{"id":4591,"date":"2019-06-01T20:00:56","date_gmt":"2019-06-01T11:00:56","guid":{"rendered":"https:\/\/pandanote.info\/?p=4591"},"modified":"2022-08-07T12:14:35","modified_gmt":"2022-08-07T03:14:35","slug":"legendre%e5%a4%9a%e9%a0%85%e5%bc%8f%e3%81%ae%e7%9b%b4%e4%ba%a4%e6%80%a7%e3%82%92%e6%9c%80%e9%ab%98%e6%ac%a1%e3%81%ae%e9%a0%85%e3%81%ae%e4%bf%82%e6%95%b0%e3%81%a0%e3%81%91%e3%82%92%e8%a8%88%e7%ae%97","status":"publish","type":"post","link":"https:\/\/pandanote.info\/?p=4591","title":{"rendered":"Legendre\u591a\u9805\u5f0f\u306e\u76f4\u4ea4\u6027\u3092\u6700\u9ad8\u6b21\u306e\u9805\u306e\u4fc2\u6570\u3060\u3051\u3092\u8a08\u7b97\u3059\u308b\u3053\u3068\u306b\u3088\u308a\u8a3c\u660e\u3057\u3066\u307f\u305f\u3002"},"content":{"rendered":"

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

\u30bf\u30a4\u30c8\u30eb\u3060\u3051\u898b\u308b\u3068\u305f\u3060\u306e\u624b\u629c\u304d\u8a08\u7b97\u306e\u3088\u3046\u306b\u898b\u3048\u306a\u304f\u3082\u306a\u3044\u3067\u3059\u304c\u3001$n$\u6b21\u306eLegendre\u591a\u9805\u5f0f$P_n(x)$\u306b\u3064\u3044\u3066\u4ee5\u4e0b\u306e(\\ref{eq:orthogonality})\u5f0f\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092(\\ref{eq:orthogonality})\u5f0f\u306e\u88ab\u7a4d\u5206\u95a2\u6570\u306e\u6700\u9ad8\u6b21\u4ee5\u5916\u306e\u9805\u306e\u4fc2\u6570\u304c\u3059\u3079\u30660\u306b\u306a\u308b\u3053\u3068\u3092\u793a\u3057\u3066\u304b\u3089\u3001\u6700\u9ad8\u6b21\u306e\u9805\u306e\u4fc2\u6570\u3092\u8a08\u7b97\u3059\u308b\u3053\u3068\u306b\u3088\u3063\u3066\u793a\u3057\u307e\u3059\u3002
\n\\begin{align}
\n\\int_{-1}^{1} P_m(x)P_n(x)dx &= \\frac{2}{2n+1}\\delta_{mn} \\label{eq:orthogonality}
\n\\end{align}
\n\u306a\u304a\u3001$\\delta_{mn}$\u306fCronecker\u306e\u30c7\u30eb\u30bf\u3092\u8868\u3057\u307e\u3059\u3002<\/p>\n

\u30b5\u30af\u30b5\u30af\u3068\u8a3c\u660e\u3057\u307e\u3059\u3002<\/h2>\n

Legendre\u591a\u9805\u5f0f\u30922\u901a\u308a\u306e\u65b9\u6cd5\u3067\u8868\u3057\u307e\u3059\u3002<\/h3>\n

(\\ref{eq:orthogonality})\u5f0f\u306e\u88ab\u7a4d\u5206\u95a2\u6570\u306b\u542b\u307e\u308c\u308b$P_m(x)$\u53ca\u3073$P_n(x)$\u306f\u5165\u308c\u66ff\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u306e\u3067\u3001\u4e00\u822c\u6027\u3092\u5931\u3046\u3053\u3068\u306a\u304f$m \\le n$\u3068\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n

