Cauchy Residue Theorem’s Application in Improper integrals

Abstract

Definite integrals are an essential tool for understanding and calculating many aspects of the natural world. An improper integral, one type of definite integral, has either an infinite interval or an integrand that is not defined at one or more points within the interval of integration. In this research, improper integrals are the main concepts that will be discussed. The function can be expressed in terms of its Laurent series expansion–a general form or representation for analytic functions that includes both negative and positive power series of (z – singularity)–about each of its isolated singularities within the contour. And then, finding the singularities that are inside the contour based on the contour function. Calculating the residue of the singularities. Then, substituting the residue with the calculated value, adding all the residue together. Lastly, the result multiplies by, which is the result, namely the integral of the original functions. With the help of the other two methods, Keyhole Contour, and principal values, it is possible to evaluate the integrals and determine the domain of the function. Keyhole Contour separates the function’s domain and evaluates real integrals. And principal values can be used for determining the specific range of the function, so a single value can be chosen.