{"id":3657,"date":"2019-01-02T13:15:42","date_gmt":"2019-01-02T04:15:42","guid":{"rendered":"https:\/\/pandanote.info\/?p=3657"},"modified":"2022-08-07T12:04:03","modified_gmt":"2022-08-07T03:04:03","slug":"%e6%a8%99%e6%ba%96%e6%ad%a3%e8%a6%8f%e5%88%86%e5%b8%83%e3%81%ab%e5%be%93%e3%81%862%e5%80%8b%e3%81%ae%e7%a2%ba%e7%8e%87%e5%a4%89%e6%95%b0%e3%81%ae%e7%a9%8d%e3%81%8c%e5%be%93%e3%81%86%e7%a2%ba%e7%8e%87","status":"publish","type":"post","link":"https:\/\/pandanote.info\/?p=3657","title":{"rendered":"\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\u3002"},"content":{"rendered":"

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

\u304b\u306a\u308a\u524d\u306e\u8a18\u4e8b<\/a>\u3067\u3001\u3061\u3087\u3044\u3068\u91ce\u66ae\u7528\u3067\u6b63\u898f\u5206\u5e03\u306b\u5f93\u30462\u500b\u306e\u78ba\u7387\u5909\u6570\u306e\u548c\u3068\u5dee\u304c\u5f93\u3046\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3092\u8a08\u7b97\u3057\u3066\u307f\u307e\u3057\u305f\u3002<\/p>\n

\u3053\u306e\u8a18\u4e8b\u306f\u305d\u306e\u7d9a\u7de8\u306e\u3088\u3046\u306a\u3082\u306e\u3067\u3059\u3002<\/p>\n

\u8abf\u5b50\u306b\u4e57\u3063\u3066\u3042\u307e\u308a\u4f7f\u308f\u306a\u3044\u304b\u3082\u3057\u308c\u306a\u3044<\/del>\u6a19\u6e96\u6b63\u898f\u5206\u5e03\u95a2\u6570\u306b\u5f93\u30462\u500b\u306e\u78ba\u7387\u5909\u6570\u306e\u7a4d\u3092\u8a08\u7b97\u3057\u3088\u3046\u3068\u8a66\u307f\u305f\u308f\u3051\u3067\u3059\u304c\u3001\u672cWeb\u30b5\u30a4\u30c8\u306e\u7ba1\u7406\u4eba\u305f\u308bpanda\u306e\u8a08\u7b97\u306e\u80fd\u529b\u304c\u672a\u719f\u306a\u305b\u3044\u3082\u3042\u3063\u3066\u304b\u30012018\u5e74\u306e\u5e74\u672b\u306b\u3082\u306e\u3059\u3054\u3044\u52e2\u3044\u3067\u30cf\u30de\u3063\u3066\u3057\u307e\u3063\u305f\u306e\u3067\u3001\u30e1\u30e2\u3063\u3066\u304a\u304f\u3053\u3068\u306b\u3057\u307e\u3057\u305f\u3002\u270d<\/p>\n

\u554f\u984c\u306e\u5b9a\u7fa9<\/h2>\n

\u307e\u305a\u3001\u89e3\u304f\u3079\u304d\u554f\u984c\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3057\u307e\u3059\u3002<\/p>\n

\u6a19\u6e96\u6b63\u898f\u5206\u5e03$N(0,1)$\u306b\u5f93\u30462\u500b\u306e\u78ba\u7387\u5bc6\u5ea6\u5909\u6570$X,Y$\u304c\u3042\u308b\u3002\u3053\u306e\u3068\u304d\u3001$XY = Z$\u306b\u3088\u3063\u3066\u5b9a\u7fa9\u3055\u308c\u308b\u78ba\u7387\u5909\u6570\u304c\u5f93\u3046\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3092\u6c42\u3081\u3088\u3002\u305f\u3060\u3057\u3001$Z > 0$\u3068\u3059\u308b\u3002<\/p><\/blockquote>\n

