{"id":4874,"date":"2019-07-10T20:55:18","date_gmt":"2019-07-10T11:55:18","guid":{"rendered":"https:\/\/pandanote.info\/?p=4874"},"modified":"2022-08-07T12:30:38","modified_gmt":"2022-08-07T03:30:38","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%9d%a1%e4%bb%b6%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","status":"publish","type":"post","link":"https:\/\/pandanote.info\/?p=4874","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\u306e2)\u3002"},"content":{"rendered":"<h2>\u306f\u3058\u3081\u306b<\/h2>\n<p>\u3053\u306e\u8a18\u4e8b\u3067\u306f<a href=\"https:\/\/pandanote.info\/?p=4757\">\u3061\u3087\u3063\u3068\u524d\u306e\u8a18\u4e8b<\/a>\u306b\u5f15\u304d\u7d9a\u304d\u3001Legendre\u306e\u966a\u591a\u9805\u5f0f<br \/>\n\\begin{align}<br \/>\nP_l^m(x) &#038;= \\frac{(-1)^m}{2^ll!}\\frac{d^{m+l}}{dx^{m+l}}(x^2-1)^l \\label{eq:associatedlegendre}<br \/>\n\\end{align}<br \/>\n\u304c\u6e80\u305f\u3059\u76f4\u4ea4\u6761\u4ef6<br \/>\n\\begin{align}<br \/>\n\\int_{-1}^{1}\\frac{P_l^m(x)P_l^n(x)}{1-x^2}dx &#038;= \\begin{cases}<br \/>\n0 &#038; (m \\ne n) \\cr<br \/>\n\\displaystyle\\frac{(l+m)!}{m(l-m)!} &#038; (m = n \\ne 0) \\cr<br \/>\n\\infty &#038; (m = n = 0)<br \/>\n\\end{cases} \\label{eq:orthogonality}<br \/>\n\\end{align}<br \/>\n\u3092\u8a3c\u660e\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n<p>\u306a\u304a\u3001\u5076\u95a2\u6570\u53ca\u3073\u5947\u95a2\u6570\u3067\u3042\u308b\u3053\u3068\u3092\u5229\u7528\u3059\u308b\u90e8\u5206\u306e\u8a08\u7b97\u65b9\u6cd5\u306b\u3064\u3044\u3066\u306f\u53c2\u8003\u6587\u732e\u306e\u30b5\u30a4\u30c8\u3092\u5927\u5e45\u306b\u53c2\u8003\u306b\u3055\u305b\u3066\u3044\u305f\u3060\u3044\u3066\u304a\u308a\u307e\u3059\u3002<\/p>\n<h2>\u524d\u63d0\u3068\u306a\u308b\u4e8b\u5b9f\u7b49<\/h2>\n<p>\u3053\u3053\u304b\u3089\u306e\u8a08\u7b97\u3067\u306f\u3001\u4ee5\u4e0b\u306e\u4e8b\u5b9f\u306b\u3064\u3044\u3066\u306f\u8a3c\u660e\u306a\u3069\u3092\u7701\u7565\u3057\u3066\u5229\u7528\u3057\u307e\u3059\u3002\u306a\u304a\u3001$(x^2-1)^l$\u306e\u5fae\u5206\u306b\u3064\u3044\u3066\u306f<a href=\"https:\/\/sidestory.pandanote.info\/xsquareminusone_diff.html\">\u3053\u3061\u3089<\/a>\u3092\u3054\u53c2\u7167\u3044\u305f\u3060\u3051\u308b\u3068\u5e78\u3044\u3067\u3059\u3002<\/p>\n<ul>\n<li>$m \\lt l$\u306e\u3068\u304d\u306b<br \/>\n\\begin{align}<br \/>\nf(x) &#038;= \\frac{d^m}{dx^m}(x^2-1)^l \\label{eq:ml}<br \/>\n\\end{align}<br \/>\n\u3068\u304a\u304f\u3068\u3001$f(x)$\u306b\u306f$x^2-1$\u304c\u56e0\u5b50\u3068\u3057\u3066\u542b\u307e\u308c\u308b\u305f\u3081\u306b\u3001$f(\\pm 1) = 0$\u3068\u306a\u308b\u3053\u3068\u3002<\/li>\n<li>$m \\ge l$\u306e\u3068\u304d\u306b\u306f\u3001$f(x)$\u306f$x^2-1$\u3092\u56e0\u5b50\u3068\u3057\u3066\u542b\u307e\u306a\u3044\u305f\u3081\u306b\u3001$f(\\pm 1) \\ne 0$\u3068\u306a\u308b\u3053\u3068\u3002<\/li>\n<\/ul>\n<h2>\u4f8b\u306b\u3088\u3063\u3066\u30b5\u30af\u30b5\u30af\u3068\u8a08\u7b97\u3002<\/h2>\n<h3>\u3068\u308a\u3042\u3048\u305a\u4ee3\u5165\u3002<\/h3>\n<p>(\\ref{eq:orthogonality})\u5f0f\u306e\u5de6\u8fba\u306f(\\ref{eq:associatedlegendre})\u5f0f\u3092\u7528\u3044\u3066\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5909\u5f62\u3067\u304d\u307e\u3059\u3002\u306a\u304a\u3001\u9014\u4e2d\u3067\u767b\u5834\u3059\u308b$P_l(x)$\u306fLegendre\u306e\u591a\u9805\u5f0f\u3092\u8868\u3057\u307e\u3059\u3002<br \/>\n\\begin{align}<br \/>\n\\int_{-1}^{1}\\frac{P_l^m(x)P_l^n(x)}{1-x^2}dx &#038;= \\int_{-1}^{1}\\frac{(-1)^{m+n}}{(2^ll!)^2}(1-x^2)^{\\frac{m+n}{2}-1}\\frac{d^m}{dx^m}P_l(x)\\frac{d^n}{dx^n}P_l(x)dx \\nonumber \\cr<br \/>\n&#038;= \\frac{(-1)^{m+n}}{(2^ll!)^2}\\int_{-1}^{1}(1-x^2)^{\\frac{m+n}{2}-1}\\cdot\\nonumber\\cr<br \/>\n&#038;{} \\frac{d^{m+l}}{dx^{m+l}}(x^2-1)^l\\frac{d^{n+l}}{dx^{n+l}}(x^2-1)^ldx \\label{eq:alofirst}<br \/>\n\\end{align}<\/p>\n<h3>\u3053\u3053\u3067\u6700\u521d\u306e\u5834\u5408\u5206\u3051\u3002<\/h3>\n<p>(\\ref{eq:alofirst})\u5f0f\u306e\u53f3\u8fba\u3092\u898b\u308b\u3068\u3001$(1-x^2)^{\\frac{m+n}{2}-1}$\u3068\u3044\u3046$m+n$\u304c\u5947\u6570\u3060\u3068\u5f0f\u5168\u4f53\u304c\u53b3\u5bc6\u306a\u610f\u5473\u3067\u306e\u591a\u9805\u5f0f\u306b\u306a\u3089\u306a\u304f\u306a\u3063\u3066\u3057\u307e\u3046\u9805\u304c\u542b\u307e\u308c\u3066\u3044\u3066\u3001\u304b\u306a\u308a\u5acc\u306a\u4e88\u611f\u304c\u3057\u307e\u3059\u3002<\/p>\n<p>$m+n$\u304c\u5947\u6570\u306e\u3068\u304d\u306b\u306f$(1-x^2)^{\\frac{m+n}{2}-1}$\u304c\u5076\u95a2\u6570\u306b\u306a\u308a\u307e\u3059\u304c\u3001\u305d\u308c\u4ee5\u5916\u306e\u90e8\u5206\u304c\u5947\u95a2\u6570\u306b\u306a\u308b\u3068(\\ref{eq:alofirst})\u5f0f\u306e\u53f3\u8fba\u306e\u88ab\u7a4d\u5206\u95a2\u6570\u304c\u5947\u95a2\u6570\u306b\u306a\u308a\u307e\u3059\u306e\u3067\u3001\u533a\u9593$[-1,1]$\u3067\u7a4d\u5206\u3092\u3059\u308b\u3068\u3001\u305d\u306e\u5024\u304c0\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