A Numerical Analytic Resolution of a Hidden Asymptotic Branch in an Oscillatory Integral

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 We analyze the oscillatory integral I(α, β) = ReZ∞ 0 eiαu2−βu usin u (1+u2) log(1 + u2)du, α > 0, β ≥0, which couples quadratic oscillation, exponential damping, and a logarithmic weight with a branch point at u = i . A stationary-phase analysis yields the full contribution from the origin. High-precision numerical quadrature shows that when β = 0 the integral possesses an additional oscillatory contribution of order O(α−1) arising from the logarithmic singularity. This produces a two-front structure: I(α, β)∼Iendpoint(α, β) + Ibranch(α, β), α → ∞, with Ibranch exponentially suppressed for β > 0 yet dominant when β = 0. We derive the endpoint asymptotics rigorously, characterize the branch front numerically, and formulate the associated saddle branch asymptotic problem as a well defined open question in logarithmic-uniform analysis. Keywords: oscillatory integrals, steepest descents, logarithmic branch points, Stokes phenomena, asymptotics. MSC (2020): 41A60, 41A80, 30E15. 1 Introduction We study the oscillatory integral I(α, β) = ReZ∞ 0 eiαu2−βu usin u (1+u2) log(1 + u2)du, (1) which features two analytic mechanisms: •a stationary point of the phase iαu2−βu at u= 0, •a logarithmic branch point of the kernel at u=i. For β > 0, damping suppresses the upper half plane, and only the stationary point contributes asymptotically. When β = 0, this suppression vanishes, and numerical evidence shows that the logarithmic singularity at u = i generates a second oscillatory contribution comparable in magnitude to the endpoint term. Thus I(α, β) exhibits a two-front asymptotic structure. 1 2 The Endpoint Contribution Define K(u) = usin u (1+u2) log(1 + u2), Iendpoint(α, β) = ReZ∞ 0 χ(u)K(u)eiαu2−βu du, with χsmooth and supported near the origin. A Taylor expansion gives K(u) = 1 2u2+O(u4), u →0. 2.1 Asymptotic evaluation The classical integral Z∞ 0 u2eiαu2du =eiπ/4√π 4α−3/2 and a first-order integration by parts yield the following. Theorem 1. For fixed β≥0, Iendpoint(α, β)∼ −1 4rπ 2α−3/2+2β α2+O(α−5/2), α → ∞.(2) 2.2 Numerical confirmation High-precision quadrature (160-digit arithmetic) confirms (2) for β > 0. For β= 0, the difference I(α, 0) −Iendpoint(α, 0) is oscillatory with envelope O(α−1), indicating an additional asymptotic front. 3 Activation of the Logarithmic Branch Point The kernel in (1) has a branch point at u = i . For β > 0, the contour of integration cannot be deformed upward without encountering damping; thus the singularity is asymptotically inactive. At β = 0, the damping disappears and the branch point contributes an oscillatory term. This transition is consistent with Stokes-type activation of a subdominant component. 4 Numerical Form of the Branch Contribution Define the residual Ibranch(α, 0) := I(α, 0) −Iendpoint(α, 0). Numerical data show that Ibranch ( α, 0) = O ( α−1 ) and oscillatory. Among all candidate asymptotic forms tested, only Ibranch(α, 0) ∼C1log α+C2cos α+C3sin α+C4 α, α → ∞,(3) fits all datasets. Measured coefficients: C1≈0.0266, C2≈ −0.0141, C3≈0.0329, C4≈ −0.0565. 2 5 Local Analysis Near the Logarithmic Singularity Let s=u−i. A symbolic expansion yields K(i+s)=A s +B s log s+O(s2log s), s →0, with B=−2isinh(1), A =4 sinh(1)π−ilog 4. Locally the phase becomes iαs2−2αs + constant. This motivates the canonical model Icanon(α)∼ZH (A+Blog s)s eiαs2−2αs ds, (4) where H is a steepest-descent contour crossing the logarithmic branch cut. The four channels in (3) are compatible with the asymptotics of (4). 6 An Open Problem in Logarithmic Uniform Asymptotics Open Problem. Determine the asymptotic expansion of the canonical integral (4) as α→ ∞ and compute the coefficients ( C1, C2, C3, C4 ) in (3) in terms of the local data (A, B). This problem requires uniform treatment of quadratic saddles, logarithmic singularities, and Stokes switching, and appears to lie beyond currently established steepest-descent methods. 7 Conclusion The integral (1) exhibits a two-front asymptotic structure consisting of: •a monotone algebraic contribution from the saddle at u= 0, •an oscillatory O(α−1) contribution from the branch point at u=i. The endpoint contribution is rigorously derived in Theorem 1. The oscillatory front is numerically characterized by (3) and explained via the canonical model (4). Computing the associated Stokestype multipliers remains a natural and challenging open problem. 3 References [1] N. Bleistein and R. A. 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