{"id":4022,"date":"2019-03-02T23:47:39","date_gmt":"2019-03-02T14:47:39","guid":{"rendered":"https:\/\/pandanote.info\/?p=4022"},"modified":"2022-08-07T12:08:24","modified_gmt":"2022-08-07T03:08:24","slug":"%e6%95%b4%e6%95%b0%e6%ac%a1%e6%95%b0%e3%81%ae%e7%ac%ac2%e7%a8%ae%e5%a4%89%e5%bd%a2bessel%e9%96%a2%e6%95%b0%e3%81%ae%e7%b4%9a%e6%95%b0%e8%a1%a8%e7%8f%be%e3%82%92%e8%a8%88%e7%ae%97%e3%81%97%e3%81%a6","status":"publish","type":"post","link":"https:\/\/pandanote.info\/?p=4022","title":{"rendered":"\u6574\u6570\u6b21\u6570\u306e\u7b2c2\u7a2e\u5909\u5f62Bessel\u95a2\u6570\u306e\u7d1a\u6570\u8868\u73fe\u3092\u8a08\u7b97\u3057\u3066\u307f\u305f\u3002"},"content":{"rendered":"

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

\u3061\u3087\u3063\u3068\u524d\u306e\u8a18\u4e8b<\/a>\u3067\u3001\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u304c\u6a19\u6e96\u6b63\u898f\u5206\u5e03\u306b\u5f93\u30462\u500b\u306e\u78ba\u7387\u5909\u6570\u306e\u7a4d\u304c\u5f93\u3046\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3092\u8a08\u7b97\u3057\u3066\u307f\u305f\u3068\u3053\u308d\u3001\u7b2c2\u7a2e\u5909\u5f62Bessel\u95a2\u6570\u304c\u7a4d\u5206\u8868\u793a\u3067\u51fa\u73fe\u3057\u305f\u4ef6\u306b\u3064\u3044\u3066\u66f8\u304d\u307e\u3057\u305f\u3002<\/p>\n

Bessel\u95a2\u6570\u3068\u3044\u3048\u3070\u7279\u6b8a\u95a2\u6570\u3068\u3057\u3066\u306f\u304b\u306a\u308a\u306e\u6709\u540d\u3069\u3053\u308d\u3067\u3001\u7a4d\u5206\u8868\u793a\u306e\u4ed6\u306b\u3082\u3044\u308d\u3044\u308d\u306a\u8868\u793a\u65b9\u6cd5\u304c\u3042\u308a\u3001\u7d1a\u6570\u8868\u73fe\u3082\u3042\u308a\u307e\u3059\u3002<\/p>\n

\u305f\u3060\u3001<\/p>\n

\u300c\u7b2c2\u7a2e\u300d<\/strong><\/p>\n

\u305d\u308c\u3082<\/p>\n

\u300c\u5909\u5f62\u300d<\/strong><\/p>\n

\u3068\u3044\u3046\u3053\u3068\u306b\u306a\u308b\u3068\u3001\u76f8\u5f53\u306a\u66f2\u8005\u611f\u304c\u6f02\u3044\u307e\u3059\u3002\u6570\u5b66\u3084\u7269\u7406\u3092\u5c02\u9580\u306b\u3057\u3066\u3044\u308b\u306a\u3089\u307e\u3060\u3057\u3082\u3001\u7d71\u8a08\u306e\u52c9\u5f37\u3092\u3057\u3066\u3044\u308b\u6700\u4e2d\u306b\u3053\u3093\u306a\u95a2\u6570\u304c\u73fe\u308c\u308b\u3068\u304b\u306a\u308a\u9762\u98df\u3089\u3046\u3082\u306e\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n

\u8a71\u3092\u672c\u984c\u306b\u623b\u3057\u307e\u3059\u3002<\/p>\n

\u5148\u8ff0\u306e\u3061\u3087\u3063\u3068\u524d\u306e\u8a18\u4e8b<\/a>\u3067\u306f\u30010\u6b21\u306e\u7b2c2\u7a2e\u5909\u5f62Bessel\u95a2\u6570\u304c\u767b\u5834\u3057\u307e\u3057\u305f\u306e\u3067\u3001\u3053\u306e\u8a18\u4e8b\u3067\u306f\u6271\u3046\u7bc4\u56f2\u3092\u3061\u3087\u3063\u3068\u3060\u3051\u5e83\u3052\u3066\u6574\u6570\u6b21\u306e\u7b2c2\u7a2e\u5909\u5f62Bessel\u95a2\u6570\u306e\u7d1a\u6570\u8868\u73fe\u306b\u3064\u3044\u3066(\u5f0f\u5909\u5f62\u306a\u3069\u3092\u3042\u307e\u308a\u7701\u7565\u3057\u306a\u3044\u3067)\u8a08\u7b97\u3057\u3066\u307f\u308b\u3053\u3068\u306b\u3057\u307e\u3057\u305f\u3002<\/p>\n

\u30b9\u30bf\u30fc\u30c8\u5730\u70b9\u306e\u8a2d\u5b9a\u3002<\/h2>\n

\u300c\u8a08\u7b97\u3059\u308b\u300d\u3068\u306f\u3044\u3046\u3082\u306e\u306e\u3001\u3053\u306e\u8a18\u4e8b\u3067\u306f$\\mu$\u6b21\u306e\u7b2c1\u7a2e\u5909\u5f62Bessel\u95a2\u6570$I_{\\mu}(x)$\u304c\u4ee5\u4e0b\u306e(\\ref{eq:mbff})\u5f0f\u3067\u8868\u3055\u308c\u308b\u3053\u3068\u306f\u65e2\u77e5\u3068\u3059\u308b\u3053\u3068\u306b\u3057\u307e\u3059\u3002
\n\\begin{align}
\nI_{\\mu}(x) &= \\left(\\frac{x}{2}\\right)^{\\mu}\\sum_{p=0}^{\\infty}\\frac{1}{p!\\Gamma(\\mu + p + 1)}\\left(\\frac{x}{2}\\right)^{2p} \\label{eq:mbff}
\n\\end{align}<\/p>\n

