Elliptic Alpha Function

Elliptic alpha functions relate the complete elliptic integrals of the first and second kinds at elliptic integral singular values according to

			(1)
			(2)
			(3)

where is a Jacobi theta function and

			(4)
			(5)

and is the elliptic lambda function. The elliptic alpha function is related to the elliptic delta function by

(6)

It satisfies

(7)

and has the limit

(8)

(Borwein et al. 1989). A few specific values (Borwein and Borwein 1987, p. 172) are

			(9)
			(10)
			(11)
			(12)
			(13)
			(14)
			(15)
			(16)
			(17)
			(18)
			(19)
			(20)
			(21)
			(22)
			(23)
			(24)
			(25)
			(26)

			(27)
			(28)
			(29)
			(30)
			(31)
			(32)
			(33)

J. Borwein has written an algorithm which uses lattice basis reduction to provide algebraic values for .