Applications of Cauchy’s Residue Theorem in Computing Improper Integral

Abstract

An improper integral is a definite integral that either has an infinite interval or has the integrand that is not defined on some points in the interval. Many improper integrals are difficult to compute by using real analysis methods, especially those containing infinity. By contrast, introducing the complex methods and applying Cauchy’s residue theorem can give a much more simplified solution. In order to apply Cauchy’s residue theorem, the residues of the integrand at the singularities that are interior to the contour are first to be found, then the integral along the whole simple closed contour can be evaluated. These contours always consist of line segments and sectors of circle. In most cases, only the part of contour on the real axis is related to the real definite integral, and other parts should be eliminated by proving it tendency to a certain value where most commonly it is zero. This is always done by considering the property of the integrand or using Jordan lemma.