Residues of Complex Functions with Definite and Infinite Poles on X-axis

Problem statement: One of the most popular areas in the mathematics is the computational complex analysis. In this study several computational complex techniques were investigated and implemented numerically. Objective: This study produced new procedures to compute the residues of complex functions by changing their numerator from a constant number to either even or odd function. Approach: In this project we studied the functions that had finite and infinite poles Z_i, i greater than one of order greater or equal one, also we found new relation between residues at the poles Z_i and residues at the poles -Z_i, i greater than one and we had used these relations to solve improper integrals of this type. The project needed the knowledge of computing the complex improper integrations. Results: Our numerical results in computing the residues for improper integrals of definite and infinite poles on the x-axis were well defined. Conclusion: In this study, we had concluded that the residues of the complex functions had definite and infinite poles of higher order with constant numerator. A general form of residues of these functions of high orders were also investigated.