A Lambert W Route to the Perimeter of an Ellipse near the Degenerate Limit

The Lambert W function is shown to be the natural language for describing the perimeter of an ellipse near the degenerate limit b/a → 0. Starting from the power mean representation P = 4(aⁿ + bⁿ)^(1/n), the exponent n(x) required to match the true perimeter satisfies n(x) ≈ 1 + W(4/x)/ln(4/x) as x = b/a → 0, a structural result following directly from the asymptotic expansion of the complete elliptic integral E(k) near k → 1. A two-term asymptotic for n(x) is derived and confirmed numerically to 100,000-digit precision at exponents up to x = 10⁻⁵⁰⁰⁰⁰⁰⁰. A gated composite formula — applying the W-based approximation for x < 0.01487 and Ramanujan's second approximation elsewhere — achieves a maximum error of −0.019% across the full domain, with one-signed error approaching zero at both endpoints, halving the worst-case error of Ramanujan's second approximation alone. Part of a series: AGM · Nonlinear pendulum · Quartic anharmonic oscillator · Asymmetric cubic oscillator · Relativistic harmonic oscillator