\u306a\u304a\u3001\u306a\u305c$Z > 0$\u3068\u3057\u305f\u304b\u3068\u3044\u3046\u3068\u3001<\/p>\n

\u300c\u5834\u5408\u5206\u3051\u304c\u9762\u5012\u3060\u304b\u3089\u3067\u3059\u3002\u300d<\/strong><\/p>\n

\u3068\u6f54\u304f(\u672cWeb\u30b5\u30a4\u30c8\u6bd4)\u8a00\u3044\u5207\u3063\u3066\u304a\u304f\u3053\u3068\u306b\u3057\u307e\u3059\u3002<\/p>\n

[2019\/08\/17 \u8ffd\u8a18] $Z < 0$\u306e\u5834\u5408\u306b\u3064\u3044\u3066\u3082\u5225\u9014\u66f8\u304d\u307e\u3057\u305f\u3002\u3053\u3061\u3089<\/a>\u3092\u3054\u53c2\u7167\u3044\u305f\u3060\u3051\u308b\u3068\u5e78\u3044\u3067\u3059\u3002<\/p>\n

\u30b5\u30af\u30b5\u30af\u3068\u8a08\u7b97\u3002<\/h2>\n

$Y$\u306e\u7b26\u53f7\u306b\u3088\u308b\u5834\u5408\u5206\u3051\u3002<\/h3>\n

\u554f\u984c\u3092\u5b9a\u7fa9\u3057\u305f\u3068\u3053\u308d\u3067\u3001\u30b5\u30af\u30b5\u30af\u3068\u8a08\u7b97\u3057\u307e\u3059\u3002<\/p>\n

\u307e\u305a\u3001\u7a4d\u5206\u306e\u9818\u57df\u3092\u5b9a\u3081\u307e\u3059\u3002<\/p>\n

\u7a4d\u5206\u3092\u5b9f\u884c\u3059\u308b$X,Y$\u306e\u7bc4\u56f2\u30922\u6b21\u5143\u5e73\u9762\u4e0a\u306b\u30d7\u30ed\u30c3\u30c8\u3059\u308b\u3068\u3001\u4ee5\u4e0b\u306e\u56f3\u306e\u6c34\u8272\u3067\u793a\u3057\u305f\u90e8\u5206\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n

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

\u3053\u306e\u8a18\u4e8b\u306e\u6700\u521d\u306e\u65b9\u3067$Z > 0$\u3068\u3044\u304d\u306a\u308a\u5909\u6570\u306e\u7bc4\u56f2\u3092\u9650\u5b9a\u3057\u305f\u306b\u3082\u304b\u304b\u308f\u3089\u305a\u3001$Y$\u306e\u7b26\u53f7\u306b\u3088\u3063\u3066\u5834\u5408\u5206\u3051\u3092\u3059\u308b\u5fc5\u8981\u304c\u3042\u308a\u305d\u3046\u3067\u3059\u3002(\u00b4\u30fb\u03c9\u30fb\uff40)<\/p>\n

$Y$\u306e\u7b26\u53f7\u3092\u8003\u616e\u3057\u3064\u3064\u3001$X,Y$\u306e\u7a4d\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u78ba\u7387\u5909\u6570$Z$\u306e\u3068\u308b\u5024\u304c$z$\u4ee5\u4e0b\u3068\u306a\u308b\u78ba\u7387$P\\{XY\\le z\\}$\u306f(\\ref{eq:Pxy})\u5f0f\u3067\u8868\u3055\u308c\u307e\u3059\u3002<\/p>\n

[2019\/01\/12\u8ffd\u8a18]\u4e0d\u7b49\u53f7\u306e\u5411\u304d\u304c\u9593\u9055\u3063\u3066\u3044\u305f\u306e\u3067\u3001\u4fee\u6b63\u3057\u307e\u3057\u305f\u3002
\n\\begin{align}
\nP\\{XY\\le z\\} &= P\\{XY\\le z,Y \\ge 0\\} + P\\{XY\\le z, Y \\le 0\\}\\label{eq:Pxy}
\n\\end{align}<\/p>\n

