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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 Zeta function (operator) ： ウィキペディア英語版 Zeta function (operator) The zeta function of a mathematical operator $\backslash mathcal\; O$ is a function defined as :$\backslash zeta\_(s)\; =\; \backslash operatorname\; \backslash ;\; \backslash mathcal\; O^$ for those values of ''s'' where this expression exists, and as an analytic continuation of this function for other values of ''s''. Here "tr" denotes a functional trace. The zeta function may also be expressible as a spectral zeta function〔Lapidus & van Frankenhuijsen (2006) p.23〕 in terms of the eigenvalues $\backslash lambda\_i$ of the operator $\backslash mathcal\; O$ by :$\backslash zeta\_(s)\; =\; \backslash sum\_\; \backslash lambda\_i^$. It is used in giving a rigorous definition to the functional determinant of an operator, which is given by :$\backslash det\; \backslash mathcal\; O\; :=\; e^\; \backslash ;.$ The Minakshisundaram–Pleijel zeta function is an example, when the operator is the Laplacian of a compact Riemannian manifold. One of the most important motivations for Arakelov theory is the zeta functions for operators with the method of heat kernels generalized algebro-geometrically. ==References== * * 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「Zeta function (operator)」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.