The Contour Integral Approach to Several Improper Integrals with Hyperbolic Functions

Abstract

In the field of Complex Analysis, it is acknowledged that Cauchy’s Residue Theorem plays an essential role, which allows the calculation of complex integrals by adding up the residues of singularities in the complex plane. Many mathematicians have developed various theorems out of Cauchy’s Residue Theorem and have solved numerous problems using Cauchy’s Residue Theorem, but there are still a lot more studies needed. Thus, this paper focuses on examining Residue Theorem deeply by introducing singularity point and residue, combining Laurent series and complex integral, then deducting Cauchy’s Residue Theorem. This paper then concentrates on solving four unique complex integrals to illustrate Cauchy’s Residue Theorem by analyzing the graph of integration, reformatting the integrals, applying theorems or tricks, integrating the reformatted integrals, and simplifying the results. As a result, this paper not only presented a deeper analysis of the deduction of Cauchy’s Residue Theorem, but also presented the solutions towards four previously unsolved complex integrals. The deduction of Cauchy’s Residue Theorem and the four complex analysis problems have important applications for dealing with integrals associated with hyperbolic functions and lead to future research in other areas of mathematics and physics.

Applications of Residue Theorem to Some Selected Integrals

“…𝑓(𝑥) just has two singularities a 0 and a 1 in the upper half complex plane. Then 𝒇(𝒙)𝒆 𝒊𝒎𝒙 -type integral In this part, the example is[10] From point −𝑎 to point 𝑎, it travels in a true straight line before turning counterclockwise along a semicircle centered on point −𝑎 to point 𝑎. In order for the hypothetical unit i to be contained within the curve, author takes a to be greater than 1.…”