Mémoire sur les intégrales définies, prises entre des limites imaginaires.

Paris: Chez de Bure Frères, 1825.

First edition of this rare memoir which contains the first publication of many of the ideas and results of modern complex function theory, notably the Cauchy integral theorem and the residue theorem. “This paper is considered by many his most important, and one of the most beautiful in the history of science” (Kline, Mathematical Thought from Ancient to Modern Times, p. 637). “The discoveries with which Cauchy’s name is most firmly associated in the minds of both pure and applied mathematicians are without doubt his fundamental theorems on complex functions” (DSB). “Between 1814 and 1831, the great French mathematician A. L. Cauchy created practically single-handedly a new branch of pure mathematics. Complex function theory was and remains of central importance, and its creation marked the start of one of the most exciting periods in the development of mathematics” (Smithies). “Particular complex functions had been studied by Euler, if not earlier. In hydrodynamics d’Alembert had developed what are now called the Cauchy-Riemann differential equations and had solved them by complex functions. Yet even at the beginning of the nineteenth century complex numbers were not yet unanimously accepted; functions like the multivalued logarithm aroused long-winded discussions. The geometrical interpretation of complex numbers, although familiar to quite a few people, was made explicit by Gauss as late as 1830 … Gauss’s proofs of the fundamental theorem of algebra, although reinterpreted in the real domain, implicitly presupposed some facts from complex function theory. The most courageous ventures in complex functions up to that time were the rash ideas of Euler and Laplace of shifting real integration paths in the complex domain to get new formulas for definite integrals, then an entirely unjustified procedure. People sometimes ask why Newton or Leibniz or the Bernoullis did not discover Cauchy’s integral theorem and integral formula. Historically, however, such a discovery should depend first on some geometrical idea on complex numbers and second on some more sophisticated ideas on definite integrals. As long as these conditions were not fulfilled, it was hardly possible to imagine integration along complex paths and theorems about such kinds of integrals. Even Cauchy moved slowly from his initial hostility toward complex integration to the apprehension of the theorems that now bear his name. The first comprehensive theory of complex numbers is found in Cauchy’s Cours d’analyse of 1821. There he justified the algebraic and limit operations on complex numbers, considered absolute values, and defined continuity for complex functions. He did not touch complex integration although in a sense it had been the subject of his mémoire submitted to the French Academy in 1814. It is clear from its introduction that this mémoire was written in order to justify such rash but fruitful procedures as those of Euler and Laplace mentioned above. But Cauchy still felt uneasy in the complex domain. He interpreted complex functions as pairs of real functions of two variables to which the Cauchy-Riemann differential equations apply. This meant bypassing rather than justifying the complex method” (DSB). It was only in the 1825 memoir that he gave this justification by introducing geometrical methods into the study of complex functions. ABPC/RBH records the sale of only two copies, both in modern bindings.

Towards the end of the 18th century, there was considerable interest in finding ways to evaluate definite integrals. Both Euler and later Laplace evaluated certain integrals by using substitutions involving complex functions. “From 1809 onwards Laplace used this method of ‘imaginary substitutions’ several times, mainly for integrals arising in his work on probability theory. This triggered a lively exchange of views with S.D. Poisson (1781–1840), who maintained in a number of papers that the method should be regarded only as a kind of induction, useful for discovering new results, but that these should be confirmed more directly; Laplace defended the method by appealing to the widely accepted principle of the generality of analysis, but eventually had to admit that direct confirmation was desirable” (Landmark Writings, p. 378).

Cauchy’s first significant work on complex function theory was Mémoire sur les intégrales définies, submitted to the Académie des Sciences on 22 August 1814 but not published until 1827. “It seems not unlikely that Cauchy’s 1814 memoir was stimulated by a suggestion from Laplace that Cauchy should investigate the method of imaginary substitutions. In his introduction Cauchy refers to the use of the method by Euler and Laplace as a kind of induction from the real to the imaginary, and announces that he proposes to establish it by a direct and rigorous analysis” (ibid.).

