class=\"p\">(<\/span><span class=\"nv\">x<\/span><span class=\"p\">)<\/span>, <span class=\"p\">[<\/span><span class=\"nv\">x<\/span>, <span class=\"o\">-<\/span>3<span class=\"o\">*<\/span><span class=\"nv\">%pi<\/span>, 3<span class=\"o\">*<\/span><span class=\"nv\">%pi<\/span><span class=\"p\">])<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_svg output_subarea \">\n<p><!--?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?--><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-large wp-image-6149\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/mmathc404-1.svg\" alt=\"\" width=\"640\" height=\"481\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u5c11\u3057\u30aa\u30d7\u30b7\u30e7\u30f3\u3092\u8a2d\u5b9a\u3057\u3066\u63cf\u304f\u4f8b\uff1a<\/p>\n<p>\u4ee5\u4e0b\u306e\u4f8b\u3067\u306f\uff0c<\/p>\n<ol>\n<li><code>legend<\/code> \u3067\u51e1\u4f8b\u3092\u8a2d\u5b9a\u3057\uff0c<\/li>\n<li><code>[y, -2, 12]<\/code> \u3067\u7e26\u8ef8\u306e\u8868\u793a\u7bc4\u56f2\u3092\u8a2d\u5b9a\u3057\uff0c<\/li>\n<li><code>grid2d<\/code> \u3067\u30b0\u30ea\u30c3\u30c9\uff08\u683c\u5b50\u7dda\uff09\u3092\u8868\u793a\u3055\u305b\u308b\u8a2d\u5b9a\u3092\u3057\uff0c<\/li>\n<li><code>[ylabel, \"\"]<\/code> \u3067 $y$ \u8ef8\u306e\u30e9\u30d9\u30eb\u3092\u30ab\u30e9\u306b\u3057\uff0c<\/li>\n<li><code>[xtics, %pi]<\/code> \u3067 $x$ \u8ef8\u306e\u76ee\u76db\u308a\u3092 $\\pi$ \u3054\u3068\u306b\u3057\uff0c<\/li>\n<li>$x$ \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<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[7]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"nf\">plot2d<\/span><span class=\"p\">(<\/span><span class=\"nf\">f<\/span><span class=\"p\">(<\/span><span class=\"nv\">x<\/span><span class=\"p\">)<\/span>, <span class=\"p\">[<\/span><span class=\"nv\">x<\/span>, <span class=\"o\">-<\/span>3<span class=\"o\">*<\/span><span class=\"nv\">%pi<\/span>, 3<span class=\"o\">*<\/span><span class=\"nv\">%pi<\/span><span class=\"p\">]<\/span>, <span class=\"p\">[<\/span><span class=\"nv\">legend<\/span>,<span class=\"s\">\"f(x)\"<\/span><span class=\"p\">]<\/span>, <span class=\"p\">[<\/span><span class=\"nv\">y<\/span>, <span class=\"o\">-<\/span><span class=\"mi\">2<\/span>, 12<span class=\"p\">]<\/span>, \r\n       <span class=\"nv\">grid2d<\/span>, <span class=\"p\">[<\/span><span class=\"nv\">ylabel<\/span>, <span class=\"s\">\"\"<\/span><span class=\"p\">]<\/span>, \r\n       <span class=\"p\">[<\/span><span class=\"nv\">xtics<\/span>, <span class=\"nv\">%pi<\/span><span class=\"p\">]<\/span>, <span class=\"p\">[<\/span><span class=\"nv\">gnuplot_preamble<\/span>, <span class=\"s\">\"set format x '%4.1P \u03c0'\"<\/span><span class=\"p\">]<\/span>\r\n<span class=\"p\">)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_svg output_subarea \">\n<p><!--?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?--><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-large wp-image-6150\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/mmathc405-1.svg\" alt=\"\" width=\"640\" height=\"481\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u53c2\u8003\u307e\u3067\u306b\uff0c<code>draw2d()<\/code> \u3067\u63cf\u304f\u4f8b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[8]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"nf\">draw2d<\/span><span class=\"p\">(<\/span>\r\n  <span class=\"nv\">yrange<\/span> <span class=\"o\">=<\/span> <span class=\"p\">[<\/span><span class=\"o\">-<\/span><span class=\"mi\">2<\/span>, 12<span class=\"p\">]<\/span>, <span class=\"nv\">grid<\/span> <span class=\"o\">=<\/span> <span class=\"no\">true<\/span>,\r\n  <span class=\"nv\">xtics<\/span> <span class=\"o\">=<\/span> <span class=\"nv\">%pi<\/span>, <span class=\"nv\">user_preamble<\/span> <span class=\"o\">=<\/span> <span class=\"s\">\"set format x '%4.1P \u03c0'\"<\/span>,\r\n  \r\n  <span class=\"nv\">key<\/span> <span class=\"o\">=<\/span> <span class=\"s\">\"f(x)\"<\/span>, \r\n  <span class=\"nf\">explicit<\/span><span class=\"p\">(<\/span><span class=\"nf\">f<\/span><span class=\"p\">(<\/span><span class=\"nv\">x<\/span><span class=\"p\">)<\/span>, <span class=\"nv\">x<\/span>,  <span class=\"o\">-<\/span>3<span class=\"o\">*<\/span><span class=\"nv\">%pi<\/span>, 3<span class=\"o\">*<\/span><span class=\"nv\">%pi<\/span><span class=\"p\">)<\/span>\r\n<span class=\"p\">)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_svg output_subarea \">\n<p><!--?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?--><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-large wp-image-6151\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/mmathc406-1.svg\" alt=\"\" width=\"640\" height=\"481\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u307e\u3057\u305f\u3002<\/p>\n<p>$$ f(x) = \\frac{a_0}{2} + \\sum_{n=1}^{\\infty} \\bigl( a_n \\cos n x + b_n \\sin nx \\bigr) $$<\/p>\n<p>\u3053\u3053\u3067\uff0c\u30d5\u30fc\u30ea\u30a8\u4fc2\u6570 $a_n, b_n$ \u306f<\/p>\n<p>$$a_n = \\frac{1}{\\pi} \\int_{-\\pi}^{\\pi} f(x) \\cos nx \\, dx $$$$b_n = \\frac{1}{\\pi} \\int_{-\\pi}^{\\pi} f(x) \\sin nx \\, dx $$<\/p>\n<p>Maxima \u306b\u306f\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u3092\u6271\u3046\u30d1\u30c3\u30b1\u30fc\u30b8\uff08<code>load(fourie)$<\/code>\uff09\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<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[9]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"cm\">\/* \u30d5\u30fc\u30ea\u30a8\u4fc2\u6570\u306e\u5b9a\u7fa9\u3068\u8a08\u7b97\u3002\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u306e\u5b9a\u7fa9\u3002*\/<\/span>\r\n\r\n<span class=\"nf\">a<\/span><span class=\"p\">(<\/span><span class=\"nv\">n<\/span><span class=\"p\">)<\/span><span class=\"o\">:=<\/span> 1<span class=\"o\">\/<\/span><span class=\"nv\">%pi<\/span> <span class=\"o\">*<\/span> <span class=\"nf\">integrate<\/span><span class=\"p\">(<\/span><span class=\"nf\">f0<\/span><span class=\"p\">(<\/span><span class=\"nv\">x<\/span><span class=\"p\">)<\/span><span class=\"o\">*<\/span><span class=\"nf\">cos<\/span><span class=\"p\">(<\/span><span class=\"nv\">n<\/span><span class=\"o\">*<\/span><span class=\"nv\">x<\/span><span class=\"p\">)<\/span>, <span class=\"nv\">x<\/span>, <span class=\"o\">-<\/span><span class=\"nv\">%pi<\/span>, <span class=\"nv\">%pi<\/span><span class=\"p\">)<\/span>;\r\n<span class=\"nf\">b<\/span><span class=\"p\">(<\/span><span class=\"nv\">n<\/span><span class=\"p\">)<\/span><span class=\"o\">:=<\/span> 1<span class=\"o\">\/<\/span><span class=\"nv\">%pi<\/span> <span class=\"o\">*<\/span> <span class=\"nf\">integrate<\/span><span class=\"p\">(<\/span><span class=\"nf\">f0<\/span><span class=\"p\">(<\/span><span