Elliptic Alpha Function

The elliptic alpha function , defined for , relates the complete elliptic integrals of the first and second kinds at elliptic integral singular values according to

			(1)
			(2)
			(3)

where , , is the elliptic lambda function, and and are Jacobi theta functions. Here and use the elliptic modulus , and is the complementary complete elliptic integral of the second kind. The elliptic alpha function is related to the elliptic delta function by

(4)

It satisfies

(5)

and has the asymptotic behavior

(6)

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

			(7)
			(8)
			(9)
			(10)
			(11)
			(12)
			(13)
			(14)
			(15)
			(16)
			(17)
			(18)
			(19)
			(20)
			(21)
			(22)
			(23)
			(24)

			(25)
			(26)
			(27)
			(28)
			(29)
			(30)
			(31)

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