{"id":2368,"date":"2022-02-26T10:53:04","date_gmt":"2022-02-26T01:53:04","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2368"},"modified":"2025-07-24T17:41:54","modified_gmt":"2025-07-24T08:41:54","slug":"%e8%a4%87%e7%b4%a0%e3%83%95%e3%83%bc%e3%83%aa%e3%82%a8%e7%b4%9a%e6%95%b0","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6c\/%e3%83%95%e3%83%bc%e3%83%aa%e3%82%a8%e8%a7%a3%e6%9e%90\/%e8%a4%87%e7%b4%a0%e3%83%95%e3%83%bc%e3%83%aa%e3%82%a8%e7%b4%9a%e6%95%b0\/","title":{"rendered":"\u8907\u7d20\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570"},"content":{"rendered":"

\\(\\cos \\) \u3068\\(\\sin \\) \u306e\u5225\u3005\u306e\u91cd\u306d\u5408\u308f\u305b\uff08\u8db3\u3057\u5408\u308f\u305b\uff09\u3067\u8868\u3055\u308c\u308b\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u3092\uff0c\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f\u3092\u4f7f\u3063\u30661\u3064\u306b\u307e\u3068\u3081\u308b\u3002
\n<\/p>\n

\n

\u5148\u306b\uff0c\u8907\u7d20\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u306e\u307e\u3068\u3081<\/h3>\n

 <\/p>\n

\u5148\u306b\u7b54\u3048\u3092\u66f8\u3044\u3066\u304a\u304f\u3002<\/p>\n

\u533a\u9593 \\( -L \\le x \\le L\\) \u3067\u5b9a\u7fa9\u3055\u308c\u305f\u95a2\u6570 \\(f(x)\\) \u304c\u533a\u9593\u5916\u3067\u306f\u5468\u671f \\(2 L\\) \u306e\u5468\u671f\u95a2\u6570\u3067\u3042\u308b\u5834\u5408\uff0c\u305d\u306e\u8907\u7d20\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570<\/b><\/span>\u306f
\n$$\\color{red}{f(x) = \\sum_{n = -\\infty}^{\\infty} c_n \\,\\exp\\left({i \\frac{n\\pi x}{L}}\\right)}$$ \u3067\u3042\u308a\uff0c\u305d\u306e\u8907\u7d20\u30d5\u30fc\u30ea\u30a8\u4fc2\u6570<\/b><\/span>\u306f
\n\\begin{eqnarray}\\color{red}
\nc_n &\\color{red}=&\\color{red}\u00a0 \\frac{1}{2L} \\int_{-L}^{L} f(x)\\,\\biggl(\\exp\\left(i\\frac{n\\pi x}{L}\\right)\\biggr)^* \\, dx \\\\
\n&\\color{red}=&\\color{red}\u00a0 \\frac{1}{2L} \\int_{-L}^{L} f(x)\\,\\exp\\left(-i\\frac{n\\pi x}{L}\\right) \\, dx
\n\\end{eqnarray}<\/p>\n

\u306e\u3088\u3046\u306b\u3057\u3066\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u3053\u3053\u3067 ${}^*$ \u306f\u8907\u7d20\u5171\u5f79\u3092\u3042\u3089\u308f\u3059\u3002<\/p>\n

\u4ee5\u4e0b\u306f\uff0c\u3053\u3046\u306a\u308b\u7406\u7531\u3002<\/p>\n

 <\/p>\n

\u5468\u671f \\( 2L\\) \u306e\u5468\u671f\u95a2\u6570<\/h3>\n

 <\/p>\n

\u533a\u9593 \\( -L \\le x \\le L\\) \u3067\u5b9a\u7fa9\u3055\u308c\u305f\u95a2\u6570 \\(f(x)\\) \u304c\u533a\u9593\u5916\u3067\u306f\u5468\u671f \\(2 L\\) \u306e\u5468\u671f\u95a2\u6570\u3067\u3042\u308b\u5834\u5408\uff0c\u305d\u306e\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u5c55\u958b\u306f
\n$$ f(x) = \\frac{a_0}{2} + \\sum_{n=1}^{\\infty} \\biggl( a_n \\cos \\left(\\frac{n\\pi x}{L}\\right)\u00a0 + b_n \\sin \\left(\\frac{n\\pi x}{L}\\right)\u00a0 \\biggr) $$
\n$$a_n = \\frac{1}{L} \\int_{-L}^{L} f(x) \\,\\cos \\left(\\frac{n\\pi x}{L}\\right)\u00a0 \\, dx $$
\n$$b_n = \\frac{1}{L} \\int_{-L}^{L} f(x) \\,\\sin \\left(\\frac{n\\pi x}{L}\\right)\u00a0 \\, dx $$ \u3068\u66f8\u3051\u308b\u306e\u3067\u3042\u3063\u305f\u3002<\/p>\n

