Paris: Chez de Bure Frères, 1825.
First edition of this very rare memoir in which “Cauchy began to lay the foundations of the general theory of analytic functions of a complex variable and their integration by residues.” (Landmark Writtings in Western Mathematics, ch. 28). “This paper is considered by many as his most important, and one of the most beautiful in the history of science” (Kline, Mathematical Thought from Ancient to Modern Times, p. 637);
Cauchy begins the present work 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.
“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... 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 two 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the device of splitting the function in this way appears to come from Ostrogradsky, who had used a similar device in an unpublished paper dated 24 July 1824 and was working closely with Cauchy at the time.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... 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 he had mistrusted its use, feeling that it would tend to disguise the general validity of analytic methods... 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” (Landmark Writings, p. 388).
The present memoir must be distinguished from Cauchy’s Mémoire sur les intégrales définies, submitted to the Académie des Sciences on 22 August 1814 but not published until 1827. 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.
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 – falsely 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, Ch. 28 (by F. Smithies); for a very detailed analysis of the work, see Smithies, Cauchy and the Creation of Complex Function Theory, Ch. 4.
4to (247 x 197 mm), contemporary half cloth over marbled boards, manuscript paper label to spine, blind stamp of the Berlin State Library to the front board, rubber stamp of the same to verso of the title, stamped out and sold as a duplicate in 1936, pp. [iv] 68.