(\\ref{eq:Pxy})\u5f0f\u306e\u53f3\u8fba\u306e\u7b2c1\u9805\u306f(\\ref{eq:Pxypositive})\u5f0f\u306e\u3088\u3046\u306b\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002\u4e0a\u56f3\u306e(a)\u306e\u5411\u304d\u306b\u6700\u521d\u306b\u7a4d\u5206\u3059\u308b\u30a4\u30e1\u30fc\u30b8\u306b\u306a\u308a\u307e\u3059\u3002\u306a\u304a\u3001(\\ref{eq:Pxypositive})\u5f0f\u306e\u53f3\u8fba\u3067\u306f$Y$\u3092$y$\u3068\u66f8\u304d\u63db\u3048\u3066\u3044\u307e\u3059\u3002<\/p>\n

\\begin{align}
\nP\\{XY\\le z,Y \\ge 0\\} &= \\frac{1}{2\\pi}\\int^{\\infty}_{0}e^{-\\frac{y^2}{2}}dy\\int^{\\frac{z}{y}}_{-\\infty}e^{-\\frac{x^2}{2}}dx
\n\\label{eq:Pxypositive}
\n\\end{align}<\/p>\n

\u3053\u3053\u3067\u3001\u3061\u3087\u3063\u3068\u6c17\u306b\u306a\u308b\u306e\u306f$Y=0$\u306e\u5834\u5408\u306e\u3068\u3053\u308d\u3067\u3059\u304c\u3001\u3053\u306e\u5834\u5408\u306f(\\ref{eq:Pxypositive})\u5f0f\u306e\u3046\u3061\u5909\u6570$x$\u306b\u3088\u308b\u7a4d\u5206\u533a\u9593\u304c$[-\\infty,\\infty]$\u306b\u306a\u308b\u306e\u3067\u3001
\n\\begin{align}
\n\\int^{\\infty}_{-\\infty}e^{-\\frac{x^2}{2}}dx&=\\sqrt{2\\pi} \\label{eq:gaussian}
\n\\end{align}
\n\u3068\u8a08\u7b97\u3067\u304d\u3066\u7a4d\u5206\u53ef\u80fd\u3067\u3059\u306e\u3067\u3001(\\ref{eq:Pxypositive})\u5f0f\u306f$Y=0$\u306e\u5834\u5408\u306b\u3082\u6210\u308a\u7acb\u3064\u3068\u8003\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n

\u3055\u3089\u306b\u3001(\\ref{eq:Pxypositive})\u5f0f\u3067$x = \\displaystyle{\\frac{u}{y}}$\u3068\u7f6e\u304d\u307e\u3059\u3002\u3059\u308b\u3068\u3001(\\ref{eq:Pxypositive})\u5f0f\u306e\u53f3\u8fba\u306f\u3001
\n\\begin{align}
\nP\\{XY\\le z,Y \\ge 0\\} &= \\frac{1}{2\\pi}\\int^{\\infty}_{0}e^{-\\frac{y^2}{2}}dy\\int^{z}_{-\\infty}\\frac{1}{y}e^{-\\frac{u^2}{2y^2}}du
\n\\label{eq:Pxypositivefirst}
\n\\end{align}
\n\u3068\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002<\/p>\n

\u6b21\u306b\u3001(\\ref{eq:Pxy})\u5f0f\u306e\u53f3\u8fba\u306e\u7b2c2\u9805\u3092\u8a08\u7b97\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n