>\u305d\u3053\u3067\u3001(\\ref{eq:alofirst})\u5f0f\u306e\u53f3\u8fba\u306e\u88ab\u7a4d\u5206\u95a2\u6570\u306e$(1-x^2)^{\\frac{m+n}{2}-1}$\u4ee5\u5916\u306e\u56e0\u5b50\u306e\u7a4d\u304c\u5947\u95a2\u6570\u306b\u306a\u3089\u306a\u3044\u304b\u3069\u3046\u304b\u8abf\u3079\u3066\u307f\u307e\u3059\u3002<\/p>\n<p>$m+n$\u304c\u5947\u6570\u306b\u306a\u308b\u5834\u5408\u306f$m$\u307e\u305f\u306f$n$\u306e\u3044\u305a\u308c\u304b\u4e00\u65b9\u304c\u5947\u6570\u3067\u3082\u3046\u4e00\u65b9\u304c\u5076\u6570\u306e\u5834\u5408\u3067\u3059\u3002$m$\u304c\u5076\u6570\u306e\u5834\u5408\u306b\u306f$l$\u3082\u5076\u6570\u306e\u5834\u5408\u306f$m+l$\u53ca\u3073$n+l$\u306f\u305d\u308c\u305e\u308c\u5076\u6570\u3001\u5947\u6570\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>\u307e\u305f\u3001$l$\u304c\u5947\u6570\u306e\u5834\u5408\u306f$m+l$\u53ca\u3073$n+l$\u306e\u5076\u5947\u306f\u5165\u308c\u66ff\u308f\u308a\u307e\u3059\u304c\u3001\u3069\u3061\u3089\u304b\u4e00\u65b9\u304c\u5947\u6570\u3067\u3082\u3046\u4e00\u65b9\u304c\u5076\u6570\u306b\u306a\u308b\u3068\u3044\u3046\u7d44\u307f\u5408\u308f\u305b\u306b\u5909\u5316\u306f\u3042\u308a\u307e\u305b\u3093\u3002<\/p>\n<p>\u540c\u69d8\u306b\u3001$m$\u304c\u5947\u6570\u306e\u5834\u5408\u306f$l$\u304c\u5076\u6570\u306e\u5834\u5408\u306f$m+l$\u53ca\u3073$n+l$\u306f\u305d\u308c\u305e\u308c\u5947\u6570\u3001\u5076\u6570\u3068\u306a\u308a\u3001$l$\u304c\u5947\u6570\u306e\u5834\u5408\u306f$m+l$\u53ca\u3073$n+l$\u306f\u305d\u308c\u305e\u308c\u5076\u6570\u3001\u5947\u6570\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>\u3059\u306a\u308f\u3061\u3001(\\ref{eq:alofirst})\u5f0f\u306e\u53f3\u8fba\u306e\u88ab\u7a4d\u5206\u95a2\u6570\u306f\u2026<\/p>\n<ul>\n<li>$(1-x^2)^{\\frac{m+n}{2}-1}$\u304c\u5076\u95a2\u6570\u3002<\/li>\n<li>$(x^2-1)$\u3092\u5947\u6570\u56de\u5fae\u5206\u3059\u308b\u95a2\u6570(\u5947\u95a2\u6570)\u3068$(x^2-1)$\u3092\u5076\u6570\u56de\u5fae\u5206\u3059\u308b\u95a2\u6570(\u5076\u95a2\u6570)\u306e\u7a4d($(x^2-1)$\u3092$l$\u56de\u5fae\u5206\u3057\u305f\u7d50\u679c\u306b\u3064\u3044\u3066\u306f<a href=\"https:\/\/sidestory.pandanote.info\/xsquareminusone_diff.html\">\u3053\u3061\u3089\u306e\u8a18\u4e8b<\/a>\u53c2\u7167)\u3002<\/li>\n<\/ul>\n<p>\u306e\u7a4d\u3068\u306a\u308a\u3001\u5f8c\u8005\u306f\u5947\u95a2\u6570\u3067\u3042\u308b\u305f\u3081\u3001(\\ref{eq:alofirst})\u5f0f\u306e\u53f3\u8fba\u306e\u88ab\u7a4d\u5206\u95a2\u6570\u5168\u4f53\u3068\u3057\u3066\u306f\u5947\u95a2\u6570\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>\u3057\u305f\u304c\u3063\u3066\u3001$m+n$\u304c\u5947\u6570\u306e\u5834\u5408\u306b\u306f\u5947\u95a2\u6570\u3092\u533a\u9593$[-1,1]$\u306b\u304a\u3044\u3066\u7a4d\u5206\u3059\u308b\u3053\u3068\u3068\u306a\u308a\u307e\u3059\u306e\u3067\u3001(\\ref{eq:alofirst})\u5f0f\u306e\u53f3\u8fba\u306e\u5024\u306f0\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n<h3>$m+n$\u304c\u5076\u6570\u306e\u3068\u304d<\/h3>\n<p>(\\ref{eq:alofirst})\u5f0f\u306e\u53f3\u8fba\u306e\u88ab\u7a4d\u5206\u95a2\u6570\u306f$m,n$\u306e\u3064\u3044\u3066\u306e\u5bfe\u79f0\u5f0f\u3068\u306a\u3063\u3066\u3044\u308b\u3053\u3068\u3068\u3001Legendre\u306e\u966a\u591a\u9805\u5f0f\u306e\u5b9a\u7fa9\u3088\u308a$l$\u306f$m,n$\u306e\u3046\u3061\u306e\u5927\u304d\u3044\u65b9\u306e\u5024\u4ee5\u4e0a\u306e\u5024\u3067\u3042\u308b\u305f\u3081\u3001\u4e00\u822c\u6027\u3092\u5931\u3046\u3053\u3068\u306a\u304f$m \\le n \\le l$\u3068\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n<p>\u3053\u3053\u304b\u3089\u306f\u7d19\u9762\u306e\u9762\u7a4d\u306e\u90fd\u5408\u4e0a\u3001(\\ref{eq:orthogonality})\u5f0f\u306e\u5de6\u8fba\u306e\u3046\u3061\u3001\u88ab\u7a4d\u5206\u95a2\u6570\u306e\u90e8\u5206\u3092$I_{l}^{mn}$\u3068\u304a\u304d\u307e\u3059\u3002\u305d\u3057\u3066\u3001$m+n$\u304c\u5076\u6570\u3067\u3042\u308b\u3053\u3068\u304b\u3089$(-1)^{m+n}=1$\u3067\u3042\u308b\u3053\u3068\u7b49\u3082\u8003\u616e\u3057\u3064\u3064\u3001(\\ref{eq:alofirst})\u5f0f\u3092\u90e8\u5206\u7a4d\u5206\u3092\u7528\u3044\u3066\u5909\u5f62\u3059\u308b\u3068\u2026<br \/>\n\\begin{align}<br \/>\nI_{l}^{mn} &#038;= \\int_{-1}^{1}\\left[\\frac{d^{m+l-1}}{dx^{m+l-1}}(x^2-1)^l\\right]^{\\prime}(1-x^2)^{\\frac{m+n}{2}-1}\\frac{d^{n+l}}{dx^{n+l}}(x^2-1)^ldx \\nonumber \\cr<br \/>\n&#038;= \\left[ (1-x^2)^{\\frac{m+n}{2}-1}\\frac{d^{m+l-1}}{dx^{m+l-1}}(x^2-1)^l\\frac{d^{n+l}}{dx^{n+l}}(x^2-1)^l \\right]_{-1}^{1} \\nonumber \\cr<br \/>\n&#038;{} &#8211; \\int_{-1}^{1}\\frac{d^{m+l-1}}{dx^{m+l-1}}(x^2-1)^l\\frac{d}{dx}\\left[(1-x^2)^{\\frac{m+n}{2}-1}\\frac{d^{n+l}}{dx^{n+l}}(x^2-1)^l\\right]dx \\label{eq:alosecond}<br \/>\n\\end{align}<br \/>\n(\\ref{eq:alosecond})\u5f0f\u306e\u53f3\u8fba\u7b2c1\u9805\u306e\u62ec\u5f27\u5185\u306b\u306f$1\u2212x^2$\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:alosecond})\u5f0f\u306e\u53f3\u8fba\u7b2c1\u9805\u306f0\u3068\u306a\u308a\u307e\u3059\u3002\u3088\u3063\u3066\u3001(\\ref{eq:alothird})\u5f0f\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<br \/>\n\\begin{align}<br \/>\nI_{l}^{mn} &#038;= \\, -\\int_{-1}^{1}\\frac{d^{m+l-1}}{dx^{m+l-1}}(x^2-1)^l\\frac{d}{dx}\\left[(1-x^2)^{\\frac{m+n}{2}-1}\\frac{d^{n+l}}{dx^{n+l}}(x^2-1)^l\\right]dx\\label{eq:alothird}<br \/>\n\\end{align}<br \/>\n\u3053\u306e\u7bc0\u306e\u3053\u3053\u307e\u3067\u306e\u5909\u5f62\u3068\u540c\u69d8\u306e\u5909\u5f62\u3092\u3042\u3068$m\u22122$\u56de\u7e70\u308a\u8fd4\u3059\u3068\u3001<br \/>\n\\begin{align}<br \/>\nI_{l}^{mn} &#038;= (-1)^{m-1}\\int_{-1}^{1}\\frac{d^{l+1}}{dx^{l+1}}(x^2-1)^l\\frac{d^{m-1}}{dx^{m-1}}\\left[(1-x^2)^{\\frac{m+n}{2}-1}\\frac{d^{n+l}}{dx^{n+l}}(x^2-1)^l\\right]dx \\label{eq:alofourth}<br \/>\n\\end{align}<br \/>\n\u3068\u5909\u5f62\u3067\u304d\u307e\u3059\u3002<\/p>\n<h3>$m$\u56de\u76ee\u306e\u90e8\u5206\u7a4d\u5206\u3002<\/h3>\n<p>$m-1$\u56de\u76ee\u306e\u90e8\u5206\u7a4d\u5206\u306e\u6b21\u306f$m$\u56de\u76ee\u306e\u90e8\u5206\u7a4d\u5206\u306b\u306a\u308a\u307e\u3059\u306e\u3067\u3001(\\ref{eq:alofourth})\u5f0f\u3092\u7528\u3044\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5909\u5f62\u3057\u3066\u307f\u307e\u3059\u3002<br \/>\n\\begin{align}<br \/>\nI_{l}^{mn} &#038;= \\left[(-1)^{m-1}\\frac{d^{l}}{dx^{l}}(x^2-1)^l\\frac{d^{m-1}}{dx^{m-1}}\\left[(1-x^2)^{\\frac{m+n}{2}-1}\\frac{d^{n+l}}{dx^{n+l}}(x^2-1)^l\\right]\\right]_{-1}^{1} \\nonumber \\cr<br \/>\n&#038;+(-1)^m\\int_{-1}^{1}\\frac{d^{l}}{dx^{l}}(x^2-1)^l\\frac{d^{m}}{dx^{m}}\\left[(1-x^2)^{\\frac{m+n}{2}-1}\\frac{d^{n+l}}{dx^{n+l}}(x^2-1)^l\\right]dx \\label{eq:aloatmthpartial}<br \/>\n\\end{align}<br \/>\n(\\ref{eq:aloatmthpartial})\u5f0f\u306e\u53f3\u8fba\u7b2c1\u9805\u306e\u56e0\u5b50\u306e\u3046\u3061\u3001$\\displaystyle\\frac{d^{l}}{dx^{l}}(x^2-1)^l$\u306f$x = \\pm 1$\u306b\u304a\u3044\u30660\u3068\u306f\u306a\u308a\u307e\u305b\u3093\u304c\u3001$m+n$\u304c\u5076\u6570\u3067\u3042\u308b\u3053\u3068\u304b\u3089$m$\u53ca\u3073$n$\u306f\u3068\u3082\u306b\u5076\u6570\u3067\u3042\u308b\u304b\u3001\u307e\u305f\u306f\u5947\u6570\u3067\u3042\u308b\u304b\u306e\u3044\u305a\u308c\u304b\u3067\u3059\u3002<\/p>\n<p>\u3055\u3089\u306b$m \\le n$\u3068\u3057\u3066\u3044\u307e\u3059\u306e\u3067\u3001$\\displaystyle\\frac{m+n}{2}-1 \\ge m-1$\u3067\u3059\u304c\u3001(\\ref{eq:aloatmthpartial})\u5f0f\u306e\u53f3\u8fba\u7b2c1\u9805\u306e\u56e0\u5b50\u306e\u3046\u3061\u3001$m-1$\u56de\u5fae\u5206\u306e\u5bfe\u8c61\u3068\u306a\u308b\u5f0f\u306f$\\displaystyle\\frac{m+n}{2}-1 \\gt m-1$\u3067\u3042\u308b\u5834\u5408\u306b\u306f\u3001$m-1$\u56de\u5fae\u5206\u3057\u305f\u7d50\u679c\u5f97\u3089\u308c\u308b\u5f0f\u304c$(1-x^2)^{\\frac{m+n}{2}-1}$\u7531\u6765\u306e$1-x^2$\u3092\u56e0\u5b50\u3068\u3057\u3066\u542b\u3080\u305f\u3081\u3001$x = \\pm 1$\u306b\u304a\u3044\u30660\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>\u4e00\u65b9\u3001$\\displaystyle\\frac{m+n}{2}-1 = m-1$\u3001\u3059\u306a\u308f\u3061$m=n$\u306e\u5834\u5408\u306b\u306f$1-x^2$\u306f\u56e0\u5b50\u3068\u3057\u3066\u542b\u307e\u308c\u305a\u3001\u304b\u3064$\\displaystyle\\frac{d^{n+l}}{dx^{n+l}}(x^2-1)^l$\u3082$x^2-1$\u3092\u56e0\u5b50\u3068\u3057\u3066\u542b\u307e\u306a\u3044\u305f\u3081\u3001$x = \\pm 1$\u306b\u304a\u3044\u30660\u4ee5\u5916\u306e\u5024\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>\u305d\u3053\u3067\u3001(\\ref{eq:aloatmthpartial})\u5f0f\u306fKronecker\u306e\u30c7\u30eb\u30bf\u3092\u7528\u3044\u3066\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<br \/>\n\\begin{align}<br \/>\nI_{l}^{mn} &#038;= \\delta_{mn}\\left[(-1)^{m-1}\\frac{d^{l}}{dx^{l}}(x^2-1)^l\\frac{d^{m-1}}{dx^{m-1}}\\left[(1-x^2)^{m-1}\\frac{d^{m+l}}{dx^{m+l}}(x^2-1)^l\\right]\\right]_{-1}^{1} \\nonumber \\cr<br \/>\n&#038;+(-1)^m\\int_{-1}^{1}\\frac{d^{l}}{dx^{l}}(x^2-1)^l\\frac{d^{m}}{dx^{m}}\\left[(1-x^2)^{\\frac{m+n}{2}-1}\\frac{d^{n+l}}{dx^{n+l}}(x^2-1)^l\\right]dx \\label{eq:aloatmthpartialdelta}<br \/>\n\\end{align}<\/p>\n<h3>$m+1$\u56de\u76ee\u4ee5\u964d\u306e\u90e8\u5206\u7a4d\u5206\u3002<\/h3>\n<p>(\\ref{eq:aloatmthpartialdelta})\u5f0f\u306e\u53f3\u8fba\u7b2c1\u9805\u306f<br \/>\n\\begin{align}<br \/>\nJ_{l}^{m} &#038;= \\left[(-1)^{m-1}\\frac{d^{l}}{dx^{l}}(x^2-1)^l\\frac{d^{m-1}}{dx^{m-1}}\\left[(1-x^2)^{m-1}\\frac{d^{m+l}}{dx^{m+l}}(x^2-1)^l\\right]\\right]_{-1}^{1} \\label{eq:jmn}<br \/>\n\\end{align}<br \/>\n\u3068\u304a\u3044\u3066\u8a08\u7b97\u81ea\u4f53\u306f\u5f8c\u56de\u3057\u306b\u3057\u3064\u3064\u3001(\\ref{eq:aloatmthpartialdelta})\u5f0f\u306e\u53f3\u8fba\u7b2c2\u9805\u30921\u56de\u3060\u3051\u90e8\u5206\u7a4d\u5206\u3057\u307e\u3059\u3002<\/p>\n<p>\u3059\u308b\u3068\u2026<br \/>\n\\begin{align}<br \/>\nI_{l}^{mn} &#038;= \\delta_{mn}J_{l}^{m} \\nonumber \\cr<br \/>\n&#038;+\\left[(-1)^m\\frac{d^{l-1}}{dx^{l-1}}(x^2-1)^l\\frac{d^{m}}{dx^{m}}\\left[(1-x^2)^{\\frac{m+n}{2}-1}\\frac{d^{n+l}}{dx^{n+l}}(x^2-1)^l\\right]\\right]_{-1}^{1} \\nonumber \\cr<br \/>\n&#038;+(-1)^{m+1}\\int_{-1}^{1}\\frac{d^{l-1}}{dx^{l-1}}(x^2-1)^l\\frac{d^{m+1}}{dx^{m+1}}\\left[(1-x^2)^{\\frac{m+n}{2}-1}\\frac{d^{n+l}}{dx^{n+l}}(x^2-1)^l\\right]dx \\label{eq:aloataftermthpartial}<br \/>\n\\end{align}<br \/>\n\u3068\u306a\u308a\u307e\u3059\u304c\u3001(\\ref{eq:aloataftermthpartial})\u5f0f\u306e\u53f3\u8fba\u7b2c1\u9805\u306e$\\displaystyle\\frac{d^{l-1}}{dx^{l-1}}(x^2-1)^l$\u306b\u306f$x^2-1$\u304c\u56e0\u5b50\u3068\u3057\u3066\u542b\u307e\u308c\u307e\u3059\u306e\u3067\u3001$x = \\pm 1$\u306b\u304a\u3044\u30660\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>\u3057\u305f\u304c\u3063\u3066\u2026<br \/>\n\\begin{align}<br \/>\nI_{l}^{mn} &#038;= \\delta_{mn}J_{l}^{m} \\nonumber \\cr<br \/>\n&#038;+(-1)^{m+1}\\int_{-1}^{1}\\frac{d^{l-1}}{dx^{l-1}}(x^2-1)^l\\frac{d^{m+1}}{dx^{m+1}}\\left[(1-x^2)^{\\frac{m+n}{2}-1}\\frac{d^{n+l}}{dx^{n+l}}(x^2-1)^l\\right]dx \\label{eq:aloataftermthpartialresult}<br \/>\n\\end{align}<br \/>\n\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>\u4e0a\u8a18\u3068\u540c\u69d8\u306e\u90e8\u5206\u7a4d\u5206\u306f$l-1$\u56de\u7e70\u308a\u8fd4\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u306e\u3067\u3001<br \/>\n\\begin{align}<br \/>\nI_{l}^{mn} &#038;= \\delta_{mn}J_{l}^{m} \\nonumber \\cr<br \/>\n&#038;+(-1)^{l+m}\\int_{-1}^{1}(x^2-1)^l\\frac{d^{l+m}}{dx^{l+m}}\\left[(1-x^2)^{\\frac{m+n}{2}-1}\\frac{d^{n+l}}{dx^{n+l}}(x^2-1)^l\\right]dx \\label{eq:aloatafterlplusmthpartial}<br \/>\n\\end{align}<br \/>\n\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n<h3>\u3053\u3053\u3067\u3001Leibniz\u306e\u516c\u5f0f\u306e\u767b\u5834\u3067\u3059\u3002<\/h3>\n<p>(\\ref{eq:aloatafterlplusmthpartial})\u5f0f\u306e\u53f3\u8fba\u7b2c2\u9805\u306fLeibniz\u306e\u516c\u5f0f(<a href=\"https:\/\/pandanote.info\/?p=4757\">\u3053\u306e\u8a18\u4e8b<\/a>\u306e(7)\u5f0f)\u3092\u5229\u7528\u3059\u308b\u3068\u3001<br \/>\n\\begin{align}<br \/>\nI_{l}^{mn} &#038;= \\delta_{mn}J_{l}^{m} \\nonumber \\cr<br \/>\n&#038;+ (-1)^{m+l}\\int_{-1}^{1}(x^2-1)^l\\cdot\\nonumber\\cr<br \/>\n&#038;{} \\left[{\\color[RGB]{30,62,138}\\sum_{k=0}^{m+l}\\begin{pmatrix}<br \/>\nm+l\\cr<br \/>\nk<br \/>\n\\end{pmatrix}\\frac{d^{k}}{dx^{k}}(1-x^2)^{\\frac{m+n}{2}-1}\\frac{d^{n+m+2l-k}}{dx^{n+m+2l-k}}(x^2-1)^l}\\right]dx\\label{eq:alofifth}<br \/>\n\\end{align}<br \/>\n(\\ref{eq:alofifth})\u5f0f\u306e\u62ec\u5f27\u5185\u306e\u90e8\u5206(\u9752\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\u3001<br \/>\n\\begin{align}<br \/>\nk &#038;\\le m+n-2 \\label{eq:alononzerofirst}<br \/>\n\\end{align}<br \/>\n\u53ca\u3073\u3001<br \/>\n\\begin{align}<br \/>\nn+m+2l-k &#038;\\le 2l \\label{eq:alononzerosecond}<br \/>\n\\end{align}<br \/>\n\u3067\u3059\u306e\u3067\u3001(\\ref{eq:alononzerofirst})\u5f0f\u53ca\u3073(\\ref{eq:alononzerosecond})\u5f0f\u3092$k$\u306b\u3064\u3044\u3066\u89e3\u3044\u3066\u307e\u3068\u3081\u308b\u3068\u3001<br \/>\n\\begin{align}<br \/>\nn+m &#038;\\le k \\le m+n-2 \\label{eq:alononzerosolution}<br \/>\n\\end{align}<br \/>\n\u3068\u306a\u308a\u307e\u3059\u304c\u3001\u3053\u308c\u3092\u6e80\u305f\u3059$k$\u306f\u5b58\u5728\u3057\u307e\u305b\u3093\u306e\u3067\u3001(\\ref{eq:aloatafterlplusmthpartial})\u5f0f\u306e\u53f3\u8fba\u7b2c2\u9805\u306f0\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>\u3088\u3063\u3066\u3001<br \/>\n\\begin{align}<br \/>\nI_{l}^{mn} &#038;= \\delta_{mn}J_{l}^{m} \\label{eq:alosixth}<br \/>\n\\end{align}<br \/>\n\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n<p>\u3061\u3087\u3063\u3068\u7c21\u5358\u306b\u306a\u308a\u307e\u3057\u305f\u306d\u3002\ud83d\udc3c<\/p>\n<h3>$J_{l}^{m}$\u306e\u8a08\u7b97\u3002<\/h3>\n<p>\u3053\u3053\u304b\u3089\u306f\u3001(\\ref{eq:jmn})\u5f0f\u306e\u53f3\u8fba\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002<\/p>\n<p>$(x^2-1)^l$\u3092$n$\u56de\u5fae\u5206\u3057\u305f\u7d50\u679c\u5f97\u3089\u308c\u308b\u591a\u9805\u5f0f\u306f$n$\u306e\u5076\u5947\u306b\u3088\u308a\u3001\u4ee5\u4e0b\u306e\u3044\u305a\u308c\u304b\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n<ul>\n<li>$n$\u304c\u5076\u6570\u306e\u5834\u5408\u306b\u306f\u3001\u6700\u9ad8\u6b21\u306e\u9805\u306e\u6b21\u6570\u304c\u5076\u6570\u3067\u3001\u304b\u3064\u5076\u6570\u6b21\u306e\u9805\u3057\u304b\u6301\u305f\u306a\u3044\u591a\u9805\u5f0f\u3002<\/li>\n<li>$n$\u304c\u5947\u6570\u306e\u5834\u5408\u306b\u306f\u3001\u6700\u9ad8\u6b21\u306e\u9805\u306e\u6b21\u6570\u304c\u5947\u6570\u3067\u3001\u304b\u3064\u5947\u6570\u6b21\u306e\u9805\u3057\u304b\u6301\u305f\u306a\u3044\u591a\u9805\u5f0f\u3002<\/li>\n<\/ul>\n<p>\u307e\u305f\u3001\u5076\u6570\u6b21\u306e\u9805\u307e\u305f\u306f\u5947\u6570\u6b21\u306e\u9805\u3057\u304b\u6301\u305f\u306a\u3044\u591a\u9805\u5f0f\u540