{"id":11190,"date":"2023-09-21T21:07:24","date_gmt":"2023-09-21T12:07:24","guid":{"rendered":"https:\/\/pandanote.info\/?p=11190"},"modified":"2023-09-21T21:11:05","modified_gmt":"2023-09-21T12:11:05","slug":"x1%e3%81%ae%e5%a0%b4%e5%90%88%e3%81%aearctanx%e3%81%ae%e5%a4%9a%e9%a0%85%e5%bc%8f%e5%b1%95%e9%96%8b","status":"publish","type":"post","link":"https:\/\/pandanote.info\/?p=11190","title":{"rendered":"|x|>1\u306e\u5834\u5408\u306eArctan(x)\u306e\u591a\u9805\u5f0f\u5c55\u958b"},"content":{"rendered":"

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

$|x|<1$\u306e\u3068\u304d\u306e${\\rm Arctan}x$\u306e\u591a\u9805\u5f0f\u5c55\u958b\u306e\u5f0f\u306f\u3088\u304f\u898b\u304b\u3051\u308b\u306e\u3067\u3059\u304c\u3001$|x|>1$\u306e\u6642\u306e\u5c55\u958b\u5f0f\u306f\u3042\u307e\u308a\u898b\u304b\u3051\u308b\u6a5f\u4f1a\u304c\u3042\u308a\u307e\u305b\u3093\u3002<\/p>\n

\u305d\u3053\u3067\u3001\u8a08\u7b97\u3057\u3066\u307f\u308b\u3053\u3068\u306b\u3057\u307e\u3057\u305f\u3002<\/p>\n

y = Arctan(x)\u306e\u30b0\u30e9\u30d5<\/h2>\n

Inkscape 1.3\u3092\u4f7f\u3063\u3066$y = {\\rm Arctan}(x)$\u306e\u30b0\u30e9\u30d5\u3092\u63cf\u304f\u3068\u2026<\/p>\n

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

\u306e\u3088\u3046\u306b\u63cf\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u3059($|x| \\le 5$\u306e\u7bc4\u56f2\u306e\u307f\u63cf\u3044\u3066\u3044\u307e\u3059)\u3002<\/p>\n

\u53ce\u675f\u534a\u5f84<\/h2>\n

\u306a\u305c$|x|>1$\u306e\u6642\u306e\u5c55\u958b\u5f0f\u3092\u8003\u3048\u308b\u306e\u304b\u3068\u3044\u3046\u3068\u3001${\\rm Arctan} x$\u306e\u30de\u30af\u30ed\u30fc\u30ea\u30f3\u5c55\u958b\u306e\u5f0f
\n\\begin{align}
\n{\\rm Arctan}\\,x &= \\sum_{n=0}^{\\infty}\\frac{(-1)^n}{2n+1}x^{2n+1}\\label{eq:maclaurinseriesofarctan}
\n\\end{align}
\n\u306e\u53f3\u8fba\u306e\u5f0f\u306e\u53ce\u675f\u534a\u5f84\u304c1\u306a\u306e\u3067\u3001$|x|>1$\u306e\u3068\u304d\u306b\u306f(\\ref{eq:maclaurinseriesofarctan})\u5f0f\u3092\u76f4\u63a5\u5229\u7528\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u306a\u3044\u305f\u3081\u3067\u3059\u3002<\/p>\n

\u3053\u3053\u304b\u3089\u304c\u672c\u984c<\/h2>\n

\u305d\u3053\u3067\u3001\u3044\u3063\u305f\u3093<\/p>\n

\\begin{align}
\n{\\rm Arctan}\\,x &= y \\label{eq:y}
\n\\end{align}<\/p>\n

\u306e\u3088\u3046\u306b\u7f6e\u3044\u3066\u307f\u307e\u3059\u3002<\/p>\n

(\\ref{eq:y})\u5f0f\u3088\u308a\u3001$x = \\tan y$\u3068\u3044\u3046\u3053\u3068\u306b\u306a\u308b\u306e\u3067\u3001\u3055\u3089\u306b$x = 1\/u$\u3068\u304a\u3044\u3066\u3001$u$\u306b\u3064\u3044\u3066\u89e3\u3044\u3066\u307f\u307e\u3059\u3002<\/p>\n

\u3059\u308b\u3068\u3001$u$\u306f$y$\u3092\u7528\u3044\u3066\u3001<\/p>\n

\\begin{align}
\nu &= \\frac{1}{\\tan y}\\nonumber\\cr
\n&= \\tan\\left( \\frac{\\pi}{2} – y \\right)\\label{eq:uy}
\n\\end{align}<\/p>\n

\u3068\u8868\u305b\u308b\u306e\u3067\u3001<\/p>\n

\\begin{align}
\n{\\rm Arctan}\\,u &= \\frac{\\pi}{2} – y \\label{eq:yy}
\n\\end{align}<\/p>\n