class=\"nv\">x<\/span><span class=\"p\">)<\/span><span class=\"o\">*<\/span><span class=\"nf\">sin<\/span><span class=\"p\">(<\/span><span class=\"nv\">n<\/span><span class=\"o\">*<\/span><span class=\"nv\">x<\/span><span class=\"p\">)<\/span>, <span class=\"nv\">x<\/span>, <span class=\"o\">-<\/span><span class=\"nv\">%pi<\/span>, <span class=\"nv\">%pi<\/span><span class=\"p\">)<\/span>;\r\n\r\n<span class=\"nf\">Fourier<\/span><span class=\"p\">(<\/span><span class=\"nv\">n<\/span>, <span class=\"nv\">x<\/span><span class=\"p\">)<\/span><span class=\"o\">:=<\/span> <span class=\"nf\">a<\/span><span class=\"p\">(<\/span><span class=\"mi\">0<\/span><span class=\"p\">)<\/span><span class=\"o\">\/<\/span><span class=\"mi\">2<\/span> <span class=\"o\">+<\/span> <span class=\"nf\">sum<\/span><span class=\"p\">(<\/span><span class=\"nf\">a<\/span><span class=\"p\">(<\/span><span class=\"nv\">i<\/span><span class=\"p\">)<\/span><span class=\"o\">*<\/span><span class=\"nf\">cos<\/span><span class=\"p\">(<\/span><span class=\"nv\">i<\/span><span class=\"o\">*<\/span><span class=\"nv\">x<\/span><span class=\"p\">)<\/span> <span class=\"o\">+<\/span> <span class=\"nf\">b<\/span><span class=\"p\">(<\/span><span class=\"nv\">i<\/span><span class=\"p\">)<\/span><span class=\"o\">*<\/span><span class=\"nf\">sin<\/span><span class=\"p\">(<\/span><span class=\"nv\">i<\/span><span class=\"o\">*<\/span><span class=\"nv\">x<\/span><span class=\"p\">)<\/span>, <span class=\"nv\">i<\/span>, <span class=\"mi\">1<\/span>, <span class=\"nv\">n<\/span><span class=\"p\">)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[9]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{15}$}a\\left(n\\right):=\\frac{1}{\\pi}\\,{\\it integrate}\\left(f_{0}\\left(x\\right)\\,\\cos \\left(n\\,x\\right) , x , -\\pi , \\pi\\right)\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[9]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{16}$}b\\left(n\\right):=\\frac{1}{\\pi}\\,{\\it integrate}\\left(f_{0}\\left(x\\right)\\,\\sin \\left(n\\,x\\right) , x , -\\pi , \\pi\\right)\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[9]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{17}$}{\\it Fourier}\\left(n , x\\right):=\\frac{a\\left(0\\right)}{2}+{\\it sum}\\left(a\\left(i\\right)\\,\\cos \\left(i\\,x\\right)+b\\left(i\\right)\\,\\sin \\left(i\\,x\\right) , i , 1 , n\\right)\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u3061\u306a\u307f\u306b\uff0c\u4e0a\u306e\u30bb\u30eb\u306e\u5b9a\u7fa9\u3067\u4f7f\u308f\u308c\u3066\u3044\u308b\u95a2\u6570 <code>integrate()<\/code> \u306f\u7a4d\u5206\u3092\u5b9f\u884c\u3057\uff0c <code>sum()<\/code> \u306f\u548c\u3092\u3068\u308a\uff0c\u8aac\u660e\u306f\u4ee5\u4e0b\u306e\u901a\u308a\u3067\u3059\u3002<\/p>\n<p>$\\displaystyle \\int_a^b f(x)\\, dx = $ <code>integrate(f(x), x, a, b);<\/code><\/p>\n<p>$\\displaystyle \\sum_{i = 1}^n a_i = $ <code>sum(a(i), i, 1, n);<\/code><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>$f(x) = x^2$ \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&#8230;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[10]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"nf\">declare<\/span><span class=\"p\">(<\/span><span class=\"nv\">n<\/span>, <span class=\"nv\">integer<\/span><span class=\"p\">)<\/span>$\r\n\r\n<span class=\"nf\">b<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"p\">)<\/span>;\r\n<span class=\"nf\">b<\/span><span class=\"p\">(<\/span><span class=\"mi\">2<\/span><span class=\"p\">)<\/span>;\r\n<span class=\"nf\">b<\/span><span class=\"p\">(<\/span><span class=\"mi\">3<\/span><span class=\"p\">)<\/span>;\r\n\r\n<span class=\"nf\">b<\/span><span class=\"p\">(<\/span><span class=\"nv\">n<\/span><span class=\"p\">)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[10]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{19}$}0\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[10]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{20}$}0\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[10]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{21}$}0\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[10]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{22}$}0\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[11]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"nf\">a<\/span><span class=\"p\">(<\/span><span class=\"mi\">0<\/span><span class=\"p\">)<\/span>;\r\n<span class=\"nf\">a<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"p\">)<\/span>;\r\n<span class=\"nf\">a<\/span><span class=\"p\">(<\/span><span class=\"mi\">2<\/span><span class=\"p\">)<\/span>;\r\n<span class=\"nf\">a<\/span><span class=\"p\">(<\/span><span class=\"mi\">3<\/span><span class=\"p\">)<\/span>;\r\n\r\n<span class=\"nf\">a<\/span><span class=\"p\">(<\/span><span class=\"nv\">n<\/span><span class=\"p\">)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[11]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{23}$}\\frac{2\\,\\pi^2}{3}\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[11]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{24}$}-4\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[11]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{25}$}1\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[11]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{26}$}-\\frac{4}{9}\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[11]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{27}$}\\frac{4\\,\\left(-1\\right)^{n}}{n^2}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>$n = 1, 2, 3, 4$ \u307e\u3067\u306e\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u5c55\u958b\u3092\u305d\u308c\u305e\u308c\uff0c<code>f1<\/code>, <code>f2<\/code>, <code>f3<\/code>, <code>f4<\/code> \u3068\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[12]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"nv\">f1<\/span><span class=\"o\">:<\/span> <span class=\"nf\">Fourier<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span>, <span class=\"nv\">x<\/span><span class=\"p\">)<\/span>;\r\n<span class=\"nv\">f2<\/span><span class=\"o\">:<\/span> <span class=\"nf\">Fourier<\/span><span class=\"p\">(<\/span><span class=\"mi\">2<\/span>, <span class=\"nv\">x<\/span><span class=\"p\">)<\/span>;\r\n<span class=\"nv\">f3<\/span><span class=\"o\">:<\/span> <span class=\"nf\">Fourier<\/span><span class=\"p\">(<\/span><span class=\"mi\">3<\/span>, <span class=\"nv\">x<\/span><span class=\"p\">)<\/span>;\r\n<span class=\"nv\">f4<\/span><span class=\"o\">:<\/span> <span class=\"nf\">Fourier<\/span><span class=\"p\">(<\/span><span class=\"mi\">4<\/span>, <span class=\"nv\">x<\/span><span