翻訳と辞書 Words near each other ・ Functional disconnection ・ Functional discourse grammar ・ Functional disorder ・ Functional divergence ・ Functional diversity ・ Functional diversity (disability) ・ Functional diversity (organizational) ・ Functional ecology ・ Functional Ecology (journal) ・ Functional electrical stimulation ・ Functional element SNPs database ・ Functional encryption ・ Functional endoscopic sinus surgery ・ Functional English ・ Functional equation ・ Functional equation (L-function) ・ Functional extinction ・ Functional finance ・ Functional fixedness ・ Functional flow block diagram ・ Functional food ・ Functional Food Centre at Oxford Brookes University ・ Functional gastrointestinal disorder ・ Functional generative description ・ Functional genomics ・ Functional grammar ・ Functional group ・ Functional group (disambiguation) ・ Functional group (ecology) ・ Functional holography Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Functional equation (L-function) ： ウィキペディア英語版 Functional equation (L-function) In mathematics, the L-functions of number theory are expected to have several characteristic properties, one of which is that they satisfy certain functional equations. There is an elaborate theory of what these equations should be, much of which is still conjectural. == Introduction == A prototypical example, the Riemann zeta function has a functional equation relating its value at the complex number ''s'' with its value at 1 − ''s''. In every case this relates to some value ζ(''s'') that is only defined by analytic continuation from the infinite series definition. That is, writingas is conventionalσ for the real part of ''s'', the functional equation relates the cases :σ > 1 and σ < 0, and also changes a case with :0 < σ < 1 in the ''critical strip'' to another such case, reflected in the line σ = ½. Therefore use of the functional equation is basic, in order to study the zeta-function in the whole complex plane. The functional equation in question for the Riemann zeta function takes the simple form :$Z(s)\; =\; Z(1-s)\; \backslash ,$ where ''Z''(''s'') is ζ(''s'') multiplied by a ''gamma-factor'', involving the gamma function. This is now read as an 'extra' factor in the Euler product for the zeta-function, corresponding to the infinite prime. Just the same shape of functional equation holds for the Dedekind zeta function of a number field ''K'', with an appropriate gamma-factor that depends only on the embeddings of ''K'' (in algebraic terms, on the tensor product of ''K'' with the real field). There is a similar equation for the Dirichlet L-functions, but this time relating them in pairs: :$\backslash Lambda(s,\backslash chi)=\backslash varepsilon\backslash Lambda(1-s,\backslash chi^$ *) with χ a primitive Dirichlet character, χ * its complex conjugate, Λ the L-function multiplied by a gamma-factor, and ε a complex number of absolute value 1, of shape :$G(\backslash chi)\; \backslash over$ where ''G''(χ) is a Gauss sum formed from χ. This equation has the same function on both sides if and only if χ is a ''real character'', taking values in . Then ε must be 1 or −1, and the case of the value −1 would imply a zero of ''Λ''(''s'') at ''s'' = ½. According to the theory (of Gauss, in effect) of Gauss sums, the value is always 1, so no such ''simple'' zero can exist (the function is ''even'' about the point). 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「Functional equation (L-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.