ABSTRACT
The analysis of functions of a complex variable can be envisaged at three successive levels. At the first local level (part 2) a holomorphic function is defined by the existence of derivate at a point (Chapter 11); this definition involves only the neighborhood of the point, and allows the consideration of the primitive and the deduction of solutions of the Laplace equation in the plane, that is, harmonic functions (Section 23.1). At the second level (part 3) of a region, an analytic function is defined as a function holomorphic at all points of the region; by means of contour and loop integrals, it can be shown that a such a function is infinitely differentiable (Chapter 15), that is, smooth, and that it has a Taylor series (Chapter 25) with radius of convergence determined by the nearest singularity. At the third or global level (part 4), a set of functions analytic in overlapping regions defines a single monogenic function, if the functions coincide in the overlapping subregions; a monogenic function provides the analytic continuation of each of its component functions beyond the region where it was originally defined. The analytic continuation allows the study of a function over the maximum possible extent of the complex plane, including special points, such as zeros and singularities; the singularities of complex analytic function are either poles (Chapter 31) or essential singularities (Chapter 39). The analytic continuation of a monogenic function (Section 31.1) can be used to extend its domain, for example, if it is real on the real axis by reflection (Section 31.2). The analytic continuation is not possible for lacunary functions, whose singularities are dense on the boundary of convergence (Section 31.1); analytic extension may be possible across an arc for a function analytic on both sides with a finite discontinuity (Section 31.3). The analytic continuation depends on the location of poles that can be determined, together with zeros via Cauchy’s fourth integral theorem (Section 31.4). The zeros relate to: (i) the factorization of polynomials (Sections 31.5-31.7); (ii) the decomposition of rational function into partial fractions (Sections 31.8 and 31.9) based on their poles. The location of the zeros of a polynomial, even without their precise determination, can be sufficient to ensure the stability of a system, for example, if all zeros of the characteristic polynomial are on the r.h.s. half-plane (Chapter 2).