$P_m(x)$\u306f$x$\u306b\u3064\u3044\u3066\u306e$m$\u6b21\u306e\u591a\u9805\u5f0f\u306a\u306e\u3067\u3001\u9069\u5f53\u306a\u4fc2\u6570$a_l(0 \\le l \\le m)$\u3092\u7528\u3044\u3066\u3001
\n\\begin{align}
\nP_m(x) &= \\sum_{l=0}^{m} a_l x^l \\label{eq:pmpolynomial}
\n\\end{align}
\n\u3068\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u305d\u3053\u3067\u3001(\\ref{eq:pmpolynomial})\u5f0f\u3092(\\ref{eq:orthogonality})\u5f0f\u306e\u5de6\u8fba\u306b\u4ee3\u5165\u3059\u308b\u3068\u3001\u548c\u3068\u7a4d\u5206\u306e\u9806\u5e8f\u306f\u4ea4\u63db\u3067\u304d\u307e\u3059\u306e\u3067\u3001
\n\\begin{align}
\n\\int_{-1}^{1} P_m(x)P_n(x)dx &= \\int_{-1}^{1} P_n(x) \\sum_{l=0}^{m} a_l x^l dx \\nonumber \\cr
\n&= \\sum_{l=0}^{m} a_l \\int_{-1}^{1} x^l P_n(x) dx \\label{eq:pmpolynomialsubstitute}
\n\\end{align}
\n\u3068\u5909\u5f62\u3067\u304d\u307e\u3059\u3002<\/p>\n

\u4e00\u65b9\u3067\u3001$P_n(x)$\u306f\u3001
\n\\begin{align}
\nP_n(x) &= \\frac{1}{2^n n!} \\frac{d^n}{dx^n} [(x^2-1)^n] \\label{eq:legendredef}
\n\\end{align}
\n\u3068\u8868\u3059\u3053\u3068\u3082\u3067\u304d\u307e\u3059\u3002<\/p>\n

\u90e8\u5206\u7a4d\u5206\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002<\/h3>\n

\u3053\u3053\u3067\u3001(\\ref{eq:pmpolynomialsubstitute})\u5f0f\u53ca\u3073(\\ref{eq:legendredef})\u5f0f\u304b\u3089\u7a4d\u5206\u8a08\u7b97\u306b\u95a2\u4fc2\u3059\u308b\u90e8\u5206\u3092\u62bd\u51fa\u3057\u3001
\n\\begin{align}
\nI_l &= \\int_{-1}^{1} x^l \\frac{d^n}{dx^n} [(x^2-1)^n] dx \\label{eq:integraltozero}
\n\\end{align}
\n\u3068\u304a\u3044\u3066\u8a08\u7b97\u3057\u307e\u3059\u3002\u7a4d\u5206\u8a08\u7b97\u306b\u95a2\u4fc2\u3057\u306a\u3044(\\ref{eq:legendredef})\u5f0f\u306e$\\displaystyle\\frac{1}{2^n n!}$\u306b\u3064\u3044\u3066\u306f\u7a4d\u5206\u8a08\u7b97\u5f8c\u306b\u518d\u5ea6\u307e\u3068\u3081\u76f4\u3057\u307e\u3059\u3002<\/p>\n

(\\ref{eq:integraltozero})\u5f0f\u3092\u90e8\u5206\u7a4d\u5206\u3092\u7528\u3044\u3066\u5909\u5f62\u3059\u308b\u3068\u2026
\n\\begin{align}
\nI_l &= \\left[ x^l\\frac{d^{n-1}}{dx^{n-1}}[(x^2-1)^n]\\right]_{-1}^{1} – \\int_{-1}^{1} lx^{l-1}\\frac{d^{n-1}}{dx^{n-1}}[(x^2-1)^n]dx \\label{eq:integraltozeroatfirst}
\n\\end{align}
\n\u3068\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002<\/p>\n