class=\"p\">)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[12]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{28}$}\\frac{\\pi^2}{3}-4\\,\\cos x\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[12]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{29}$}\\cos \\left(2\\,x\\right)-4\\,\\cos x+\\frac{\\pi^2}{3}\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[12]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{30}$}-\\frac{4\\,\\cos \\left(3\\,x\\right)}{9}+\\cos \\left(2\\,x\\right)-4\\,\\cos x+\\frac{\\pi^2}{3}\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[12]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{31}$}\\frac{\\cos \\left(4\\,x\\right)}{4}-\\frac{4\\,\\cos \\left(3\\,x\\right)}{9}+\\cos \\left(2\\,x\\right)-4\\,\\cos x+\\frac{\\pi^2}{3}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\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<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[13]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"nf\">plot2d<\/span><span class=\"p\">([<\/span><span class=\"nf\">f<\/span><span class=\"p\">(<\/span><span class=\"nv\">x<\/span><span class=\"p\">)<\/span>, <span class=\"nv\">f1<\/span><span class=\"p\">]<\/span>, <span class=\"p\">[<\/span><span class=\"nv\">x<\/span>, <span class=\"o\">-<\/span>3<span class=\"o\">*<\/span><span class=\"nv\">%pi<\/span>, 3<span class=\"o\">*<\/span><span class=\"nv\">%pi<\/span><span class=\"p\">]<\/span>, <span class=\"p\">[<\/span><span class=\"nv\">legend<\/span>,<span class=\"s\">\"f(x)\"<\/span>, <span class=\"s\">\"n=1\"<\/span><span class=\"p\">]<\/span>, \r\n       <span class=\"p\">[<\/span><span class=\"nv\">y<\/span>, <span class=\"o\">-<\/span><span class=\"mi\">2<\/span>, 12<span class=\"p\">]<\/span>, \r\n       <span class=\"nv\">grid2d<\/span>, <span class=\"p\">[<\/span><span class=\"nv\">ylabel<\/span>, <span class=\"s\">\"\"<\/span><span class=\"p\">]<\/span>, \r\n       <span class=\"p\">[<\/span><span class=\"nv\">xtics<\/span>, <span class=\"nv\">%pi<\/span><span class=\"p\">]<\/span>, <span class=\"p\">[<\/span><span class=\"nv\">gnuplot_preamble<\/span>, <span class=\"s\">\"set format x '%4.1P \u03c0'\"<\/span><span class=\"p\">]<\/span>\r\n<span class=\"p\">)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_svg output_subarea \">\n<p><!--?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?--><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-large wp-image-6152\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/mmathc407-1.svg\" alt=\"\" width=\"640\" height=\"481\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>$n = 2, 3, 4, \\dots$ \u306b\u3064\u3044\u3066\u3082\uff0c\u4e0a\u306e\u3088\u3046\u306a\u30b0\u30e9\u30d5\u3092\u63cf\u3044\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[14]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"nf\">plot2d<\/span><span class=\"p\">([<\/span><span class=\"nf\">f<\/span><span class=\"p\">(<\/span><span class=\"nv\">x<\/span><span class=\"p\">)<\/span>, <span class=\"nv\">f4<\/span>, <span class=\"nv\">f3<\/span>, <span class=\"nv\">f2<\/span>, <span class=\"nv\">f1<\/span><span class=\"p\">]<\/span>, <span class=\"p\">[<\/span><span class=\"nv\">x<\/span>, <span class=\"o\">-<\/span>3<span class=\"o\">*<\/span><span class=\"nv\">%pi<\/span>, 3<span class=\"o\">*<\/span><span class=\"nv\">%pi<\/span><span class=\"p\">]<\/span>, \r\n       <span class=\"p\">[<\/span><span class=\"nv\">legend<\/span>,<span class=\"s\">\"f(x)\"<\/span>, <span class=\"s\">\"n=4\"<\/span>, <span class=\"s\">\"n=3\"<\/span>, <span class=\"s\">\"n=2\"<\/span>, <span class=\"s\">\"n=1\"<\/span><span class=\"p\">]<\/span>, \r\n       <span class=\"p\">[<\/span><span class=\"nv\">y<\/span>, <span class=\"o\">-<\/span><span class=\"mi\">2<\/span>, 12<span class=\"p\">]<\/span>, \r\n       <span class=\"nv\">grid2d<\/span>, <span class=\"p\">[<\/span><span class=\"nv\">ylabel<\/span>, <span class=\"s\">\"\"<\/span><span class=\"p\">]<\/span>, \r\n       <span class=\"p\">[<\/span><span class=\"nv\">xtics<\/span>, <span class=\"nv\">%pi<\/span><span class=\"p\">]<\/span>, <span class=\"p\">[<\/span><span class=\"nv\">gnuplot_preamble<\/span>, <span class=\"s\">\"set format x '%4.1P \u03c0'\"<\/span><span class=\"p\">]<\/span>\r\n<span class=\"p\">)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_svg output_subarea \">\n<p><!--?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?--><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-large wp-image-6153\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/mmathc408-1.svg\" alt=\"\" width=\"640\" height=\"481\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h3 id=\"\u4efb\u610f\u306e\u5468\u671f\u3092\u3082\u3064\u95a2\u6570\u306e\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u5c55\u958b\">\u4efb\u610f\u306e\u5468\u671f\u3092\u3082\u3064\u95a2\u6570\u306e\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u5c55\u958b<\/h3>\n<p>\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&#8230;<\/p>\n<p>$$ f(x) = \\frac{a_0}{2} + \\sum_{n=1}^{\\infty} \\bigl( a_n \\cos \\frac{n\\pi x}{L} + b_n \\sin \\frac{n\\pi x}{L} \\bigr) $$$$a_n = \\frac{1}{L} \\int_{-L}^{L} f(x) \\cos \\frac{n\\pi x}{L} \\, d{x} $$$$b_n = \\frac{1}{L} \\int_{-L}^{L} f(x) \\sin \\frac{n\\pi x}{L} \\, d{x} $$<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h4 id=\"\u4f8b\u984c\">\u4f8b\u984c<\/h4>\n<p>\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<p>\u4e0a\u8a18\u306e\u5f0f\u3067 $L = 1$ \u3068\u3059\u308c\u3070\u3088\u3044\u304b\u3089&#8230;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[15]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"nf\">kill<\/span><span class=\"p\">(<\/span><span class=\"nv\">x<\/span>, <span class=\"nv\">L<\/span><span class=\"p\">)<\/span>$\r\n<span class=\"nf\">xcyc<\/span><span class=\"p\">(<\/span><span class=\"nv\">x<\/span>, <span class=\"nv\">L<\/span><span class=\"p\">)<\/span><span class=\"o\">:=<\/span> <span class=\"nv\">x<\/span> <span class=\"o\">-<\/span> 2<span class=\"o\">*<\/span><span class=\"nv\">L<\/span><span class=\"o\">*<\/span><span class=\"nf\">floor<\/span><span class=\"p\">((<\/span><span class=\"nv\">x<\/span><span class=\"o\">+<\/span><span class=\"nv\">L<\/span><span class=\"p\">)<\/span><span class=\"o\">\/<\/span><span class=\"p\">(<\/span>2<span class=\"o\">*<\/span><span class=\"nv\">L<\/span><span class=\"p\">))<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[15]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{37}$}{\\it xcyc}\\left(x , L\\right):=x-2\\,L\\,\\left \\lfloor \\frac{x+L}{2\\,L} \\right \\rfloor\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>1\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<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[16]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"nf\">f0<\/span><span