This study discusses thickening annulus in Cauchy’s integral theorem for matrix function via Mobius transformation. We introduce a Jacobi type elliptic integral for arc length using Mobius transformation leading to optimization in the contour integral wherein Trapezoidal rule and Runge-Kutta fourth order method are used yielding approximate solutions to the integral problem. We filtered out noise from solution space using Tikhonov regularization method. This leads to computing for the path integral, path length and size of the path for the Cauchy integral. We have established that integral operators used have analytic continuation in their paths. As a further insight into what was studied is the computation of action matrix on Cauchy integral for each of Trapezoidal rule and Runge Kutta method.