\u7b2c1\u7a2e\u5909\u5f62Bessel\u95a2\u6570\u304b\u3089\u7b2c2\u7a2e\u5909\u5f62Bessel\u95a2\u6570\u3092\u6c42\u3081\u308b\u3002<\/h2>\n

\u7b2c2\u7a2e\u5909\u5f62Bessel\u95a2\u6570\u306e\u5b9a\u7fa9\u3002<\/h3>\n

\u3082\u306e\u306eWikipedia\u306b\u3088\u308a\u307e\u3059\u3068\u3001\u7b2c2\u7a2e\u5909\u5f62Bessel\u95a2\u6570$J_{\\mu}(x)$\u306f(\\ref{eq:mbfs})\u5f0f\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u307e\u3059\u3002
\n\\begin{align}
\nJ_{\\mu}(x) &= \\frac{\\pi}{2} \\frac{I_{-\\mu}(x)-I_{\\mu}(x)}{\\sin(\\mu\\pi)} \\label{eq:mbfs}
\n\\end{align}<\/p>\n

\u6574\u6570\u6b21\u306e\u5834\u5408\u3067\u3059\u3068$\\mu$\u304c\u6574\u6570\u306b\u306a\u308a\u307e\u3059\u304b\u3089\u3001\u5c11\u306a\u304f\u3068\u3082\u5206\u6bcd\u304c0\u306b\u306a\u308a\u305d\u3046\u3067\u3059\u3002\u307e\u305f\u3001\u5206\u5b50\u3082$\\mu$\u304c\u6574\u6570\u306e\u3068\u304d\u306b\u306f0\u306b\u306a\u308a\u307e\u3059(\u3053\u308c\u306b\u3064\u3044\u3066\u306f\u5225\u9014\u8a08\u7b97\u3057\u3066\u307f\u307e\u3057\u305f\u3002\u3053\u306e\u8a18\u4e8b<\/a>\u53c2\u7167)\u3002\u305d\u3053\u3067\u3001$\\mu$\u304c\u6574\u6570\u306e\u5834\u5408\u306b\u306f$\\mu$\u304c\u305d\u306e\u6574\u6570\u3078\u8fd1\u3065\u304f\u3068\u304d\u306e\u6975\u9650\u3068\u3057\u3066\u7b2c2\u7a2e\u5909\u5f62Bessel\u95a2\u6570\u3092\u5b9a\u7fa9\u3057\u307e\u3059\u3002\u3059\u306a\u308f\u3061(\\ref{eq:mbfsint})\u5f0f\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3057\u307e\u3059\u3002
\n\\begin{align}
\nJ_{\\mu}(x) &= \\lim_{k \\to \\mu} \\frac{\\pi}{2} \\frac{I_{-k}(x)-I_{k}(x)}{\\sin(k\\pi)} \\label{eq:mbfsint}
\n\\end{align}<\/p>\n

\u8a08\u7b97\u306e\u65b9\u91dd\u3002<\/h3>\n

(\\ref{eq:mbfsint})\u5f0f\u306e\u6975\u9650\u3092\u305d\u306e\u307e\u307e\u8a08\u7b97\u3059\u308b\u306e\u306f\u5206\u6bcd\u306b$\\sin$\u3068\u304b\u304c\u73fe\u308c\u305f\u308a\u3057\u3066\u3044\u3066\u304b\u306a\u308a\u7e41\u96d1\u305d\u3046\u306a\u306e\u3067\u3001\u3082\u3046\u5c11\u3057\u7c21\u5358\u306b\u8a08\u7b97\u3067\u304d\u306a\u3044\u304b\u8003\u3048\u3066\u307f\u307e\u3059\u3002<\/p>\n

\u5206\u6bcd\u306f\u4e09\u89d2\u95a2\u6570\u3067\u3059\u304b\u3089\u3001\u305d\u308c\u3092$k$\u3067\u5fae\u5206\u3057\u305f\u7d50\u679c\u5f97\u3089\u308c\u308b\u95a2\u6570$\\pi\\cos(k\\pi)$\u306f$k \\in \\left(\\mu-\\displaystyle\\frac{1}{2},\\mu+\\displaystyle\\frac{1}{2}\\right)$\u3067\u3042\u308c\u3070$\\pi\\cos(k\\pi) \\ne 0$\u3067\u3042\u308a\u3001
\n\\begin{align}
\n\\lim_{k \\to \\mu} \\left[I_{-k}(x)-I_{k}(x)\\right] &= 0 \\label{eq:numerator} \\cr
\n\\lim_{k \\to \\mu} \\sin(k\\pi) &= 0 \\label{eq:denominator}
\n\\end{align}
\n\u3067\u3059\u304b\u3089\u3001(\\ref{eq:numerator})\u5f0f\u3092$k$\u3067\u5fae\u5206\u3057\u305f\u5f0f\u3092(\\ref{eq:denominator})\u5f0f\u3092$k$\u3067\u5fae\u5206\u3057\u305f\u5f0f\u3067\u5272\u3063\u305f\u7d50\u679c\u5f97\u3089\u308c\u308b\u95a2\u6570$\\eta(k)$(\u306a\u3093\u3060\u304b\u3088\u304f\u308f\u304b\u3089\u306a\u3044\u306e\u3067\u3001\u3068\u308a\u3042\u3048\u305a\u3053\u306e\u3088\u3046\u306b\u7f6e\u304d\u307e\u3059\u3002)\u306e\u5909\u6570$k$\u304c$\\mu$\u306b\u8fd1\u3065\u304f\u3068\u304d\u306e\u6975\u9650\u5024\u304c\u5b58\u5728\u3059\u308c\u3070\u3001\u305d\u306e\u5024\u304c(\\ref{eq:mbfsint})\u5f0f\u306e\u6975\u9650\u5024\u306b\u7b49\u3057\u304f\u306a\u308a\u307e\u3059(\u30ed\u30d4\u30bf\u30eb\u306e\u5b9a\u7406)\u3002<\/p>\n