class=\"p\">(<\/span><span class=\"nv\">x<\/span><span class=\"p\">)<\/span><span class=\"o\">:=<\/span> <span class=\"nv\">x<\/span>;\r\n<span class=\"nf\">f<\/span><span class=\"p\">(<\/span><span class=\"nv\">x<\/span><span class=\"p\">)<\/span><span class=\"o\">:=<\/span> <span class=\"nf\">f0<\/span><span class=\"p\">(<\/span><span class=\"nf\">xcyc<\/span><span class=\"p\">(<\/span><span class=\"nv\">x<\/span>, <span class=\"mi\">1<\/span><span class=\"p\">))<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[16]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{38}$}f_{0}\\left(x\\right):=x\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[16]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{39}$}f\\left(x\\right):=f_{0}\\left({\\it xcyc}\\left(x , 1\\right)\\right)\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[17]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"nf\">plot2d<\/span><span class=\"p\">(<\/span><span class=\"nf\">f<\/span><span class=\"p\">(<\/span><span class=\"nv\">x<\/span><span class=\"p\">)<\/span>, <span class=\"p\">[<\/span><span class=\"nv\">x<\/span>, <span class=\"o\">-<\/span><span class=\"mi\">5<\/span>, 5<span class=\"p\">]<\/span>, <span class=\"p\">[<\/span><span class=\"nv\">xtics<\/span>, 1<span class=\"p\">])<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_svg output_subarea \">\n<p><!--?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?--><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-large wp-image-6154\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/mmathc409-1.svg\" alt=\"\" width=\"640\" height=\"481\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[18]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"nf\">a<\/span><span class=\"p\">(<\/span><span class=\"nv\">n<\/span><span class=\"p\">)<\/span><span class=\"o\">:=<\/span> 1<span class=\"o\">\/<\/span><span class=\"nv\">L<\/span> <span class=\"o\">*<\/span> <span class=\"nf\">integrate<\/span><span class=\"p\">(<\/span><span class=\"nf\">f0<\/span><span class=\"p\">(<\/span><span class=\"nv\">x<\/span><span class=\"p\">)<\/span><span class=\"o\">*<\/span><span class=\"nf\">cos<\/span><span class=\"p\">(<\/span><span class=\"nv\">n<\/span><span class=\"o\">*<\/span><span class=\"nv\">%pi<\/span><span class=\"o\">*<\/span><span class=\"nv\">x<\/span><span class=\"o\">\/<\/span><span class=\"nv\">L<\/span><span class=\"p\">)<\/span>, <span class=\"nv\">x<\/span>, <span class=\"o\">-<\/span><span class=\"nv\">L<\/span>, <span class=\"nv\">L<\/span><span class=\"p\">)<\/span>;\r\n<span class=\"nf\">b<\/span><span class=\"p\">(<\/span><span class=\"nv\">n<\/span><span class=\"p\">)<\/span><span class=\"o\">:=<\/span> 1<span class=\"o\">\/<\/span><span class=\"nv\">L<\/span> <span class=\"o\">*<\/span> <span class=\"nf\">integrate<\/span><span class=\"p\">(<\/span><span class=\"nf\">f0<\/span><span class=\"p\">(<\/span><span class=\"nv\">x<\/span><span class=\"p\">)<\/span><span class=\"o\">*<\/span><span class=\"nf\">sin<\/span><span class=\"p\">(<\/span><span class=\"nv\">n<\/span><span class=\"o\">*<\/span><span class=\"nv\">%pi<\/span><span class=\"o\">*<\/span><span class=\"nv\">x<\/span><span class=\"o\">\/<\/span><span class=\"nv\">L<\/span><span class=\"p\">)<\/span>, <span class=\"nv\">x<\/span>, <span class=\"o\">-<\/span><span class=\"nv\">L<\/span>, <span class=\"nv\">L<\/span><span class=\"p\">)<\/span>;\r\n\r\n<span class=\"nf\">Fourier<\/span><span class=\"p\">(<\/span><span class=\"nv\">n<\/span>, <span class=\"nv\">x<\/span><span class=\"p\">)<\/span><span class=\"o\">:=<\/span> <span class=\"nf\">a<\/span><span class=\"p\">(<\/span><span class=\"mi\">0<\/span><span class=\"p\">)<\/span><span class=\"o\">\/<\/span><span class=\"mi\">2<\/span> <span class=\"o\">+<\/span> <span class=\"nf\">sum<\/span><span class=\"p\">(<\/span><span class=\"nf\">a<\/span><span class=\"p\">(<\/span><span class=\"nv\">i<\/span><span class=\"p\">)<\/span><span class=\"o\">*<\/span><span class=\"nf\">cos<\/span><span class=\"p\">(<\/span><span class=\"nv\">i<\/span><span class=\"o\">*<\/span><span class=\"nv\">%pi<\/span><span class=\"o\">*<\/span><span class=\"nv\">x<\/span><span class=\"o\">\/<\/span><span class=\"nv\">L<\/span><span class=\"p\">)<\/span> <span class=\"o\">+<\/span> \r\n                             <span class=\"nf\">b<\/span><span class=\"p\">(<\/span><span class=\"nv\">i<\/span><span class=\"p\">)<\/span><span class=\"o\">*<\/span><span class=\"nf\">sin<\/span><span class=\"p\">(<\/span><span class=\"nv\">i<\/span><span class=\"o\">*<\/span><span class=\"nv\">%pi<\/span><span class=\"o\">*<\/span><span class=\"nv\">x<\/span><span class=\"o\">\/<\/span><span class=\"nv\">L<\/span><span class=\"p\">)<\/span>, <span class=\"nv\">i<\/span>, <span class=\"mi\">1<\/span>, <span class=\"nv\">n<\/span><span class=\"p\">)<\/span>;\r\n<span class=\"nv\">L<\/span><span class=\"o\">:<\/span> 1$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[18]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{42}$}a\\left(n\\right):=\\frac{1}{L}\\,{\\it integrate}\\left(f_{0}\\left(x\\right)\\,\\cos \\left(\\frac{n\\,\\pi\\,x}{L}\\right) , x , -L , L\\right)\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[18]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{43}$}b\\left(n\\right):=\\frac{1}{L}\\,{\\it integrate}\\left(f_{0}\\left(x\\right)\\,\\sin \\left(\\frac{n\\,\\pi\\,x}{L}\\right) , x , -L , L\\right)\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[18]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{44}$}{\\it Fourier}\\left(n , x\\right):=\\frac{a\\left(0\\right)}{2}+{\\it sum}\\left(a\\left(i\\right)\\,\\cos \\left(\\frac{i\\,\\pi\\,x}{L}\\right)+b\\left(i\\right)\\,\\sin \\left(\\frac{i\\,\\pi\\,x}{L}\\right) , i , 1 , n\\right)\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>$f_0(x) = x$ \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&#8230;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[19]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"nf\">a<\/span><span class=\"p\">(<\/span><span class=\"mi\">0<\/span><span class=\"p\">)<\/span>;\r\n<span class=\"nf\">a<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"p\">)<\/span>;\r\n<span class=\"nf\">a<\/span><span class=\"p\">(<\/span><span class=\"mi\">2<\/span><span class=\"p\">)<\/span>;\r\n<span class=\"nf\">a<\/span><span class=\"p\">(<\/span><span class=\"mi\">3<\/span><span class=\"p\">)<\/span>;\r\n\r\n<span class=\"nf\">a<\/span><span class=\"p\">(<\/span><span class=\"nv\">n<\/span><span