\u6700\u521d\u306e \\(a_0\\) \u3060\u3051 \\(\\frac{1}{2}\\) \u304c\u304b\u304b\u3063\u3066\u3044\u308b\u3068\u304b\uff0c\\(\\displaystyle\\cos \\left(\\frac{n\\pi x}{L} \\right)\\) \u3068 \\(\\displaystyle\\sin \\left(\\frac{n\\pi x}{L}\\right) \\) \u306e\u5225\u3005\u306e\u91cd\u306d\u5408\u308f\u305b\u306b\u306a\u3063\u3066\u3044\u305f\u308a\u3057\u3066\u4f55\u304b\u3059\u3063\u304d\u308a\u3057\u306a\u3044\u306a\u3041\u3068\u3044\u3046\u3068\u3053\u308d\u3092\uff0c\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f
\n$$ e^{i \\theta} = \\cos \\theta + i \\sin \\theta$$ \u3092\u4f7f\u3063\u3066\uff0c\u30b3\u30f3\u30d1\u30af\u30c8\u306b\u307e\u3068\u3081\u308b\uff0c\u3068\u3044\u3046\u8a71\u3002<\/p>\n

\u307e\u305a\uff0c\u7b54\u3048\u3092\u5148\u306b\u66f8\u304f\u3002\u8907\u7d20\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570<\/b><\/u><\/span>\u306f
\n$$\\color{red}{f(x) = \\sum_{n = -\\infty}^{\\infty} c_n \\,\\exp\\left({i \\frac{n\\pi x}{L}}\\right)}$$ \u3068\u66f8\u3051\u308b\u3002\uff08\u3068\u3066\u3082\u7f8e\u3057\u3044\u5f0f\u306a\u306e\u3067\uff0c\u8272\u3092\u3064\u3051\u3066\u307f\u307e\u3057\u305f\u3002\uff09<\/p>\n

\u4ee5\u4e0b\u3067\u306f\uff0c\u8907\u7d20\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u306e\u5f0f<\/strong><\/span><\/p>\n

$$\\color{red}f(x) = \\sum_{n = -\\infty}^{\\infty} c_n \\,\\exp\\left({i \\frac{n\\pi x}{L}}\\right)$$<\/p>\n

\u3092\u5909\u5f62\u3057\u3066\u3044\u304f\u3068\uff0c\uff08\u5b9f\uff09\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u306e\u5f0f<\/strong><\/span><\/p>\n

$$\\color{blue} f(x) = \\frac{a_0}{2} + \\sum_{n=1}^{\\infty} \\biggl(a_n\\cos \\left(\\frac{n\\pi x}{L}\\right) + b_n \\sin \\left(\\frac{n\\pi x}{L}\\right) \\biggr)$$<\/p>\n

\u306b\u306a\u308b\uff0c\u3068\u3044\u3046\u65b9\u5411\u3067\u8a3c\u660e\u3059\u308b\u3002<\/p>\n

