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{{short description|Symmetric holomorphic function}}
[[File:Modular lambda function in range -3 to 3.png|thumb|Modular lambda function in the complex plane.]]
In [[mathematics]], the '''modular lambda''' function λ(τ)<ref group="note><math>\lambda(\tau)</math> is not a [[Modular form#Modular functions|modular function]] (per the Wikipedia definition), but every modular function is a [[rational function]] in <math>\lambda(\tau)</math>. Some authors use a non-equivalent definition of "modular functions".</ref> is a highly symmetric [[holomorphic function]] on the complex [[upper half-plane]]. It is invariant under the fractional linear action of the [[congruence subgroup|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]] <math>\mathbb{C}/\langle 1, \tau \rangle</math>, where the map is defined as the quotient by the [−1] involution.
The q-expansion, where <math>q = e^{\pi i \tau}</math> is the [[Nome (mathematics)|nome]], is given by:
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which is the ''j''-invariant of the elliptic curve of [[Legendre form]] <math>y^2=x(x-1)(x-\lambda)</math>
Given <math>m\in\mathbb{C}\setminus\{0,1\}</math>, let
:<math>\tau=i\frac{K\{1-m\}}{K\{m\}}</math>
where <math>K</math> is the [[Elliptic integral#Complete elliptic integral of the first kind|complete elliptic integral of the first kind]] with parameter <math>m=k^2</math>.
Then
:<math>\lambda (\tau)=m.</math>
==Modular equations==
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:<math>\lambda^*(x \in \mathbb{Q}^+) \in \mathbb{A}^+.</math>
<math>K(\lambda^*(x))</math> and <math>E(\lambda^*(x))</math> (the [[Elliptic integral#Complete elliptic integral of the second kind|complete elliptic integral of the second kind]]) can be expressed in closed form in terms of the [[gamma function]] for any <math>x\in\mathbb{Q}^+</math>, as Selberg and Chowla proved in 1949.<ref>{{Cite
The following expression is valid for all <math>n \in \mathbb{N}</math>:
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:<math>\lambda^*(x) - \lambda^*(9x) = 2[\lambda^*(x)\lambda^*(9x)]^{1/4} - 2[\lambda^*(x)\lambda^*(9x)]^{3/4}</math>
<math display=block>\begin{align}
▲:<math>a^{8}+b^{8}-7a^4b^4 = 2\sqrt{2}ab+2\sqrt{2}a^7b^7\, \left(a = \left[\frac{2\lambda^*(x)}{1-\lambda^*(x)^2}\right]^{1/12}\right) \left(b = \left[\frac{2\lambda^*(49x)}{1-\lambda^*(49x)^2}\right]^{1/12}\right) </math>
▲:<math>a^{12}-c^{12} = 2\sqrt{2}(ac+a^3c^3)(1+3a^2c^2+a^4c^4)(2+3a^2c^2+2a^4c^4)\, \left(a = \left[\frac{2\lambda^*(x)}{1-\lambda^*(x)^2}\right]^{1/12}\right) \left(c = \left[\frac{2\lambda^*(121x)}{1-\lambda^*(121x)^2}\right]^{1/12}\right) </math>
\end{align}
</math>
{{Collapse top|title=Special values}}
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