A Unified Angular–Hyperbolic Representation of the Laplace Transform

This paper introduces a unified angular–hyperbolic representation of the Laplace transform, aimed at providing a geometric interpretation of Laplace- induced kernels . The proposed framework distinguishes between bounded oscillatory kernels and unbounded kernels like exponential or hyperbolic through their geometric realization in the parameter space (t, b) This approach illus- trates how the angular geometric representation simplifies the understanding of kernel behaviors and provides new insights into Laplace transform applica- tions In this paper, the term Laplace-induced kernels refers to the input functions within the Laplace integral. We use this term to denote those functions whose geometric behavior is analyzed within the framework of the transform.