\u307e\u305a\uff0c\\(\\theta\\equiv \\dfrac{\\pi x}{L}\\) \u3068\u304a\u3044\u3066\\begin{eqnarray}
\nf(x) &=& \\sum_{n = -\\infty}^{\\infty} c_n \\,\\exp\\left({i \\frac{n\\pi x}{L}}\\right) \\\\
\n&=& \\sum_{n = -\\infty}^{\\infty} c_n \\,e^{i n\\theta} \\\\
\n&=& c_0 + c_{+1} \\,e^{+ i \\theta} + c_{+2} \\,e^{+2i \\theta}+ c_{+3} \\,e^{+3i\\theta} +\\cdots\\\\
\n&& \\quad + c_{-1} \\,e^{-i\\theta} + c_{-2} \\,e^{-2i\\theta}+ c_{-3} \\,e^{-3i\\theta} +\\cdots
\n\\end{eqnarray}
\n\u3068\u306a\u308a\uff0c
\n$$ c_0 \\equiv \\frac{a_0}{2}, \\quad c_{\\pm n} \\equiv \\frac{1}{2} \\left(a_n \\pm \\frac{1}{i} b_n\\right)$$
\n\u3068\u3059\u308b\u3068\uff0c
\n\\begin{eqnarray}
\nf(x) &=& c_0 + c_{+1} \\,e^{+ i \\theta} + c_{-1} \\,e^{-i\\theta}\\\\
\n&& \\quad + c_{+2} \\,e^{+2i \\theta}+ c_{-2} \\,e^{-2i\\theta}\\\\
\n&& \\quad + c_{+3} \\,e^{+3i\\theta} +c_{-3} \\,e^{-3i\\theta} + \\cdots \\\\
\n&=& \\frac{a_0}{2} + \\frac{1}{2}\\left(a_1 + \\frac{1}{i} b_1\\right) \\,e^{+ i \\theta} + \\frac{1}{2}\\left(a_1 -\\frac{1}{i} b_1\\right) \\,e^{-i \\theta} \\\\
\n&&\\quad + \\frac{1}{2}\\left(a_2 + \\frac{1}{i} b_2\\right) \\,e^{+2 i \\theta} + \\frac{1}{2}\\left(a_2 -\\frac{1}{i} b_2\\right) \\,e^{-i 2\\theta} \\\\
\n&&\\quad + \\frac{1}{2}\\left(a_3 + \\frac{1}{i} b_3\\right) \\,e^{+3 i \\theta} + \\frac{1}{2}\\left(a_3 -\\frac{1}{i} b_3\\right) \\,e^{-i 3\\theta}+\\cdots \\\\
\n&=& \\frac{a_0}{2} + a_1 \\frac{e^{+ i \\theta} + e^{-i\\theta}}{2} + b_1 \\frac{e^{+ i \\theta} -e^{-i\\theta}}{2 i} \\\\
\n&&\\quad +a_2 \\frac{e^{+ 2i \\theta} + e^{-2i\\theta}}{2} + b_2 \\frac{e^{+ 2i \\theta} -e^{-2i\\theta}}{2 i}\\\\
\n&&\\quad +a_3 \\frac{e^{+ 3i \\theta} + e^{-3i\\theta}}{2} + b_3 \\frac{e^{+ 3i \\theta} -e^{-3i\\theta}}{2 i} + \\cdots\\\\
\n&=& \\frac{a_0}{2} + a_1 \\cos \\theta + b_1 \\sin \\theta \\\\
\n&&\\quad + a_2 \\cos 2\\theta + b_2 \\sin 2\\theta \\\\
\n&&\\quad + a_3 \\cos 3\\theta + b_3\\sin 3\\theta+ \\cdots\\\\
\n&=& \\frac{a_0}{2} + \\sum_{n=1}^{\\infty} \\left(a_n\\cos n\\theta + b_n \\sin n\\theta\\right) \\\\
\n&=& \\frac{a_0}{2} + \\sum_{n=1}^{\\infty} \\biggl(a_n\\cos \\left(\\frac{n\\pi x}{L} \\right)+ b_n \\sin \\left(\\frac{n\\pi x}{L}\\right) \\biggr)
\n\\end{eqnarray} \u3068\uff0c\u7121\u4e8b\uff0c\uff08\u5b9f\uff09\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u306e\u5f0f\u306b\u306a\u308a\u307e\u3057\u305f\u3002<\/p>\n

\u307e\u305f\uff0c\u305d\u306e\u4fc2\u6570\u3067\u3042\u308b\u8907\u7d20\u30d5\u30fc\u30ea\u30a8\u4fc2\u6570<\/b><\/u><\/span>\u306f
\n\\begin{eqnarray}
\nc_n &=& \\frac{1}{2} \\left(a_n -i b_n\\right) \\\\
\n&=&\\frac{1}{L}\\int_{-L}^{L} f(x) \\, \\frac{1}{2}\\biggl(\\cos \\left(\\frac{n\\pi x}{L}\\right)\u00a0 -i \\sin \\left(\\frac{n\\pi x}{L}\\right) \\biggr) \\, dx \\\\
\n&=& \\frac{1}{2L} \\int_{-L}^{L} f(x)\\,\\exp\\left(-i\\frac{n\\pi x}{L}\\right) \\, dx\\\\
\n&=& \\frac{1}{2L} \\int_{-L}^{L} f(x)\\,\\biggl(\\exp\\left(i\\frac{n\\pi x}{L}\\right)\\biggr)^* \\, dx
\n\\end{eqnarray} \u3068\u3057\u3066\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u6700\u5f8c\u306e \\({}^*\\) \u306f\u8907\u7d20\u5171\u5f79\u3092\u3042\u3089\u308f\u3059\u3002<\/p>\n

\u8907\u7d20\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u306e\u307e\u3068\u3081<\/h3>\n