\u6b21\u306b\u3001(\\ref{eq:integraltozeroatfirst})\u5f0f\u306e\u53f3\u8fba\u7b2c1\u9805\u3092\u8a08\u7b97\u3057\u3066\u307f\u307e\u3059\u3002\u307e\u305a\u3001
\n\\begin{align}
\nf_k(x) &= \\frac{d^{k}}{dx^{k}}[(x^2-1)^n] \\label{eq:leibniz}
\n\\end{align}
\n\u3068\u7f6e\u304f\u3068\u3001$k \\lt n$\u306e\u3068\u304d\u306b\u3001
\n\\begin{align}
\nf_k(x) &= \\frac{d^{k}}{dx^{k}}[(x+1)^n(x-1)^n]\u3000\\nonumber \\cr
\n&= \\sum_{\\mu = 0}^{k}
\n\\begin{pmatrix}
\nk \\cr
\n\\mu
\n\\end{pmatrix}
\n\\frac{d^{k-\\mu}}{dx^{k-\\mu}}[(x+1)^n]\\frac{d^{\\mu}}{dx^{\\mu}}[(x-1)^n] \\nonumber \\cr
\n&= \\sum_{\\mu = 0}^{k}
\n\\begin{pmatrix}
\nk \\cr
\n\\mu
\n\\end{pmatrix}
\n\\frac{n!}{(n-k+\\mu)!}[(x+1)^{n-k+\\mu}]\\frac{n!}{(n-\\mu)!}[(x-1)^{n-\\mu}]
\n\\label{eq:leibnizformula}
\n\\end{align}
\n\u3068\u306a\u308a\u307e\u3059(Leibniz\u306e\u516c\u5f0f\u3092\u5229\u7528)\u3002<\/p>\n

\u3053\u3053\u3067\u3001$\\mu \\le k \\lt n$\u3067\u3042\u308b\u3053\u3068\u304b\u3089\u3001$n-k+\\mu \\gt 0$ \u53ca\u3073 $n-\\mu\\gt 0$\u3067\u3042\u308b\u306e\u3067\u3001$f_k(1)=f_k(-1)=0$\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n

\u3088\u3063\u3066\u3001(\\ref{eq:integraltozeroatfirst})\u5f0f\u306e\u53f3\u8fba\u7b2c1\u9805\u306f0\u306b\u306a\u308a\u307e\u3059\u306e\u3067\u3001
\n\\begin{align}
\nI_l &=\\, -\\int_{-1}^{1} lx^{l-1}\\frac{d^{n-1}}{dx^{n-1}}[(x^2-1)^n]dx \\label{eq:integraltozeroatsecond}
\n\\end{align}
\n\u3068\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002<\/p>\n

\u3053\u3053\u307e\u3067\u306e\u90e8\u5206\u7a4d\u5206\u3092\u3042\u3068$l-1$\u56de\u7e70\u308a\u8fd4\u3059\u3068\u3001\u90e8\u5206\u7a4d\u5206\u3092\u5b9f\u884c\u3059\u308b\u3054\u3068\u306b(\\ref{eq:integraltozeroatfirst})\u5f0f\u306e\u53f3\u8fba\u7b2c1\u9805\u306b\u76f8\u5f53\u3059\u308b\u90e8\u5206\u304c\u73fe\u308c\u3001\u304b\u3064\u305d\u308c\u3089\u304c\u3059\u3079\u30660\u306b\u306a\u308a\u307e\u3059\u306e\u3067\u2026
\n\\begin{align}
\nI_l &= (-1)^l l! \\int_{-1}^{1}\\frac{d^{n-l}}{dx^{n-l}}[(x^2-1)^n]dx \\label{eq:integraltozeroatthird}
\n\\end{align}
\n\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n

$n \\gt l$\u306e\u5834\u5408\u306b\u306f(\\ref{eq:integraltozeroatthird})\u5f0f\u306f\u3055\u3089\u306b\u7a4d\u5206\u3067\u304d\u307e\u3059\u3002\u305d\u3053\u3067\u3001(\\ref{eq:leibnizformula})\u5f0f\u306e\u7d50\u679c\u3092\u5229\u7528\u3059\u308b\u3068\u3001$l = n-1$\u306e\u5834\u5408\u3082\u542b\u3081\u3066$f_{n-l-1}(1)=f_{n-l-1}(-1)=0$\u3068\u306a\u308a\u307e\u3059\u306e\u3067\u3001
\n\\begin{align}
\nI_l &= (-1)^l l! \\left[ \\frac{d^{n-l-1}}{dx^{n-l-1}}[(x^2-1)^n] \\right]_{-1}^{1} \\nonumber \\cr
\n&= 0 \\label{eq:integraltozeroatfourth}
\n\\end{align}
\n\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n