c\u58eb\u306e\u7a4d\u306f\u5076\u6570\u6b21\u306e\u9805\u3057\u304b\u6301\u305f\u306a\u3044\u591a\u9805\u5f0f\u306b\u306a\u308a\u3001\u5076\u6570\u6b21\u306e\u9805\u3057\u304b\u6301\u305f\u306a\u3044\u591a\u9805\u5f0f\u3068\u5947\u6570\u6b21\u306e\u9805\u3057\u304b\u6301\u305f\u306a\u3044\u591a\u9805\u5f0f\u306e\u7a4d\u306f\u5947\u6570\u6b21\u306e\u9805\u3057\u304b\u6301\u305f\u306a\u3044\u591a\u9805\u5f0f\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>\u305d\u3053\u3067\u3001(\\ref{eq:jmn})\u5f0f\u306e\u53f3\u8fba\u306e\u6700\u9ad8\u6b21\u306e\u9805\u306e\u6b21\u6570\u3092\u8abf\u3079\u308b\u3068\u3001<\/p>\n<ul>\n<li>$(1-x^2)^{m-1}\\displaystyle\\frac{d^{m+l}}{dx^{m+l}}(x^2-1)^l$\u306e\u6700\u9ad8\u6b21\u306e\u9805\u306e\u6b21\u6570\u306f$l+m-2$\u3067\u3001\u3053\u308c\u3092$m-1$\u56de\u5fae\u5206\u3059\u308b\u3068\u3001\u6700\u9ad8\u6b21\u306e\u9805\u306e\u6b21\u6570\u306f$l-1$\u3002<\/li>\n<li>\u524d\u9805\u306e\u5f0f\u306b\u3001$(x^2-1)^l$\u3092\u304b\u3051\u3066\u304b\u3089$l$\u56de\u5fae\u5206\u3057\u305f\u7d50\u679c\u5f97\u3089\u308c\u308b\u591a\u9805\u5f0f\u306e\u6700\u9ad8\u6b21\u306e\u9805\u306e\u6b21\u6570\u306f$l-1+2l-l=2l-1$\u3002<\/li>\n<\/ul>\n<p>\u3068\u306a\u308a\u307e\u3059\u306e\u3067\u3001(\\ref{eq:jmn})\u5f0f\u306e\u53f3\u8fba\u306e\u6700\u9ad8\u6b21\u306e\u9805\u306e\u6b21\u6570\u306f\u5947\u6570\u3067\u3042\u308a\u3001\u5f0f\u81ea\u4f53\u3082\u5947\u95a2\u6570\u3068\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n<p>\u3057\u305f\u304c\u3063\u3066\u3001$J_{l}^m$\u306f<br \/>\n\\begin{align}<br \/>\ng(x) &#038;= (-1)^{m-1}{\\color[RGB]{30,62,138}\\frac{d^{l}}{dx^{l}}(x^2-1)^l}{\\color[RGB]{0,127,0}\\frac{d^{m-1}}{dx^{m-1}}\\left[(1-x^2)^{m-1}\\frac{d^{m+l}}{dx^{m+l}}(x^2-1)^l\\right]} \\label{eq:jmngx}<br \/>\n\\end{align}<br \/>\n\u3068\u304a\u304f\u3068\u3001<br \/>\n\\begin{align}<br \/>\nJ_{l}^m &#038;= 2g(1) \\label{eq:jmngone}<br \/>\n\\end{align}<br \/>\n\u3068\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n<p>(\\ref{eq:jmngx})\u5f0f\u306f$\\displaystyle\\frac{d^{l}}{dx^{l}}(x^2-1)^l$(\u9752\u8272\u306e\u90e8\u5206)\u3068$\\displaystyle\\frac{d^{m-1}}{dx^{m-1}}\\left[(1-x^2)^{m-1}\\displaystyle\\frac{d^{m+l}}{dx^{m+l}}(x^2-1)^l\\right]$(\u7dd1\u8272\u306e\u90e8\u5206)\u306e\u7a4d\u3067\u3059\u306e\u3067\u3001\u3053\u308c\u3089\u306f\u5225\u3005\u306b\u8a08\u7b97\u3057\u3066\u3001\u305d\u308c\u3089\u306e\u7a4d\u3092\u8003\u3048\u308b\u3053\u3068\u3067$g(1)$\u306e\u5024\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n<p>\u9752\u8272\u306e\u90e8\u5206\u306b\u3064\u3044\u3066\u306f\u2026<br \/>\n\\begin{align}<br \/>\n\\frac{d^{l}}{dx^{l}}(x^2-1)^l &#038;= \\frac{d^{l}}{dx^{l}}(x+1)^l(x-1)^l \\nonumber \\cr<br \/>\n&#038;= \\sum_{k=0}^{l}\\begin{pmatrix}<br \/>\nl\\cr<br \/>\nk<br \/>\n\\end{pmatrix}\\frac{d^{l-k}}{dx^{l-k}}(x+1)^l\\frac{d^{k}}{dx^{k}}(x-1)^l \\nonumber \\cr<br \/>\n&#038;= \\sum_{k=0}^{l}\\begin{pmatrix}<br \/>\nl\\cr<br \/>\nk<br \/>\n\\end{pmatrix}\\frac{l!}{k!}(x+1)^k\\frac{l!}{(l-k)!}(x-1)^{l-k}\\label{eq:jmnblue}<br \/>\n\\end{align}<br \/>\n\u3068\u306a\u308a\u307e\u3059\u304c\u3001(\\ref{eq:jmnblue})\u5f0f\u306e\u7dcf\u548c\u306e\u8a08\u7b97\u306e\u5bfe\u8c61\u3068\u306a\u308b\u9805\u306e\u3046\u3061\u3001$l=k$\u3067\u306a\u3044\u9805\u306f$x-1$\u3092\u56e0\u5b50\u306b\u6301\u3064\u305f\u3081\u3001$x=1$\u306e\u3068\u304d\u306b\u306f\u3059\u3079\u30660\u3068\u306a\u308a\u307e\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001<br \/>\n\\begin{align}<br \/>\n\\left.\\frac{d^{l}}{dx^{l}}(x^2-1)^l\\right|_{x=1} &#038;= 2^ll! \\label{eq:jmnblueatone}<br \/>\n\\end{align}<br \/>\n\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>\u6b21\u306b\u7dd1\u8272\u306e\u90e8\u5206\u3067\u3059\u304c\u2026<br \/>\n\\begin{align}<br \/>\n\\frac{d^{m-1}}{dx^{m-1}}\\left[(1-x^2)^{m-1}\\frac{d^{m+l}}{dx^{m+l}}(x^2-1)^l\\right] &#038;= \\sum_{k=0}^{m-1}\\left[ \\begin{pmatrix}<br \/>\nm-1\\cr<br \/>\nk<br \/>\n\\end{pmatrix}\\frac{d^{k}}{dx^{k}}(1-x^2)^{m-1}\\right. \\nonumber\\cr<br \/>\n&#038;{} \\left.\\frac{d^{2m+l-1-k}}{dx^{2m+l-1-k}}(x^2-1)^l \\right] \\label{eq:jmngreen}<br \/>\n\\end{align}<br \/>\n\u3068\u5909\u5f62\u3067\u304d\u307e\u3059\u3002\u3053\u3053\u3067\u3001\u7dcf\u548c\u306e\u8a08\u7b97\u306e\u5bfe\u8c61\u3068\u306a\u308b\u9805\u3092\u3088\u30fc\u304f\u898b\u308b\u3068\u3001$k \\lt m-1$\u306e\u5834\u5408\u306b\u306f$\\left.