The Application of Residue Theorem in The Calculations of Real Integrals

Complex analysis focuses on the functions of complex variables. The research on complex analysis began in 19th century by mainly Cauchy, Riemann, and Weierstrass. In 1825, Cauchy established Cauchy’s integral theorem, indicating that the value of a complex function at a point within a closed contour is entirely dependent on values of the functions at points on the contour . Cauchy’s integral theorem is a significant theorem in complex analysis, which was then further developed into Cauchy’s residue theorem. In this paper, the authors introduce the fundamentals of complex analysis. In Introduction, the authors discuss the history of complex analysis and residue theorem and summarizes the content of published articles and books. In Method, the authors explain the definitions and fundamental properties of complex functions. In Result, the authors use examples of different forms of real integrals to demonstrate the application of residue theorem in the calculation of real integrals.