{"id":4757,"date":"2019-06-23T19:53:29","date_gmt":"2019-06-23T10:53:29","guid":{"rendered":"https:\/\/pandanote.info\/?p=4757"},"modified":"2022-08-07T12:17:03","modified_gmt":"2022-08-07T03:17:03","slug":"legendre%e3%81%ae%e9%99%aa%e5%a4%9a%e9%a0%85%e5%bc%8f%e3%81%ae%e7%9b%b4%e4%ba%a4%e6%80%a7%e3%81%ae%e8%a8%bc%e6%98%8e%e3%81%ae%e3%81%9f%e3%82%81%e3%81%ae%e8%a8%88%e7%ae%97%e3%82%92%e3%81%97%e3%81%a6","status":"publish","type":"post","link":"https:\/\/pandanote.info\/?p=4757","title":{"rendered":"Legendre\u306e\u966a\u591a\u9805\u5f0f\u306e\u76f4\u4ea4\u6761\u4ef6\u306e\u8a3c\u660e\u306e\u305f\u3081\u306e\u8a08\u7b97\u3092\u3057\u3066\u307f\u305f(\u305d\u306e1)\u3002"},"content":{"rendered":"

\u306f\u3058\u3081\u306b<\/h2>\n

\u524d\u306e\u8a18\u4e8b<\/a>\u3067\u3001Legendre\u306e\u591a\u9805\u5f0f\u306e\u76f4\u4ea4\u6761\u4ef6\u306e\u8a3c\u660e\u306e\u305f\u3081\u306e\u8a08\u7b97\u3092\u884c\u3063\u3066\u307f\u307e\u3057\u305f\u3002<\/p>\n

\u591a\u9805\u5f0f\u306e\u76f4\u4ea4\u6027\u306e\u6b21\u306f\u966a\u591a\u9805\u5f0f\u306e\u76f4\u4ea4\u6761\u4ef6\u306e\u8a3c\u660e\u306e\u305f\u3081\u306e\u8a08\u7b97\u3092\u884c\u3046\u306e\u304c\u81ea\u7136\u306a\u6210\u308a\u884c\u304d\u3060\u308d\u3046\u3068\u601d\u3046\u306e\u3067\u3001\u3053\u306e\u8a18\u4e8b\u3067\u306fLegendre\u306e\u966a\u591a\u9805\u5f0f
\n\\begin{align}
\nP_l^m(x) &= \\frac{(-1)^m}{2^l l!}(1-x^2)^{\\frac{m}{2}}\\frac{d^{l+m}}{dx^{l+m}}\\left[(x^2-1)^l\\right]\\label{eq:alp}
\n\\end{align}<\/p>\n

\u306e\u76f4\u4ea4\u6761\u4ef6\u3092\u793a\u3059\u5f0f
\n\\begin{align}
\n\\int_{-1}^{1} P_k^m(x)P_l^m(x)dx &= \\frac{2(l+m)!}{(2l+1)(l-m)!}\\delta_{kl}\\label{eq:orthogonality}
\n\\end{align}
\n($\\delta_{kl}$\u306fCronecker\u306edelta\u3067\u3059\u3002)\u306e\u8a08\u7b97\u3092\u308f\u304b\u308a\u306b\u304f\u3044\u3068\u3053\u308d(\u6b21\u9805\u3067\u6319\u3052\u308b\u3082\u306e\u306f\u9664\u304d\u307e\u3059\u3002)\u306f\u3067\u304d\u308b\u3060\u3051\u7701\u7565\u3057\u306a\u3044\u3067\u8a08\u7b97\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n

\u524d\u63d0\u3068\u306a\u308b\u4e8b\u5b9f\u7b49<\/h2>\n

(\\ref{eq:orthogonality})\u5f0f\u306e\u8a08\u7b97\u306b\u306f\u3001\u4ee5\u4e0b\u306e\u4e8b\u5b9f(\u8a73\u7d30\u306a\u8a08\u7b97\u3084\u8a3c\u660e\u306f\u3053\u306e\u8a18\u4e8b\u3067\u306f\u7701\u7565\u3057\u307e\u3059\u3002)\u306a\u3069\u3092\u4f7f\u7528\u3057\u307e\u3059\u3002<\/p>\n

