Elliptic Lambda Function

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

(1)

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

(2)

and are theta functions.

It is implemented as the Mathematica command ModularLambda[tau].

The elliptic lambda function satisfies the functional equations

			(3)
			(4)

has the series expansion

(5)

(Sloane's A014972), and has the series expansion

(6)

(Sloane's 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 K(k) are related by

(7)

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

(8)

where

(9)

and is a Jacobi theta function. is related to by

(10)

For all rational r, 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 r include

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

			(27)
			(28)

where

(29)

The algebraic orders of these are given by 2, 2, 4, 2, 8, 4, 4, 4, 8, 4, 12, 4, 8, 8, 8, 4, ... (Sloane's A084540). is the first "hard" value, with an algebraic order apparently larger than 32.