Mathematical Methods in Science and Engineering, 2nd Edition

Chapter 12Complex Integrals and Series

In this chapter, we first introduce 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, we show how these theorems can be used to evaluate some of the frequently encountered definite integrals in science and engineering. In conjunction with our discussion of definite integrals, we also introduce the gamma and the beta functions, which frequently appear in applications. Next, we introduce complex power series, that is, the Taylor and the Laurent series and discuss classification of singular points. Finally, we discuss the integral representations of special functions.

12.1 Complex Integral Theorems

12.1.1 Cauchy–Goursat Theorem

Let be a closed contour in a simply connected domain (Figure 12.1). If a given function, , is analytic within and on this contour, then the following integral is true:

12.1

Proof

If we write the function as , the above integral [Eq. (12.1)] becomes

12.2

Using the Stokes theorem: ...