    \n
  1. \u95a2\u6570
    \n\\begin{align}
    \nf_{+}(x) &= (x^2-1)^k \\label{eq:xsquareminusone}
    \n\\end{align}
    \n\u53ca\u3073
    \n\\begin{align}
    \nf_{-}(x) &= (1-x^2)^k \\label{eq:oneminusxsquare}
    \n\\end{align}
    \n\u306e$n$\u56de\u5fae\u5206\u306f$x$\u306b\u3064\u3044\u3066\u306e$2k-n$\u6b21\u306e\u591a\u9805\u5f0f\u3067\u3042\u308b\u3053\u3068\u3002<\/li>\n
  2. \u7279\u306b\u3001(\\ref{eq:xsquareminusone})\u5f0f\u53ca\u3073(\\ref{eq:oneminusxsquare})\u5f0f\u306e\u4e21\u8fba\u3092$2k$\u56de\u5fae\u5206\u3059\u308b\u3068\u4ee5\u4e0b\u306e\u5f0f\u304c\u6210\u308a\u7acb\u3061\u3001$x$\u306b\u3088\u3089\u306a\u3044\u5b9a\u6570\u3068\u306a\u308b\u3053\u3068\u3002
    \n\\begin{align}
    \n\\frac{d^{2k}}{dx^{2k}}f_{+}(x) &= (2k)! \\label{eq:xsquareminusonediffktimes} \\cr
    \n\\frac{d^{2k}}{dx^{2k}}f_{-}(x) &= (-1)^k(2k)! \\label{eq:oneminusxsquarediffktimes}
    \n\\end{align}\n<\/li>\n
  3. Leibniz\u306e\u516c\u5f0f((\\ref{eq:leibniz})\u5f0f)\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3002
    \n\\begin{align}
    \n\\frac{d^n}{dx^n}\\left[f(x)g(x)\\right] &= \\sum_{k=0}^{n}\\begin{pmatrix}
    \nn \\cr
    \nk
    \n\\end{pmatrix}f^{(k)}(x)g^{(n-k)}(x) \\label{eq:leibniz}
    \n\\end{align}\n<\/li>\n
  4. Wallis\u7a4d\u5206<\/a>\u3002<\/li>\n
  5. \u6574\u6570$n$\u306e\u4e8c\u91cd\u968e\u4e57\u306b\u3064\u3044\u3066\u3001\u4ee5\u4e0b\u306e\u5f0f\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3002
    \n\\begin{align}
    \n(2n)!! &= 2^nn! \\label{eq:doublefactorial} \\cr
    \n(2n)!!(2n+1)!! &= (2n+1)! \\label{eq:doublefactorialandfactorial}
    \n\\end{align}\n<\/li>\n<\/ol>\n

    \u8a08\u7b97\u3057\u3066\u307f\u307e\u3059\u3002<\/h2>\n

    \u307e\u305a\u306f\u4ee3\u5165\u304b\u3089\u2026<\/h3>\n

    \u4f8b\u306b\u3088\u3063\u3066\u30b5\u30af\u30b5\u30af\u3068\u8a08\u7b97\u3057\u307e\u3059\u3002<\/p>\n

    (\\ref{eq:orthogonality})\u5f0f\u306b(\\ref{eq:alp})\u5f0f\u3092\u4ee3\u5165\u3059\u308b\u3068\u2026
    \n\\begin{align}
    \n\\int_{-1}^{1} P_k^m(x)P_l^m(x)dx &= {\\color[RGB]{30,62,138}\\int_{-1}^{1}}{\\color[RGB]{0,128,0}\\frac{1}{2^{l+k}l!k!}}\\color[RGB]{30,62,138}{(1-x^2)^m}{\\color[RGB]{30,62,138}\\frac{d^{l+m}}{dx^{l+m}}(x^2-1)^l\\frac{d^{k+m}}{dx^{k+m}}(x^2-1)^kdx}\\label{eq:alpinnerproduct}
    \n\\end{align}
    \n\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n