\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n

$u=1\/x$\u3067\u3042\u308b\u3053\u3068\u306b\u7559\u610f\u3057\u3064\u3064\u3001(\\ref{eq:yy})\u5f0f\u3092$y$\u306b\u3064\u3044\u3066\u89e3\u304f\u3068\u2026<\/p>\n

\\begin{align}
\ny &= \\frac{\\pi}{2} – {\\rm Arctan}\\,u\\nonumber\\cr
\n&= \\frac{\\pi}{2} – {\\rm Arctan}\\left( \\frac{1}{x} \\right)\\label{eq:finalform}
\n\\end{align}<\/p>\n

\u3068\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n

$|x| > 1$\u306a\u3089\u3070$\\displaystyle\\left| \\frac{1}{x} \\right| < 1$\u306a\u306e\u3067\u3001(\\ref{eq:finalform})\u5f0f\u306e\u53f3\u8fba\u7b2c2\u9805\u306f(\\ref{eq:maclaurinseriesofarctan})\u5f0f\u3092\u5229\u7528\u3057\u3066\u2026\n\n\\begin{align}\ny &= \\frac{\\pi}{2} - \\sum_{n=0}^{\\infty}\\frac{(-1)^n}{2n+1}\\left(\\frac{1}{x}\\right)^{2n+1}\\nonumber\\cr\n&= \\frac{\\pi}{2} - \\sum_{n=0}^{\\infty}\\frac{(-1)^n}{2n+1}x^{-(2n+1)}\\label{eq:finalexpandedform}\n\\end{align}\n\n\u3068\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002$\\blacksquare$\n\n\n

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

\u3053\u3053\u307e\u3067\u306e\u5185\u5bb9\u304c\u7406\u89e3\u3067\u304d\u308c\u3070\u3001\u6025\u306b\u8857\u4e2d\u3067${\\rm Arctan}(3\/2)$\u3092\u8a08\u7b97\u3057\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u304f\u306a\u3063\u305f\u5834\u5408\u3067\u3082\u5b89\u5fc3\u3067\u3059\u3002<\/p>\n

$x=1$\u306e\u3068\u304d\u306b\u306f${\\rm Arctan(1)}=\\displaystyle\\frac{\\pi}{4}$\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3082\u899a\u3048\u3066\u304a\u304f\u3068\u306a\u304a\u826f\u3044\u3068\u601d\u3044\u307e\u3059\u3002<\/p>\n

\u3068\u3053\u308d\u3067\u3001Arctan\u3063\u3066$\\LaTeX$\u306e\u6570\u5f0f\u74b0\u5883\u3067\u306f<\/p>\n

{\\\\rm Arctan}<\/p>\n

\u3088\u308a\u3082\u30a8\u30ec\u30ac\u30f3\u30c8\u306a\u66f8\u304d\u65b9\u3063\u3066\u306a\u3044\u306e\u304b\u306a\u2026 \u3068\u3044\u3064\u3082\u601d\u3046\u306e\u3067\u3059\u304c\u3001\u306a\u3044\u3082\u306e\u306a\u306e\u3067\u3059\u304b\u306d\u2026 (\u00b4\uff65\u03c9\uff65`)<\/p>\n

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

\u306f\u3058\u3081\u306b $|x|1$\u306e\u6642\u306e\u5c55\u958b\u5f0f\u306f\u3042\u307e\u308a\u898b\u304b\u3051\u308b\u6a5f\u4f1a\u304c\u3042\u308a\u307e\u305b\u3093\u3002 \u305d\u3053\u3067\u3001\u8a08\u7b97\u3057\u3066\u307f\u308b\u3053\u3068\u306b\u3057\u307e\u3057\u305f\u3002 y = Arctan(x)\u306e\u30b0\u30e9\u30d5 Inkscape 1.3\u3092\u4f7f\u3063\u3066$y = {\\rm Arctan}(x)$\u306e\u30b0\u30e9\u30d5\u3092\u63cf\u304f\u3068\u2026 \u306e\u3088\u3046\u306b\u63cf\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u3059($|x| \\le 5$\u306e\u7bc4\u56f2\u306e\u307f\u63cf\u3044\u3066\u3044\u307e\u3059)\u3002 \u53ce\u675f\u534a\u5f84 \u306a\u305c$|x|>1$\u306e\u6642\u306e\u5c55\u958b\u5f0f\u3092\u8003\u3048\u308b\u306e\u304b\u3068\u3044\u3046\u3068\u3001${\\rm Arcta\u2026 Read More »<\/a><\/span><\/p>\n","protected":false},"author":1,"featured_media":11193,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[31,13],"tags":[],"class_list":["post-11190","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-inkscape","category-13"],"_links":{"self":[{"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/posts\/11190","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=11190"}],"version-history":[{"count":8,"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/posts\/11190\/revisions"}],"predecessor-version":[{"id":11199,"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/posts\/11190\/revisions\/11199"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/media\/11193"}],"wp:attachment":[{"href":"https:\/\/pandanote.info\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11190"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/pandanote.info\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11190"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/pandanote.info\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11190"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}