Contour Integral and Consequences of Cauchy Integral Theorem in Differential Equations

Ismaila IO, Lawal AA and Zuwaira B

Published on: 2026-01-27

Abstract

This study investigates the role of contour integration and the Cauchy integral theorem in the theory and practice of ordinary and partial differential equations. We develop rigorous foundations from complex analysis, derive central consequences such as residue calculus, analytic continuation, and integral representations, and apply these to solve boundary-value problems, evaluate inverse Laplace transforms, and analyze asymptotic expansions of solutions. Numerical aspects (Bromwich inversion, deformation of contours for numerical stability) are presented, and illustrative figures demonstrate core geometric and computational ideas. Finally our results connect contour methods to fundamental solution representations and modern numerical inverse transforms.