    \u3053\u3053\u3067\u3001\u7a4d\u5206\u5909\u6570$x$\u3092\u542b\u3080\u90e8\u5206((\\ref{eq:alpinnerproduct})\u5f0f\u306e\u9752\u8272\u306e\u90e8\u5206)\u3092\u53d6\u308a\u51fa\u3057\u3001
    \n\\begin{align}
    \nI_{kl}^m &= \\int_{-1}^{1}(1-x^2)^m\\frac{d^{l+m}}{dx^{l+m}}(x^2-1)^l\\frac{d^{k+m}}{dx^{k+m}}(x^2-1)^kdx\\label{eq:imkldef}
    \n\\end{align}
    \n\u3068\u304a\u304d\u307e\u3059\u3002\u306a\u304a\u3001$l$\u3068$k$\u3092\u5165\u308c\u66ff\u3048\u3066\u3082\u4e00\u822c\u6027\u306f\u5931\u308f\u306a\u3044\u306e\u3067\u3001$l \\le k$\u3068\u3057\u307e\u3059\u3002\u307e\u305f\u3001Legendre\u306e\u966a\u591a\u9805\u5f0f\u306e\u5b9a\u7fa9\u3088\u308a$m$\u306f$k$\u307e\u305f\u306f$l$\u306e\u3046\u3061\u5c0f\u3055\u3044\u65b9\u4ee5\u4e0b\u306e\u5024\u306e\u5834\u5408\u306e\u307f\u5b9a\u7fa9\u3055\u308c\u308b\u3053\u3068\u304b\u3089(\\ref{eq:mkl})\u5f0f\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002
    \n\\begin{align}
    \nm &\\le l \\le k \\label{eq:mkl}
    \n\\end{align}<\/p>\n

    \u90e8\u5206\u7a4d\u5206\u3092\u5229\u7528\u3057\u3066\u5909\u5f62\u3057\u307e\u3059\u3002<\/h3>\n

    (\\ref{eq:imkldef})\u5f0f\u3092\u90e8\u5206\u7a4d\u5206\u3092\u7528\u3044\u3066\u5909\u5f62\u3057\u307e\u3059\u3002\u3059\u308b\u3068\u3001(\\ref{eq:imklpartialoncefirst})\u5f0f\u306e\u3088\u3046\u306b\u5909\u5f62\u3067\u304d\u307e\u3059\u3002
    \n\\begin{align}
    \nI_{kl}^m &= \\int_{-1}^{1}\\left[\\frac{d^{k+m-1}}{dx^{k+m-1}}(x^2-1)^l\\right]^{\\prime}(1-x^2)^m \\frac{d^{l+m}}{dx^{l+m}}(x^2-1)^ldx \\nonumber\\cr
    \n&= \\left[(1-x^2)^m\\frac{d^{k+m-1}}{dx^{k+m-1}}(x^2-1)^k\\frac{d^{l+m}}{dx^{l+m}}(x^2-1)^l\\right]_{-1}^{1} \\nonumber \\cr
    \n&{} -\\int_{-1}^{1}\\frac{d^{k+m-1}}{dx^{k+m-1}}(x^2-1)^k\\frac{d}{dx}\\left[(1-x^2)^m\\frac{d^{l+m}}{dx^{l+m}}(x^2-1)^l\\right]dx \\label{eq:imklpartialoncefirst}
    \n\\end{align}
    \n(\\ref{eq:imklpartialoncefirst})\u5f0f\u306e\u53f3\u8fba\u7b2c1\u9805\u306e\u62ec\u5f27\u5185\u306b\u306f$1-x^2$\u306e\u3079\u304d\u4e57\u304c\u542b\u307e\u308c\u3066\u3044\u3066\u3001$x=\\pm 1$\u306e\u3068\u304d\u306b\u306f\u3053\u308c\u304c0\u306b\u306a\u308b\u3053\u3068\u304b\u3089\u3001(\\ref{eq:imklpartialoncefirst})\u5f0f\u306e\u53f3\u8fba\u7b2c1\u9805\u306f0\u3068\u306a\u308a\u307e\u3059\u3002\u3088\u3063\u3066\u3001(\\ref{eq:imklpartialoncesecond})\u5f0f\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002
    \n\\begin{align}
    \nI_{kl}^m &= -\\int_{-1}^{1}\\frac{d^{k+m-1}}{dx^{k+m-1}}(x^2-1)^k\\frac{d}{dx}\\left[(1-x^2)^m\\frac{d^{l+m}}{dx^{l+m}}(x^2-1)^l\\right]dx \\label{eq:imklpartialoncesecond}
    \n\\end{align}
    \n\u3053\u306e\u7bc0\u306e\u3053\u3053\u307e\u3067\u306e\u5909\u5f62\u3068\u540c\u69d8\u306e\u5909\u5f62\u3092\u3042\u3068$k+m-1$\u56de\u7e70\u308a\u8fd4\u3059\u3068\u3001
    \n\\begin{align}
    \nI_{kl}^m &= (-1)^{k+m}\\int_{-1}^{1}(x^2-1)^k\\frac{d^{k+m}}{dx^{k+m}}\\left[(1-x^2)^m\\frac{d^{l+m}}{dx^{l+m}}(x^2-1)^l\\right]dx \\label{eq:imklpartialoncethird}
    \n\\end{align}
    \n\u3068\u5909\u5f62\u3067\u304d\u307e\u3059\u3002<\/p>\n