\\displaystyle\\frac{d^{k}}{dx^{k}}(1-x^2)^{m-1}\\right|_{x=1}=0$\u306b\u306a\u308a\u307e\u3059\u3002\u3088\u3063\u3066(\\ref{eq:jmngreen})\u5f0f\u306f\u3001<br \/>\n\\begin{align}<br \/>\n\\frac{d^{m-1}}{dx^{m-1}}\\left[(1-x^2)^{m-1}\\frac{d^{m+l}}{dx^{m+l}}(x^2-1)^l\\right] &#038;= \\frac{d^{m-1}}{dx^{m-1}}(1-x^2)^{m-1}\\frac{d^{m+l}}{dx^{m+l}}(x^2-1)^l \\label{eq:jmngreentwo}<br \/>\n\\end{align}<br \/>\n\u3068\u5909\u5f62\u3067\u304d\u307e\u3059\u3002<\/p>\n<p>(\\ref{eq:jmngreentwo})\u5f0f\u3078\u306e\u5909\u5f62\u306f$k = m-1$\u306e\u5834\u5408\u306e\u5177\u4f53\u7684\u306a$x$\u306e\u5024\u306b\u3064\u3044\u3066\u306f\u691c\u8a0e\u3057\u306a\u3044\u307e\u307e\u884c\u3046\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u304c\u3001(\\ref{eq:jmnblueatone})\u5f0f\u3068($x^2$\u306e\u7b26\u53f7\u306b\u306f\u6ce8\u610f\u304c\u5fc5\u8981\u3067\u3059\u3002)\u540c\u69d8\u306e\u8b70\u8ad6\u306b\u3088\u308a\u3001$x=1$\u306e\u6642\u306b\u306f(\\ref{eq:jmngreentwo})\u5f0f\u53f3\u8fba\u306e\u6700\u521d\u306e\u56e0\u5b50\u306b\u3064\u3044\u3066\u306f\u3001<br \/>\n\\begin{align}<br \/>\n\\left.\\frac{d^{m-1}}{dx^{m-1}}(1-x^2)^{m-1}\\right|_{x=1} &#038;= (-1)^{m-1}2^{m-1}(m-1)! \\label{eq:jmngreenone}<br \/>\n\\end{align}<br \/>\n\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>\u6700\u5f8c\u306b\u6b8b\u308b\u306f$\\displaystyle\\frac{d^{m+l}}{dx^{m+l}}(x^2-1)^l$\u306e\u8a08\u7b97\u3067\u3059\u304c\u3001Leibniz\u306e\u516c\u5f0f\u3092\u7528\u3044\u3066\u3001<br \/>\n\\begin{align}<br \/>\n\\frac{d^{m+l}}{dx^{m+l}}(x^2-1)^l &#038;= \\frac{d^{m+l}}{dx^{m+l}}\\left[(x+1)^l(x-1)^l\\right] \\nonumber \\cr<br \/>\n&#038;= \\sum_{k=0}^{m+l}\\left[\\begin{pmatrix}<br \/>\nm+l\\cr<br \/>\nk<br \/>\n\\end{pmatrix}\\frac{d^{m+l-k}}{dx^{m+l-k}}(x+1)^l\\frac{d^{k}}{dx^{k}}(x-1)^l\\right] \\label{eq:jmngreenthree}<br \/>\n\\end{align}<\/p>\n<p>\u3068\u5c55\u958b\u3067\u304d\u307e\u3059\u3002\u3053\u3053\u3067\u518d\u3073(\\ref{eq:jmngreenthree})\u5f0f\u3092\u3058\u30fc\u3063\u3068\u898b\u308b\u3068\u3001$k \\gt l$\u306e\u5404\u9805\u306f\u3059\u3079\u30660\u3068\u306a\u308b\u3053\u3068\u3068\u3001$k \\lt l$\u306e\u9805\u306f$x=1$\u306e\u3068\u304d\u306b\u306f\u3059\u3079\u30660\u3068\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n<p>\u305d\u3093\u306a\u308f\u3051\u3067\u3001$x=1$\u306e\u3068\u304d\u306b\u306f$k=l$\u306e\u9805\u3060\u3051\u304c\u6b8b\u308a\u307e\u3059\u306e\u3067\u3001(\\ref{eq:jmngreenthree})\u5f0f\u306f\u6b21\u306e\u3088\u3046\u306b\u66f8\u304d\u63db\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n<p>\\begin{align}<br \/>\n\\left.\\frac{d^{m+l}}{dx^{m+l}}(x^2-1)^l\\right|_{x=1} &#038;= \\left.\\begin{pmatrix}<br \/>\nm+l\\cr<br \/>\nl<br \/>\n\\end{pmatrix}\\frac{d^{m}}{dx^{m}}(x+1)^l\\frac{d^{l}}{dx^{l}}(x-1)^l\\right|_{x=1} \\nonumber \\cr<br \/>\n&#038;= \\frac{(l+m)!}{l!m!}\\frac{l!}{(l-m)!}2^{l-m}(l!) \\nonumber \\cr<br \/>\n&#038;= \\frac{(l+m)!l!2^{l-m}l!}{l!m!(l-m)!} \\nonumber \\cr<br \/>\n&#038;= \\frac{(l+m)!l!2^{l-m}}{m!(l-m)!} \\label{eq:jmngreentfourth}<br \/>\n\\end{align}<\/p>\n<p>\u3068\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002<\/p>\n<h3>\u3053\u3053\u3067\u3044\u3063\u305f\u3093\u4e2d\u7de0\u3081\u3002<\/h3>\n<p>\u3053\u3053\u307e\u3067\u306e\u7d50\u679c\u3092\u307e\u3068\u3081\u308b\u3068\u3001(\\ref{eq:jmngx})\u3001(\\ref{eq:jmngone})\u3001(\\ref{eq:jmnblueatone})\u3001(\\ref{eq:jmngreenone})\u53ca\u3073(\\ref{eq:jmngreentfourth})\u5f0f\u3088\u308a\u3001$J_{l}^m$\u306f\u3001<\/p>\n<p>\\begin{align}<br \/>\nJ_{l}^m &#038;= 2g(1) \\nonumber \\cr<br \/>\n&#038;= 2^ll!2(-1)^{m-1}\\cdot(-1)^{m-1}2^{m-1}(m-1)!\\frac{(l+m)!l!2^{l-m}}{m!(l-m)!} \\nonumber \\cr<br \/>\n&#038;= 2^{2l-m+m-1+1}\\frac{(l+m)!(l!)^2(m-1)!}{m!(l-m)!} \\nonumber \\cr<br \/>\n&#038;= 2^{2l}\\frac{(l+m)!(l!)^2}{m(l-m)!} \\label{eq:jmnfinal}<br \/>\n\\end{align}<br \/>\n\u3068\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n<p>$J_{l}^m$\u306f$I_{l}^{mn}$\u3067$m=n$\u3068\u7f6e\u3044\u305f\u3068\u304d\u306e\u5024\u3067\u3042\u308a\u3001$I_{l}^{mn}$\u306f$\\displaystyle\\int_{-1}^{1}\\frac{P_l^m(x)P_l^n(x)}{1-x^2}dx$\u306b$2^{2l}(l!)^2$\u3092\u304b\u3051\u305f\u3082\u306e\u306b\u76f8\u5f53\u3057\u307e\u3059\u306e\u3067\u3001<br \/>\n\\begin{align}<br \/>\n\\left.\\int_{-1}^{1}\\frac{P_l^m(x)P_l^n(x)}{1-x^2}dx\\right|_{m=n} &#038;= \\frac{J_{l}^m}{2^{2l}(l!)^2} \\nonumber \\cr<br \/>\n&#038;= \\frac{2^{2l}(l+m)!(l!)^2}{2^{2l}(l!)^2m(l-m)!} \\nonumber \\cr<br \/>\n&#038;= \\frac{(l+m)!}{m(l-m)!} \\label{eq:pmfinal}<br \/>\n\\end{align}<br \/>\n\u3068\u306a\u308a\u3001(\\ref{eq:orthogonality})\u5f0f\u53f3\u8fba\u306e$m = n \\ne 0$\u3068\u4e00\u81f4\u3059\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n<p>\u3044\u308d\u3044\u308d\u3068\u767b\u5834\u3057\u307e\u3057\u305f\u304c\u3001\u304b\u306a\u308a\u7c21\u5358\u306a\u5f0f\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<h3>$m=n=0$\u306e\u5834\u5408<\/h3>\n<h4>\u307e\u305a\u3001\u4ee3\u5165\u3057\u3066\u5f0f\u5909\u5f62\u3057\u3066\u307f\u307e\u3059\u3002<\/h4>\n<p>(\\ref{eq:alofirst})\u5f0f\u306b$m=n=0$\u3092\u4ee3\u5165\u3059\u308b\u3068\u2026<br \/>\n\\begin{align}<br \/>\n\\int_{-1}^{1}\\frac{[P_l^0(x)]^2}{1-x^2}dx &#038;= \\frac{1}{(2^ll!)