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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 Modular lambda function ： ウィキペディア英語版 Modular lambda function In mathematics, the elliptic modular lambda function λ(τ) is a highly symmetric holomorphic function on the complex upper half-plane. It is invariant under the fractional linear action of the congruence group Γ(2), and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the modular curve ''X''(2). Over any point τ, its value can be described as a cross ratio of the branch points of a ramified double cover of the projective line by the elliptic curve $\backslash mathbb/\backslash langle\; 1,\; \backslash tau\; \backslash rangle$, where the map is defined as the quotient by the () involution. The q-expansion, where $q\; =\; e^$ is the nome, is given by: : $\backslash lambda(\backslash tau)\; =\; 16q\; -\; 128q^2\; +\; 704\; q^3\; -\; 3072q^4\; +\; 11488q^5\; -\; 38400q^6\; +\; \backslash dots$. By symmetrizing the lambda function under the canonical action of the symmetric group ''S''3 on ''X''(2), and then normalizing suitably, one obtains a function on the upper half-plane that is invariant under the full modular group $SL\_2(\backslash mathbb)$, and it is in fact Klein's modular j-invariant. ==Modular properties== The function $\backslash lambda(\backslash tau)$ is invariant under the group generated by〔Chandrasekharan (1985) p.115〕 :$\backslash tau\; \backslash mapsto\; \backslash tau+2\; \backslash \; ;\backslash \; \backslash tau\; \backslash mapsto\; \backslash frac\; \backslash \; .$ The generators of the modular group act by〔Chandrasekharan (1985) p.109〕 :$\backslash tau\; \backslash mapsto\; \backslash tau+1\; \backslash \; :\backslash \; \backslash lambda\; \backslash mapsto\; \backslash frac\; \backslash ,\; ;$ :$\backslash tau\; \backslash mapsto\; -\backslash frac\; \backslash \; :\backslash \; \backslash lambda\; \backslash mapsto\; 1\; -\; \backslash lambda\; \backslash \; .$ Consequently, the action of the modular group on $\backslash lambda(\backslash tau)$ is that of the anharmonic group, giving the six values of the cross-ratio:〔Chandrasekharan (1985) p.110〕 :$\backslash left\backslash lbrace\; ,\; \backslash frac,\; \backslash frac,\; \backslash frac,\; 1-\backslash lambda\; \}\; \backslash right\backslash rbrace\; \backslash \; .$ 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「Modular lambda 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.