    \u3053\u3053\u3067\u3001Leibniz\u306e\u516c\u5f0f\u306e\u767b\u5834\u3067\u3059\u3002<\/h3>\n

    (\\ref{eq:imklpartialoncethird})\u5f0f\u306fLeibniz\u306e\u516c\u5f0f((\\ref{eq:leibniz})\u5f0f)\u3092\u5229\u7528\u3059\u308b\u3068\u3001
    \n\\begin{align}
    \nI_{kl}^m &= (-1)^{k+m}\\int_{-1}^{1}(x^2-1)^k\\cdot\\nonumber\\cr
    \n&{} \\left[\\color[RGB]{0,192,0}\\sum_{p=0}^{k+m}\\begin{pmatrix}
    \nk+m \\cr
    \np
    \n\\end{pmatrix}\\frac{d^{p}}{dx^{p}}(1-x^2)^m\\frac{d^{l+2m+k-p}}{dx^{l+2m+k-p}}(x^2-1)^l\\color[RGB]{0,0,0}\\right]dx\\label{eq:imklpartialoncefourth}
    \n\\end{align}
    \n\u3068\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002<\/p>\n

    (\\ref{eq:imklpartialoncefourth})\u5f0f\u306e\u62ec\u5f27\u5185\u306e\u90e8\u5206(\u7dd1\u8272\u3067\u793a\u3057\u305f\u90e8\u5206)\u3067\u548c\u3092\u8a08\u7b97\u3057\u3066\u3044\u307e\u3059\u304c\u3001\u548c\u306e\u5404\u9805\u304c0\u4ee5\u5916\u306e\u5024\u306b\u306a\u308b\u6761\u4ef6\u306f(\\ref{eq:xsquareminusone})\u5f0f\u3088\u308a\u3001
    \n\\begin{align}
    \np &\\le 2m \\label{eq:pmdouble}
    \n\\end{align}
    \n\u53ca\u3073\u3001
    \n\\begin{align}
    \nl+2m+k-p &\\le 2l \\label{eq:lmkp}
    \n\\end{align}
    \n\u3067\u3059\u306e\u3067\u3001(\\ref{eq:pmdouble})\u5f0f\u53ca\u3073(\\ref{eq:lmkp})\u5f0f\u3092$p$\u306b\u3064\u3044\u3066\u89e3\u3044\u3066\u307e\u3068\u3081\u308b\u3068\u3001
    \n\\begin{align}
    \n2m+k-l &\\le p \\le 2m \\label{eq:mklp}
    \n\\end{align}
    \n\u3068\u306a\u308a\u307e\u3059\u3002\u3068\u3053\u308d\u304c\u3001$k$\u53ca\u3073$l$\u306e\u9593\u306b\u306f(\\ref{eq:mkl})\u5f0f\u306e\u95a2\u4fc2\u304c\u3042\u308a\u307e\u3059\u306e\u3067\u3001$l \\lt k$\u306e\u5834\u5408\u306b\u306f$2m \\lt 2m+k-l$\u3068\u306a\u308b\u306e\u3067\u3001(\\ref{eq:mklp})\u5f0f\u3092\u6e80\u305f\u3059$p$\u304c\u5b58\u5728\u3057\u306a\u304f\u306a\u3063\u3066\u3057\u307e\u3044\u307e\u3059\u3002<\/p>\n

    \u3057\u305f\u304c\u3063\u3066\u3001(\\ref{eq:mklp})\u5f0f\u3092\u6e80\u305f\u3059$p$\u304c\u5b58\u5728\u3059\u308b\u5834\u5408\u306f$l=k$\u306e\u5834\u5408\u306b\u9650\u3089\u308c\u308b\u3053\u3068\u3068\u3001$l \\lt k$\u306e\u5834\u5408\u306b\u306f(\\ref{eq:imklpartialoncefourth})\u5f0f\u306e\u548c\u304c0\u306b\u306a\u308b\u305f\u3081\u306b\u3001
    \n\\begin{align}
    \nI_{kl}^m &= 0 \\label{eq:lltk}
    \n\\end{align}
    \n\u3068\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n

