A Parameter-Free Closed-Form Approximation for the Quartic Anharmonic Oscillator Period

A closed-form approximation for the period of the quartic anharmonic oscillator is derived from first principles. The key step is an exact reduction: introducing the harmonic energy fraction k² = x₊²/2 at the turning point transforms the period integral into T/T₀ = (2k/π)K(k²−1), where K is the complete elliptic integral evaluated at a negative argument. A quintic polynomial for K(−(1−k²)) is then constructed via Hermite interpolation, anchored by six conditions — values, first derivatives, and second derivatives at both endpoints — derived from the power series of K(m) at m=0 and exact evaluation at m=−1 via classical identities. The approximation achieves a maximum error of 0.009990% near ε ≈ 0.452, is positive throughout, and returns to zero exactly at both endpoints. All constants are composed exclusively of π and Γ(1/4), with no free parameters. Part of a series: AGM · Ellipse perimeter · Nonlinear pendulum · Asymmetric cubic oscillator · Relativistic harmonic oscillator