\u3082\u3046\u982d\u3082\u60aa\u304f\u306a\u3063\u3066\u4e45\u3057\u3044\u304c\uff0c\u8150\u308a\u304b\u3051\u306e\u8133\u7d30\u80de\u3067\u3082\uff0c\u305f\u307e\u306b\u306f\u826f\u3044\u3053\u3068\u3092\u601d\u3044\u3064\u304f\u3063\u3066\u3053\u3068\u3067\u30e1\u30e2\u3002<\/p>\n
\u30c1\u30a7\u30d3\u30b7\u30a7\u30d5\u306e\u591a\u9805\u5f0f$T_n(x)$\u306b\u3064\u3044\u3066\uff0c$x>1$\u306e\u3068\u304d$T_n(x)>1$\u3092\u793a\u305b\uff0c\u3068\u3044\u3046\u306e\u304c\u554f\u984c\u3002$-1\\leqq x\\leqq 1$\u3067\u5897\u6e1b\u3057\u307e\u304f\u3063\u3066\u308b\u3093\u3060\u304b\u3089\uff0c$x>1$\u3067\u306f\u3082\u3046\u5358\u8abf\u5897\u52a0\u3057\u304b\u306a\u3044\u3088\u306a\u3042\u30fb\u30fb\u30fb\u3068\u6f20\u7136\u3068\u8003\u3048\u3066\u3044\u308b\u3068\uff0c\u305d\u306e\u65b9\u5411\u3067\u306e\u8a3c\u660e\u3092\u601d\u3044\u3064\u3044\u305f\u3002<\/p>\n
$T_n(\\cos\\theta)=\\cos n\\theta$\u3092$\\theta$\u3067\u5fae\u5206\u3059\u308b\u3068\uff0cChain Rule\u306b\u3088\u3063\u3066\uff0c
\n\\begin{gather*}
\n T_n'(\\cos\\theta) (-\\sin\\theta) = -n\\sin n\\theta \\\\
\n T_n'(\\cos\\theta) = \\frac{n\\sin n\\theta}{\\sin\\theta}
\n\\end{gather*}
\n\u3088\u3063\u3066\uff0c$T_n'(x)=0$ \u306f $(n-1)$\u500b\u306e\u6839 $x=\\cos\\frac{k\\pi}{n}$ ($k=1$, $2$, $\\cdots$, $n-1$) \u3092\u6301\u3064\u304c\uff0c$T_n'(x)$\u306f$(n-1)$\u6b21\u5f0f\u3060\u304b\u3089\uff0c\u3053\u308c\u304c\u6839\u306e\u3059\u3079\u3066\u3067\u3042\u308b\u3002<\/p>\n
\u5f93\u3063\u3066\uff0c
\n\\[ T_n'(x)=n2^{n-1} \\prod_{k=1}^{n-1} \\left(x-\\cos\\frac{k\\pi}{n}\\right) \\]
\n\u3068\u56e0\u6570\u5206\u89e3\u3055\u308c\u308b\u3002\u3053\u308c\u304b\u3089\uff0c$x\\geqq 1$\u306b\u304a\u3044\u3066 $T_n'(x)>0$ \u3068\u306a\u308b\u3053\u3068\uff0c\u3059\u306a\u308f\u3061\uff0c$T_n(x)$\u304c\u5358\u8abf\u5897\u52a0\u306b\u306a\u308b\u3053\u3068\u306f\u660e\u3089\u304b\u3067\u3042\u308b\u3002$T_n(1)=1$\u3067\u3042\u308b\u304b\u3089\uff0c$x>1$\u306e\u3068\u304d\uff0c$T_n(x)>T_n(1)=1$ \u3067\u3042\u308b\u3002QED<\/p>\n","protected":false},"excerpt":{"rendered":"
\u3082\u3046\u982d\u3082\u60aa\u304f\u306a\u3063\u3066\u4e45\u3057\u3044\u304c\uff0c\u8150\u308a\u304b\u3051\u306e\u8133\u7d30\u80de\u3067\u3082\uff0c\u305f\u307e\u306b\u306f\u826f\u3044\u3053\u3068\u3092\u601d\u3044\u3064\u304f\u3063\u3066\u3053\u3068\u3067\u30e1\u30e2\u3002 \u30c1\u30a7\u30d3\u30b7\u30a7\u30d5\u306e\u591a\u9805\u5f0f$T_n(x)$\u306b\u3064\u3044\u3066\uff0c$x>1$\u306e\u3068\u304d$T_n(x)>1$\u3092\u793a\u305b\uff0c\u3068\u3044\u3046\u306e\u304c\u554f\u984c\u3002$-1\\leqq x […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[],"class_list":["post-2338","post","type-post","status-publish","format-standard","hentry","category-5"],"_links":{"self":[{"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/posts\/2338","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/comments?post=2338"}],"version-history":[{"count":14,"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/posts\/2338\/revisions"}],"predecessor-version":[{"id":2351,"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/posts\/2338\/revisions\/2351"}],"wp:attachment":[{"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/media?parent=2338"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/categories?post=2338"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/tags?post=2338"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}