\u4e00\u65b9$n = l$\u306e\u5834\u5408\u306b\u306f$x = \\cos\\theta$\u3068\u304a\u3044\u3066\u304b\u3089\u5076\u95a2\u6570\u3067\u3042\u308b\u3053\u3068\u3092\u5229\u7528\u3057\u3066\u5f0f\u3092\u5909\u5f62\u3059\u308b\u3068\u3001Wallis\u7a4d\u5206\u304c\u73fe\u308c\u3066\u3001
\n\\begin{align}
\n\\int_{-1}^{1}(x^2-1)^n dx &= \\int_{-\\frac{\\pi}{2}}^{\\frac{\\pi}{2}}(-\\cos^2\\theta)^n \\cos\\theta\\,d\\theta \\nonumber \\cr
\n&= (-1)^n\\int_{-\\frac{\\pi}{2}}^{\\frac{\\pi}{2}}\\cos^{2n+1}\\theta\\,d\\theta \\nonumber \\cr
\n&= 2(-1)^n\\int_{0}^{\\frac{\\pi}{2}}\\cos^{2n+1}\\theta\\,d\\theta \\nonumber \\cr
\n&= 2(-1)^n\\frac{(2n)!!}{(2n+1)!!} \\label{eq:integraltozeroatfifth}
\n\\end{align}
\n\u3068\u8a08\u7b97\u3067\u304d\u307e\u3059\u306e\u3067\u3001(\\ref{eq:integraltozeroatthird})\u5f0f\u53ca\u3073(\\ref{eq:integraltozeroatfifth})\u5f0f\u3088\u308a\u3001
\n\\begin{align}
\nI_n &= (-1)^n n! \\cdot 2(-1)^n\\frac{(2n)!!}{(2n+1)!!} \\nonumber \\cr
\n&= 2\\,\\frac{n!(2n)!!}{(2n+1)!!} \\label{eq:integraltoorthogonal}
\n\\end{align}
\n\u3068\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002<\/p>\n

\u3055\u3089\u306b\u3001(\\ref{eq:integraltozeroatfourth})\u5f0f\u53ca\u3073(\\ref{eq:integraltoorthogonal})\u5f0f\u306f\u307e\u3068\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u3066\u3001
\n\\begin{align}
\nI_l &= \\left\\{
\n\\begin{array}{ll}
\n0 & (l \\lt n) \\cr
\n2\\,\\displaystyle\\frac{n!(2n)!!}{(2n+1)!!} & (l = n)
\n\\end{array} \\right. \\label{eq:integralatfinal}
\n\\end{align}
\n\u3068\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n

\u90e8\u5206\u7a4d\u5206\u306e\u7d50\u679c\u3092\u5229\u7528\u3057\u3066\u3001\u8a08\u7b97\u3092\u7d9a\u884c\u3057\u307e\u3059\u3002<\/h3>\n