(\\ref{eq:Pxy})\u5f0f\u306e\u53f3\u8fba\u306e\u7b2c2\u9805\u306f\u3001
\n\\begin{align}
\nP\\{XY\\le z,Y \\le 0\\} &= \\frac{1}{2\\pi}\\int^{0}_{-\\infty}e^{-\\frac{y^2}{2}}dy\\int^{\\infty}_{\\frac{z}{y}}e^{-\\frac{x^2}{2}}dx
\n\\label{eq:Pxynegative}
\n\\end{align}
\n\u3068\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002\u4e0a\u56f3\u306e(b)\u306e\u5411\u304d\u306b\u6700\u521d\u306b\u7a4d\u5206\u3059\u308b\u30a4\u30e1\u30fc\u30b8\u306b\u306a\u308a\u307e\u3059\u3002\u3053\u308c\u3092(\\ref{eq:Pxypositivefirst})\u5f0f\u3068\u4f3c\u305f\u3088\u3046\u306a\u5f0f\u306b\u5909\u5f62\u3067\u304d\u306a\u3044\u304b\u8003\u3048\u3066\u307f\u307e\u3059\u3002<\/p>\n

\u307e\u305a\u3001$e^{-\\frac{x^2}{2}}$\u306f\u5076\u95a2\u6570\u3067\u3059\u306e\u3067\u3001(\\ref{eq:Pxynegative})\u5f0f\u306e\u53f3\u8fba\u306f(\\ref{eq:Pxynegativefirst})\u5f0f\u306e\u3088\u3046\u306b\u66f8\u304d\u63db\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002
\n\\begin{align}
\nP\\{XY\\le z,Y \\le 0\\} &= \\frac{1}{2\\pi}\\int^{0}_{-\\infty}e^{-\\frac{y^2}{2}}dy\\int^{-\\frac{z}{y}}_{-\\infty}e^{-\\frac{x^2}{2}}dx
\n\\label{eq:Pxynegativefirst}
\n\\end{align}
\n\u6b21\u306b\u3001$y = -\\xi$\u3068\u7f6e\u304d\u307e\u3059\u3002\u3059\u308b\u3068\u3001(\\ref{eq:Pxynegativefirst})\u5f0f\u306e\u53f3\u8fba\u306f\u3001
\n\\begin{align}
\nP\\{XY\\le z,Y \\le 0\\} &= -\\frac{1}{2\\pi}\\int^{0}_{\\infty}e^{-\\frac{\\xi^2}{2}}d\\xi\\int^{\\frac{z}{\\xi}}_{-\\infty}e^{-\\frac{x^2}{2}}dx
\n\\label{eq:Pxynegativesecond}
\n\\end{align}
\n\u3068\u306a\u308a\u307e\u3059\u306e\u3067\u3001\u3055\u3089\u306b$x = \\displaystyle{\\frac{u}{\\xi}}$\u3068\u7f6e\u304d\u307e\u3059\u3002\u3059\u308b\u3068\u3001(\\ref{eq:Pxynegativesecond})\u5f0f\u306e\u53f3\u8fba\u306f\u3001
\n\\begin{align}
\nP\\{XY\\le z,Y \\le 0\\} &= -\\frac{1}{2\\pi}\\int^{0}_{\\infty}e^{-\\frac{\\xi^2}{2}}d\\xi\\int^{z}_{-\\infty}\\frac{1}{\\xi}e^{-\\frac{u^2}{2\\xi^2}}du \\nonumber \\\\
\n&= \\frac{1}{2\\pi}\\int^{\\infty}_{0}e^{-\\frac{\\xi^2}{2}}d\\xi\\int^{z}_{-\\infty}\\frac{1}{\\xi}e^{-\\frac{u^2}{2\\xi^2}}du
\n\\label{eq:Pxynegativethird}
\n\\end{align}
\n\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n

(\\ref{eq:Pxynegativethird})\u5f0f\u306e$\\xi$\u3092$y$\u306b\u7f6e\u304d\u63db\u3048\u308b\u3068(\\ref{eq:Pxypositivefirst})\u5f0f\u306e\u53f3\u8fba\u3068\u540c\u3058\u5f0f\u304c\u5f97\u3089\u308c\u307e\u3059\u3002<\/p>\n

