Application of Reidues in the inverse Laplace Transform of composite functions

Abstract

The inverse Laplace transform is a pivotal mathematical tool in solving differential equations and analyzing engineering and physical systems. However, traditional methods such as partial fractions fail when dealing with composite and non-algebraic functions. This research aims to investigate and apply the Residue Theorem in complex analysis to compute the inverse Laplace transform of composite functions via the Bromwich integral formula. The study covers the theoretical framework of identifying poles and singularities, proceeds to the practical computation of residues for simple and multiple poles, and extends to advanced techniques for handling functions containing roots and logarithms using modified contour integration paths (such as the keyhole contour). The findings demonstrate that the residue theorem serves as a highly powerful and precise analytical tool, overcoming the limitations of conventional algebraic methods and providing accurate time-domain solutions for complex engineering problems such as diffusion and heat transfer systems, thereby recommending its adoption as an advanced methodology in applied mathematical studies.