ABSTRACT
The proof of Cauchy’s first theorem (1821) given before (Section 13.6) uses the concept of primitive and the Cauchy-Riemann conditions, and makes only indirect use of the definitions of holomorphic function (Section 19.2) and Riemann integral (Sections 19.1). It is possible to give a totally distinct proof of the same theorem (Goursat, 1900) that uses directly the fundamental concepts of derivate and integral, without other intermediaries (Sections 19.3 and 19.4). This proof has the advantage of allowing two extensions, that is, the condition that the function be holomorphic in a closed region for Cauchy’s theorem to apply, can be relaxed (Section 19.5) on the boundary to uniform continuity (Goursat, 1900); then (Section 19.6) isolated singularities of order less than a pole are also allowed (Littlewood, 1944) on the boundary. These conditions were called Cauchy conditions (Section 15.1), since they are sufficient conditions for Cauchy’s theorem to hold; it can be shown that they are minimal conditions, since violation of any of them will cause the theorem to fail. Obtaining minimal conditions for Cauchy’s theorem to hold is important, since it extends the range of cases in which the many useful consequences of this theorem, for example, calculus of residues (part 2) and power series (part 3) can be applied; the notion of singularity with order less than a pole, is an instance of the indication of order of an infinitesimal (Section 19.7). This notion is useful in resolving indeterminate limits of the type 0/0, for example, by L’ Hoˆspital rule (Section 19.8). The latter provide the method III of calculation of the residues of certain classes of functions at poles (Section 19.9). The infinitesimals, are “smaller than anything else, but not zero, and could only be an invention of the devil,” according to bishop Berkeley, in his critique of Newton’s method of fluxions (1670). In science rather than theology they are the basis of the calculus and a convenient way to calculate limits, for example, a velocity, a direction, or a density, when they exist; and otherwise, to prove that a tentative concept is zero or infinity.
