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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 Hölder's theorem ： ウィキペディア英語版 Hölder's theorem In mathematics, Hölder's theorem states that the gamma function does not satisfy any algebraic differential equation whose coefficients are rational functions. The result was first proved by Otto Hölder in 1887; several alternative proofs have subsequently been found.〔Bank, Steven B. & Kaufman, Robert "(A Note on Hölder's Theorem Concerning the Gamma Function )". ''Mathematische Annalen'', vol 232, 1978.〕 The theorem also generalizes to the q-gamma function. ==Statement of the Theorem== There is no non-constant polynomial $P(x;\backslash ;y\_0,\backslash ;y\_1,\backslash ldots,\backslash ;y\_n)$ such that :$\backslash ,P\backslash left(x;\backslash ;\backslash Gamma(x),\backslash ;\backslash Gamma\text{'}(x),\backslash ;\backslash ldots\backslash ;,\backslash ;\backslash Gamma^(x)\backslash right)\backslash equiv\; 0.\backslash !$ where $y\_0,\backslash ;y\_1,\backslash ldots,\backslash ;y\_n,$ are functions of ''x'', Γ(''x'') is the gamma function, and ''P'' is a polynomial in $y\_0,\backslash ;y\_1,\backslash ldots,\backslash ;y\_n,$ with coefficients drawn from the ring of polynomials in ''x''. That is, :$\backslash ,\; P(x;\backslash ;y\_0,\backslash ;y\_1,\backslash ldots,\backslash ;y\_n)=\backslash sum\_\; A\_(x)\backslash cdot(y\_0)^\backslash cdot(y\_1)^\backslash cdot\backslash ldots\backslash cdot(y\_n)^\backslash !$ where $\backslash ,(a\_0,\backslash ;a\_1,\backslash ldots,\backslash ;a\_n)\backslash ,$ indexes all possible terms of the polynomial and $\backslash ,\; A\_(x)\backslash ,$ are polynomials in ''x'' acting as coefficients of the polynomial ''P''. The $\backslash ,\; A\_(x)\backslash ,$ may be constants or zero. For example, if $\backslash ,\; P(x;\backslash ;\; y\_0,\; \backslash ;y\_1,\backslash ;\; y\_2)=\; x^2\; y\_2+x\; y\_1+(x^2-\backslash nu^2)y\_0\backslash ,$ then $\backslash ,\; A\_(x)=x^2\backslash ,$, $\backslash ,\; A\_(x)=x\backslash ,$ and $\backslash ,\; A\_(x)=(x^2-\backslash nu^2)\backslash ,$ where ν is a constant. All the other coefficients in the summation are zero. Then :$\backslash ,\; P(z;\backslash ;f,\backslash ;f\text{'},\backslash ;f\text{'}\text{'})=\; x^2\; f\text{'}\text{'}+x\; f\text{'}+(x^2-\backslash nu^2)f=0\backslash ,$ is an ''algebraic differential equation'' which, in this example, has solutions $\backslash ,\; f=J\_\backslash nu(x)\backslash ,$ and $\backslash ,\; f=Y\_\backslash nu(x)\backslash ,$, the Bessel functions of either the first or second kind. So :$\backslash ,\; P\backslash left(x;\backslash ;J\_\backslash nu(x),\backslash ;J\_\backslash nu\; \text{'}(x),\backslash ;J\_\backslash nu\; \text{'}\text{'}(x)\backslash right)\backslash equiv\; 0.\backslash !$ and therefore both $\backslash ,\; J\_\backslash nu(x)\; \backslash ,$ and $\backslash ,\; Y\_\backslash nu(x)\backslash ,$ are ''differentially algebraic'' (also ''algebraically transcendental''). Most of the familiar special functions of mathematical physics are differentially algebraic. All algebraic combinations of differentially algebraic functions are also differentially algebraic. Also, all compositions of differentially algebraic functions are differentially algebraic. Hölder's Theorem simply states that the gamma function, Γ(''x'') is not differentially algebraic and is, therefore, ''transcendentally transcendental.''〔 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「Hölder's theorem」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.