\u3057\u305f\u304c\u3063\u3066\u3001$\\eta(k)$\u3092\u8a08\u7b97\u3059\u308c\u3070\u6574\u6570\u6b21\u6570\u306e\u7b2c2\u7a2e\u5909\u5f62Bessel\u95a2\u6570\u304c\u6c42\u307e\u308b\u3053\u3068\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n

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

\u5206\u6bcd\u306e\u5fae\u5206\u306e\u6975\u9650<\/h3>\n

\u6700\u521d\u306b\u7c21\u5358\u306a\u65b9\u304b\u3089\u8a08\u7b97\u3057\u307e\u3059\u3002<\/p>\n

\u5206\u6bcd\u306e\u5fae\u5206\u3057\u305f\u7d50\u679c\u5f97\u3089\u308c\u308b\u95a2\u6570\u306e$k$\u304c$\\mu$\u306b\u8fd1\u3065\u304f\u3068\u304d\u306e\u6975\u9650\u5024\u306f$\\mu$\u304c\u6574\u6570\u3067\u3042\u308b\u3053\u3068\u306b\u7740\u76ee\u3059\u308b\u3068\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002
\n\\begin{align}
\n\\lim_{k \\to \\mu}\\frac{d}{dk}\\sin(k\\pi) &= \\lim_{k \\to \\mu}\\pi\\cos(k\\pi) \\nonumber \\cr
\n&= (-1)^{\\mu}\\pi \\label{eq:mbfsintdenominator}
\n\\end{align}<\/p>\n

\u5206\u5b50\u306e\u5fae\u5206\u306e\u6975\u9650<\/h3>\n

\u6b21\u306b\u5206\u5b50\u306e\u5fae\u5206\u306e\u6975\u9650\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002<\/p>\n

\u6700\u521d\u306f1\u9805\u305a\u3064\u5fae\u5206\u3057\u307e\u3059\u3002<\/h4>\n

\u305f\u3060\u5206\u6bcd\u3068\u6bd4\u3079\u308b\u3068\u5c11\u3005\u3069\u3053\u308d\u304b\u3001\u304b\u306a\u308a\u7e41\u96d1\u306a\u306e\u3067\u3001\u6700\u521d\u306f$I_{-k}(x)$\u306e\u90e8\u5206\u3068$I_{-k}(x)$\u306e\u90e8\u5206\u3068\u306b\u5206\u3051\u3066\u5fae\u5206\u3057\u307e\u3059\u3002<\/p>\n

\u307e\u305a\u3001$I_{-k}(x)$\u3092$k$\u3067\u5fae\u5206\u3057\u307e\u3059\u3002\u3059\u308b\u3068\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002
\n\\begin{align}
\n\\frac{\\partial}{\\partial k} I_{-k}(x) &= -I_{-k}(x) \\log\\frac{x}{2} + \\left(\\frac{x}{2}\\right)^{-k} \\sum_{p=0}^{\\infty} \\frac{\\Gamma^{\\prime}(-k+p+1)}{p!\\left[\\Gamma(-k+p+1)\\right]^2} \\left(\\frac{x}{2}\\right)^{2p} \\label{eq:partialfirst}
\n\\end{align}<\/p>\n

(\\ref{eq:partialfirst})\u5f0f\u306b\u306f$\\Gamma$\u95a2\u6570\u306e\u5fae\u5206\u306a\u3093\u3066\u3044\u3046\u3082\u306e\u3084$\\Gamma$\u95a2\u6570\u306e\u81ea\u4e57\u3068\u3044\u3046\u5c11\u3005\u6271\u3044\u306b\u304f\u305d\u3046\u306a\u3082\u306e\u304c\u767b\u5834\u3057\u307e\u3059\u304c\u3001\u3053\u308c\u306f\u3001
\n\\begin{align}
\n\\psi(z) &= \\frac{d}{dz}\\log\\Gamma(z) \\nonumber \\cr
\n&= \\frac{\\Gamma^{\\prime}(z)}{\\Gamma(z)}\\label{eq:digamma}
\n\\end{align}
\n\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u30c7\u30a3\u30ac\u30f3\u30de\u95a2\u6570$\\psi(z)$\u3092\u4f7f\u3046\u3068\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5909\u5f62\u3067\u304d\u307e\u3059\u3002
\n\\begin{align}
\n\\frac{\\partial}{\\partial k} I_{-k}(x) &= -I_{-k}(x) \\log\\frac{x}{2} + \\left(\\frac{x}{2}\\right)^{-k} \\sum_{p=0}^{\\infty} \\frac{\\psi(-k+p+1)}{p!\\Gamma(-k+p+1)} \\left(\\frac{x}{2}\\right)^{2p} \\label{eq:partialfirstdigamma}
\n\\end{align}
\n$\\Gamma$\u95a2\u6570\u306e\u5fae\u5206\u3084\u81ea\u4e57\u306e\u9805\u304c\u6d88\u3048\u3066\u3001\u5c11\u3057\u3059\u3063\u304d\u308a\u3057\u305f\u611f\u3058\u306b\u306a\u308a\u307e\u3057\u305f\u3002\ud83d\udc3c<\/p>\n

