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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 Minakshisundaram–Pleijel zeta function ： ウィキペディア英語版 Minakshisundaram–Pleijel zeta function The Minakshisundaram–Pleijel zeta function is a zeta function encoding the eigenvalues of the Laplacian of a compact Riemannian manifold. It was introduced by . The case of a compact region of the plane was treated earlier by . ==Definition== For a compact Riemannian manifold ''M'' of dimension ''N'' with eigenvalues $\backslash lambda\_1,\; \backslash lambda\_2,\; \backslash ldots$ of the Laplace–Beltrami operator Δ the zeta function is given for $\backslash operatorname(s)$ sufficiently large by :$Z(s)\; =\; \backslash operatorname(\backslash Delta^)\; =\; \backslash sum\_^\; \backslash vert\; \backslash lambda\_\; \backslash vert^.$ (where if an eigenvalue is zero it is omitted in the sum). The manifold may have a boundary, in which case one has to prescribe suitable boundary conditions, such as Dirichlet or Neumann boundary conditions. More generally one can define :$Z(P,\; Q,\; s)\; =\; \backslash sum\_^\; \backslash frac$ for ''P'' and ''Q'' on the manifold, where the ''f''''n'' are normalized eigenfunctions. This can be analytically continued to a meromorphic function of ''s'' for all complex ''s'', and is holomorphic for ''P''≠''Q''. The only possible poles are simple poles at the points ''s'' = ''N''/2, ''N''/2−1, ''N''/2−2,..., 1/2,−1/2, −3/2,... for ''N'' odd, and at the points ''s'' = ''N''/2, ''N''/2−1, ''N''/2−2, ...,2, 1 for ''N'' even. If ''N'' is odd then ''Z''(''P'',''P'',''s'') vanishes at ''s'' = 0, −1, −2,... If ''N'' is even its values can be explicitly by Wiener-Ikehara theorem as a corollary the relation :$\backslash displaystyle\; Z(P,P,s)\backslash sim\backslash frac)^N\backslash Gamma(N/2+1)\}$ where the sign ~ indicates that the quotient of both the sides tend to 1 when T tends to +∞. The function ''Z''(''s'') can be recovered from this by integrating ''Z''(''P'', ''P'', ''s'') over the whole manifold ''M'': :$\backslash displaystyle\; Z(s)\; =\; \backslash int\_M\; Z(P,P,s)dP$ 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「Minakshisundaram–Pleijel zeta 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.