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

A closed-form approximation for the complete elliptic integral of the first kind K(k) is derived without curve fitting or free parameters. The key step is identifying the normalized function h(x) = K(k)/ln(4/x), where x = k′ = √(1−k²), which interpolates smoothly between the exact boundary values h(0) = 1 (from the logarithmic divergence of K near k = 1) and h(1) = π/(2 ln 4) (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 ≈ 1.8845 from the slope condition at x = 1 derived from K'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 −0.0984% at k ≈ 0.8734, vanishing 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 second kind E(k).