Applications of Residue Theorem to Some Selected Integrals

A complex variable is a quantity that contains both a real part and an imaginary part. Complex analysis is an application of mathematics that analyzes the features of functions of complex variables. Physics, engineering, and computer science are just a few of the scientific disciplines that benefit from complex analysis. In order to tackle issues that are challenging or impossible to resolve using only real variables, complex analysis is crucial. This paper introduces an important theorem in complex analysis, which is Cauchy’s residue theorem. A relative general expression for a complex integral along a simple closed contour is provided by Cauchy's residue theorem. Cauchy's residue theorem can be used to select an acceptable closed contour for the calculation of some unusual definite integrals that may be highly complex and challenging to solve using traditional methods. The residue at the function's isolated singularities is then established. In the formula that is subtracted in Cauchy's residue theorem, the values of the residues are extracted. The integral along a simple closed contour can then be represented in two pieces, in which one along the real axis and the other along the circle.