The fundamental theorems of complex analysis

Holomorphic functions differ fundamentally from real differentiable functions: they are infinitely often (real and complex) differentiable (II.3, II.7), they even admit power series expansions (II.4), their entire behaviour is determined by their values on arbitrarily small open sets (II.4, II.7), and they satisfy powerful convergence theorems and estimates (II.5). All of these properties are consequences of the Cauchy integral theorem and the integral representations that arise from it (II.1–3). Meromorphic functions extend the class of holomorphic functions (II.6); their study leads to the notion of isolated singularities and to generalizations of power series obtained by allowing negative powers (Laurent series). In addition to the phenomena that occur in the theory of functions of one complex variable, a fundamentally new phenomenon enters the picture in higher dimensions: the simultaneous holomorphic continuation of all holomorphic functions from a given domain to a larger one (II.7). Here the Cauchy integral formula (in one variable!) is again the decisive tool.