Closed-form evaluation of the spinor norm integral I0

Determine a closed-form expression for the constant I0 defined by the two equivalent integral representations I0 = (2/π) ∫_{D} [x y / √((1−x)(1−y)(x^2 + y^2 − 1))] dx dy = (4√2/π) ∫_0^1 [(1+√z)/(1+z)^3] [2E(z) − (1−z)K(z)] dz, where D = {(x,y) : x^2 + y^2 ≥ 1, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1} and K(z), E(z) denote the complete elliptic integrals of the first and second kind, respectively.

Background

The paper constructs a basis for spinors in three-dimensional Euclidean space using differential forms and develops an inner product for fractional-order differential forms via both formal power series and multiple integrals. In computing the norm density for the covariant spinor basis, only one integral contribution, denoted I0, remains nonzero.

I0 is derived in two equivalent forms: a two-dimensional integral over a domain D in the unit square and a one-dimensional integral involving complete elliptic integrals K(z) and E(z). The authors numerically estimate I0 ≈ 1.774 but explicitly state that a closed-form expression is not known.