\u3053\u3053\u3067\u672c\u984c\u306b\u623b\u308a\u307e\u3059\u3002<\/p>\n

(\\ref{eq:pmpolynomialsubstitute})\u5f0f(\\ref{eq:legendredef})\u5f0f\u53ca\u3073(\\ref{eq:integralatfinal})\u5f0f\u3088\u308a\u3001$l = n$\u306e\u3068\u304d\u306f\u3001
\n\\begin{align}
\n\\int_{-1}^{1} x^l P_n(x) dx &= \\frac{1}{2^n n!}I_n \\nonumber \\cr
\n&= 2\\,\\frac{1}{2^n n!}\\frac{n!(2n)!!}{(2n+1)!!} \\nonumber \\cr
\n&= \\frac{1}{2^{n-1}}\\frac{(2n)!!}{(2n+1)!!} \\label{eq:integralatn}
\n\\end{align}
\n$l \\ne n$\u306e\u3068\u304d\u306f
\n\\begin{align}
\n\\int_{-1}^{1} x^l P_n(x) dx &= 0 \\label{eq:integralatl}
\n\\end{align}
\n\u3068\u306a\u308a\u307e\u3059\u3002$P_m(x)$\u306f$m$\u6b21\u306e\u591a\u9805\u5f0f\u3067\u3042\u308b\u3053\u3068\u3068\u3001(\\ref{eq:integralatl})\u5f0f\u306e\u7d50\u679c\u3088\u308a\u3001(\\ref{eq:pmpolynomialsubstitute})\u5f0f\u306e\u53f3\u8fba\u3067\u7dcf\u548c\u306e\u8a08\u7b97\u306e\u5bfe\u8c61\u3068\u306a\u3063\u3066\u3044\u308b\u5404\u9805\u306f$m \\lt n$\u306e\u5834\u5408\u306b\u306f\u3059\u3079\u30660\u3068\u306a\u308a\u307e\u3059\u3002\u3088\u3063\u3066\u3001
\n\\begin{align}
\n\\int_{-1}^{1} P_m(x)P_n(x)dx &= 0 \\label{eq:resultmnen}
\n\\end{align}
\n\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n

\u307e\u305f\u3001$m = n$\u306e\u5834\u5408\u3082\u7dcf\u548c\u306e\u8a08\u7b97\u306e\u5bfe\u8c61\u3068\u306a\u3063\u3066\u3044\u308b\u5404\u9805\u306e\u3046\u3061\u3001$m \\lt n$\u306e\u9805\u306f\u3059\u3079\u30660\u3068\u306a\u308a\u307e\u3059\u306e\u3067\u3001(\\ref{eq:integralatn})\u5f0f\u306e\u7d50\u679c\u3092\u5229\u7528\u3057\u3066\u3001
\n\\begin{align}
\n\\int_{-1}^{1} \\left\\{ P_n(x) \\right\\}^2 dx &= \\int_{-1}^{1} P_n(x) a_n x^n dx \\nonumber \\cr
\n&= a_n\\frac{1}{2^{n-1}}\\frac{(2n)!!}{(2n+1)!!} \\label{eq:pnsquareatfirst}
\n\\end{align}
\n\u3068\u8a08\u7b97\u3067\u304d\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n

\u3042\u3068\u306f$a_n$\u3092\u6c42\u3081\u308b\u3068\u3001(\\ref{eq:pnsquareatfirst})\u5f0f\u306e\u5024\u304c\u5b9a\u307e\u308b\u3053\u3068\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n