\u3053\u3053\u307e\u3067\u306e\u8b70\u8ad6\u3068\u540c\u69d8\u306e\u6d41\u308c\u3067$I_{k}(x)$\u3082$k$\u3067\u5fae\u5206\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u3066\u3001
\n\\begin{align}
\n\\frac{\\partial}{\\partial k} I_{k}(x) &= I_{k}(x) \\log\\frac{x}{2} – \\left(\\frac{x}{2}\\right)^{k} \\sum_{p=0}^{\\infty} \\frac{\\psi(k+p+1)}{p!\\Gamma(k+p+1)} \\left(\\frac{x}{2}\\right)^{2p} \\label{eq:partialseconddigamma}
\n\\end{align}
\n\u3068\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002<\/p>\n

\u5206\u5b50\u306e\u6975\u9650\u5024\u3092\u6c42\u3081\u307e\u3059\u3002<\/h4>\n

(\\ref{eq:partialfirstdigamma})\u5f0f\u53ca\u3073(\\ref{eq:partialseconddigamma})\u5f0f\u3092\u307e\u3068\u3081\u3066\u3001$k$\u304c$\\mu$\u306b\u8fd1\u3065\u304f\u6642\u306e\u6975\u9650\u5024\u3092\u6c42\u3081\u3066\u307f\u307e\u3059\u3002<\/p>\n

\u307e\u305a\u3001\u5358\u7d14\u306b(\\ref{eq:partialfirstdigamma})\u5f0f\u53ca\u3073(\\ref{eq:partialseconddigamma})\u5f0f\u306e\u53f3\u8fba\u4e26\u3073\u306b\u5de6\u8fba\u3069\u3046\u3057\u3092\u5f15\u3044\u3066\u3001$k$\u304c$\\mu$\u306b\u8fd1\u3065\u304f\u6642\u306e\u6975\u9650\u3092\u3068\u308b\u3068(\\ref{eq:mbfsintnumeratorfirst})\u5f0f\u306e\u3088\u3046\u306b\u66f8\u3051\u307e\u3059\u3002
\n\\begin{align}
\n\\lim_{k \\to \\mu} \\frac{\\partial}{\\partial k} \\left[I_{-k}(x) – I_{k}(x)\\right] &= \\lim_{k \\to \\mu} \\left[ -I_{-k}(x) \\log\\frac{x}{2} + \\left(\\frac{x}{2}\\right)^{-k} \\sum_{p=0}^{\\infty} \\frac{\\psi(-k+p+1)}{p!\\Gamma(-k+p+1)} \\left(\\frac{x}{2}\\right)^{2p} \\right. \\nonumber \\cr
\n&{} \\left. – I_{k}(x) \\log\\frac{x}{2} + \\left(\\frac{x}{2}\\right)^{k} \\sum_{p=0}^{\\infty} \\frac{\\psi(k+p+1)}{p!\\Gamma(k+p+1)} \\left(\\frac{x}{2}\\right)^{2p} \\right] \\label{eq:mbfsintnumeratorfirst}
\n\\end{align}<\/p>\n

(\\ref{eq:mbfsintnumeratorfirst})\u5f0f\u306e\u62ec\u5f27\u306e\u4e2d\u306e\u53f3\u8fba\u306e\u7b2c1\u9805\u53ca\u3073\u7b2c3\u9805\u306f$\\mu$\u304c\u6574\u6570\u306e\u3068\u304d\u306b\u306f\u3053\u306e\u8a18\u4e8b<\/a>\u306e\u8a08\u7b97\u7d50\u679c\u3088\u308a$I_{-\\mu}(x) = I_{\\mu}(x)$\u306b\u306a\u308a\u307e\u3059\u3002\u305d\u3053\u3067\u3001(\\ref{eq:mbfsintnumeratorfirst})\u5f0f\u306e\u62ec\u5f27\u306e\u4e2d\u306e\u53f3\u8fba\u306e\u7b2c1\u9805\u53ca\u3073\u7b2c3\u9805\u3092\u307e\u3068\u3081\u308b\u3068\u2026
\n\\begin{align}
\n\\lim_{k \\to \\mu} \\frac{\\partial}{\\partial k} \\left[I_{-k}(x) – I_{k}(x)\\right] &= -2I_{\\mu}(x) \\log\\frac{x}{2} \\nonumber \\cr
\n&{} + \\lim_{k \\to \\mu} \\left[ \\left(\\frac{x}{2}\\right)^{-k} \\sum_{p=0}^{\\infty} \\frac{\\psi(-k+p+1)}{p!\\Gamma(-k+p+1)} \\left(\\frac{x}{2}\\right)^{2p} \\right. \\nonumber \\cr
\n&{} \\left. + \\left(\\frac{x}{2}\\right)^{k} \\sum_{p=0}^{\\infty} \\frac{\\psi(k+p+1)}{p!\\Gamma(k+p+1)} \\left(\\frac{x}{2}\\right)^{2p} \\right] \\label{eq:mbfsintnumeratorsecond}
\n\\end{align}
\n\u3068\u306a\u308a\u307e\u3059(\u307e\u3068\u3081\u305f\u90e8\u5206\u306f\u5148\u306b\u6975\u9650\u5024\u3092\u6c42\u3081\u305f\u3053\u3068\u306b\u3057\u3066\u3057\u307e\u3044\u307e\u3057\u305f)\u3002<\/p>\n