The 1814/1827 memoir was an attempt to justify methods that had been used by Euler, Laplace and others to evaluate certain definite integrals, methods which (again in modern terminology) involved shifting the path of integration from the real into the complex domain. Cauchy’s main result was a primitive version of the integral theorem which allowed integration only along rectangular paths with sides parallel to the real or imaginary axes, since the limits on the definite integrals he considered were always real numbers. Moreover, the integrands in the 1814/1827 paper were real functions – complex functions had to be dealt with by splitting them into their real and imaginary parts. Finally, geometrical language was nowhere used in the 1814/1827 paper. “Genuine complex integration is still lacking in the 1814 mémoire, and even in 1823 Poisson’s reflections on complex integration were bluntly rejected by Cauchy. But they were a thorn in his side; and while Poisson did not work out this idea, Cauchy soon did. In a mémoire détaché of 1825 [offered here], he took a long step toward what is now called Cauchy’s integral theorem” (DSB).

Cauchy begins the 1825 memoir by giving a definition of a definite integral for which the limits are complex rather than real numbers, attempting to imitate the definition he had given for real limits in his Résumé des Leçons (1823). This led him to define the integral of a complex function f(z) over an arbitrary path in the complex plane (in modern terminology). The main theorem, which he derived from the Cauchy-Riemann differential equations by means of variational calculus, asserted that in a domain of regularity of f(z) such an integral depended only on the endpoints of the path, and not on the path itself. This is the famous Cauchy integral theorem, the foundation of modern complex function theory. In more modern terms, his argument seems to involve the continuous deformation of the path of integration.

“Cauchy next attacks the problem of what happens when f(z) has a singularity at a point lying between the two paths along which the integral is evaluated … Cauchy deals first with the case where the singularity is a simple pole giving it two separate treatments. In the first of these he uses an approximation argument, getting involved with some complicated, expressions, which he ploughs through in his usual manner. In the second treatment he splits f(z) into two parts, one of which is a simple rational expression, reproducing the behaviour of f(z) near the singularity, and the other a well-behaved function, which makes no contribution to the difference between the integrals … the device of splitting the function in this way appears to come from Ostrogradsky, who had used a similar device in an unpublished paper dated 24 July 1824 and was working closely with Cauchy at the time.

“Cauchy then goes on to deal with the case where the pole of f(z) between the paths is a multiple one (arts. 6-7). Again he gives two treatments; in the first one he again uses Ostrogradsky’s device of splitting a function into a rational principal part and a well-behaved function, and in the second he gives an alternative approximation argument of ferocious complexity … In every case he concludes that the difference between the integrals is ±2πif0, where f0 is precisely what he was later to call the ‘residue’ of f(z) at the singularity.

“Shortly after proving these results Cauchy suddenly introduces, without any prior warning except a brief mention in his preliminary abstract, some geometrical language in describing his results (art. 9) … up to this point Cauchy had carefully avoided it; as with Lagrange, he had mistrusted its use, feeling that it would tend to disguise the general validity of analytic methods. From this point onward, he began to use it, occasionally to start with, and in later papers more and more freely. He seems to feel, indeed, that some of the ideas appearing in the present context can be expressed more concisely by admitting some geometrical terminology; it enables him, for instance, to say that a singularity lies ‘between’ two paths.

“[In] the remainder of the memoir … he gives numerous illustrative examples of the evaluation of particular definite integrals and the derivation of identities between them. He also examines some cases where the function f(z) has an infinite number of singularities, using them for the summation of certain infinite series.

“We see that in this memoir Cauchy has achieved primitive versions of all the basic results of complex function theory” (Landmark Writings, pp. 388-9).

The present memoir was presented to the Académie on 28 February 1825, in a manuscript which is still preserved in the Académie’s pochette for that meeting. It came out in August in a print run of 500 copies. It may be regarded as the herald of his journal Exercices de mathématiques, which began to appear from De Bure in 1826. Over the decades its form of publication made it inaccessible, so that it received a reprint in the Bulletin des Sciences Mathématiques in 1874 (a version which some historians – e.g. Kline – mistakenly take to be its first publication). (See Grattan-Guinness, Convolutions in French Mathematics 1800-1840, Vol. II. p. 766.)

DSB III: 137-9; Grattan-Guinness, Landmark Writings in Western Mathematics 1640-1940, 2005 (see Ch. 28). Smithies, Cauchy and the Creation of Complex Function Theory, 1997 (see Ch. 4).



4to (247 x 197 mm), pp. [iv] 68 (ink stamp of Berlin State Library to title verso, cancelled indicating the book was sold as a duplicate in 1936). Contemporary half-cloth over marbled boards, manuscript paper label to spine (blind stamp of Berlin State Library to front board).

Item #3163

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