    $l = k$\u306e\u5834\u5408\u306b\u3064\u3044\u3066\u3055\u3089\u306b\u8a08\u7b97\u3057\u307e\u3059\u3002<\/h3>\n

    \u6b21\u306b\u3001$l = k$\u306e\u5834\u5408\u306b\u3064\u3044\u3066\u8003\u3048\u307e\u3059\u3002<\/p>\n

    \u524d\u7bc0\u3067\u691c\u8a0e\u3057\u305f\u901a\u308a\u3001(\\ref{eq:imklpartialoncefourth})\u5f0f\u306e\u548c\u3092\u8a08\u7b97\u3059\u308b\u90e8\u5206\u306f$l = k$\u306e\u5834\u5408\u3082$p=2m$\u3067\u306a\u3044\u9805\u306f0\u3068\u306a\u308b\u3053\u3068\u306b\u6ce8\u610f\u3057\u3066\u3001(\\ref{eq:imklpartialoncefourth})\u5f0f\u306b$l = k$\u3092\u4ee3\u5165\u3057\u3001$k$\u3092\u6d88\u53bb\u3059\u308b\u3068\u2026
    \n\\begin{align}
    \nI_{ll}^m &= (-1)^{l+m}\\int_{-1}^{1}(x^2-1)^l\\cdot\\nonumber\\cr
    \n&{} \\left[\\begin{pmatrix}
    \nl+m \\cr
    \n2m
    \n\\end{pmatrix}\\frac{d^{2m}}{dx^{2m}}(1-x^2)^m\\frac{d^{2l}}{dx^{2l}}(x^2-1)^l\\right]dx\\label{eq:imklpartialoncefifth}
    \n\\end{align}
    \n(\\ref{eq:imklpartialoncefifth})\u5f0f\u306e\u53f3\u8fba\u306e\u62ec\u5f27\u5185\u306b(\\ref{eq:xsquareminusonediffktimes})\u5f0f\u53ca\u3073(\\ref{eq:oneminusxsquarediffktimes})\u3092\u7528\u3044\u308b\u3068\u3001
    \n\\begin{align}
    \nI_{ll}^m &= (-1)^{l+m}\\int_{-1}^{1}(x^2-1)^l\\begin{pmatrix}
    \nl+m \\cr
    \n2m
    \n\\end{pmatrix}(2m)!(2l)!dx \\nonumber \\cr
    \n&= (-1)^{l+m} \\frac{(l+m)!}{(l-m)!(2m)!}(-1)^m(2m)!(2l)!\\int_{-1}^{1}(x^2-1)^ldx \\nonumber \\cr
    \n&= (-1)^{l} \\frac{(l+m)!(2l)!}{(l-m)!}{\\color[RGB]{30,62,138} \\int_{-1}^{1}(x^2-1)^ldx} \\label{eq:imklpartialfirst}
    \n\\end{align}
    \n\u3068\u5909\u5f62\u3067\u304d\u3001\u7a4d\u5206\u5909\u6570$x$\u304c\u95a2\u4e0e\u3057\u3066\u3044\u308b\u56e0\u5b50\u306f${\\color[RGB]{30,62,138} (x^2-1)^l}$\u306e\u90e8\u5206(\u6fc3\u3044\u9752\u8272\u3067\u793a\u3057\u305f\u90e8\u5206)\u306e\u307f\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n