$|k-\\mu|$\u304c\u5341\u5206\u306b\u5c0f\u3055\u3044\u3068\u304d\u3001(\\ref{eq:mbfsintnumeratorsecond})\u5f0f\u53f3\u8fba\u306e$\\lim$\u306e\u4e2d\u306b\u6b8b\u3063\u3066\u3044\u308b\u9805\u306e\u3046\u3061\u3001\u6700\u521d\u306e\u9805\u306e\u7121\u9650\u7d1a\u6570\u306e\u6700\u521d\u306e$\\mu$\u9805($p = 0 \\cdots \\mu-1$)\u306b\u3064\u3044\u3066\u306f\u5206\u6bcd\u306b\u3044\u308b$\\Gamma$\u95a2\u6570\u53ca\u3073\u30c7\u30a3\u30ac\u30f3\u30de\u95a2\u6570\u306e\u5f15\u6570\u304c\u6b63\u3067\u306a\u3044\u6574\u6570\u306b\u8fd1\u3065\u304f\u3082\u306e\u304c\u3042\u308a\u307e\u3059($\\mu=0$\u306e\u5834\u5408\u306b\u9650\u308a\u3001\u5206\u6bcd\u306b\u3044\u308b$\\Gamma$\u95a2\u6570\u53ca\u3073\u30c7\u30a3\u30ac\u30f3\u30de\u95a2\u6570\u306e\u5f15\u6570\u304c\u6b63\u3067\u306a\u3044\u6574\u6570\u306b\u8fd1\u3065\u304f\u3082\u306e\u304c\u3042\u308a\u307e\u305b\u3093\u304c\u3001\u3053\u306e\u5834\u5408\u306b\u3064\u3044\u3066\u306f\u5f8c\u8ff0\u3057\u307e\u3059)\u3002<\/p>\n

\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u611f\u3058\u306e\u30a4\u30e1\u30fc\u30b8\u3067\u3059\u2193<\/p>\n

\"\"<\/a><\/p>\n

\u305d\u3053\u3067\u3001
\n\\begin{align}
\n\\left(\\frac{x}{2}\\right)^{-k} \\sum_{p=0}^{\\infty} \\frac{\\psi(-k+p+1)}{p!\\Gamma(-k+p+1)} \\left(\\frac{x}{2}\\right)^{2p} &= \\left(\\frac{x}{2}\\right)^{-k} \\sum_{p=0}^{\\mu-1} \\frac{\\psi(-k+p+1)}{p!\\Gamma(-k+p+1)} \\left(\\frac{x}{2}\\right)^{2p}\\nonumber \\cr
\n&{} + \\left(\\frac{x}{2}\\right)^{-k} \\sum_{p=\\mu}^{\\infty} \\frac{\\psi(-k+p+1)}{p!\\Gamma(-k+p+1)} \\left(\\frac{x}{2}\\right)^{2p} \\label{eq:iminusk}
\n\\end{align}
\n\u306e\u3088\u3046\u306b\u5206\u5272\u3057\u3066\u304b\u3089$k$\u304c$\\mu$\u306b\u8fd1\u3065\u304f\u6642\u306e\u6975\u9650\u3092\u3068\u308b\u3068\u3001(\\ref{eq:iminusk})\u5f0f\u306e\u53f3\u8fba\u7b2c1\u9805\u306b\u3064\u3044\u3066\u306f\u3053\u306e\u8a18\u4e8b<\/a>\u3067\u793a\u3057\u305f\u5f0f
\n\\begin{align}
\n\\lim_{z \\to -n}\\frac{\\psi(z)}{\\Gamma(z)} &= (-1)^{n-1}(z) \\label{eq:digammagamma}
\n\\end{align}
\n\u304c\u4f7f\u3048\u307e\u3059\u3002<\/p>\n

\u307e\u305f\u3001(\\ref{eq:iminusk})\u5f0f\u306e\u53f3\u8fba\u7b2c2\u9805\u306b\u3064\u3044\u3066\u306f$p$\u3092$p+\\mu$\u306b\u7f6e\u304d\u63db\u3048\u308b\u3068\u3001\u7121\u9650\u7d1a\u6570\u3067\u4f7f\u7528\u3057\u3066\u3044\u308b\u5909\u6570\u306e\u958b\u59cb\u4f4d\u7f6e\u30920\u306b\u79fb\u52d5\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u306e\u3067\u3001$k$\u304c$\\mu$\u306b\u8fd1\u3065\u304f\u6642\u306e\u6975\u9650\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002
\n\\begin{align}
\n\\lim_{k \\to \\mu} \\left[ \\left(\\frac{x}{2}\\right)^{-k} \\sum_{p=0}^{\\infty} \\frac{\\psi(-k+p+1)}{p!\\Gamma(-k+p+1)} \\left(\\frac{x}{2}\\right)^{2p} \\right] &= \\left(\\frac{x}{2}\\right)^{-\\mu} \\sum_{p=0}^{\\mu-1} \\frac{(-1)^{\\mu-p}(\\mu-p-1)!}{p!} \\left(\\frac{x}{2}\\right)^{2p}\\nonumber \\cr
\n&{} + \\left(\\frac{x}{2}\\right)^{-\\mu} \\sum_{p=0}^{\\infty} \\frac{\\psi(p+1)}{(p+\\mu)!\\Gamma(p+1)} \\left(\\frac{x}{2}\\right)^{2(p+\\mu)} \\nonumber \\cr
\n&= \\left(\\frac{x}{2}\\right)^{-\\mu} \\sum_{p=0}^{\\mu-1} \\frac{(-1)^{\\mu-p}(\\mu-p-1)!}{p!} \\left(\\frac{x}{2}\\right)^{2p}\\nonumber \\cr
\n&{} + \\sum_{p=0}^{\\infty} \\frac{\\psi(p+1)}{(p+\\mu)!\\Gamma(p+1)} \\left(\\frac{x}{2}\\right)^{2p+\\mu} \\label{eq:iminusksecond}
\n\\end{align}<\/p>\n

