Correction and Numerical Clarification to "A Numerical Analytic Resolution of a Hidden Asymptotic Branch in an Oscillatory Integral"

Correction and Numerical Clarification to A Numerical Analytic Resolution of a Hidden Asymptotic Branch in an Oscillatory Integral T. A. Ziad Independent Researcher, Analytic Number Theory and Mathematical Physics ORCID: 0009-0001-7304-3533 Abstract This note corrects the interpretation of numerical results reported in the original preprint. Further analytic and numerical investigation demonstrates that the previously reported oscillatory O(α−1) remainder arises from truncation-induced boundary artifacts in real-axis quadrature rather than from an intrinsic asymptotic contribution. The endpoint asymptotics derived in the original work remain valid. We clarify the mechanism responsible for the numerical instability and distinguish the present case from settings in which genuine algebraic oscillatory terms are known to occur. In the original manuscript, high-precision numerical quadrature of the oscillatory integral I(α, 0) = ℜZ∞ 0 eiαu2K(u)du, K(u)=usin u(1+u2) log(1 + u2), was interpreted as exhibiting an additional oscillatory contribution of order O(α−1) beyond the rigorously derived endpoint asymptotics. This interpretation is incorrect. The apparent oscillatory remainder is not an intrinsic asymptotic feature of I(α, 0) but arises from numerical truncation of the oscillatory tail. For large u, the kernel has polynomial–logarithmic envelope growth, |K(u)|≲u3log u(u→ ∞), and the improper integral defining I(α, 0) converges only through delicate cancellation in the highly oscillatory tail. Direct real-axis quadrature of truncated integrals IR(α) = ℜZR 0 eiαu2K(u)du is therefore numerically unstable. As Rincreases, computed values can exhibit rapid growth and alternating sign rather than stabilization. This behavior is explained by a single integration-by-parts identity for the tail: Z∞ R eiαu2K(u)du =K(u) 2iαu eiαu2∞ u=R −1 2iα Z∞ R eiαu2d duK(u) udu. 1 Truncation introduces a boundary term ℜK(R) 2iαR eiαR2, whose magnitude is of order ∼|K(R)| αR ∼R2log R α, thereby mimicking algebraic oscillatory contributions of apparent order O(α−1) in naive truncationbased quadrature. These contributions depend sensitively on the truncation parameter Rand do not represent genuine asymptotic structure. Algebraic oscillatory contributions of order O(α−1) are known to arise in oscillatory integrals under specific analytic conditions, such as genuine endpoint singularities, saddle–endpoint coalescence, or branch points accessible by admissible steepest-descent contours. In the present problem the endpoint is sufficiently smooth to enforce a half-integer asymptotic hierarchy, and the logarithmic branch point at u=ilies on an anti-Stokes boundary that cannot be reached without entering regions of exponential growth. Consequently, no intrinsic algebraic oscillatory contribution is expected. We therefore withdraw the claim of a genuine oscillatory O(α−1) branch contribution in I(α, 0). The endpoint asymptotic expansion derived in the original manuscript remains valid. The principal lesson is that oscillatory integrals with polynomially growing kernels can exhibit severe numerical instability under naive real-axis quadrature, and apparent algebraic remainders may arise purely from truncation artifacts. 2