ABSTRACT
Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 6.13.1 Homotopic Closed Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 6.13.2 The Homotopic Version of Cauchy’s Theorem . . . . . . . . . 390
6.14 Expansion of Analytic Functions as Power Series . . . . . . . . . . . . . . . 392 6.14.1 Taylor’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 6.14.2 Laurent’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
6.15 Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
In this chapter, we derive results which are fundamental in the study of analytic functions. These results constitute one of the pillars of mathematics and have far-ranging applications. Notice that many important properties of analytic functions are very difficult to prove without use of complex integrations. For instance, the existence of higher derivatives of analytic functions is a striking property of this type. There occur real integrals in applications that can be evaluated by complex integration. We now turn our attention to the question of integration of the complex valued function.
