The Abu-Ghuwaleh Hyper-Residue Calculus for Improper Integrals Phase-Lifted Spectral Solvers, Finite-Part Regularization, Branch Kernels, and Continuous Spectra

We develop a new calculus for improper integrals in which the decisive object is not the meromorphic structure of the integrand in the physical variable, but a spectral measure μ generating the kernel through a Laplace orbit and a Mellin symbol Γ(z)Zμ(z;ω). The first main result is an exact phaselifted solver for oscillatory and non-oscillatory improper integrals. The second is the Abu-Ghuwaleh hyper-residue theorem, which yields meromorphic continuation for gapped spectra and explicit pole coefficients at the negative integers. The third is a finite-part regularization theorem that turns divergent endpoint integrals into regular values of the same symbol. The framework then extends to discrete entire kernels, branch and logarithmic kernels, multiscale analytic germs, fractional lattices, continuous spectral densities, and annihilator theorems that convert spectral support constraints into ordinary or infinite-order differential equations. A benchmark section evaluates hard examples one by one, including essential-singularity kernels, oscillatory u−1 problems, finite-part u−2 integrals, branch kernels, nonlinear phases, Mittag–Leffler and Airy kernels, polylogarithmic kernels, and algebraic continuous-spectrum kernels. The resulting method is exact, constructive, and specifically adapted to a large family of improper integrals for which contour geometry is not the natural language.