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Elliptic Lambda Function

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The elliptic lambda function lambda(tau) is a lambda-modular function defined on the upper half-plane by

lambda(tau)=(theta_2^4(0,q))/(theta_3^4(0,q)),
(1)

where tau is the half-period ratio, q is the nome

q=e^(ipitau)
(2)

and theta_i(z,q) are Jacobi theta functions.

The elliptic lambda function is essentially the same as the inverse nome, the difference being that elliptic lambda function is a function of the half-period ratio tau, while the inverse nome is a function of the nome q, where q is itself a function of tau.

It is implemented as the Wolfram Language function ModularLambda[tau].

Writing S for the Schwarzian derivative, lambda satisfies the nonlinear third-order differential equation

S(lambda)(tau)=-(1-lambda(tau)+[lambda(tau)]^2)/(2[lambda(tau)]^2[1-lambda(tau)]^2)[lambda^'(tau)]^2.
(3)

This is the inverse form of the equation for the Schwarzian derivative of lnQ(m), where Q is the nome expressed in the parameter m. Write tau(m)=lnQ(m)/(ipi) and apply the chain rule to the local inverse function m=lambda(tau). For 0<m<1, the branch on the positive imaginary axis is tau=iK(sqrt(1-m))/K(sqrt(m)), with K the complete elliptic integral of the first kind in the elliptic modulus convention.

The elliptic lambda function lambda(tau) satisfies the functional equations

lambda(tau+2)=lambda(tau)
(4)
lambda(tau/(2tau+1))=lambda(tau).
(5)

lambda(tau) has the series expansion

lambda(tau)=16q-128q^2+704q^3-3072q^4+11488q^5+...
(6)

(OEIS A115977), and 16/lambda(tau) has the series expansion

(16)/(lambda(tau))=1/q+8+20q-62q^3+216q^5-641q^7+...
(7)

(OEIS A029845; Conway and Norton 1979; Borwein and Borwein 1987, p. 117).

For r>0, lambda^*(r) gives the value of the elliptic modulus k_r for which the complementary K^'(k)=K(sqrt(1-k^2)) and normal complete elliptic integrals of the first kind K(k) are related by

(K^'(k_r))/(K(k_r))=sqrt(r),
(8)

i.e., the elliptic integral singular value for r. It can be computed from

lambda^*(r)=k(q_r)=(theta_2^2(0,q_r))/(theta_3^2(0,q_r)),
(9)

where

q_r=e^(-pisqrt(r))
(10)

and theta_i is a Jacobi theta function. lambda(tau) is related to lambda^*(r) by

lambda^*(r)=sqrt(lambda(isqrt(r))).
(11)

For all positive rational r, K(lambda^*(r)) and E(lambda^*(r)) are known as elliptic integral singular values, and can be expressed in terms of a finite number of gamma functions (Selberg and Chowla 1967). Values of lambda^*(r) for small r include

lambda^*(1)=1/2sqrt(2)
(12)
lambda^*(2)=sqrt(2)-1
(13)
lambda^*(3)=1/4sqrt(2)(sqrt(3)-1)
(14)
lambda^*(4)=3-2sqrt(2)
(15)
lambda^*(5)=sqrt(1/2-sqrt(sqrt(5)-2))
(16)
lambda^*(6)=(2-sqrt(3))(sqrt(3)-sqrt(2))
(17)
lambda^*(7)=1/8sqrt(2)(3-sqrt(7))
(18)
lambda^*(8)=(sqrt(2)+1-sqrt(2sqrt(2)+2))^2
(19)
lambda^*(9)=1/2(sqrt(2)-3^(1/4))(sqrt(3)-1)
(20)
lambda^*(10)=(sqrt(10)-3)(sqrt(2)-1)^2
(21)
lambda^*(11) approx 0.021849722611
(22)
lambda^*(12)=(sqrt(3)-sqrt(2))^2(sqrt(2)-1)^2
(23)
lambda^*(13)=1/2(sqrt(5sqrt(13)-17)-sqrt(19-5sqrt(13)))
(24)
lambda^*(14)=-11-8sqrt(2)-2(sqrt(2)+2)sqrt(5+4sqrt(2))+sqrt(11+8sqrt(2))(2+2sqrt(2)+sqrt(2)sqrt(5+4sqrt(2)))
(25)
lambda^*(15)=1/(16)sqrt(2)(3-sqrt(5))(sqrt(5)-sqrt(3))(2-sqrt(3))
(26)
lambda^*(16)=33+24sqrt(2)-4sqrt(140+99sqrt(2))
(27)
lambda^*(18)=(sqrt(2)-1)^3(2-sqrt(3))^2.
(28)

For r=1, 2, ..., the degrees of the minimal polynomials of lambda^*(r) begin 2, 2, 4, 2, 8, 4, 4, 4, 8, 4, 12, 4, 8, 8, 8, 4, ... (OEIS A084540).

