Elliptic Lambda Function

Min   Max       Re       Im

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

(1)

where  is the half-period ratio,  is the nome

(2)

and  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 , while the inverse nome is a function of the nome , where  is itself a function of .

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

The elliptic lambda function  satisfies the functional equations

			(3)
			(4)

 has the series expansion

(5)

(OEIS A115977), and  has the series expansion

(6)

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

 gives the value of the elliptic modulus  for which the complementary  and normal complete elliptic integrals of the first kind  are related by

(7)

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

(8)

where

(9)

and  is a Jacobi theta function.  is related to  by

(10)

For all rational ,  and  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  for small  include

			(11)
			(12)
			(13)
			(14)
			(15)
			(16)
			(17)
			(18)
			(19)
			(20)
			(21)
			(22)
			(23)
			(24)
			(25)
			(26)
			(27)

where

(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

			(29)
			(30)
			(31)
			(32)
			(33)
			(34)

Exact values can also be found for rational , including

			(35)
			(36)
			(37)
			(38)
			(39)
			(40)
			(41)
			(42)
			(43)
			(44)

where  is a polynomial root.

 is related to the Ramanujan g- and G-functions by

			(45)
			(46)

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.