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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].

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

lambda(tau+2)=lambda(tau)
(3)
lambda(tau/(2tau+1))=lambda(tau).
(4)

lambda(tau) has the series expansion

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

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

(16)/(lambda(tau))=1/q+8+20q-62q^3+216q^5-641q^7+...
(6)

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

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),
(7)

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)),
(8)

where

q_r=e^(-pisqrt(r))
(9)

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

lambda^*(r)=sqrt(lambda(isqrt(r))).
(10)

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

where

x_(11)=(17+3sqrt(33))^(1/3).
(28)

The algebraic orders of these are given by 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)

Exact values can also be found for 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(23-7sqrt(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 Modulus, Elliptic Integral Singular Value, Inverse Nome, j-Function, Jacobi Theta Functions, Klein's Absolute Invariant, Modular Function, Modular Group Lambda, Ramanujan g- and G-Functions, Weber Functions

Related Wolfram sites

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

Explore with Wolfram|Alpha

References

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.

Referenced on Wolfram|Alpha

Elliptic Lambda Function

Cite this as:

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

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