The Real and Complex Approaches to a Class of Dirichlet's Sinc Integrals

Abstract

Integration is a powerful tool to solve many problems in different areas, such as probability theory, physics, economics, etc. This paper deals with the integration of an important function, Dirichlet’s sinc functions of the form . Sinc function is a significant function that is used in many different fields, particularly in engineering and optics. In this article, by using the iteration method, a general solution about the integrals of the sinc functions with different powers is found, and several examples are given to support this idea. Meanwhile, Cauchy’s residue theorem and contour integration are the fundamental methods in the complex analysis, and they are chosen to compute the integrals of sinc functions with power one, two, and three. The integrals found in the examples correspond to the general solutions. The findings of the paper suggest that the real methods have the advantage to discover the recursion formula, while the complex methods are powerful in dealing with specific integrals.