Integral formulas, residues, and applications

Abstract

The concept of the winding number allows a general formulation of the Cauchy integral theorems (IV.1), which is indispensable for everything that follows. IV.2 presents a generalization of the Cauchy integral formula to real differentiable functions; it will play a basic role in Chapter VI. With the Laurent series expansion (IV.3) and the residue theorem (IV.4), further essential tools of complex analysis are at our disposal. They will be used to evaluate complicated integrals (IV.5) and then to study the equation f(z) = w, where f is a holomorphic function (IV.6). If one makes the integral formulas from sections IV.3 and IV.6 dependent on parameters, then one obtains the Weierstrass preparation theorem (IV.7), which gives fundamental information about the zeros of holomorphic functions of several variables.