class=\"p\">)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[19]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{46}$}0\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[19]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{47}$}0\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[19]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{48}$}0\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[19]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{49}$}0\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[19]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{50}$}0\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[20]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"nf\">b<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"p\">)<\/span>;\r\n<span class=\"nf\">b<\/span><span class=\"p\">(<\/span><span class=\"mi\">2<\/span><span class=\"p\">)<\/span>;\r\n<span class=\"nf\">b<\/span><span class=\"p\">(<\/span><span class=\"mi\">3<\/span><span class=\"p\">)<\/span>;\r\n\r\n<span class=\"nf\">b<\/span><span class=\"p\">(<\/span><span class=\"nv\">n<\/span><span class=\"p\">)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[20]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{51}$}\\frac{2}{\\pi}\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[20]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{52}$}-\\frac{1}{\\pi}\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[20]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{53}$}\\frac{2}{3\\,\\pi}\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[20]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{54}$}-\\frac{2\\,\\left(-1\\right)^{n}}{\\pi\\,n}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[21]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"nv\">f1<\/span><span class=\"o\">:<\/span> <span class=\"nf\">Fourier<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span>, <span class=\"nv\">x<\/span><span class=\"p\">)<\/span>;\r\n<span class=\"nv\">f5<\/span><span class=\"o\">:<\/span> <span class=\"nf\">Fourier<\/span><span class=\"p\">(<\/span><span class=\"mi\">5<\/span>, <span class=\"nv\">x<\/span><span class=\"p\">)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[21]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{55}$}\\frac{2\\,\\sin \\left(\\pi\\,x\\right)}{\\pi}\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[21]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{56}$}\\frac{2\\,\\sin \\left(5\\,\\pi\\,x\\right)}{5\\,\\pi}-\\frac{\\sin \\left(4\\,\\pi\\,x\\right)}{2\\,\\pi}+\\frac{2\\,\\sin \\left(3\\,\\pi\\,x\\right)}{3\\,\\pi}-\\frac{\\sin \\left(2\\,\\pi\\,x\\right)}{\\pi}+\\frac{2\\,\\sin \\left(\\pi\\,x\\right)}{\\pi}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[22]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"nf\">plot2d<\/span><span class=\"p\">([<\/span><span class=\"nf\">f<\/span><span class=\"p\">(<\/span><span class=\"nv\">x<\/span><span class=\"p\">)<\/span>, <span class=\"nv\">f1<\/span><span class=\"p\">]<\/span>, <span class=\"p\">[<\/span><span class=\"nv\">x<\/span>, <span class=\"o\">-<\/span><span class=\"mi\">5<\/span>, 5<span class=\"p\">]<\/span>, <span class=\"p\">[<\/span><span class=\"nv\">legend<\/span>, <span class=\"s\">\"f(x)\"<\/span>, <span class=\"s\">\"n = 1\"<\/span><span class=\"p\">]<\/span>, \r\n       <span class=\"p\">[<\/span><span class=\"nv\">y<\/span>, <span class=\"o\">-<\/span><span class=\"mf\">1.3<\/span>, 1<span class=\"o\">.<\/span>3<span class=\"p\">]<\/span>, <span class=\"nv\">grid2d<\/span>\r\n<span class=\"p\">)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_svg output_subarea \">\n<p><!--?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?--><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-large wp-image-6155\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/mmathc410-1.svg\" alt=\"\" width=\"640\" height=\"481\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>$n = 2, 3, \\dots$ \u306e\u5834\u5408\u306b\u3064\u3044\u3066\u3082\u30b0\u30e9\u30d5\u3092\u63cf\u3044\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[23]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"nf\">plot2d<\/span><span class=\"p\">([<\/span><span class=\"nf\">f<\/span><span class=\"p\">(<\/span><span class=\"nv\">x<\/span><span class=\"p\">)<\/span>, <span class=\"nf\">Fourier<\/span><span class=\"p\">(<\/span><span class=\"mi\">3<\/span>, <span class=\"nv\">x<\/span><span class=\"p\">)<\/span>, <span class=\"nf\">Fourier<\/span><span class=\"p\">(<\/span><span class=\"mi\">2<\/span>,<span class=\"nv\">x<\/span><span class=\"p\">)<\/span>, <span class=\"nf\">Fourier<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span>, <span class=\"nv\">x<\/span><span class=\"p\">)]<\/span>, \r\n       <span class=\"p\">[<\/span><span class=\"nv\">x<\/span>, <span class=\"o\">-<\/span><span class=\"mi\">5<\/span>, 5<span class=\"p\">]<\/span>, \r\n       <span class=\"p\">[<\/span><span class=\"nv\">legend<\/span>, <span class=\"s\">\"f(x)\"<\/span>, <span class=\"s\">\"n = 3\"<\/span>, <span class=\"s\">\"n = 2\"<\/span>, <span class=\"s\">\"n = 1\"<\/span><span class=\"p\">]<\/span>, \r\n       <span class=\"p\">[<\/span><span class=\"nv\">y<\/span>, <span class=\"o\">-<\/span><span class=\"mf\">1.3<\/span>, 1<span class=\"o\">.<\/span>8<span class=\"p\">]<\/span>, <span class=\"nv\">grid2d<\/span>\r\n<span class=\"p\">)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_svg output_subarea \">\n<p><!--?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?--><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-large wp-image-6156\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/mmathc411.svg\" alt=\"\" width=\"640\" height=\"481\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h3 id=\"\u30d5\u30fc\u30ea\u30a8\u7a4d\u5206\u30fb\u30d5\u30fc\u30ea\u30a8\u5909\u63db\">\u30d5\u30fc\u30ea\u30a8\u7a4d\u5206\u30fb\u30d5\u30fc\u30ea\u30a8\u5909\u63db<\/h3>\n<p>\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<p>\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<p>$$ f(x) = \\frac{1}{2\\pi} \\int_{-\\infty}^{\\infty}\\ F(k)\\ e^{i k x}\\ dk, \\quad F(k) \\equiv \\int_{-\\infty}^{\\infty} f(x) \\ e^{-i k x} \\ dx $$<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h4 id=\"\u4f8b\u984c\">\u4f8b\u984c<\/h4>\n<p>\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<p>$$ f(x) = \\begin{cases}<br \/>\n\\frac{1}{a} &amp; (|x| \\leq \\frac{a}{2}) \\\\<br \/>\n0 &amp; (|x|&gt; \\frac{a}{2})<br \/>\n\\end{cases}$$<\/p>\n<p>\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<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\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\u306e\u4f8b\u3067\u3059\u3002\u5ff5\u306e\u305f\u3081 <code>block()<\/code> \u3067\u56f2\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[24]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"nf\">kill<\/span><span class=\"p\">(<\/span><span class=\"nv\">a<\/span><span class=\"p\">)<\/span>$\r\n<span class=\"nf\">f<\/span><span class=\"p\">(<\/span><span class=\"nv\">x<\/span>, <span class=\"nv\">a<\/span><span class=\"p\">)<\/span><span class=\"o\">:=<\/span> <span class=\"nf\">block<\/span><span class=\"p\">(<\/span><span class=\"k\">if<\/span> <span class=\"nf\">abs<\/span><span class=\"p\">(<\/span><span class=\"nv\">x<\/span><span class=\"p\">)<\/span> <span class=\"o\">&lt;=<\/span> <span class=\"nv\">a<\/span><span class=\"o\">\/<\/span><span class=\"mi\">2<\/span> <span class=\"k\">then<\/span> 1<span class=\"o\">\/<\/span><span class=\"nv\">a<\/span> <span class=\"k\">else<\/span> <span class=\"mi\">0<\/span><span class=\"p\">)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[24]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{62}$}f\\left(x , a\\right):=\\mathbf{block}\\;\\left(\\mathbf{if}\\;\\left| x\\right| \\leq \\frac{a}{2}\\;\\mathbf{then}\\;\\frac{1}{a}\\;\\mathbf{else}\\;0\\right)\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[25]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"cm\">\/* \u30b0\u30e9\u30d5\u3092\u63cf\u304f\u305f\u3081\u306b a = 2 \u3068\u3057\u3066\u307f\u308b\u3002*\/<\/span>\r\n<span class=\"nf\">plot2d<\/span><span class=\"p\">(<\/span><span class=\"nf\">f<\/span><span class=\"p\">(<\/span><span class=\"nv\">x<\/span>, <span class=\"mi\">2<\/span><span class=\"p\">)<\/span>, <span class=\"p\">[<\/span><span class=\"nv\">x<\/span>, <span class=\"o\">-<\/span><span class=\"mi\">5<\/span>, 5<span class=\"p\">]<\/span>, <span class=\"p\">[<\/span><span class=\"nv\">y<\/span>, <span class=\"o\">-<\/span><span class=\"mf\">0.5<\/span>, 1<span class=\"p\">])<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_svg output_subarea \">\n<p><!--?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?--><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-large wp-image-6157\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/mmathc412.svg\" alt=\"\" width=\"640\" height=\"481\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>$$ F(k) = \\int_{-\\infty}^{\\infty} f(x) \\ e^{-i k x} \\ dx = \\int_{-\\frac{a}{2}}^{\\frac{a}{2}} \\frac{1}{a}\\ e^{-i k x} \\ dx$$\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<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[26]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"nv\">F<\/span><span class=\"o\">:<\/span> <span class=\"nf\">integrate<\/span><span class=\"p\">(<\/span>1<span class=\"o\">\/<\/span><span class=\"nv\">a<\/span> <span class=\"o\">*<\/span> <span class=\"nf\">exp<\/span><span class=\"p\">(<\/span><span class=\"o\">-<\/span><span class=\"nv\">%i<\/span><span class=\"o\">*<\/span><span class=\"nv\">k<\/span><span class=\"o\">*<\/span><span class=\"nv\">x<\/span><span class=\"p\">)<\/span>, <span class=\"nv\">x<\/span>, <span class=\"o\">-<\/span><span class=\"nv\">a<\/span><span class=\"o\">\/<\/span><span class=\"mi\">2<\/span>, <span class=\"nv\">a<\/span><span class=\"o\">\/<\/span><span class=\"mi\">2<\/span><span class=\"p\">)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[26]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{65}$}\\frac{\\frac{i\\,e^ {- \\frac{i\\,a\\,k}{2} }}{k}-\\frac{i\\,e^{\\frac{i\\,a\\,k}{2}}}{k}}{a}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u53c2\u8003\uff1a <code>demoivre()<\/code> \u95a2\u6570\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[27]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"nf\">exp<\/span><span class=\"p\">(<\/span><span class=\"nv\">%i<\/span> <span class=\"o\">*<\/span> <span class=\"nv\">x<\/span><span class=\"p\">)<\/span> <span class=\"o\">=<\/span> <span class=\"nf\">demoivre<\/span><span class=\"p\">(<\/span><span class=\"nf\">exp<\/span><span class=\"p\">(<\/span><span class=\"nv\">%i<\/span> <span class=\"o\">*<\/span> <span class=\"nv\">x<\/span><span class=\"p\">))<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[27]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{66}$}e^{i\\,x}=i\\,\\sin x+\\cos x\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[28]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"nf\">demoivre<\/span><span class=\"p\">(<\/span><span class=\"nv\">F<\/span><span class=\"p\">)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[28]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{67}$}\\frac{\\frac{i\\,\\left(\\cos \\left(\\frac{a\\,k}{2}\\right)-i\\,\\sin \\left(\\frac{a\\,k}{2}\\right)\\right)}{k}-\\frac{i\\,\\left(i\\,\\sin \\left(\\frac{a\\,k}{2}\\right)+\\cos \\left(\\frac{a\\,k}{2}\\right)\\right)}{k}}{a}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[29]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"nv\">ans<\/span><span class=\"o\">:<\/span> <span class=\"nf\">expand<\/span><span class=\"p\">(<\/span><span class=\"nv\">%<\/span><span class=\"p\">)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[29]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{68}$}\\frac{2\\,\\sin \\left(\\frac{a\\,k}{2}\\right)}{a\\,k}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[30]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"o\">'<\/span><span class=\"nf\">limit<\/span><span class=\"p\">(<\/span><span class=\"nv\">ans<\/span>, <span class=\"nv\">a<\/span>, <span class=\"mi\">0<\/span><span class=\"p\">)<\/span> <span class=\"o\">=<\/span> <span class=\"nf\">limit<\/span><span class=\"p\">(<\/span><span class=\"nv\">ans<\/span>, <span class=\"nv\">a<\/span>, <span class=\"mi\">0<\/span><span class=\"p\">)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[30]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{69}$}\\frac{2\\,\\left(\\lim_{a\\rightarrow 0}{\\frac{\\sin \\left(\\frac{a\\,k}{2}\\right)}{a}}\\right)}{k}=1\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u7b97\u6570\u306e\u554f\u984c\u3068\u3057\u3066\uff0c<\/p>\n<p>$$ \\lim_{a \\rightarrow 0} F(k) = \\lim_{a \\rightarrow 0} \\frac{\\sin\\frac{ak}{2}}{\\frac{ak}{2}} = 1$$<\/p>\n<p>\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<p>\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<p>$\\displaystyle \\int_{-\\infty}^{\\infty} f(x)\\, dx =\\int_{-\\frac{a}{2}}^{\\frac{a}{2}} \\frac{1}{a} \\,dx = 1$ \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<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[31]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"nf\">plot2d<\/span><span class=\"p\">([<\/span><span class=\"nf\">f<\/span><span class=\"p\">(<\/span><span class=\"nv\">x<\/span>, <span class=\"mf\">0.1<\/span><span class=\"p\">)<\/span>, <span class=\"nf\">f<\/span><span