Residue Theorem and Its Application

Sometimes it can be frustrating to get the anti-derivatives of some functions. Some definite integrals are difficult, or even impossible, to calculate using the elementary calculus method, especially in the field of complex analysis. In such cases, Cauchy's Residue Theorem might be proper. In this article, we state the difference between Cauchy's Integral Theorem and the Residue Theorem, followed by the definitions of isolated singular point and residue with some examples given after. These are prerequisites for understanding the theorem. A short statement of the content of Cauchy's Residue Theorem has been made, according to which the integral of a function along a positively oriented simple closed contour is related to the residues, which can be obtained through Laurent expansion, of all the limited isolated singular points inside the contour. In addition, using the Cauchy-Goursat theorem, we developed the derivation of CR Theorem. It denotes that the value of a function around a simple closed contour is zero if the function is analytic at all points interior to and on that contour. Then we launch an application and accomplish it in two ways. The first method makes use of the Fourier Matrix, while the second is related to power series. In both of these approaches, we feel the efficacy of Cauchy's Residue Theorem.