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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Stirling's approximation ： ウィキペディア英語版 Stirling's approximation In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials. It is a very powerful approximation, leading to accurate results even for small values of ''n''. It is named after James Stirling, though it was first stated by Abraham de Moivre.〔 〕〔.〕 The formula as typically used in applications is :$\backslash ln(n!)\; =\; n\backslash ln(n)\; -\; n\; +O(\backslash ln(n))$ (in big O notation). The next term in the ''O''(ln(''n'')) is (1/2)ln(2π''n''); a more precise variant of the formula is therefore :$n!\; \backslash sim\; \backslash sqrt\backslash left(\backslash frac\backslash right)^n.$ It is also possible to give a version of Stirling's formula with bounds valid for all positive integers ''n'', rather than asymptotics: one has :$\backslash sqrt\backslash \; n^e^\; \backslash le\; n!\; \backslash le\; e\backslash \; n^e^$ for all positive integers ''n''. Thus the ratio $\backslash frac\}$ is always between $\backslash sqrt\; =\; 2.5066\backslash ldots$ and $e\; =\; 2.71828\backslash ldots$. As an asymptotic formula, Stirling's approximation has the property that the ratio :$\backslash frac\backslash right)^n\}$ approaches 1 as ''n'' grows to infinity. == Derivation == The formula, together with precise estimates of its error, can be derived as follows. Instead of approximating ''n''!, one considers its natural logarithm as this is a slowly varying function: :$\backslash ln(n!)\; =\; \backslash ln(1)\; +\; \backslash ln(2)\; +\; \backslash cdots\; +\; \backslash ln(n).$ The right-hand side of this equation minus :$\backslash tfrac(\backslash ln(1)+\backslash ln(n))\; =\; \backslash tfrac\backslash ln(n),$ is the approximation by the trapezoid rule of the integral :$\backslash ln(n!)\; -\; \backslash tfrac\backslash ln(n)\; \backslash approx\; \backslash int\_1^n\; \backslash ln(x)\backslash ,x\; =\; n\; \backslash ln(n)\; -\; n\; +\; 1,$ and the error in this approximation is given by the Euler–Maclaurin formula: :$\backslash begin$ \ln (n!) - \tfrac\ln(n) & = \tfrac\ln(1) + \ln(2) + \ln(3) + \cdots + \ln(n-1) + \tfrac\ln(n)\\ & = n \ln(n) - n + 1 + \sum_^ \frac \left( \frac, \end where ''B''''k'' is a Bernoulli number and ''R''''m'',''n'' is the remainder term in the Euler–Maclaurin formula. Take limits to find that :$\backslash lim\_\; \backslash left(\; \backslash ln(n!)\; -\; n\; \backslash ln(n)\; +\; n\; -\; \backslash tfrac\backslash ln(n)\; \backslash right)\; =\; 1\; -\; \backslash sum\_^\; \backslash frac\; +\; \backslash lim\_\; R\_.$ Denote this limit by ''y''. Because the remainder ''R''''m'',''n'' in the Euler–Maclaurin formula satisfies :$R\_\; =\; \backslash lim\_\; R\_\; +\; O\; \backslash left(\; \backslash frac\; \backslash right),$ where we use Big-O notation, combining the equations above yields the approximation formula in its logarithmic form: :$\backslash ln(n!)\; =\; n\; \backslash ln\; \backslash left(\; \backslash frac\; \backslash right)\; +\; \backslash tfrac\backslash ln(n)\; +\; y\; +\; \backslash sum\_^\; \backslash frac\; \backslash right).$ Taking the exponential of both sides, and choosing any positive integer ''m'', we get a formula involving an unknown quantity ''e''''y''. For ''m'' = 1, the formula is :$n!\; =\; e^\; \backslash sqrt\; \backslash left(\; \backslash frac\; \backslash right)^n\; \backslash left(\; 1\; +\; O\; \backslash left(\; \backslash frac\; \backslash right)\; \backslash right).$ The quantity ''e''''y'' can be found by taking the limit on both sides as ''n'' tends to infinity and using Wallis' product, which shows that $e^y\; =\; \backslash sqrt$. Therefore, we get Stirling's formula: :$n!\; =\; \backslash sqrt\; \backslash left(\; \backslash frac\; \backslash right)^n\; \backslash left(\; 1\; +\; O\; \backslash left(\; \backslash frac\; \backslash right)\; \backslash right).$ The formula may also be obtained by repeated integration by parts, and the leading term can be found through Laplace's method. Stirling's formula, without the factor $\backslash sqrt$ that is often irrelevant in applications, can be quickly obtained by approximating the sum :$\backslash ln(n!)\; =\; \backslash sum\_^n\; \backslash ln(j)$ with an integral: :$\backslash sum\_^n\; \backslash ln(j)\; \backslash approx\; \backslash int\_1^n\; \backslash ln(x)\; \backslash ,x\; =\; n\backslash ln(n)\; -\; n\; +\; 1.$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Stirling's approximation」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.