class=\"p\">(<\/span><span class=\"nv\">x<\/span>, <span class=\"mf\">0.2<\/span><span class=\"p\">)<\/span>, <span class=\"nf\">f<\/span><span class=\"p\">(<\/span><span class=\"nv\">x<\/span>, <span class=\"mf\">0.5<\/span><span class=\"p\">)<\/span>, <span class=\"nf\">f<\/span><span class=\"p\">(<\/span><span class=\"nv\">x<\/span>, <span class=\"mi\">2<\/span><span class=\"p\">)]<\/span>, \r\n       <span class=\"p\">[<\/span><span class=\"nv\">x<\/span>, <span class=\"o\">-<\/span><span class=\"mi\">5<\/span>, 5<span class=\"p\">]<\/span>, <span class=\"p\">[<\/span><span class=\"nv\">y<\/span>, <span class=\"o\">-<\/span><span class=\"mf\">0.5<\/span>, 10<span class=\"p\">]<\/span>, \r\n       <span class=\"p\">[<\/span><span class=\"nv\">legend<\/span>, <span class=\"s\">\"a=0.1\"<\/span>, <span class=\"s\">\"a=0.2\"<\/span>, <span class=\"s\">\"a=0.5\"<\/span>, <span class=\"s\">\"a=2\"<\/span><span class=\"p\">]<\/span>\r\n<span class=\"p\">)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_svg output_subarea \">\n<p><!--?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?--><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-large wp-image-6158\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/mmathc413.svg\" alt=\"\" width=\"640\" height=\"481\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\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<p>$$\\delta(x) \\equiv \\frac{1}{2\\pi} \\int_{-\\infty}^{\\infty} e^{i k x}\\ dk$$<\/p>\n<p>\u30c7\u30eb\u30bf\u95a2\u6570\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u6027\u8cea\u3092\u6301\u3061\u307e\u3059\u3002<\/p>\n<p>$$\\delta(x) = 0, \\ \\mbox{if $x \\neq 0$}$$$$\\int_{-\\infty}^{\\infty} g(x) \\delta(x &#8211; c)\\, dx = g(a)$$<\/p>\n<p>\u7279\u306b\uff0c<\/p>\n<p>$$\\int_{-\\infty}^{\\infty} \\delta(x)\\, dx = 1$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h3 id=\"\u53c2\u8003\uff1afourie-\u30d1\u30c3\u30b1\u30fc\u30b8\u3092\u4f7f\u3046\u5834\u5408\">\u53c2\u8003\uff1afourie \u30d1\u30c3\u30b1\u30fc\u30b8\u3092\u4f7f\u3046\u5834\u5408<\/h3>\n<p>Maxima \u306e <code>fourie<\/code> \u30d1\u30c3\u30b1\u30fc\u30b8\u3092\u4f7f\u3063\u3066\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u5c55\u958b\u3092\u884c\u3046\u306b\u306f\uff0c\u6700\u521d\u306b\u4e00\u5ea6\uff0c<code>load()<\/code> \u3057\u307e\u3059\u3002<\/p>\n<ul>\n<li>\u53c2\u8003\uff1a<a href=\"https:\/\/maxima.osdn.jp\/maxima_28.html#SEC161\">28.4 Introduction to Fourier series<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[32]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"nf\">load<\/span><span class=\"p\">(<\/span><span class=\"s\">\"fourie\"<\/span><span class=\"p\">)<\/span>$ <span class=\"cm\">\/* \u30d1\u30c3\u30b1\u30fc\u30b8\u540d\u6ce8\u610f\uff01 fourier \u3067\u306f\u306a\u3044\u3002*\/<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p><code>fourie<\/code> \u30d1\u30c3\u30b1\u30fc\u30b8\u3067\u306f $\\cos$ \u306e\u4fc2\u6570\u306e $a_0$ \u306e\u5b9a\u7fa9\u304c\u4ee5\u4e0b\u306e\u3088\u3046\u306b $1\/2$ \u3060\u3051\u7570\u306a\u308b\u3053\u3068\u306b\u6ce8\u610f\u3002<\/p>\n<p>$$ f(x) = a_0 + \\sum_{n=1}^{\\infty} \\bigl( a_n \\cos n x + b_n \\sin nx \\bigr) $$<\/p>\n<pre><code>-- Function: fourier (&lt;f&gt;, &lt;x&gt;, &lt;p&gt;)\r\n\r\n     Returns a list of the Fourier coefficients of '&lt;f&gt;(&lt;x&gt;)' defined on\r\n     the interval '[-p, p]'.<\/code><\/pre>\n<hr \/>\n<pre><code>-- Function: totalfourier (&lt;f&gt;, &lt;x&gt;, &lt;p&gt;)\r\n\r\n     Returns 'fourexpand (foursimp (fourier (&lt;f&gt;, &lt;x&gt;, &lt;p&gt;)), &lt;x&gt;, &lt;p&gt;,\r\n     'inf)'.<\/code><\/pre>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h4 id=\"\u4f8b\u984c\">\u4f8b\u984c<\/h4>\n<p>\u307e\u305a\uff0c1\u5468\u671f\u5206\u306e\u95a2\u6570 $f_0(x)$ \u3092\u5b9a\u7fa9\u3059\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[33]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"nf\">f0<\/span><span class=\"p\">(<\/span><span class=\"nv\">x<\/span><span class=\"p\">)<\/span><span class=\"o\">:=<\/span> <span class=\"nv\">x<\/span><span class=\"o\">**<\/span><span class=\"mi\">2<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[33]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{73}$}f_{0}\\left(x\\right):=x^2\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>$[-\\pi: \\pi]$ \u3067\u5b9a\u7fa9\u3055\u308c\u305f$f_0(x)$ \u304c\uff0c\u533a\u9593\u5916\u3067\u306f\u5468\u671f $2\\pi$ \u306e\u5468\u671f\u95a2\u6570\u3068\u3057\u305f\u3068\u304d\u306e\u30d5\u30fc\u30ea\u30a8\u4fc2\u6570\u3092\u6c42\u3081\u308b\u306b\u306f\uff0c\u95a2\u6570 <code>fourier(f0(x), x, %pi)<\/code> \u3092\u4f7f\u3046\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[34]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"cm\">\/* \u3059\u3067\u306b\u5ba3\u8a00\u6e08\u307f\u3060\u304c\uff0c\u3042\u3089\u305f\u3081\u3066 *\/<\/span>\r\n<span class=\"nf\">declare<\/span><span class=\"p\">(<\/span><span class=\"nv\">n<\/span>, <span class=\"nv\">integer<\/span><span class=\"p\">)<\/span>$\r\n\r\n<span class=\"cm\">\/* \u30d5\u30fc\u30ea\u30a8\u4fc2\u6570\u3092\u6c42\u3081\u308b *\/<\/span>\r\n<span class=\"nv\">ans1<\/span><span class=\"o\">:<\/span> <span class=\"nf\">fourier<\/span><span class=\"p\">(<\/span><span class=\"nf\">f0<\/span><span class=\"p\">(<\/span><span class=\"nv\">x<\/span><span class=\"p\">)<\/span>, <span class=\"nv\">x<\/span>, <span class=\"nv\">%pi<\/span><span class=\"p\">)<\/span>$\r\n\r\n<span class=\"cm\">\/* \u221e\u307e\u3067\u306e\u5168\u5c55\u958b\u3092\u8868\u793a\u3002fourier \u3068\u306e\u91cd\u8907\u8868\u793a\u3082\u3002 *\/<\/span>\r\n<span class=\"nf\">totalfourier<\/span><span class=\"p\">(<\/span><span class=\"nf\">f0<\/span><span class=\"p\">(<\/span><span class=\"nv\">x<\/span><span class=\"p\">)<\/span>, <span class=\"nv\">x<\/span>, <span class=\"nv\">%pi<\/span><span class=\"p\">)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_latex output_subarea \">\\[\\tag{${\\it \\%t}_{75}$}a_{0}=\\frac{\\pi^2}{3}\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre><\/pre>\n<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_latex output_subarea \">\\[\\tag{${\\it \\%t}_{76}$}a_{n}=\\frac{4\\,\\left(-1\\right)^{n}}{n^2}\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre><\/pre>\n<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_latex output_subarea \">\\[\\tag{${\\it \\%t}_{77}$}b_{n}=0\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre><\/pre>\n<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_latex output_subarea \">\\[\\tag{${\\it \\%t}_{78}$}a_{0}=\\frac{\\pi^2}{3}\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre><\/pre>\n<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_latex