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:''This article describes holomorphic Eisenstein series in dimension 1; for the non-holomorphic case see real analytic Eisenstein series and for the higher dimensional case see Siegel Eisenstein series'' Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly. Originally defined for the modular group, Eisenstein series can be generalized in the theory of automorphic forms. == Eisenstein series for the modular group == Let τ be a complex number with strictly positive imaginary part. Define the holomorphic Eisenstein series ''G''2''k''(τ) of weight 2''k'', where ''k'' ≥ 2 is an integer, by the following series: : and ''G''2''k'' is therefore a modular form of weight 2''k''. Note that it is important to assume that ''k'' ≥ 2, otherwise it would be illegitimate to change the order of summation, and the SL(2, Z)-invariance would not hold. In fact, there are no nontrivial modular forms of weight 2. Nevertheless, an analogue of the holomorphic Eisenstein series can be defined even for ''k'' = 1, although it would only be a quasimodular form. 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「Eisenstein series」の詳細全文を読む スポンサード リンク
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