\u5168\u4f53\u306e\u6975\u9650\u5024\u3092\u6c42\u3081\u307e\u3059\u3002<\/h3>\n

\u2026\u3068\u3044\u3046\u308f\u3051\u3067\u3001(\\ref{eq:mbfsintdenominator}),(\\ref{eq:mbfsintnumeratorsecond})\u53ca\u3073(\\ref{eq:iminusksecond})\u5f0f\u3092\u307e\u3068\u3081\u308b\u3068\u3001
\n\\begin{align}
\n\\lim_{k \\to \\mu}\\frac{\\pi}{2} \\frac{\\displaystyle\\frac{\\partial}{\\partial k}\\left[I_{-k}(x) – I_{k}(x)\\right]}{\\displaystyle\\frac{d}{dk}\\sin(k\\pi)} &= \\frac{\\pi}{2\\cdot(-1)^{\\mu}\\cdot\\pi} \\left[ -2I_{\\mu}(x) \\log\\frac{x}{2} \\right. \\nonumber \\cr
\n&{} + \\left(\\frac{x}{2}\\right)^{-\\mu} \\sum_{p=0}^{\\mu-1} \\frac{(-1)^{\\mu-p}(\\mu-p-1)!}{p!} \\left(\\frac{x}{2}\\right)^{2p} \\nonumber \\cr
\n&{} + \\sum_{p=0}^{\\infty} \\frac{\\psi(p+1)}{(p+\\mu)!\\Gamma(p+1)} \\left(\\frac{x}{2}\\right)^{2p+\\mu} \\nonumber \\cr
\n&{} \\left. + \\left(\\frac{x}{2}\\right)^{\\mu} \\sum_{p=0}^{\\infty} \\frac{\\psi(\\mu+p+1)}{p!\\Gamma(\\mu+p+1)} \\left(\\frac{x}{2}\\right)^{2p} \\right] \\nonumber \\cr
\n&= \\frac{(-1)^{\\mu}}{2} \\left[ -2I_{\\mu}(x) \\log\\frac{x}{2} \\right. \\nonumber \\cr
\n&{} + \\left(\\frac{x}{2}\\right)^{-\\mu} \\sum_{p=0}^{\\mu-1} \\frac{(-1)^{\\mu-p}(\\mu-p-1)!}{p!} \\left(\\frac{x}{2}\\right)^{2p} \\nonumber \\cr
\n&{} + \\sum_{p=0}^{\\infty} \\frac{\\psi(p+1)}{(p+\\mu)!\\Gamma(p+1)} \\left(\\frac{x}{2}\\right)^{2p+\\mu} \\nonumber \\cr
\n&{} \\left. + \\left(\\frac{x}{2}\\right)^{\\mu} \\sum_{p=0}^{\\infty} \\frac{\\psi(\\mu+p+1)}{p!\\Gamma(\\mu+p+1)} \\left(\\frac{x}{2}\\right)^{2p} \\right]
\n\\label{eq:combined}
\n\\end{align}
\n\u3068\u306a\u3063\u3066\u3001\u6975\u9650\u304c\u5b58\u5728\u3057\u305d\u3046\u306a\u611f\u3058\u306e\u5f0f\u306b\u306a\u308a\u307e\u3059\u3002(\\ref{eq:combined})\u5f0f\u3092\u3082\u3046\u5c11\u3057\u307e\u3068\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u306a\u3044\u304b\u8003\u3048\u3066\u307f\u307e\u3059\u3002<\/p>\n

(\\ref{eq:combined})\u5f0f\u306e\u53f3\u8fba\u306f\u6975\u9650\u3092\u3068\u3063\u305f\u3042\u3068\u306e\u5024\u3067\u3042\u308a\u3001$\\Gamma$\u95a2\u6570\u306e\u5f15\u6570\u306f\u3059\u3079\u3066\u6b63\u306e\u6574\u6570\u3067\u3059\u306e\u3067\u3001\u968e\u4e57\u306e\u5f62\u3067\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u305d\u3046\u3067\u3059\u3002\u307e\u305f\u3001(\\ref{eq:combined})\u5f0f\u306e\u53f3\u8fba\u306e\u7b2c3\u9805\u53ca\u3073\u7b2c4\u9805\u306b\u3064\u3044\u3066\u306f$x$\u306e\u51aa\u306e\u3068\u3053\u308d\u304c\u4f3c\u305f\u5f62\u306b\u306a\u3063\u3066\u3044\u307e\u3059\u306e\u3067\u3001\u3082\u3057\u304b\u3059\u308b\u3068\u307e\u3068\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u304b\u3082\u3057\u308c\u307e\u305b\u3093\u3002<\/p>\n