\u3057\u305f\u304c\u3063\u3066\u3001$P\\{XY\\le z\\}$\u306f(\\ref{eq:Pxyfinal})\u5f0f\u3067\u8868\u3055\u308c\u307e\u3059($u$\u3092$x$\u306b\u7f6e\u304d\u63db\u3048\u3066\u3044\u307e\u3059)\u3002
\n\\begin{align}
\nP\\{XY\\le z\\} &= \\frac{1}{\\pi}\\int^{\\infty}_{0}e^{-\\frac{y^2}{2}}dy\\int^{z}_{-\\infty}\\frac{1}{y}e^{-\\frac{x^2}{2y^2}}dx \\label{eq:Pxyfinal}
\n\\end{align}<\/p>\n

(\\ref{eq:Pxyfinal})\u5f0f\u306e$x$\u53ca\u3073$y$\u306b\u3064\u3044\u3066\u306e\u7a4d\u5206\u306e\u9806\u5e8f\u306f\u5165\u308c\u66ff\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u305d\u3053\u3067\u3001\u7a4d\u5206\u306e\u9806\u5e8f\u3092\u5165\u308c\u66ff\u3048\u3066\u304b\u3089$z$\u3067\u5fae\u5206\u3059\u308b\u3068\u3001$z$\u306b\u304a\u3051\u308b\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570$p(z)$\u304c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002
\n\\begin{align}
\np(z) &= \\frac{1}{\\pi}\\int^{\\infty}_{0}\\frac{e^{-\\frac{y^2}{2}-\\frac{z^2}{2y^2}}}{y}dy \\label{eq:pzfinal}
\n\\end{align}<\/p>\n

\u3055\u3089\u306b\u5909\u6570\u5909\u63db\u3002<\/h3>\n

\u3053\u3053\u304b\u3089\u306f\u7d50\u8ad6\u304c\u3042\u308b\u7a0b\u5ea6\u898b\u3048\u3066\u3044\u306a\u3044\u3068\u8a08\u7b97\u3092\u3059\u308b\u30e2\u30c1\u30d9\u30fc\u30b7\u30e7\u30f3\u304c\u8d77\u3053\u308a\u306b\u304f\u3044\u3068\u3053\u308d\u3067\u306f\u3042\u308a\u307e\u3059\u304c\u3001(\\ref{eq:pzfinal})\u5f0f\u3092\u3059\u3067\u306b\u3088\u304f\u77e5\u3089\u308c\u305f\u5f62\u306b\u5909\u5f62\u3067\u304d\u306a\u3044\u304b\u8003\u3048\u3066\u307f\u307e\u3059\u3002<\/p>\n

(\\ref{eq:pzfinal})\u5f0f\u3067$t=\\displaystyle\\frac{y^2}{2}$\u3068\u304a\u304d\u307e\u3059\u3002\u3059\u308b\u3068\u3001$\\displaystyle\\frac{dy}{dt}=\\displaystyle\\frac{1}{\\sqrt{2t}}$\u3068\u306a\u308a\u307e\u3059\u306e\u3067(\\ref{eq:pandt})\u5f0f\u306e\u3088\u3046\u306b\u5909\u5f62\u3067\u304d\u307e\u3059\u3002
\n\\begin{align}
\np(z) &= \\frac{1}{\\pi}\\int^{\\infty}_{0}\\frac{1}{\\sqrt{2t}}e^{-2t-\\frac{z^2}{4t}}\\frac{1}{\\sqrt{2t}}dt \\nonumber \\\\
\n&= \\frac{1}{2\\pi}\\int^{\\infty}_{0}\\frac{1}{t}e^{-2t-\\frac{z^2}{4t}}dt\\label{eq:pandt}
\n\\end{align}<\/p>\n

