ABSTRACT
A continuous function (Chapter 21) real or complex, may fail to be differentiable. For a real function, the existence of derivate of a given order does not (Section 27.1) imply the existence, or continuity, of derivates of higher order, and the existence of derivates of all orders does not guarantee the convergence of the Taylor series; in contrast, for a complex function, if it is holomorphic, that is, has first-order derivate (Chapter 21), then (Section 27.2) it has derivates of all orders (Chapter 13), and a convergent Taylor series (Section 25.7 and 27.2), and it is a harmonic function (Section 23.1) as well. Another difference for complex functions is that when they fail to be holomorphic at a point, z = a, the associated Laurent series allows the classification of the singularity (Section 27.3) into two types: (i) the pole that is always an isolated singularity; (ii) the essential singularity that may or may not be isolated. The concept of residue at a pole (Chapter 15) may be extended to essential singularities (Section 27.4), while retaining its usefulness for the evaluation of loop integrals. A complex function may have a singularity at infinity (Section 27.5), that is, either a pole or an essential singularity, just as at any other point of the complex plane. The type of singularities of a function (Table 27.1) on the finite part of the complex plane and at infinity, can be used to classify (Table 27.2) the function as (i) a constant (Section 27.6); (ii) a polynomial, or a rational function (Section 27.7); (iii) an integral, a rational-integral, a meromorphic or a polymorphic function (Section 27.9). The latter set (iii) has at least one essential singularity (Section 27.8). The classification of functions in classes allows a listing of the properties associated to each class; it may also serve for the reconstruction of a function from partial data, thus completing the solution of a problem by specifying some desirable properties (e.g., the hodograph method with singularities in Section 38.8 and Example 40.16).
