Complex Inversion Formula for the Laplace-Carson Transform
Fasiyoddin I. Momin
Research Center in Mathematics, Maulana Azad College of Arts Science and Commerce, Chhatrapati Sambhajinagar, Maharashtra, India.
Sadikali L. Shaikh
Research Center in Mathematics, Maulana Azad College of Arts Science and Commerce, Chhatrapati Sambhajinagar, Maharashtra, India.
Anil D. Chindhe *
Department of Mathematics, Balbhim Arts, Science and Commerce College, Beed, Maharashtra, India.
*Author to whom correspondence should be addressed.
Abstract
Aims/ Objectives: To present a systematic study of the complex inversion formula for the Laplace–Carson transform.To establish convergence criteria and study detailed computational methodologies for both finite and infinite poles.
Study Design: Analytical study.
Place and Duration of Study: Research Center in Mathematics,Maulana Azad College of Arts, Science and Commerce, Chh. Sambhajinagar, Maharashtra,India. june 2025 to march 2026.
Methodology: The complex inversion formula is an extremely powerful method for calculating the inverse of an LC-transform. A practical post-inversion formula and a new inversion method, analogous to classical Laplace inversion techniques, are introduced. The LC-transform is a modified version of the classical Laplace transform and it offers distinct analytical advantages over the Laplace transform in various applied mathematical fields. Additionally, we discuss functions possessing branch points and deal with meromorphic functions containing infinitely many poles.
Results: Several illustrative examples and applications to partial differential equations demonstrate the analytical advantages of the Laplace–Carson transform over the classical Laplace transform.
Conclusion: In this paper,the complex inversion formula for LC-transform transform is rigorously established and its computational application extended to complex branch points and meromorphic functions with various poles.
Keywords: Laplace transform, Laplace–Carson transform, complex inversion formula, meromorphic functions, damped harmonic oscillator, heat equation, wave equation.