Complex Integrals and Series - Bayin, Selcuk S (Mathematical Methods in Science and Engineering, 2018)
This chapter introduces the complex integral theorems. Despite their simplicity, these theorems are incredibly powerful and establish the basis of complex techniques in applied mathematics. Using analytic continuation, the chapter shows how these theorems can be used to evaluate some of the frequently encountered definite integrals in science and engineering. When discussing harmonic functions and mappings, one saw that analytic functions have very interesting properties. It is for this reason that it is very important to determine the region where a function is analytic and, if possible, to extend this region to other parts of the z‐plane. In conjunction with the discussion of definite integrals, the chapter also introduces the gamma and the beta functions, which frequently appear in applications. It further introduces complex power series, that is, the Taylor and the Laurent series and discusses classification of singular points. Finally, the chapter explains the integral representations of special functions.