A Unified Algebraic Approach to Laplace Inversion: Application of the Extended Kyungu Formula to Singular, Fractional and Distributional Kernels

This work introduces the Extended Kyungu Formula (KL-Formula), a unified algebraic framework for the inversion of Laplace transforms in the setting of regular functions, fractional kernels, and Schwartz distributions. The method replaces the classical Bromwich contour integration by a local algebraic expansion based on the auxiliary transformation: φ(x) = F(1/x) The framework systematically classifies auxiliary exponents into regular, impulsive, and distributional components, allowing direct computation of inverse Laplace transforms without contour integration. Applications developed in this work include: - Fractional kernels- Branch-point singularities- Mittag-Leffler dynamics- Bessel-type inversions- Exponentially growing kernels- Ultra-distributions- Distributional Laplace inversion The resulting operational structure is naturally compatible with symbolic computation and Computer Algebra Systems (CAS).