\u5b9f\u969b\u306b\u8a08\u7b97\u3057\u3066\u307f\u308b\u3068\u2026
\n\\begin{align}
\n\\lim_{k \\to \\mu}\\frac{\\pi}{2} \\frac{\\displaystyle\\frac{\\partial}{\\partial k}\\left[I_{-k}(x) – I_{k}(x)\\right]}{\\displaystyle\\frac{d}{dk}\\sin(k\\pi)} &=
\n(-1)^{\\mu+1}I_{\\mu}(x) \\log\\frac{x}{2} \\nonumber \\cr
\n&{} + \\frac{(-1)^{\\mu}}{2} \\left[ \\left(\\frac{x}{2}\\right)^{-\\mu} \\sum_{p=0}^{\\mu-1} \\frac{(-1)^{\\mu-p}(\\mu-p-1)!}{p!} \\left(\\frac{x}{2}\\right)^{2p} \\right. \\nonumber \\cr
\n&{} + \\sum_{p=0}^{\\infty} \\frac{\\psi(p+1)}{(p+\\mu)!p!} \\left(\\frac{x}{2}\\right)^{2p+\\mu} \\nonumber \\cr
\n&{} \\left. + \\sum_{p=0}^{\\infty} \\frac{\\psi(\\mu+p+1)}{p!(\\mu+p)!} \\left(\\frac{x}{2}\\right)^{2p+\\mu} \\right] \\nonumber \\cr
\n&= (-1)^{\\mu+1}I_{\\mu}(x) \\log\\frac{x}{2} \\nonumber \\cr
\n&{} + \\frac{1}{2} \\left(\\frac{x}{2}\\right)^{-\\mu} \\sum_{p=0}^{\\mu-1} \\frac{(-1)^p(\\mu-p-1)!}{p!} \\left(\\frac{x}{2}\\right)^{2p} \\nonumber \\cr
\n&{} + \\frac{1}{2} \\left(-\\frac{x}{2}\\right)^{\\mu} \\sum_{p=0}^{\\infty} \\frac{\\psi(p+1)+\\psi(\\mu+p+1)}{(p+\\mu)!p!} \\left(\\frac{x}{2}\\right)^{2p} \\label{eq:combinedandcalc}
\n\\end{align}
\n\u3068\u3044\u3046\u611f\u3058\u3067\u307e\u3068\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059($(-1)^{a} = (-1)^{-a}$\u3067\u3042\u308b\u3053\u3068\u306f\u7279\u306b\u660e\u8a18\u305b\u305a\u306b\u4f7f\u3063\u3066\u3044\u307e\u3059)\u3002<\/p>\n

\u306a\u304a\u3001(\\ref{eq:combinedandcalc})\u5f0f\u306e\u53f3\u8fba\u7b2c3\u9805\u304c\u53ce\u675f\u3059\u308b\u306e\u304b\u3069\u3046\u304b\u304c\u6c17\u306b\u306a\u308b\u3068\u3053\u308d\u3067\u3059\u304c\u3001\u53ce\u675f\u3057\u307e\u3059(\u8a3c\u660e\u306b\u3064\u3044\u3066\u306f\u3053\u3061\u3089<\/a>\u306b\u66f8\u304d\u307e\u3057\u305f(panda\u5927\u5b66\u7fd2\u5e33\u5916\u4f1d\u306e\u30da\u30fc\u30b8\u3067\u3059))\u3002<\/p>\n

\u3057\u305f\u304c\u3063\u3066\u3001(\\ref{eq:combinedandcalc})\u5f0f\u306e\u53f3\u8fba\u306f\u53ce\u675f\u3059\u308b\u306e\u3067\u3001\u305d\u306e\u5024\u304c\u305d\u306e\u307e\u307e$\\mu$\u6b21\u306e\u7b2c2\u7a2e\u5909\u5f62Bessel\u95a2\u6570$J_{\\mu}(x)$\u306e\u7d1a\u6570\u8868\u73fe\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n

$\\mu = 0$\u306e\u5834\u5408\u306b\u306f\u3001(\\ref{eq:iminusk})\u5f0f\u306e\u53f3\u8fba\u7b2c1\u9805\u306f\u73fe\u308c\u306a\u3044\u306e\u3067\u3001(\\ref{eq:iminusk})\u5f0f\u306e\u53f3\u8fba\u7b2c1\u9805\u3092\u5909\u5f62\u3057\u305f\u7d50\u679c\u5f97\u3089\u308c\u308b(\\ref{eq:combinedandcalc})\u5f0f\u306e\u53f3\u8fba\u7b2c2\u9805\u306f\u5b58\u5728\u3057\u307e\u305b\u3093\u3002\u3088\u3063\u3066\u3001$\\mu = 0$\u306e\u5834\u5408\u306b\u306f\u3001\u7b2c2\u7a2e\u5909\u5f62Bessel\u95a2\u6570$J_{\\mu}(x)$\u306e\u7d1a\u6570\u8868\u73fe\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u307e\u3059\u3002
\n\\begin{align}
\nJ_0(x) &= -I_0(x)\\log\\frac{x}{2} + \\sum_{p=0}^{\\infty}\\frac{\\psi(p+1)}{(p!)^2}\\left(\\displaystyle\\frac{x}{2}\\right)^{2p} \\label{eq:zerothorder}
\n\\end{align}<\/p>\n

(\\ref{eq:combinedandcalc})\u5f0f\u306e\u53f3\u8fba\u53ca\u3073(\\ref{eq:zerothorder})\u5f0f\u304c0\u6b21\u306e\u5834\u5408\u3082\u542b\u3081\u305f\u6574\u6570\u6b21\u6570($\\mu$\u6b21)\u306e\u7b2c2\u7a2e\u5909\u5f62Bessel\u95a2\u6570$J_{\\mu}(x)$\u306e\u7d1a\u6570\u8868\u73fe\u306b\u306a\u308a\u307e\u3059\u3002$\\qquad\\blacksquare$<\/p>\n