\u6700\u5f8c\u306e\u4ed5\u4e0a\u3052\u306b\u3001$t = \\displaystyle\\frac{ze^{\\eta}}{2}$\u3068\u304a\u304d\u307e\u3059\u3002(\\ref{eq:pandt})\u5f0f\u306e\u7a4d\u5206\u533a\u9593\u306f$[0,\\infty]$\u3067\u3059\u304c\u3001$\\eta$\u306e\u7a4d\u5206\u533a\u9593\u306f$[-\\infty,\\infty]$\u3068\u306a\u308b\u3053\u3068\u3068\u3001$\\displaystyle\\frac{dt}{d\\eta} = \\displaystyle\\frac{ze^{\\eta}}{2}$\u3067\u3042\u308b\u3053\u3068\u304b\u3089\u3001
\n\\begin{align}
\np(z) &= \\frac{1}{2\\pi}\\int^{\\infty}_{-\\infty}\\frac{1}{\\displaystyle\\frac{ze^{\\eta}}{2}}\\exp\\left[-2\\frac{ze^{\\eta}}{2}-\\frac{z^2}{4\\frac{ze^{\\eta}}{2}}\\right]\\frac{ze^{\\eta}}{2}d\\eta \\nonumber \\\\
\n&= \\frac{1}{2\\pi}\\int^{\\infty}_{-\\infty}\\exp\\left[-\\frac{ze^{\\eta}}{2}-\\frac{z}{2e^{\\eta}}\\right]d\\eta \\nonumber \\\\
\n&= \\frac{1}{2\\pi}\\int^{\\infty}_{-\\infty}\\exp\\left[-z\\frac{e^{\\eta}+e^{-\\eta}}{2}\\right]d\\eta \\nonumber \\\\
\n&= \\frac{1}{2\\pi}\\int^{\\infty}_{-\\infty}\\exp\\left[-z\\cosh(\\eta)\\right]d\\eta \\label{eq:expcosh}
\n\\end{align}
\n\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n

\u3053\u3053\u3067\u3001(\\ref{eq:expcosh})\u5f0f\u306e\u53f3\u8fba\u306e$\\exp(\\cdot)$\u306e$\\cdot$\u306e\u90e8\u5206\u3092\u3058\u30fc\u3063\u3068\u898b\u308b\u3068\u3001$z$\u306f$\\eta$\u306b\u4f9d\u5b58\u3057\u306a\u3044\u5b9a\u6570\u3067\u3042\u308a\u3001$\\cosh(\\eta)$\u306f\u5076\u95a2\u6570\u3067\u3042\u308b\u3053\u3068\u304b\u3089\u3001(\\ref{eq:expcosh})\u5f0f\u306e\u53f3\u8fba\u306f\u3001
\n\\begin{align}
\n((\\ref{eq:expcosh})\u5f0f\u306e\u53f3\u8fba) &= \\frac{1}{\\pi}\\int^{\\infty}_{0}\\exp\\left[-z\\cosh(\\eta)\\right]d\\eta \\label{eq:expcoshfinal}
\n\\end{align}
\n\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n

(\\ref{eq:expcoshfinal})\u5f0f\u306f0\u6b21\u306e\u7b2c2\u7a2e\u5909\u5f62Bessel\u95a2\u6570(Modified Bessel Function of the Second Kind, $K_0(z)$)\u306e\u7a4d\u5206\u8868\u793a\u3092$\\pi$\u3067\u5272\u3063\u305f\u3082\u306e\u306b\u7b49\u3057\u304f\u306a\u308a\u307e\u3059\u306e\u3067\u3001\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570$p(z)$\u306f(\\ref{eq:pzbessel})\u5f0f\u306e\u3088\u3046\u306b\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002
\n\\begin{align}
\np(z) &= \\frac{K_0(z)}{\\pi} \\label{eq:pzbessel}
\n\\end{align}<\/p>\n