^2}\\int_{-1}^{1}(1-x^2)^{-1}\\left[\\frac{d^{l}}{dx^{l}}(x^2-1)^l\\right]^2dx \\label{eq:alodoublezero}<br \/>\n\\end{align}<br \/>\n\u3068\u306a\u308a\u307e\u3059\u304c\u3001\u3053\u306e\u5f8c\u306e\u8a71\u306e\u5c55\u958b\u306e\u90fd\u5408\u4e0a\u3001$K_{l} = \\displaystyle\\int_{-1}^{1}\\frac{[P_l^0(x)]^2}{1-x^2}dx$\u3068\u304a\u304d\u3064\u3064\u3001<br \/>\n\\begin{align}<br \/>\nK_{l} &#038;= \\frac{1}{(2^ll!)^2}\\int_{-1}^{1}(1-x^2)^{-1}\\left[\\frac{d^{l}}{dx^{l}}(x^2-1)^l\\right]^2dx \\nonumber \\cr<br \/>\n&#038;= \\frac{1}{(2^ll!)^2}\\int_{-1}^{1}(1-x^2)^{-1}\\left[\\frac{d^{l}}{dx^{l}}[(-1)(1-x^2)]^l\\right]^2dx\\nonumber \\cr<br \/>\n&#038;= \\frac{1}{(2^ll!)^2}\\int_{-1}^{1}(1-x^2)^{-1}\\left[\\frac{d^{l}}{dx^{l}}(-1)^l(1-x^2)^l\\right]^2dx \\nonumber \\cr<br \/>\n&#038;= \\frac{1}{(2^ll!)^2}\\int_{-1}^{1}(1-x^2)^{-1}\\left[(-1)^l\\frac{d^{l}}{dx^{l}}(1-x^2)^l\\right]^2dx \\nonumber \\cr<br \/>\n&#038;= \\frac{1}{(2^ll!)^2}\\int_{-1}^{1}(1-x^2)^{-1}(-1)^{2l}\\left[\\frac{d^{l}}{dx^{l}}(1-x^2)^l\\right]^2dx \\nonumber \\cr<br \/>\n&#038;= \\frac{1}{(2^ll!)^2}\\int_{-1}^{1}(1-x^2)^{-1}\\left[\\frac{d^{l}}{dx^{l}}(1-x^2)^l\\right]^2dx \\label{eq:alodoubletwo}<br \/>\n\\end{align}<br \/>\n\u3068\u5909\u5f62\u3057\u307e\u3059\u3002<\/p>\n<p>$\\displaystyle\\frac{d^{l}}{dx^{l}}(1-x^2)^l$\u306f$l$\u306e\u5076\u5947\u306b\u3088\u308a\u5076\u95a2\u6570\u307e\u305f\u306f\u5947\u95a2\u6570\u306b\u306a\u308a\u307e\u3059\u304c\u3001\u3053\u308c\u3092\u81ea\u4e57\u3057\u305f\u3082\u306e\u306f$l$\u306e\u5076\u5947\u306b\u95a2\u4fc2\u306a\u304f\u5076\u95a2\u6570\u306b\u306a\u308a\u307e\u3059\u3002\u3088\u3063\u3066\u3001(\\ref{eq:alodoublezero})\u3001(\\ref{eq:alodoubletwo})\u306e\u5404\u5f0f\u306e\u53f3\u8fba\u306f\u3068\u3082\u306b\u5076\u95a2\u6570\u306b\u306a\u308a\u307e\u3059\u306e\u3067\u3001<br \/>\n\\begin{align}<br \/>\nK_{l} &#038;= 2\\frac{1}{(2^ll!)^2}\\int_{0}^{1}(1-x^2)^{-1}\\left[\\frac{d^{l}}{dx^{l}}(1-x^2)^l\\right]^2dx\\label{eq:eqalodoublethree}<br \/>\n\\end{align}<br \/>\n\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>(\\ref{eq:eqalodoublethree})\u5f0f\u3092\u3082\u3046\u3061\u3087\u3044\u8a08\u7b97\u3057\u3066\u307f\u307e\u3059\u3002\u3059\u308b\u3068\u2026<br \/>\n\\begin{align}<br \/>\nK_{l} &#038;= 2\\frac{1}{(2^ll!)^2}\\int_{0}^{1}(1-x^2)^{-1}\\left[\\sum_{k=0}^l\\begin{pmatrix}<br \/>\nl\\cr<br \/>\nk<br \/>\n\\end{pmatrix}\\frac{d^{k}}{dx^{k}}(x+1)^l\\frac{d^{l-k}}{dx^{l-k}}(1-x)^l\\right]^2dx \\nonumber \\cr<br \/>\n&#038;= 2\\frac{1}{(2^ll!)^2}\\int_{0}^{1}(1-x^2)^{-1}\\left[\\sum_{k=0}^l\\begin{pmatrix}<br \/>\nl\\cr<br \/>\nk<br \/>\n\\end{pmatrix}\\frac{l!}{dx^{(l-k)!}}(x+1)^{l-k}\\frac{l!}{(l-k)!}(1-x)^k\\right]^2dx \\nonumber \\cr<br \/>\n&#038;= \\frac{1}{2^{2l-1}l!}\\int_{0}^{1}(1-x^2)^{-1}\\left[\\sum_{k=0}^l\\begin{pmatrix}<br \/>\nl\\cr<br \/>\nk<br \/>\n\\end{pmatrix}^2(x+1)^{l-k}(1-x)^k\\right]^2dx \\label{eq:eqalodoublefourth}<br \/>\n\\end{align}<br \/>\n\u3068\u306a\u308a\u3001\u5fae\u5206\u306e\u8a08\u7b97\u3092\u884c\u3046\u3053\u3068\u304c\u3067\u304d\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n<h4>\u5f0f\u306e\u8a55\u4fa1\u3092\u884c\u3044\u307e\u3059\u3002<\/h4>\n<p>(\\ref{eq:eqalodoublefourth})\u5f0f\u3092\u3088\u30fc\u304f\u898b\u308b\u3068\u3001\u7dcf\u548c\u306e\u8a08\u7b97\u306e\u5bfe\u8c61\u3068\u306a\u3063\u3066\u3044\u308b\u5404\u9805\u306f$x \\in [0,1]$\u306b\u304a\u3044\u3066\u3059\u3079\u3066\u8ca0\u3067\u306a\u3044\u5024\u3092\u3068\u308a\u307e\u3059\u3002\u3088\u3063\u3066\u3001<br \/>\n\\begin{align}<br \/>\nK_{l} &#038;= \\frac{1}{2^{2l-1}l!}\\int_{0}^{1}(1-x^2)^{-1}\\left[\\sum_{k=0}^l\\begin{pmatrix}<br \/>\nl\\cr<br \/>\nk<br \/>\n\\end{pmatrix}^2(1+x)^{l-k}(1-x)^k\\right]^2dx \\nonumber \\cr<br \/>\n&#038;\\ge \\frac{1}{2^{2l-1}l!}\\int_{0}^{1}(1-x^2)^{-1}\\sum_{k=0}^l\\left[\\begin{pmatrix}<br \/>\nl\\cr<br \/>\nk<br \/>\n\\end{pmatrix}^2(1+x)^{l-k}(1-x)^k\\right]^2dx \\nonumber \\cr<br \/>\n&#038;= \\frac{1}{2^{2l-1}l!}\\int_{0}^{1}(1-x^2)^{-1}\\sum_{k=0}^l\\begin{pmatrix}<br \/>\nl\\cr<br \/>\nk<br \/>\n\\end{pmatrix}^4(1+x)^{2(l-k)}(1-x)^{2k}dx \\nonumber \\cr<br \/>\n&#038;= \\frac{1}{2^{2l-1}l!}\\int_{0}^{1}\\sum_{k=0}^l\\begin{pmatrix}<br \/>\nl\\cr<br \/>\nk<br \/>\n\\end{pmatrix}^4(1+x)^{2(l-k)-1}(1-x)^{2k-1}dx\\label{eq:eqalodoubleevaluate}<br \/>\n\\end{align}<br \/>\n\u3068\u8a55\u4fa1\u3067\u304d\u307e\u3059\u3002<br \/>\n(\\ref{eq:eqalodoubleevaluate})\u5f0f\u306e\u53f3\u8fba\u3067\u7dcf\u548c\u306e\u8a08\u7b97\u306e\u5bfe\u8c61\u3068\u306a\u3063\u3066\u3044\u308b\u9805\u306e\u3046\u3061\u3001$k\\ne 0$\u306e\u9805\u306b\u3064\u3044\u3066\u306f$x$\u306e$2l-2$\u6b21\u306e\u591a\u9805\u5f0f\u306b\u306a\u308a\u307e\u3059\u306e\u3067\u3001$x \\in [0,1]$\u306b\u304a\u3051\u308b\u7a4d\u5206\u306f\u53ce\u675f\u3057\u307e\u3059\u3002<\/p>\n<p>\u4e00\u65b9\u3067\u3001$k=0$\u306e\u5f8c\u3067\u306f$x$\u306e$2l-1$\u6b21\u306e\u591a\u9805\u5f0f\u3092$x-1$\u3067\u5272\u3063\u305f\u3082\u306e\u304c\u767b\u5834\u3057\u307e\u3059\u306e\u3067\u3001\u3053\u306e\u5f0f\u306e\u53ce\u675f\u6027\u306b\u3064\u3044\u3066\u8a55\u4fa1\u3057\u307e\u3059\u3002<\/p>\n<p>\u3068\u3044\u3063\u3066\u3082\u3001\u7a4d\u5206\u5909\u6570$x$\u306e\u7bc4\u56f2\u3092$x \\in [0,1]$\u306b\u9650\u5b9a\u3057\u3066\u3044\u307e\u3059\u306e\u3067\u3001<br \/>\n\\begin{align}<br \/>\n\\int_{0}^{1}\\frac{(1+x)^{2l-1}}{1-x}dx &#038;\\gt \\int_{0}^{1}\\frac{1}{1-x}dx\\nonumber\\cr<br \/>\n&#038;= [-\\log (1-x)]_0^1\\nonumber\\cr<br \/>\n&#038;= \\infty\\label{eq:eqalodoubleevaluatefinal}<br \/>\n\\end{align}<\/p>\n<p>\u3068\u306a\u3063\u3066\u3001\u767a\u6563\u3057\u3066\u3057\u307e\u3046\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n<p>(\\ref{eq:eqalodoubleevaluate})\u5f0f\u53ca\u3073(\\ref{eq:eqalodoubleevaluatefinal})\u5f0f\u3088\u308a$K_{l}$\u306f$\\infty$\u306b\u767a\u6563\u3059\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u304b\u3001\u3053\u308c\u306f(\\ref{eq:orthogonality})\u5f0f\u306e\u53f3\u8fba\u306e\u5024\u306e\u3046\u3061$m=n=0$\u306e\u5834\u5408\u306b\u76f8\u5f53\u3057\u307e\u3059\u3002<\/p>\n<p>\u3088\u3063\u3066\u3001\u3053\u3053\u307e\u3067\u306e\u8b70\u8ad6\u3067(\\ref{eq:orthogonality})\u5f0f\u53f3\u8fba\u306e3\u3064\u306e\u5834\u5408\u3059\u3079\u3066\u306b\u3064\u3044\u3066\u4e0e\u3048\u3089\u308c\u305f\u547d\u984c\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u304c\u793a\u305b\u305f\u3053\u3068\u306b\u306a\u308a\u307e\u3059\u3002$\\blacksquare$<\/p>\n<h2>\u307e\u3068\u3081<\/h2>\n<p>$m+n$\u306e\u5076\u5947\u3067\u306e\u5834\u5408\u5206\u3051\u3068\u304b\u6b63\u76f4\u601d\u3044\u3064\u304d\u307e\u305b\u3093\u3067\u3057\u305f\u3002\uff3f|\uffe3|\u25cb<\/p>\n<p>\u6574\u6570\u8ad6\u7684\u306a\u8b70\u8ad6\u304c\u9014\u4e2d\u3067\u767b\u5834\u3059\u308b\u306e\u304c\u8208\u5473\u6df1\u3044\u3068\u3053\u308d\u3067\u3059\u3002\u307e\u305f\u3001$1-x^2$\u304c(\\ref{eq:orthogonality})\u5f0f\u5de6\u8fba\u306e\u88ab\u7a4d\u5206\u95a2\u6570\u306e\u5206\u6bcd\u306b\u93ae\u5ea7\u3057\u3066\u3044\u308b\u3053\u3068\u306b\u3088\u308a\u3001\u90e8\u5206\u7a4d\u5206\u306e\u9014\u4e2d\u3067\u7aef\u70b9\u306b\u304a\u3051\u308b\u95a2\u6570\u5024\u3092\u4f7f\u3063\u3066\u8a08\u7b97\u3092\u3059\u308b\u9805\u306e\u5024\u304c0\u306b\u306a\u3089\u306a\u3044\u3053\u3068\u304c\u3042\u308b\u3053\u3068\u306b\u6c17\u304c\u4ed8\u304b\u306a\u3044\u307e\u307e(\\ref{eq:alofifth})\u5f0f\u306b\u305f\u3069\u308a\u7740\u3044\u3066\u3057\u307e\u3046\u3068\u30cf\u30de\u308a\u307e\u3059(\u672cWeb\u30b5\u30a4\u30c8\u306e\u7ba1\u7406\u4eba\u305f\u308bpanda\u3082\u30cf\u30de\u308a\u307e\u3057\u305f\u3001\u3063\u3066\u3044\u3046\u304b\u6c17\u304c\u4ed8\u304b\u306a\u304b\u3063\u305f\u306e\u3067\u3001\u53c2\u8003\u6587\u732e\u306e\u8a18\u8ff0\u3092\u53c2\u8003\u306b\u3055\u305b\u3066\u3044\u305f\u3060\u304d\u307e\u3057\u305f\ud83d\ude05)\u3002<\/p>\n<p>\u56e0\u6570\u5206\u89e3\u3068Leibniz\u306e\u516c\u5f0f\u3092\u4f7f\u3044\u307e\u304f\u308a\u3067\u3059\u3002<\/p>\n<p>\u56e0\u6570\u5206\u89e3\u3068\u8a00\u3048\u3070\u601d\u3044\u51fa\u3059\u8a71\u3082\u3042\u308b\u306e\u3067\u3059\u304c\u3001\u305d\u308c\u306f\u307e\u305f\u6a5f\u4f1a\u304c\u3042\u3063\u305f\u3089\u66f8\u304d\u307e\u3059\u3002<\/p>\n<p>\u3053\u306e\u8a18\u4e8b\u306f\u4ee5\u4e0a\u3067\u3059\u3002<\/p>\n<h2>References \/ \u53c2\u8003\u6587\u732e<\/h2>\n<ul>\n<li><a href=\"https:\/\/proofwiki.org\/wiki\/Orthogonality_of_Associated_Legendre_Functions\">Orthogonality of Associated Legendre Functions<\/a><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>\u306f\u3058\u3081\u306b \u3053\u306e\u8a18\u4e8b\u3067\u306f\u3061\u3087\u3063\u3068\u524d\u306e\u8a18\u4e8b\u306b\u5f15\u304d\u7d9a\u304d\u3001Legendre\u306e\u966a\u591a\u9805\u5f0f \\begin{align} P_l^m(x) &#038;= \\frac{(-1)^m}{2^ll!}\\frac{d^{m+l}}{dx^{m+l}}(x^2-1)^l \\label{eq:associatedlegendre} \\end{align} \u304c\u6e80\u305f\u3059\u76f4\u4ea4\u6761\u4ef6 \\begin{align} \\int_{-1}^{\u2026 <span class=\"read-more\"><a href=\"https:\/\/pandanote.info\/?p=4874\">Read More &raquo;<\/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-4874","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\/4874","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=4874"}],"version-history":[{"count":62,"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/posts\/4874\/revisions"}],"predecessor-version":[{"id":9375,"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/posts\/4874\/revisions\/9375"}],"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=4874"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/pandanote.info\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4874"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/pandanote.info\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4874"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}