output_subarea \">\\[\\tag{${\\it \\%t}_{79}$}a_{n}=\\frac{4\\,\\left(-1\\right)^{n}}{n^2}\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre><\/pre>\n<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_latex output_subarea \">\\[\\tag{${\\it \\%t}_{80}$}b_{n}=0\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre><\/pre>\n<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_latex output_subarea \">\\[\\tag{${\\it \\%t}_{81}$}a_{0}=\\frac{\\pi^2}{3}\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre><\/pre>\n<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_latex output_subarea \">\\[\\tag{${\\it \\%t}_{82}$}a_{n}=\\frac{4\\,\\left(-1\\right)^{n}}{n^2}\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre><\/pre>\n<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_latex output_subarea \">\\[\\tag{${\\it \\%t}_{83}$}b_{n}=0\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre><\/pre>\n<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[34]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{83}$}4\\,\\sum_{n=1}^{\\infty }{\\frac{\\left(-1\\right)^{n}\\,\\cos \\left(n\\,x\\right)}{n^2}}+\\frac{\\pi^2}{3}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u6c42\u3081\u305f\u30d5\u30fc\u30ea\u30a8\u4fc2\u6570 <code>ans1<\/code> \u3092\u4f7f\u3063\u3066\uff0c$n = 1, 2, 3, 4$ \u307e\u3067\u306e\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u5c55\u958b\u3092\u305d\u308c\u305e\u308c\uff0c<code>F1<\/code>, <code>F2<\/code>, <code>F3<\/code>, <code>F4<\/code> \u3068\u3057\u3066\u8868\u793a\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[35]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"nv\">F1<\/span><span class=\"o\">:<\/span> <span class=\"nf\">fourexpand<\/span><span class=\"p\">(<\/span><span class=\"nv\">ans1<\/span>, <span class=\"nv\">x<\/span>, <span class=\"nv\">%pi<\/span>, <span class=\"mi\">1<\/span><span class=\"p\">)<\/span>;\r\n<span class=\"nv\">F2<\/span><span class=\"o\">:<\/span> <span class=\"nf\">fourexpand<\/span><span class=\"p\">(<\/span><span class=\"nv\">ans1<\/span>, <span class=\"nv\">x<\/span>, <span class=\"nv\">%pi<\/span>, <span class=\"mi\">2<\/span><span class=\"p\">)<\/span>;\r\n<span class=\"nv\">F3<\/span><span class=\"o\">:<\/span> <span class=\"nf\">fourexpand<\/span><span class=\"p\">(<\/span><span class=\"nv\">ans1<\/span>, <span class=\"nv\">x<\/span>, <span class=\"nv\">%pi<\/span>, <span class=\"mi\">3<\/span><span class=\"p\">)<\/span>;\r\n<span class=\"nv\">F4<\/span><span class=\"o\">:<\/span> <span class=\"nf\">fourexpand<\/span><span class=\"p\">(<\/span><span class=\"nv\">ans1<\/span>, <span class=\"nv\">x<\/span>, <span class=\"nv\">%pi<\/span>, <span class=\"mi\">4<\/span><span class=\"p\">)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[35]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{84}$}\\frac{\\pi^2}{3}-4\\,\\cos x\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[35]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{85}$}\\cos \\left(2\\,x\\right)-4\\,\\cos x+\\frac{\\pi^2}{3}\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[35]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{86}$}-\\frac{4\\,\\cos \\left(3\\,x\\right)}{9}+\\cos \\left(2\\,x\\right)-4\\,\\cos x+\\frac{\\pi^2}{3}\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[35]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{87}$}\\frac{\\cos \\left(4\\,x\\right)}{4}-\\frac{4\\,\\cos \\left(3\\,x\\right)}{9}+\\cos \\left(2\\,x\\right)-4\\,\\cos x+\\frac{\\pi^2}{3}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h4 id=\"\u4f8b\u984c\">\u4f8b\u984c<\/h4>\n<p>\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<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[36]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"nf\">f0<\/span><span class=\"p\">(<\/span><span class=\"nv\">x<\/span><span class=\"p\">)<\/span><span class=\"o\">:=<\/span> <span class=\"nv\">x<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[36]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{88}$}f_{0}\\left(x\\right):=x\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[37]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"cm\">\/* \u30d5\u30fc\u30ea\u30a8\u4fc2\u6570\u3092\u6c42\u3081\u308b *\/<\/span>\r\n\r\n<span class=\"nv\">ans1<\/span><span class=\"o\">:<\/span> <span class=\"nf\">fourier<\/span><span class=\"p\">(<\/span><span class=\"nf\">f0<\/span><span class=\"p\">(<\/span><span class=\"nv\">x<\/span><span class=\"p\">)<\/span>, <span class=\"nv\">x<\/span>, <span class=\"mi\">1<\/span><span class=\"p\">)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_latex output_subarea \">\\[\\tag{${\\it \\%t}_{89}$}a_{0}=0\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre><\/pre>\n<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_latex output_subarea \">\\[\\tag{${\\it \\%t}_{90}$}a_{n}=0\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre><\/pre>\n<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_latex output_subarea \">\\[\\tag{${\\it \\%t}_{91}$}b_{n}=-\\frac{2\\,\\left(-1\\right)^{n}}{\\pi\\,n}\\]<\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre><\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[38]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-maxima\">\n<pre><span class=\"cm\">\/* \u6c42\u3081\u305f\u30d5\u30fc\u30ea\u30a8\u4fc2\u6570\u3092\u4f7f\u3063\u3066 n = 5 \u307e\u3067\u8868\u793a *\/<\/span>\r\n\r\n<span class=\"nv\">F5<\/span><span class=\"o\">:<\/span> <span class=\"nf\">fourexpand<\/span><span class=\"p\">(<\/span><span class=\"nv\">ans1<\/span>, <span class=\"nv\">x<\/span>, <span class=\"mi\">1<\/span>, <span class=\"mi\">5<\/span><span class=\"p\">)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[38]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">\\[\\tag{${\\it \\%o}_{92}$}\\frac{2\\,\\sin \\left(5\\,\\pi\\,x\\right)}{5\\,\\pi}-\\frac{\\sin \\left(4\\,\\pi\\,x\\right)}{2\\,\\pi}+\\frac{2\\,\\sin \\left(3\\,\\pi\\,x\\right)}{3\\,\\pi}-\\frac{\\sin \\left(2\\,\\pi\\,x\\right)}{\\pi}+\\frac{2\\,\\sin \\left(\\pi\\,x\\right)}{\\pi}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":6118,"menu_order":40,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-2378","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\/2378","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=2378"}],"version-history":[{"count":9,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2378\/revisions"}],"predecessor-version":[{"id":6510,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2378\/revisions\/6510"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6118"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=2378"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}