A Unified Framework for Advanced Integration: Parametric Differentiation, Diffeomorphic Substitutions, and Asymptotic Schemes

A Unified Framework for Advanced Integration: Parametric Differentiation, Diffeomorphic Substitutions, and Asymptotic Schemes,” is a research-article style synthesis that organizes several “Advanced integration” methods into a single, checkable toolkit with explicit hypotheses and error control. Advanced_Integration_AMS_Paper_… Core idea. It builds a unified framework around three pillars: Parametric differentiation under the integral sign with verifiable dominated-convergence/envelope conditions, Diffeomorphic change-of-variables structured through a coarea-style Jacobian factorization to make substitutions systematic rather than “clever,” and Laplace / steepest-descent asymptotics with explicit remainder control, including handling certain mild singular weights. Advanced_Integration_AMS_Paper_… How it’s formalized. The paper states a set of “standing hypotheses” (H1)–(H5) that encode the regularity needed for each pillar—envelope control for differentiation, measure-preserving symmetry actions, submersion/coarea structure, nondegenerate Laplace critical points, and a mild algebraic singularity model. Advanced_Integration_AMS_Paper_… Advanced_Integration_AMS_Paper_… Main results (in the paper’s terms). Theorem 2.1: a convexity-transfer/log-convexity result for parametric integral families under symmetry (Jensen/Hölder style), including an equality characterization. Advanced_Integration_AMS_Paper_… Theorem 2.2: a coarea-factorized substitution formula and a constructive principle: choose the map so fiber integrals match canonical special-function models (e.g., beta/gamma), reducing problems to a scalar outer integral. Advanced_Integration_AMS_Paper_… Theorem 2.3: a two-term Laplace expansion for I(t)=∫Ωe−tφ(x)ψ(x) dxI(t)=\int_\Omega e^{-t\varphi(x)}\psi(x)\,dxI(t)=∫Ωe−tφ(x)ψ(x)dx with an interior nondegenerate minimizer and a weak algebraic singularity, giving an explicit invariant expression for the 1/t1/t1/t correction coefficient c\*c_\*c\*. Advanced_Integration_AMS_Paper_… Advanced_Integration_AMS_Paper_… Scope and examples. The applications section sketches how the framework reproduces/derives bounds and asymptotics for beta/gamma-type integrals, Gaussian/Fresnel models, and principal-value (PV) integrals, and it positions the article as a formal consolidation of techniques connected to your longer monograph. Advanced_Integration_AMS_Paper_… Advanced_Integration_AMS_Paper_… What it leaves open. It ends with concrete extensions (boundary stationary points, oscillatory/PV stationary phase, and sharp symmetry constants). Advanced_Integration_AMS_Paper_…