Using Cauchy theorem to compute complicated real integrals

Abstract

As the central object in the theory of complex analysis, Holomorphic functions have many elegant mathematical properties. Holomorphic functions are extremely valuable because, on the one hand, they are unexpectedly common, and on the other hand, they may be used to establish extremely powerful theorems. For example, To establish theorems like the prime number theorem, analytic number theorists commonly create holomorphic or meromorphic functions that hold number-theoretic information, such as the Riemann zeta function. Given knowledge about a holomorphic function in a relatively small part of its domain, one may extract information about the function’s behavior in other a priori unrelated sections of its domain, according to Cauchy’s integral formula and the identity theorem (and this is what allows things like contour integration to work). Due to difficulties in obtaining primitives of some real function, the fundamental theorem of Calculus does not work in most cases. In this article, we review the Cauchy theorem and use it as a tool to compute several real integrals.