{"id":615,"date":"2013-02-11T11:45:42","date_gmt":"2013-02-11T02:45:42","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=615"},"modified":"2021-03-24T09:53:18","modified_gmt":"2021-03-24T00:53:18","slug":"kyr200706","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/kyr200706\/","title":{"rendered":"\u4eac\u5927\u7406\u7cfb\u4e592007\uff1a\u7b2c6\u554f"},"content":{"rendered":"
\n

\u3059\u3079\u3066\u306e\u5b9f\u6570\u3067\u5b9a\u7fa9\u3055\u308c\u4f55\u56de\u3067\u3082\u5fae\u5206\u3067\u304d\u308b\u95a2\u6570 \\(f(x)\\) \u304c \\(f(0)=0\\) , \\(f'(0)=1\\) \u3092\u6e80\u305f\u3057,\r\n\u3055\u3089\u306b\u4efb\u610f\u306e\u5b9f\u6570 \\(a , b\\) \u306b\u5bfe\u3057\u3066 \\(1+ f(a) f(b) \\neq 0\\) \u3067\u3042\u3063\u3066\r\n\\[\r\nf( a+b ) = \\dfrac{f(a) + f(b)}{1+ f(a) f(b)}\r\n\\]\r\n\u3092\u6e80\u305f\u3057\u3066\u3044\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\u4efb\u610f\u306e\u5b9f\u6570 \\(a\\) \u306b\u5bfe\u3057\u3066, \\(-1 \\lt f(a) \\lt 1\\) \u3067\u3042\u308b\u3053\u3068\u3092\u8a3c\u660e\u305b\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(y = f(x)\\) \u306e\u30b0\u30e9\u30d5\u306f \\(x \\gt 0\\) \u3067\u4e0a\u306b\u51f8\u3067\u3042\u308b\u3053\u3068\u3092\u8a3c\u660e\u305b\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n

    (1)<\/strong><\/p>\r\n

    \u6761\u4ef6\u5f0f\u306b \\(b = -a\\) \u3092\u4ee3\u5165\u3059\u308b\u3068\r\n\\[\\begin{align}\r\nf(0) = \\dfrac{f(a) +f(-a)}{1 +f(a) f(-a)} & = 0 \\\\\r\nf(a) +f(-a) & = 0 \\\\\r\n\\text{\u2234} \\quad f(-a) = -f(a) & \\quad ... [1]\r\n\\end{align}\\]\r\n\u307e\u305f\u6761\u4ef6\u3088\u308a\r\n\\[\\begin{align}\r\n1 +f(a) f(-a) & \\neq 0 \\\\\r\n1 -\\left\\{ f(a) \\right\\}^2 & \\neq 0 \\\\\r\n\\text{\u2234} \\quad f(a) & \\neq \\pm 1 \\quad ... [2]\r\n\\end{align}\\]\r\n\u95a2\u6570 \\(f(x)\\) \u306f\u3059\u3079\u3066\u306e\u5b9f\u6570\u306b\u304a\u3044\u3066\u5b9a\u7fa9\u3055\u308c, \u304b\u3064\u9023\u7d9a\u3067\u3042\u308b\u304b\u3089, \\(f(0) = 0\\) \u3068 [2] \u3088\u308a, \u95a2\u6570 \\(f(x)\\) \u306e\u5024\u57df\u306f \\(-1\\) \u3068 \\(1\\) \u306e\u9593\u306b\u542b\u307e\u308c\u308b.
    \r\n\u3088\u3063\u3066, \u4efb\u610f\u306e\u5b9f\u6570 \\(a\\) \u306b\u5bfe\u3057\u3066\r\n\\[\r\n-1 \\lt f(a) \\lt 1\r\n\\]\r\n

    (2)<\/strong><\/p>\r\n

    \\(b\\) \u3092\u5b9a\u6570\u3068\u307f\u306a\u3057\u3066, \\(f(a+b)\\) \u3092 \\(a\\) \u306b\u3064\u3044\u3066\u5fae\u5206\u3059\u308b\u3068\r\n\\[\\begin{align}\r\nf'(a+b) & = \\dfrac{f'(a) \\left\\{ 1 +f(b) f(a) \\right\\} -f(b) f'(a) \\left\\{ f(a) +f(b) \\right\\}}{\\left\\{ 1 +f(b) f(a) \\right\\}^2} \\\\\r\n& = \\dfrac{\\left[ 1 -\\left\\{ f(b) \\right\\}^2 \\right] f'(a)}{\\left\\{ 1 +f(b) f(a) \\right\\}^2}\r\n\\end{align}\\]\r\n\u3053\u308c\u306b \\(b = -a\\) \u3092\u4ee3\u5165\u3057\u3066\r\n\\[\\begin{align}\r\nf'(0) & = \\dfrac{\\left[ 1 -\\left\\{ f(a) \\right\\}^2 \\right] f'(a)}{\\left[ 1 -\\left\\{ f(a) \\right\\}^2 \\right]^2} \\\\\r\n& = \\dfrac{f'(a)}{1 -\\left\\{ f(a) \\right\\}^2} = 1 \\\\\r\n\\text{\u2234} \\quad f'(x) & = 1 -\\left\\{ f(x) \\right\\}^2 \\quad ... [3]\r\n\\end{align}\\]\r\n(1)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\r\nf'(x) \\gt 0 \\quad ... [4]\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(f(x)\\) \u306f\u5358\u8abf\u5897\u52a0\u3067\u3042\u308a, \\(x \\gt 0\\) \u306b\u5bfe\u3057\u3066\r\n\\[\r\nf(x) \\gt f(0) = 0 \\quad ... [5]\r\n\\]\r\n[3] \u3088\u308a\r\n\\[\r\nf'' (x) = -2 f'(x) f(x)\r\n\\]\r\n\u3088\u3063\u3066, \\(x \\gt 0\\) \u306b\u304a\u3044\u3066, [4] [5] \u3088\u308a\r\n\\[\r\nf'' (x) \\lt 0\r\n\\]\r\n\u3059\u306a\u308f\u3061 \\(y = f(x)\\) \u306e\u30b0\u30e9\u30d5\u306f\u4e0a\u306b\u51f8\u3067\u3042\u308b.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\u3059\u3079\u3066\u306e\u5b9f\u6570\u3067\u5b9a\u7fa9\u3055\u308c\u4f55\u56de\u3067\u3082\u5fae\u5206\u3067\u304d\u308b\u95a2\u6570 \\(f(x)\\) \u304c \\(f(0)=0\\) , \\(f'(0)=1\\) \u3092\u6e80\u305f\u3057, \u3055\u3089\u306b\u4efb\u610f\u306e\u5b9f\u6570 \\(a , b\\) \u306b\u5bfe\u3057\u3066 \\(1+ f(a) f(b) \\neq 0 … \u7d9a\u304d\u3092\u8aad\u3080 →<\/span><\/a>","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[87],"tags":[140,109],"class_list":["post-615","post","type-post","status-publish","format-standard","hentry","category-kyoto_r_2007","tag-kyoto_r","tag-109"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/615","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/comments?post=615"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/615\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=615"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=615"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=615"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}