Some additional exact values are given by

lambda^*(22)=(3sqrt(11)-7sqrt(2))(10-3sqrt(11))
(29)
lambda^*(30)=(sqrt(3)-sqrt(2))^2(2-sqrt(3))(sqrt(6)-sqrt(5))(4-sqrt(15))
(30)
lambda^*(34)=(sqrt(2)-1)^2(3sqrt(2)-sqrt(17))(sqrt(297+72sqrt(17))-sqrt(296+72sqrt(17)))
(31)
lambda^*(42)=(sqrt(2)-1)^2(2-sqrt(3))^2(sqrt(7)-sqrt(6))(8-3sqrt(7))
(32)
lambda^*(58)=(13sqrt(58)-99)(sqrt(2)-1)^6
(33)
lambda^*(210)=(sqrt(2)-1)^2(2-sqrt(3))(sqrt(7)-sqrt(6))^2(8-3sqrt(7))(sqrt(10)-3)^2(4-sqrt(15))^2(sqrt(15)-sqrt(14))(6-sqrt(35)).
(34)

Values can also be found for positive rational r, including

lambda^*(1/2)=sqrt(2(sqrt(2)-1))
(35)
lambda^*(1/3)=1/2sqrt(2+sqrt(3))
(36)
lambda^*(2/3)=(2-sqrt(3))(sqrt(2)+sqrt(3))
(37)
lambda^*(1/4)=2sqrt(3sqrt(2)-4)
(38)
lambda^*(3/4)=(x^8+3328x^6+768x^4-8192x^2+4096)_3
(39)
lambda^*(1/5)=sqrt(1/2+sqrt(sqrt(5)-2))
(40)
lambda^*(2/5)=(sqrt(10)-3)(sqrt(2)+1)^2
(41)
lambda^*(3/5)=1/4sqrt(8+sqrt(3/2(27-7sqrt(5))))
(42)
lambda^*(4/5)=(x^8-280x^7-292x^6-680x^5+2758x^4-680x^3-292x^2-280x+1)_2
(43)
lambda^*(2/(29))=(13sqrt(58)-99)(sqrt(2)+1)^6,
(44)

where (P(x))_n is a polynomial root.

lambda^*(r) is related to the Ramanujan g- and G-functions by

lambda^*(n)=1/2(sqrt(1+G_n^(-12))-sqrt(1-G_n^(-12)))
(45)
lambda^*(n)=g_n^6(sqrt(g_n^(12)+g_n^(-12))-g_n^6).
(46)

See also

Dedekind Eta Function, Elliptic Alpha Function, Elliptic Integral of the First Kind, Elliptic Integral Singular Value, Elliptic Modulus, Inverse Nome, j-Function, Jacobi Theta Functions, Klein's Absolute Invariant, Modular Function, Modular Group Lambda, Ramanujan g- and G-Functions, Schwarzian Derivative, Weber Modular Function

Related Wolfram sites

https://functions.wolfram.com/EllipticFunctions/ModularLambda/

Explore with Wolfram|Alpha

References

Bertrand, D. and Zudilin, W. "On the Transcendence Degree of the Differential Field Generated by Siegel Modular Forms." J. reine angew. Math. 554, 47-68, 2003. https://doi.org/10.1515/crll.2003.008.Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 139 and 298, 1987.Bowman, F. Introduction to Elliptic Functions, with Applications. New York: Dover, pp. 75, 95, and 98, 1961.Conway, J. H. and Norton, S. P. "Monstrous Moonshine." Bull. London Math. Soc. 11, 308-339, 1979.Selberg, A. and Chowla, S. "On Epstein's Zeta-Function." J. reine angew. Math. 227, 86-110, 1967.Sloane, N. J. A. Sequences A029845, A084540, and A115977 in "The On-Line Encyclopedia of Integer Sequences."Watson, G. N. "Some Singular Moduli (1)." Quart. J. Math. 3, 81-98, 1932.Yang, Y. "On Differential Equations Satisfied by Modular Forms." Math. Z. 246, 1-19, 2004. https://doi.org/10.1007/s00209-003-0573-4.

Referenced on Wolfram|Alpha

Elliptic Lambda Function

Cite this as:

Weisstein, Eric W. "Elliptic Lambda Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EllipticLambdaFunction.html

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