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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 , where the map is defined as the quotient by the () involution. The q-expansion, where is the nome, is given by: : . 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 , and it is in fact Klein's modular j-invariant. ==Modular properties== The function is invariant under the group generated by〔Chandrasekharan (1985) p.115〕 : The generators of the modular group act by〔Chandrasekharan (1985) p.109〕 : : Consequently, the action of the modular group on is that of the anharmonic group, giving the six values of the cross-ratio:〔Chandrasekharan (1985) p.110〕 : 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
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