    \u305d\u3053\u3067\u3001\u4e0a\u8a18\u306e\u56e0\u5b50\u306e\u90e8\u5206\u306e\u7a4d\u5206\u3092\u8a08\u7b97\u3059\u308b\u3068\u2026
    \n\\begin{align}
    \n\\int_{-1}^{1}(x^2-1)^ldx &= (-1)^l\\int_{-1}^{1}(1-x^2)^ldx\\label{eq:intxsquareminusone}
    \n\\end{align}
    \n\u3055\u3089\u306b\u3001$x=\\sin t$\u3068\u304a\u304f\u3068\u3001    Wallis\u7a4d\u5206<\/a>\u304c\u73fe\u308c\u307e\u3059\u306e\u3067\u2026
    \n\\begin{align}
    \n\\int_{-1}^{1}(x^2-1)^ldx &= (-1)^l\\int_{-\\frac{\\pi}{2}}^{\\frac{\\pi}{2}}\\cos^{2l}t\\frac{dx}{dt}dt \\nonumber \\cr
    \n&= (-1)^l\\int_{-\\frac{\\pi}{2}}^{\\frac{\\pi}{2}}\\cos^{2l}t\\cos t\\,dt \\nonumber \\cr
    \n&= (-1)^l\\int_{-\\frac{\\pi}{2}}^{\\frac{\\pi}{2}}\\cos^{2l+1}t\\,dt\\nonumber \\cr
    \n&= 2(-1)^l\\int_{0}^{\\frac{\\pi}{2}}\\cos^{2l+1}t\\,dt\\nonumber \\cr
    \n&= 2(-1)^l\\frac{(2l)!!}{(2l+1)!!}\\label{eq:intxsquareminusonefirst}
    \n\\end{align}
    \n\u3068\u5909\u5f62\u3067\u304d\u307e\u3059\u3002<\/p>\n

    \u6700\u5f8c\u306e\u4ed5\u4e0a\u3052\u3002<\/h3>\n

    (\\ref{eq:intxsquareminusonefirst})\u5f0f\u3092\u7528\u3044\u3066(\\ref{eq:imklpartialfirst})\u5f0f\u3092\u66f8\u304d\u63db\u3048\u308b\u3068\u2026
    \n\\begin{align}
    \nI_{ll}^m &= (-1)^{l} \\frac{(l+m)!(2l)!}{(l-m)!}\\cdot 2(-1)^l\\frac{(2l)!!}{(2l+1)!!}\\nonumber \\cr
    \n&= \\frac{2(l+m)!(2l)!(2l)!!}{(l-m)!(2l+1)!!}\\label{eq:imklfinalanswer}
    \n\\end{align}
    \n\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n

    \u3055\u3089\u306b\u3001(\\ref{eq:imklfinalanswer})\u5f0f\u3092(\\ref{eq:alpinnerproduct})\u5f0f\u306b\u4ee3\u5165\u3057\u3001$k = l$\u3068\u304a\u3044\u3066\u3001(\\ref{eq:doublefactorial})\u5f0f\u53ca\u3073(\\ref{eq:doublefactorialandfactorial})\u5f0f\u3092\u7528\u3044\u3066\u5909\u5f62\u3059\u308b\u3068\u2026
    \n\\begin{align}
    \n\\int_{-1}^{1} \\left\\{P_l^m(x)\\right\\}^2 dx &= \\frac{1}{2^{2l}(l!)^2}\\frac{2(l+m)!(2l)!(2l)!!}{(l-m)!(2l+1)!!} \\nonumber \\cr
    \n&= \\frac{1}{2^{2l}(l!)^2}\\frac{2(l+m)!(2l)!l!2^l}{(l-m)!(2l+1)!!} \\nonumber\\cr
    \n&= \\frac{1}{2^{l}l!}\\frac{2(l+m)!(2l)!}{(l-m)!(2l+1)!!} \\nonumber \\cr
    \n&= \\frac{1}{(2l)!!}\\frac{2(l+m)!(2l)!}{(l-m)!(2l+1)!!} \\nonumber \\cr
    \n&= \\frac{2(l+m)!}{(l-m)!}\\frac{(2l)!}{(2l+1)!} \\nonumber \\cr
    \n&= \\frac{2(l+m)!}{(2l+1)(l-m)!}\\label{eq:alpinnerproductfinalanswer}
    \n\\end{align}
    \n\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n

    (\\ref{eq:alpinnerproductfinalanswer})\u5f0f\u53ca\u3073(\\ref{eq:lltk})\u5f0f\u306e\u7d50\u679c\u3088\u308a\u3001(\\ref{eq:orthogonality})\u5f0f\u3092\u793a\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002$\\qquad\\blacksquare$<\/p>\n