\u3053\u3053\u3067\u3001$a_n$\u304c(\\ref{eq:legendredef})\u5f0f\u3067\u8868\u3055\u308c\u308b$P_n(x)$\u306e\u6700\u9ad8\u6b21($n$\u6b21)\u306e\u9805\u306e\u4fc2\u6570\u3067\u3042\u3063\u305f\u3053\u3068\u3092\u601d\u3044\u51fa\u3059\u3068\u3001
\n\\begin{align}
\na_n &= 2n\\cdots(n+1)\\cdot n\\cdot\\frac{1}{2^n n!} \\nonumber \\cr
\n&= \\frac{(2n)!}{n!}\\cdot\\frac{1}{2^n n!} \\nonumber \\cr
\n&= \\frac{1}{2^n}\\begin{pmatrix}
\n2n \\cr
\nn
\n\\end{pmatrix} \\label{eq:an}
\n\\end{align}
\n\u3067\u3059\u306e\u3067\u3001\u3053\u308c\u3092(\\ref{eq:pnsquareatfirst})\u5f0f\u306b\u4ee3\u5165\u3059\u308b\u3068\u2026
\n\\begin{align}
\n\\int_{-1}^{1} \\left\\{ P_n(x) \\right\\}^2 dx &= \\frac{1}{2^n}
\n\\begin{pmatrix}
\n2n \\cr
\nn
\n\\end{pmatrix}
\n\\frac{1}{2^{n-1}}\\frac{(2n)!!}{(2n+1)!!} \\nonumber \\cr
\n&= \\frac{1}{2^{2n-1}}
\n\\begin{pmatrix}
\n2n \\cr
\nn
\n\\end{pmatrix}
\n\\frac{(2n)!!(2n)!!}{(2n+1)!!(2n)!!} \\nonumber \\cr
\n&= \\frac{1}{2^{2n-1}}
\n\\begin{pmatrix}
\n2n \\cr
\nn
\n\\end{pmatrix}
\n\\frac{2^{2n}n!n!}{(2n+1)(2n)!} \\nonumber \\cr
\n&= 2
\n\\begin{pmatrix}
\n2n \\cr
\nn
\n\\end{pmatrix}
\n\\frac{1}{(2n+1)\\begin{pmatrix}
\n2n \\cr
\nn
\n\\end{pmatrix}} \\nonumber \\cr
\n&= \\frac{2}{2n+1}
\n\\label{eq:pnsquareatsecond}
\n\\end{align}
\n\u3068\u3001\u62cd\u5b50\u629c\u3051\u3059\u308b\u307b\u3069\u7c21\u5358\u306a\u5f0f\u306b\u306a\u308a\u307e\u3059\u3002\ud83d\ude00<\/p>\n

\u305f\u3060\u3001\u3053\u308c\u304c\u6c42\u3081\u3066\u3044\u305f\u7d50\u679c\u306a\u308f\u3051\u3067\u3001(\\ref{eq:pnsquareatsecond})\u5f0f\u53ca\u3073(\\ref{eq:resultmnen})\u5f0f\u304b\u3089(\\ref{eq:orthogonality})\u5f0f\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u304c\u8a3c\u660e\u3067\u304d\u307e\u3057\u305f\u3002$\\blacksquare$<\/p>\n

\u304a\u307e\u3051<\/h2>\n

\u305d\u306e1: Wallis\u7a4d\u5206<\/h3>\n

\u524d\u7bc0\u307e\u3067\u306e\u8a3c\u660e\u306e\u9014\u4e2d\u3067\u4f55\u306e\u65ad\u308a\u3082\u306a\u304fWallis\u7a4d\u5206\u304c\u767b\u5834\u3057\u3066\u3044\u307e\u3059\u304c\u3001\u3053\u308c\u306f$\\cos$\u53ca\u3073$\\sin$\u95a2\u6570\u306e\u6574\u6570\u4e57\u306e\u7a4d\u5206\u306b\u3064\u3044\u3066\u3001\u4ee5\u4e0b\u306e\u95a2\u4fc2\u304c\u6210\u308a\u7acb\u3064\u3068\u3044\u3046\u3082\u306e\u3067\u3059\u3002\u3053\u3053\u3067$n$\u306f\u8ca0\u3067\u306a\u3044\u6574\u6570\u3068\u3057\u307e\u3059\u3002
\n\\begin{align}
\n\\int_{0}^{\\frac{\\pi}{2}} \\cos^{2n+1}\\theta\\,d\\theta &= \\int_{0}^{\\frac{\\pi}{2}} \\sin^{2n+1}\\theta\\,d\\theta \\nonumber \\cr
\n&= \\frac{(2n)!!}{(2n+1)!!} \\nonumber \\cr
\n\\int_{0}^{\\frac{\\pi}{2}} \\cos^{2n}\\theta\\,d\\theta &= \\int_{0}^{\\frac{\\pi}{2}} \\sin^{2n}\\theta\\,d\\theta \\nonumber \\cr
\n&= \\frac{\\pi}{2}\\frac{(2n-1)!!}{(2n)!!}
\n\\label{eq:wallis}
\n\\end{align}<\/p>\n