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

\u3053\u306e\u8a18\u4e8b\u3092\u6700\u521d\u306b\u66f8\u3044\u305f\u6642\u70b9(2019\u5e741\u6708)\u3067\u306f\u5b9f\u969b\u306b\u8a08\u7b97\u3057\u3066\u307f\u305f\u308f\u3051\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u304c\u3001$z < 0$\u306e\u5834\u5408\u306f\u524d\u7bc0\u306e\u540c\u69d8\u306e\u624b\u9806\u3067(\\ref{eq:pzbesselminus})\u5f0f\u3068\u306a\u308b\u3053\u3068\u3092\u793a\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\n\n[2019\/01\/12\u8ffd\u8a18] \u3053\u3061\u3089\u306e\u30da\u30fc\u30b8<\/a>\u306b\u66f8\u304d\u307e\u3057\u305f\u3002
\n\\begin{align}
\np(z) &= \\frac{K_0(-z)}{\\pi} \\label{eq:pzbesselminus}
\n\\end{align}
\n\u305d\u3053\u3067\u3001(\\ref{eq:pzbessel})\u5f0f\u53ca\u3073(\\ref{eq:pzbesselminus})\u5f0f\u3092\u307e\u3068\u3081\u3066(\\ref{eq:pzbesselabs})\u5f0f\u3068\u66f8\u304f\u3053\u3068\u3082\u3067\u304d\u307e\u3059\u3002
\n\\begin{align}
\np(z) &= \\frac{K_0(|z|)}{\\pi} \\label{eq:pzbesselabs}
\n\\end{align}<\/p>\n

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

2019\u5e74\u306f\u65b0\u5e74\u6700\u521d\u306e\u8a18\u4e8b\u304b\u3089\u3044\u304d\u306a\u308a\u6700\u521d\u304b\u3089\u6570\u5f0f\u6e80\u8f09\u306e\u8a18\u4e8b\u3067\u304a\u9001\u308a\u3057\u3066\u307f\u307e\u3057\u305f\u3002<\/p>\n

[2019\/01\/12,2019\/02\/06\u8ffd\u8a18]\u3061\u3087\u3053\u3061\u3087\u3053\u3068\u9593\u9055\u3063\u3066\u3044\u308b\u3068\u3053\u308d\u304c\u3042\u3063\u305f\u306e\u3067\u3001\u4fee\u6b63\u3057\u307e\u3057\u305f\u3002<\/p>\n

$p(z)$\u30920\u6b21\u306e\u7b2c2\u7a2e\u5909\u5f62Bessel\u95a2\u6570\u3067\u8868\u3059\u305f\u3081\u306b\u5909\u6570\u306e\u5909\u63db\u3092\u4f55\u56de\u304b\u7e70\u308a\u8fd4\u3057\u3066\u3044\u307e\u3059\u304c\u3001\u6700\u5f8c\u306e$t = \\displaystyle\\frac{ze^{\\eta}}{2}$\u3068\u304a\u3044\u305f\u3068\u3053\u308d\u3067\u7f6e\u304d\u63db\u3048\u5f8c\u306e\u7a4d\u5206\u306e\u7bc4\u56f2\u304c\u8aa4\u3063\u3066\u3044\u305f\u3053\u3068\u306b\u6c17\u3065\u304b\u305a\u3001\u3069\u3046\u3057\u3066\u3082\u5206\u6bcd\u306e2\u304c\u53d6\u308c\u305a\u306b\u30cf\u30de\u3063\u3066\u3057\u307e\u3044\u307e\u3057\u305f\u3002<\/p>\n

\u4eca\u5f8c\u6c17\u3092\u4ed8\u3051\u307e\u3059\u3002<\/p>\n

Bessel\u95a2\u6570\u306f\u65e5\u5e38\u751f\u6d3b\u306b\u304a\u3044\u3066\u306f\u305d\u3046\u7c21\u5358\u306b\u767b\u5834\u3059\u308b\u3053\u3068\u306f\u306a\u3044\u3068\u601d\u3044\u307e\u3059\u304c\u3001\u4f55\u304b\u306e\u53c2\u8003\u306b\u3057\u3066\u3044\u305f\u3060\u3051\u308b\u3068\u5e78\u3044\u3067\u3059\u3002\ud83d\ude47\u200d\u2642\ufe0f<\/p>\n

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

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