    \u307e\u3068\u3081<\/h2>\n

    Legendre\u306e\u966a\u591a\u9805\u5f0f\u306e\u5b9a\u7fa9\u5f0f((\\ref{eq:alp})\u5f0f)\u3092\u3088\u304f\u898b\u308b\u3068\u3001$(x^2-1)^{\\frac{m}{2}}$\u3092\u56e0\u5b50\u3068\u3057\u3066\u542b\u3080\u305f\u3081\u306b\u3001$m$\u304c\u5947\u6570\u306e\u5834\u5408\u306b\u306f\u53b3\u5bc6\u306a\u610f\u5473\u3067\u306e\u591a\u9805\u5f0f\u3068\u306f\u306a\u308a\u307e\u305b\u3093\u304c\u3001(\\ref{eq:orthogonality})\u5f0f\u3067\u306f\u7a4d\u5206\u306e\u5bfe\u8c61\u3068\u306a\u308b\u966a\u591a\u9805\u5f0f\u306e\u5fae\u5206\u306e\u56de\u6570\u3092\u540c\u3058\u3068\u3057\u3066\u3044\u308b\u305f\u3081\u306b\u305d\u308c\u305e\u308c\u306e\u5f0f\u304c\u56e0\u5b50\u3068\u3057\u3066\u6301\u3063\u3066\u3044\u308b$(x^2-1)^{\\frac{m}{2}}$\u304c\u639b\u3051\u5408\u308f\u3055\u308c\u308b\u306e\u3067\u3001\u591a\u9805\u5f0f\u306e\u7a4d\u5206\u8a08\u7b97\u3068\u3057\u3066\u8a08\u7b97\u304c\u3067\u304d\u308b\u3068\u3044\u3046\u5bf8\u6cd5\u3067\u3059\u3002<\/p>\n

    \u8a08\u7b97\u306e\u7d50\u679c\u3082Legendre\u306e\u591a\u9805\u5f0f\u306e\u76f4\u4ea4\u6027\u3092\u793a\u3059\u305f\u3081\u306e\u8a08\u7b97(\u3053\u306e\u8a18\u4e8b<\/a>\u53c2\u7167\u3002)\u3068\u540c\u69d8\u306e\u65b9\u6cd5\u3067\u3067\u304d\u3066\u3001\u304b\u3064\u7d50\u679c\u304c\u6bd4\u8f03\u7684\u7c21\u5358\u306a\u5f62\u306b\u306a\u308b\u3053\u3068\u3082\u8208\u5473\u6df1\u3044\u3068\u3053\u308d\u3067\u3059\u3002<\/p>\n

    \u3053\u306e\u8a18\u4e8b\u306f\u4ee5\u4e0a\u3067\u3059\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

    \u306f\u3058\u3081\u306b \u524d\u306e\u8a18\u4e8b\u3067\u3001Legendre\u306e\u591a\u9805\u5f0f\u306e\u76f4\u4ea4\u6761\u4ef6\u306e\u8a3c\u660e\u306e\u305f\u3081\u306e\u8a08\u7b97\u3092\u884c\u3063\u3066\u307f\u307e\u3057\u305f\u3002 \u591a\u9805\u5f0f\u306e\u76f4\u4ea4\u6027\u306e\u6b21\u306f\u966a\u591a\u9805\u5f0f\u306e\u76f4\u4ea4\u6761\u4ef6\u306e\u8a3c\u660e\u306e\u305f\u3081\u306e\u8a08\u7b97\u3092\u884c\u3046\u306e\u304c\u81ea\u7136\u306a\u6210\u308a\u884c\u304d\u3060\u308d\u3046\u3068\u601d\u3046\u306e\u3067\u3001\u3053\u306e\u8a18\u4e8b\u3067\u306fLegendre\u306e\u966a\u591a\u9805\u5f0f \\begin{align} P_l^m(x) &= \\frac{(-1)^m}{2^l l!}(1-x^2)^{\\frac{m}{2}}\\frac{d^{l+\u2026 Read More »<\/a><\/span><\/p>\n","protected":false},"author":1,"featured_media":4799,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[13],"tags":[],"class_list":["post-4757","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-13"],"_links":{"self":[{"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/posts\/4757","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/pandanote.info\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=4757"}],"version-history":[{"count":43,"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/posts\/4757\/revisions"}],"predecessor-version":[{"id":9373,"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/posts\/4757\/revisions\/9373"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/media\/4799"}],"wp:attachment":[{"href":"https:\/\/pandanote.info\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4757"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/pandanote.info\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4757"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/pandanote.info\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4757"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}