Wallis\u7a4d\u5206\u306e\u6982\u8981\u7b49\u306b\u3064\u3044\u3066\u306fWikipedia<\/a>\u7b49\u3092\u3054\u53c2\u7167\u304f\u3060\u3055\u3044\u3002<\/p>\n

\u307e\u305f\u3001Wikipedia\u306e\u8a18\u4e8b\u304c\u308f\u304b\u308a\u306b\u304f\u3044\u3088\u3046\u3067\u3042\u308c\u3070\u3001\u3053\u3061\u3089<\/a>\u3082\u3054\u53c2\u7167\u3044\u305f\u3060\u3051\u308b\u3068\u5e78\u3044\u3067\u3059\u3002<\/p>\n

\u305d\u306e2: Legendre\u591a\u9805\u5f0f\u306e\u30b0\u30e9\u30d5\u306e\u63cf\u753b\u4f8b<\/h3>\n

Inkscape\u3067\u63cf\u3044\u305f$n = 0,1,2,3,4,5$\u306e\u6642\u306e$P_n(x), x \\in [-1,1]$\u306e\u30b0\u30e9\u30d5\u3092\u4ee5\u4e0b\u306b\u793a\u3057\u307e\u3059\u3002
\n\"\"<\/a><\/p>\n

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

$(x^2-1)^n$\u306e$n$\u56de\u672a\u6e80\u306e\u56de\u6570\u306e\u5fae\u5206\u3092\u8003\u3048\u308b\u3068\u3053\u308d\u304c\u6280\u5de7\u7684\u304b\u3064\u3042\u307e\u308a\u6b63\u653b\u6cd5\u3067\u306a\u3044\u3088\u3046\u306b\u3082\u898b\u3048\u307e\u3059\u304c\u3001\u6700\u9ad8\u6b21\u306e\u9805\u306e\u4fc2\u6570\u306e\u8a08\u7b97\u306b\u6ce8\u529b\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u306e\u3067\u3001\u6bd4\u8f03\u7684\u8a08\u7b97\u306e\u3057\u3084\u3059\u3044\u65b9\u6cd5\u3067\u3042\u308b\u3068\u601d\u3044\u307e\u3059(\u203b\u500b\u4eba\u306e\u611f\u60f3\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 \u30bf\u30a4\u30c8\u30eb\u3060\u3051\u898b\u308b\u3068\u305f\u3060\u306e\u624b\u629c\u304d\u8a08\u7b97\u306e\u3088\u3046\u306b\u898b\u3048\u306a\u304f\u3082\u306a\u3044\u3067\u3059\u304c\u3001$n$\u6b21\u306eLegendre\u591a\u9805\u5f0f$P_n(x)$\u306b\u3064\u3044\u3066\u4ee5\u4e0b\u306e(\\ref{eq:orthogonality})\u5f0f\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092(\\ref{eq:orthogonality})\u5f0f\u306e\u88ab\u7a4d\u5206\u95a2\u6570\u306e\u6700\u9ad8\u6b21\u4ee5\u5916\u306e\u9805\u306e\u4fc2\u6570\u304c\u3059\u3079\u30660\u306b\u306a\u308b\u3053\u3068\u3092\u793a\u3057\u3066\u304b\u3089\u3001\u6700\u9ad8\u6b21\u306e\u9805\u306e\u4fc2\u6570\u3092\u8a08\u7b97\u3059\u308b\u3053\u3068\u306b\u3088\u3063\u3066\u793a\u3057\u307e\u3059\u3002 \\begin{align} \\\u2026 Read More »<\/a><\/span><\/p>\n","protected":false},"author":1,"featured_media":4615,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[13],"tags":[],"class_list":["post-4591","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\/4591","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=4591"}],"version-history":[{"count":33,"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/posts\/4591\/revisions"}],"predecessor-version":[{"id":9371,"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/posts\/4591\/revisions\/9371"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/media\/4615"}],"wp:attachment":[{"href":"https:\/\/pandanote.info\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4591"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/pandanote.info\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4591"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/pandanote.info\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4591"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}