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

(\\ref{eq:combinedandcalc})\u5f0f\u306e\u53f3\u8fba\u53ca\u3073(\\ref{eq:zerothorder})\u5f0f\u306e\u3069\u3061\u3089\u3082$\\log$\u304c\u767b\u5834\u3057\u305f\u308a\u30c7\u30a3\u30ac\u30f3\u30de\u95a2\u6570\u304c\u767b\u5834\u3057\u305f\u308a\u3001\u6319\u53e5\u306e\u679c\u3066\u306b\u306f\u7b2c1\u7a2e\u5909\u5f62Bessel\u95a2\u6570\u304c\u767b\u5834\u3057\u305f\u308a\u3068\u304b\u306a\u308a\u8cd1\u3084\u304b\u306a\u7d1a\u6570\u5c55\u958b\u306e\u5f0f\u3068\u306a\u3063\u3066\u3044\u307e\u3059\u304c\u3001\u6b63\u306e\u6570\u3092\u5f15\u6570\u3068\u3059\u308b\u30c7\u30a3\u30ac\u30f3\u30de\u95a2\u6570\u306f\u6bd4\u8f03\u7684\u7c21\u5358\u306b\u6570\u5024\u8a08\u7b97\u304c\u3067\u304d\u307e\u3059\u306e\u3067\u3001\u7b2c1\u7a2e\u5909\u5f62Bessel\u95a2\u6570\u304c\u8a08\u7b97\u3067\u304d\u308c\u3070\u8a08\u7b97\u6a5f\u3067\u306e\u5b9f\u88c5\u304c\u3067\u304d\u305d\u3046\u306a\u96f0\u56f2\u6c17\u306f\u91b8\u3057\u51fa\u3057\u3066\u3044\u308b\u3068\u601d\u3044\u307e\u3059(\u203b\u500b\u4eba\u306e\u611f\u60f3\u3067\u3059)\u3002<\/p>\n

\u7d50\u679c\u7684\u306b\u306f\u3053\u306e\u8a18\u4e8b<\/a>\u3068\u3053\u306e\u8a18\u4e8b<\/a>\u304c\u58ee\u5927\u306a(\u672cWeb\u30b5\u30a4\u30c8\u6bd4)\u306e\u524d\u30d5\u30ea\u306b\u306a\u3063\u3066\u3044\u305f\u3053\u3068\u306b\u306a\u308a\u307e\u3059\u3002\u7279\u306b\u3053\u306e\u8a18\u4e8b<\/a>\u3067\u306f$\\Gamma$\u95a2\u6570\u3092\u542b\u3080\u95a2\u6570\u306e\u6975\u9650\u306b\u3064\u3044\u3066\u66f8\u304d\u307e\u3057\u305f\u304c\u3001\u3053\u308c\u3092\u5358\u72ec\u3067\u793a\u3055\u308c\u3066\u3082\u4e00\u4f53\u4f55\u306e\u5f79\u306b\u7acb\u3064\u306e\u3060\u308d\u3046\u7684\u306a\u7b2c\u4e00\u5370\u8c61\u3092\u6301\u305f\u308c\u308b\u3053\u3068\u306b\u306a\u308a\u304c\u3061\u3067\u3059\u3002\u3053\u3053\u307e\u3067\u306e\u8a08\u7b97\u3092\u898b\u3066\u3044\u305f\u3060\u304f\u3068\u3001\u8a08\u7b97\u6a5f\u3067\u306e\u5b9f\u88c5\u304c\u3067\u304d\u305d\u3046\u306a\u5f0f\u3078\u306e\u5909\u5f62\u306e\u9014\u4e2d\u3067\u767b\u5834\u3057\u3066\u3001\u8a08\u7b97\u5f0f\u3092\u7c21\u7565\u5316\u3059\u308b\u306e\u306b\u5f79\u7acb\u3063\u3066\u3044\u308b\u3053\u3068\u304c\u7406\u89e3\u3057\u3066\u3044\u305f\u3060\u3051\u308b\u304b\u3068\u601d\u3044\u307e\u3059\u3002<\/p>\n

Bessel\u95a2\u6570\u306fMATLAB\u3084Boost\u3084Apache Commons Math\u3084SciPy\u3001\u3055\u3089\u306b\u306fExcel\u306e\u95a2\u6570\u3068\u3057\u3066\u3082\u5b9f\u88c5\u3055\u308c\u3066\u3044\u307e\u3059\u304c\u3001\u80cc\u666f\u3092\u7406\u89e3\u3057\u3066\u3044\u306a\u3044\u3068\u4f7f\u3044\u306b\u304f\u3044\u95a2\u6570\u3067\u3082\u3042\u308a\u307e\u3059\u306e\u3067\u3001\u80cc\u666f\u304c\u77e5\u308a\u305f\u304f\u306a\u3063\u305f\u6642\u306b\u3053\u306e\u8a18\u4e8b\u3092\u30c1\u30e9\u898b\u3057\u3066\u3044\u305f\u3060\u3051\u308b\u3068\u3044\u3044\u3058\u3083\u306a\u3044\u304b\u306a\u3068\u601d\u3044\u307e\u3059\u3002\ud83d\udc3c<\/p>\n

\u3053\u306e\u8a18\u4e8b\u306f\u4ee5\u4e0a\u3067\u3059\u3002<\/p>\n

References \/ \u53c2\u8003\u6587\u732e<\/h2>\n