On the Calculation of Several Definite Integrals by Residue Theorem

Abstract

The Cauchy’s residue theorem is one of the most important theorems in complex analysis at all times, and it is demonstrated that using the residue theorem is an easier and faster method to calculate some types of real improper integrals when the targeted integrals are hard or even impossible to deal with by conventional approaches. This paper considers several types of integrals including trigonometric integrals and integrals involving logarithmic function and power function. The general methods for calculating these integrals are presented and typical examples to illustrate how to use the methods are shown. The types of integrals in the paper are useful in many fields and have applications in the engineering and scientific research. In complex analysis, the residue theorem is a powerful tool for calculating the path integrals of analytic functions along closed curves, and can also be used to calculate the integrals of real functions.