A Unified Angular–Hyperbolic Representation of the Laplace Transform

This paper introduces a unified angular–hyperbolic representation of theLaplace transform, aimed at providing a geometric interpretation of Laplaceinduced kernels . The proposed framework distinguishes between boundedoscillatory kernels and unbounded kernels like exponential or hyperbolic throughtheir geometric realization in the parameter space (t, b) This approach illustrates how the angular geometric representation simplifies the understandingof kernel behaviors and provides new insights into Laplace transform applications