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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 Nevanlinna function ： ウィキペディア英語版 Nevanlinna function :''See also Nevanlinna theory In mathematics, in the field of complex analysis, a Nevanlinna function is a complex function which is an analytic function on the open upper half-plane ''H'' and has non-negative imaginary part. A Nevanlinna function maps the upper half-plane to itself or to a real constant,〔A real number is not considered to be in the upper half-plane.〕 but is not necessarily injective or surjective. Functions with this property are sometimes also known as Herglotz, Pick or R functions. ==Integral representation== Every Nevanlinna function ''N'' admits a representation :$N(z)\; =\; C\; +\; Dz\; +\; \backslash int\_\; -\; \backslash frac\; \backslash right)\; d\backslash mu(\backslash lambda),\; \backslash quad\; z\backslash in\backslash mathbb,$ where ''C'' is a real constant, ''D'' is a non-negative constant and μ is a Borel measure on ''R'' satisfying the growth condition : Conversely, every function of this form turns out to be a Nevanlinna function. The constants in this representation are related to the function ''N'' via :$C\; =\; \backslash mathrm(N(i))\; \backslash qquad\backslash text\backslash qquad\; D\; =\; \backslash lim\_\; \backslash frac$ and the Borel measure μ can be recovered from ''N'' by employing the Stieltjes inversion formula (related to the inversion formula for the Stieltjes transformation): :$\backslash mu((\backslash lambda\_1,\backslash lambda\_2])\; =\; \backslash lim\_\; \backslash lim\_\; \backslash frac\; \backslash int\_^\; \backslash mathrm(N(\backslash lambda+i\backslash varepsilon))d\backslash lambda.$ A very similar representation of functions is also called the Poisson representation.〔See for example Section 4, "Poisson representation", of . De Branges gives a form for functions whose ''real'' part is non-negative in the upper half-plane.〕 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「Nevanlinna function」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.