A Parameter-Free Closed-Form Approximation of the Complete Elliptic Integral of the Second Kind

A closed-form approximation for the complete elliptic integral of the second kind E(k) is derived without curve fitting or free parameters. The key step is identifying the normalized function h(x) = (E(k)−1)/[(x²/2)(ln(4/x)−1/2)], where x = k′ = √(1−k²), which interpolates smoothly between the exact boundary values h(0) = 1 (from the leading term of the convergent series expansion of E near k = 1) and h(1) = 2(π/2−1)/(ln 4−1/2) (from direct evaluation at k = 0). A power-law ansatz is adopted and all parameters are fixed analytically: the amplitude from the endpoint values and the exponent β ≈ 2.255 from the slope condition at x = 1 derived from E's power series at k = 0. The resulting exponent is composed entirely of π and ln 2, with no numerical fitting at any stage. The approximation achieves a maximum error of −0.1547% at k ≈ 0.81, is one-signed throughout (a strict underestimate), and vanishes at both endpoints by construction. This paper is part of a series applying the normalization-and-anchoring method; companion papers cover the arithmetic-geometric mean and the complete elliptic integral of the first kind K(k)