\u533a\u9593 \\( -L \\le x \\le L\\) \u3067\u5b9a\u7fa9\u3055\u308c\u305f\u95a2\u6570 \\(f(x)\\) \u304c\u533a\u9593\u5916\u3067\u306f\u5468\u671f \\(2 L\\) \u306e\u5468\u671f\u95a2\u6570\u3067\u3042\u308b\u5834\u5408\uff0c\u305d\u306e\u8907\u7d20\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570<\/b><\/span>\u306f
\n$$\\color{red}{f(x) = \\sum_{n = -\\infty}^{\\infty} c_n \\,\\exp\\left({i \\frac{n\\pi x}{L}}\\right)}$$ \u3067\u3042\u308a\uff0c\u305d\u306e\u8907\u7d20\u30d5\u30fc\u30ea\u30a8\u4fc2\u6570<\/b><\/span>\u306f
\n$$\\color{red}{
\nc_n =\u00a0 \\frac{1}{2L} \\int_{-L}^{L} f(x)\\,\\biggl(\\exp\\left(i\\frac{n\\pi x}{L}\\right)\\biggr)^* \\, dx
\n}$$ \u306e\u3088\u3046\u306b\u3057\u3066\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n

\u307e\u305f\uff0c\u95a2\u6570 \\(\\exp\\left(i\\dfrac{n\\pi x}{L}\\right) \\) \u306e\u76f4\u4ea4\u6027<\/b><\/span>\u306b\u3064\u3044\u3066\u306f\uff0c\u8907\u7d20\u5171\u5f79\u3092\u304b\u3051\u3066\u7a4d\u5206<\/strong><\/span>\u3057\u3066
\n\\begin{eqnarray}
\n\\int_{-L}^L \\biggl( \\exp\\left(i\\frac{m\\pi x}{L}\\right)\\biggr)^* \\exp\\left(i\\frac{n\\pi x}{L}\\right)\\,dx &=& \\int_{-L}^L\\exp\\left(i\\frac{(n-m)\\pi x}{L}\\right)\\,dx \\\\
\n&=& \\begin{cases}
\n2L & (m = n) \\\\
\n0\u00a0 & (m \\neq n)
\n\\end{cases} = 2L \\,\\delta_{mn}
\n\\end{eqnarray}<\/p>\n

\u3042\u3068\u3067\u51fa\u3066\u304f\u308b\u30d5\u30fc\u30ea\u30a8\u7a4d\u5206\u3067\u306e\u5bfe\u5fdc\u3092\u304b\u3093\u304c\u307f\u3066\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304d\u63db\u3048\u3066\u304a\u304f\u3002<\/p>\n

\\begin{eqnarray}
\nk_n &\\equiv& \\frac{n\\pi}{L} \\\\
\nk_m &\\equiv& \\frac{m\\pi}{L}\\\\
\n\\frac{1}{2L} \\int_{-L}^L e^{i(k_n -k_m) x} \\, dx &=& \\delta_{mn}
\n\\end{eqnarray}<\/p>\n

\u3055\u3066\uff0c\u3053\u3053\u307e\u3067\u5ef6\u3005\u3068\u5f0f\u3092\u5c55\u958b\u3057\u3066\u304d\u305f\u308f\u3051\u3060\u304c\uff0c\u8907\u7d20\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u306f\uff0c\u305d\u308c\u81ea\u4f53\u306e\u6709\u52b9\u6027\u3088\u308a\u3082\uff0c\u3080\u3057\u308d\u6b21\u306e\u300c\u30d5\u30fc\u30ea\u30a8\u5909\u63db<\/b><\/span>\u300d\u3078\u3064\u306a\u304c\u308b\u70b9\u3067\u91cd\u8981\u3067\u3042\u308b\u3002\u78ba\u304b\u306b\uff0c\u8907\u7d20\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u306f \\(\\cos\\) \u3068 \\(\\sin\\) \u3092\u5225\u3005\u306b\u66f8\u3044\u3066\u91cd\u306d\u5408\u308f\u305b\u308b\u3088\u308a\u3082\u30b3\u30f3\u30d1\u30af\u30c8\u306b\u8868\u73fe\u3067\u304d\u308b\u306e\u306f\u9b45\u529b\u3067\u3042\u308b\u3051\u3069\u306d\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

\\(\\cos \\) \u3068\\(\\sin \\) \u306e\u5225\u3005\u306e\u91cd\u306d\u5408\u308f\u305b\uff08\u8db3\u3057\u5408\u308f\u305b\uff09\u3067\u8868\u3055\u308c\u308b\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u3092\uff0c\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f\u3092\u4f7f\u3063\u30661\u3064\u306b\u307e\u3068\u3081\u308b\u3002<\/p>

\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"parent":2230,"menu_order":3,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-2368","page","type-page","status-publish","hentry","nodate","item-wrap"],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t