\u8907\u7d20\u6570 \\(z\\) \u306b\u5bfe\u3057\u3066\r\n\\[\r\nf(z) = \\alpha z +\\beta\r\n\\]\r\n\u3068\u3059\u308b. \u305f\u3060\u3057, \\(\\alpha , \\beta\\) \u306f\u8907\u7d20\u6570\u306e\u5b9a\u6570\u3067 \\(\\alpha \\neq 1\\) \u3068\u3059\u308b.\r\n\\[\r\nf^1 (z) = f(z) , \\quad f^n (z) = f( f^{n-1} (z) ) \\quad ( n = 2, 3, \\cdots )\r\n\\]\r\n\u3068\u5b9a\u3081\u308b. \u6b21\u306e\u554f\u306b\u7b54\u3048\u3088.<\/p>\r\n
(1)<\/strong>\u3000\\(f^n (z)\\) \u3092 \\(\\alpha , \\beta , z , n\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(| \\alpha | \\lt 1\\) \u306e\u3068\u304d, \u3059\u3079\u3066\u306e\u8907\u7d20\u6570 \\(z\\) \u306b\u5bfe\u3057\u3066\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} \\left| f^n (z) -\\delta \\right| = 0\r\n\\]\r\n\u304c\u6210\u308a\u7acb\u3064\u3088\u3046\u306a\u8907\u7d20\u6570\u306e\u5b9a\u6570 \\(\\delta\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (3)<\/strong>\u3000\\(| \\alpha | = 1\\) \u3068\u3059\u308b. \u8907\u7d20\u6570\u306e\u5217 \\(\\{ f^n (z) \\}\\) \u306b\u5c11\u306a\u304f\u3068\u3082 \\(3\\) \u3064\u306e\u7570\u306a\u308b\u8907\u7d20\u6570\u304c\u73fe\u308c\u308b\u3068\u304d, \u3053\u308c\u3089\u306e \\(f^n (z) \\ ( n = 1, 2, \\cdots )\\) \u306f\u8907\u7d20\u6570\u5e73\u9762\u5185\u306e\u3042\u308b\u5186 \\(C _ z\\) \u4e0a\u306b\u3042\u308b. \u5186 \\(C _ z\\) \u306e\u4e2d\u5fc3\u3068\u534a\u5f84\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\(f^n(z) = \\alpha f^{n-1} (z) +\\beta\\) \u3092\u5909\u5f62\u3059\u308b\u3068\r\n\\[\r\nf^n(z) -\\dfrac{\\beta}{1 -\\alpha} = \\alpha \\left( f^{n-1} (z) -\\dfrac{\\beta}{1 -\\alpha} \\right)\r\n\\]\r\n\u306a\u306e\u3067, \u6570\u5217 \\(\\left\\{ f^n(z) -\\dfrac{\\beta}{1 -\\alpha} \\right\\}\\) \u306f, \u521d\u9805 \\(f^1(z) -\\dfrac{\\beta}{1 -\\alpha} = \\alpha z -\\dfrac{\\alpha \\beta}{1 -\\alpha}\\) , \u516c\u6bd4 \\(\\alpha\\) \u306e\u7b49\u6bd4\u6570\u5217\u306a\u306e\u3067\r\n\\[\\begin{align}\r\nf^n(z) -\\dfrac{\\beta}{1 -\\alpha} &= \\alpha^{n-1} \\left( \\alpha z -\\dfrac{\\alpha \\beta}{1 -\\alpha} \\right) \\\\\r\n\\text{\u2234} \\quad f^n(z) &= \\underline{\\alpha^n z +\\dfrac{1 -\\alpha^n}{1 -\\alpha} \\beta}\r\n\\end{align}\\]\r\n (2)<\/strong><\/p>\r\n \\(| \\alpha | \\lt 0\\) \u306e\u3068\u304d\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} | \\alpha^n | = \\displaystyle\\lim _ {n \\rightarrow \\infty} | \\alpha |^n = 0\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} \\alpha^n = 0\r\n\\]\r\n\u3053\u308c\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} \\left| f^n (z) -\\delta \\right| & = \\left| 0 \\cdot z +\\dfrac{1 -0}{1 -\\alpha} \\beta -\\delta \\right| \\\\\r\n& = \\left| \\dfrac{\\beta}{1 -\\alpha} -\\delta \\right| = 0\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\delta = \\underline{\\dfrac{\\beta}{1 -\\alpha}}\r\n\\]\r\n (3)<\/strong><\/p>\r\n \\[\\begin{align}\r\n\\left| f^n (z) -\\dfrac{\\beta}{1 -\\alpha} \\right| & =\r\n\\left| \\alpha^n z +\\dfrac{1 -\\alpha^n}{1 -\\alpha} \\beta -\\dfrac{\\beta}{1 -\\alpha} \\right| \\\\\r\n& = | \\alpha |^n \\left| z -\\dfrac{\\beta}{1 -\\alpha} \\right| \\\\\r\n& = \\left| z -\\dfrac{\\beta}{1 -\\alpha} \\right| \\quad ( \\text{\u2235} \\ | \\alpha | = 1 \\ )\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \\(f^n(z)\\) \u304c\u7570\u306a\u308b \\(3\\) \u3064\u4ee5\u4e0a\u306e\u8907\u7d20\u6570\u3092\u3068\u308c\u3070, \u3053\u308c\u3089\u3092\u901a\u308b\u305f\u3060 \\(1\\) \u3064\u306e\u5186 \\(C_z\\) \u304c\u6c7a\u307e\u308a, \u6c42\u3081\u308b\u4e2d\u5fc3\u3068\u534a\u5f84\u306f\r\n\\[\r\n\\text{\u4e2d\u5fc3} : \\ \\underline{\\dfrac{\\beta}{1 -\\alpha}} , \\ \\text{\u534a\u5f84} : \\ \\underline{z -\\dfrac{\\beta}{1 -\\alpha}}\r\n\\]\r\n (1)<\/strong><\/p>\r\n \\(f^n(z) = \\alpha^n z +\\dfrac{1 -\\alpha^n}{1 -\\alpha} \\beta\\) ... [A] \u304c\u6210\u7acb\u3059\u308b\u3053\u3068\u3092\u5e30\u7d0d\u6cd5\u3092\u7528\u3044\u3066\u793a\u3059.<\/p>\r\n 1*<\/strong>\u3000\\(n = 1\\) \u306e\u3068\u304d 2*<\/strong>\u3000\\(n = k\\) \u306e\u3068\u304d, [A] \u304c\u6210\u7acb\u3059\u308b\u3068\u4eee\u5b9a\u3059\u308b\u3068\r\n\\[\\begin{align}\r\nf^{k+1} (z) & = f \\left( f^k (z) \\right) \\\\\r\n& = \\alpha \\left( \\alpha^k z +\\dfrac{1 -\\alpha^k}{1 -\\alpha} \\beta \\right) +\\beta \\\\\r\n& = \\alpha^{k+1} z +\\dfrac{\\alpha -\\alpha^{k+1}+( 1 -\\alpha )}{1 -\\alpha} \\beta \\\\\r\n& = \\alpha^{k+1} z +\\dfrac{1 -\\alpha^{k+1}}{1 -\\alpha} \\beta\r\n\\end{align}\\]\r\n\u3086\u3048\u306b, \\(n = k+1\\) \u306e\u3068\u304d\u3082, [A] \u304c\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n<\/ol>\r\n 1*<\/strong> 2*<\/strong> \u304b\u3089, \u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u306b\u3088\u308a, \\(n \\geqq 1\\) \u306b\u3064\u3044\u3066\r\n\\[\r\nf^n(z) = \\underline{\\alpha^n z +\\dfrac{1 -\\alpha^n}{1 -\\alpha} \\beta}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u8907\u7d20\u6570 \\(z\\) \u306b\u5bfe\u3057\u3066 \\[ f(z) = \\alpha z +\\beta \\] \u3068\u3059\u308b. \u305f\u3060\u3057, \\(\\alpha , \\beta\\) \u306f\u8907\u7d20\u6570\u306e\u5b9a\u6570\u3067 \\(\\alpha \\neq 1\\) \u3068\u3059\u308b. \\[ f^ … \u7d9a\u304d\u3092\u8aad\u3080
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
\u3010 \u5225 \u89e3 \u3011<\/h2>\r\n
\r\n
\r\n\\(f(z) = \\alpha z +\\beta\\) \u306a\u306e\u3067, \u6210\u7acb\u3057\u3066\u3044\u308b.<\/p><\/li>\r\n