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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 Rankin–Selberg method ： ウィキペディア英語版 Rankin–Selberg method In mathematics, the Rankin–Selberg method, introduced by and , also known as the theory of integral representations of ''L''-functions, is a technique for directly constructing and analytically continuing several important examples of automorphic ''L''-functions. Some authors reserve the term for a special type of integral representation, namely those that involve an Eisenstein series. It has been one of the most powerful techniques for studying the Langlands program. ==History== The theory in some sense dates back to Bernhard Riemann, who constructed his zeta function as the Mellin transform of Jacobi's theta function. Riemann used asymptotics of the theta function to obtain the analytic continuation, and the automorphy of the theta function to prove the functional equation. Erich Hecke, and later Hans Maass, applied the same Mellin transform method to modular forms on the upper half-plane, after which Riemann's example can be seen as a special case. Robert Alexander Rankin and Atle Selberg independently constructed their convolution ''L''-functions, now thought of as the Langlands ''L''-function associated to the tensor product of standard representation of GL(2) with itself. Like Riemann, they used an integral of modular forms, but one of a different type: they integrated the product of two weight ''k'' modular forms ''f'', ''g'' with a real analytic Eisenstein series ''E''(τ,''s'') over a fundamental domain ''D'' of the modular group SL2(Z) acting on the upper half plane :$\backslash displaystyle\; \backslash int\_Df(\backslash tau)\backslash overlineE(\backslash tau,s)y^dxdy$. The integral converges absolutely if one of the two forms is cuspidal; otherwise the asymptotics must be used to get a meromorphic continuation like Riemann did. The analytic continuation and functional equation then boil down to those of the Eisenstein series. The integral was identified with the convolution L-function by a technique called "unfolding", in which the definition of the Eisenstein series and the range of integration are converted into a simpler expression that more readily exhibits the ''L''-function as a Dirichlet series. The simultaneous combination of an unfolding together with global control over the analytic properties, is special and what makes the technique successful. 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「Rankin–Selberg method」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.