7.1. In the last six chapters we have analysed Cauchy's work on complex function theory from 1814 to 1831, and we have indicated some of the background to the development of his ideas. In the present chapter we shall try to draw the threads together. There are several themes that recur throughout the story; we shall look at these in turn, and try to trace how Cauchy's approach to each of them changed during the period.

To begin with, we shall have to say something about the kind of functions that Cauchy regarded as admissible, and then about the development of his concept of the integral. After a brief sketch of some earlier ideas about complex functions and about techniques for the evaluation of definite integrals, we shall show how, by combining some of these ideas, he was able to obtain results equivalent to special cases of ‘Cauchy's theorem’ and the residue theorem. We shall then follow the development of his techniques, which enabled him to establish more general forms of these results, and we shall indicate the role played here by his tentative introduction and gradually increasing use of a geometrical picture of the complex plane.

I.1. During the years 1814–1831 Cauchy created the basic framework of complex function theory. The aim of the present work is to trace the details of this process, beginning with his 1814 memoir on definite integrals, and taking the story as far as the two Turin memoirs of 1831. That year marks a major break in the development of his work in this direction: it is true that for some time thereafter he was making important applications of his results, but he made no further substantial contributions to the central core of the theory until after his return to teaching in 1849.

The available evidence for Cauchy's ideas lies in his published work, practically all his manuscripts and correspondence having disappeared. I hope to show that a connected story emerges from a careful study of his publications, and that we can trace a gradual development in Cauchy's ideas about complex function theory through a number of stages of his work.

We shall concentrate our attention on the main line of development, and we shall leave a number of topics undiscussed, including, for instance, his work on Fourier transforms and differential equations, some of which does involve applications of his results on complex function theory. Important as these achievements were in their own right, they do not seem to have had any significant influence on his development of the central framework of the theory.

4.1. On 28 February 1825 Cauchy communicated to the Académie des Sciences a memoir entitled ‘Mémoire sur les intégrates définies prises entre des limites imaginaires’. He published an abstract of it in the Bulletin de Férussac [1825a] in April of the same year. The complete memoir was issued as a separate pamphlet in August [1825b]. It was republished by Darboux in the Bulletin des Sciences Mathématiques in 1874–5, and a German translation of it appeared in the series Ostwalds Klassiker der Exakten Wissenschaften in 1900, with notes and commentary by P. Stäckel.

We shall usually refer to [1825b] as the 1825 memoir; it constituted an important advance in Cauchy's development of complex function theory.

In the present chapter we shall also include some account of a paper by Cauchy, published in two parts, [1825c] and [1826f], in Gergonne's Annales de Mathématiques; it can most conveniently be regarded as a sequel to the 1825 memoir.

4.2. In his introduction to the 1825 memoir Cauchy recalls his use of singular integrals in the 1814 memoir and his results on principal values in [1822b], [1823a] and in the Analyse infinitésimale [1823b], remarking that these together had enabled him to establish numerous general results on the evaluation or transformation of definite integrals. He now proposes to apply the same principles to integrals between imaginary limits.

He recalls Laplace's use of such integrals, and mentions that Brisson had recently used them to obtain expansions of functions in exponential series.

2.1. Cauchy's contributions to the development of complex function theory began with his long memoir [1814] on definite integrals; it was presented to Classe I of the Institut de France on 22 August 1814, the day after Cauchy's 25th birthday. The memoir was examined by Legendre and Lacroix, and its publication was recommended in their report [Legendre 1814b], written by Legendre, and dated 7 November 1814. The original intention was that the memoir should appear in the Mémoires présentés à l'Institut des Sciences, Lettres et Arts, par divers savans, but no further issues of this periodical were published because of the reorganisation of the Institut after the the restoration of the Bourbon dynasty. The first volume of the replacement periodical, the Mémoires présentés par divers savans à l'Académie Royale des Sciences de l'lnstitut de France, did not appear until 1827, and did contain both this memoir and Cauchy's prize essay [1815] on the theory of waves.

Because of the long delay in the publication of the 1814 memoir. Cauchy summarised some of its results in various publications between 1814 and 1825 (see Chapter 3 below). By 1823 he was clearly wondering whether it would ever appear in the Academy's publications; in [1823a] he announced that it would probably appear in the next number of the Journal de l'Ecole Polytechnique, but this too was delayed, not being published till 1831, so this scheme did not work.

6.1. In the present chapter we shall be mainly concerned with Cauchy's investigations of the Lagrange series, and his discovery that some of the ideas used there were much more widely applicable; they led him to his important results on the convergence of the Taylor series of an analytic function and on power-series expansions by implicit functions.

We shall begin by outlining some of the early work on the Lagrange series, referring to Lagrange's discovery of the series [1770a] and his application of it to Kepler's problem [1770b], to a paper by Laplace [1779] and the criticism of it by Paoli [1788], and to the first serious investigation of the convergence of the series by Laplace [1825]. We then describe Cauchy's first studies of its convergence in his [1827b] and [1827c], and we go on to the long memoir [1831d], issued in lithograph form during Cauchy's stay in Turin, and containing his results on the convergence of power-series expansions of both explicit and implicit analytic functions, together with his ‘calculus of limits’, better known today as the method of majorants. The memoir also contains extensive applications of his results to celestial mechanics, but we shall omit any discussion of these. The part of [1831d] with which we shall be concerned first appeared in print (with some small but significant revisions) in Cauchy's Exercices d'analyse in 1841; there is also a brief preliminary abstract [1831a], published in the Bulletin de Férussac.

5.1. In 1826 Cauchy began to publish his Exercices de Mathématiques, which was essentially a mathematical periodical consisting entirely of papers written by himself; it appeared at approximately monthly intervals until 1830. He used similar series as vehicles for his publications at later stages of his career; the Nouveaux Exercices de Mathématiques were published in Prague in 1835–36 and the Exercices d'Analyse et de Physique Mathématique appeared at intervals from 1840 to 1853.

Some of the work appearing in the Exercices is expository in character, but much of it contains original research. In particular, as we shall see, his main contributions to what he called the calculus of residues are to be found there.

In the present chapter we shall describe Cauchy's definition of the notion of residue and the way in which he used it for the further development of complex function theory. We shall omit some of the applications he made of it between 1826 and 1830, in particular its use to obtain explicit expressions for solutions of differential equations, and his work on operational methods, which is more closely linked to the theory of Fourier transforms.

5.2. The notion of the residue of a function at a point where the function becomes infinite is defined in Cauchy's paper [1826a] in the first number of the Exercices. His definition is as follows.

3.1. Between the completion of his 1814 memoir and his next major contribution to complex function theory, the famous memoir [1825b] on definite integrals between imaginary limits, Cauchy made a number of minor improvements to the theory; in addition, because the publication of the 1814 memoir was held up until 1827, he restated his main results several times, in particular in the papers [1822b] and [1823a] and in his book [1823b] on the infinitesimal calculus.

To get the chronology right, it has to be noted that [1823a] falls into two parts, arising, respectively, from memoirs presented to the Academy of Sciences on 16 September 1822 and 26 May 1823; although the memoir was published in July 1823 and the Calcul infinitésimal [1823b] in August, the second part of [1823a] contains references to [1823b]. It is this second part with which we shall mainly be concerned.

We shall also have to take into account some of the supplementary notes to his prize memoir [1815] on water waves, which appear to date from about 1824, and the footnotes that he prepared about 1825 for insertion in the published version of the 1814 memoir. We shall present some evidence about the contents of his 1817 lectures at the Collège de France and an unpublished memoir summarised in [1819]; some relevant material in other papers of this period and in the Analyse algébrique will also be mentioned.