Chapter 3 Cauchy’s “Modern Analysis”

This paper is incorrectly said to have been rare since the time of its publication. Kline (1972, 636–637) even claimed that it “was not published until 1874”. Instead, there is evidence that Cauchy had it printed twice in a few months. The second edition appeared in August 1825 and differed from the first only in the number of Cauchy’s academic titles listed on the cover page, and in the improved version of the errata corrige. This second version was reprinted in 1874–1875, before being included in his Oeuvres (2) 15, 41–89.

At the time, Brisson was at the centre of an affair that involved Cauchy and Poisson on opposite sides; see Belhoste (1991, 51–52). In November 1823 Brisson had submitted a memoir on the theory of linear partial differential equations to the Académie, but the commission appointed by the Académie to evaluate it, which included Cauchy, Fourier, Laplace, and Poisson, had not been able to reach an agreement. Eventually a new commission, which excluded Poisson, was appointed and Cauchy’s report was approved by the Académie in June 1825. According to Belhoste, this affair led to a deterioration in the relationship between Cauchy and Poisson.

The Russian translation of Buniakowskii’s Dissertation was published in 1985 (see Istor.–Math. Issled. 28, 247–2609 with a commentary by Kirsanov (1985). See also Ermolaeva (1985).

As Stäckel (1900, 73) pointed out, after his return to Russia Ostrogradskii dealt with the theory of integrals with imaginary limits in a note read at St. Petersburg Academy on October 29, 1828 and published 3 years later (Ostrogradskii 1831).

As we will see, the phrases “finite and continuous” or “continuous and finite” were to be used repeatedly by Cauchy to refer to a class of appropriate complex functions that he was unable to characterise more precisely until the late 1840s.

Cauchy actually wrote ± 2π i f. Not having introduced the idea of a path in the complex domain and parameterisations of it, he had carefully to make precise how the sign had to be taken.

Thus, for instance, in the homology form it is stated as follows: Let Ω be a simply connected open set in \(\mathbb{C}\). If f is a holomorphic function in Ω, then ∫ _γ f d z = 0. for every closed curve γ which is homologous to zero in Ω.

Therefore, we cannot agree with Kline’s (1972, 637) claim that “here x + i y is definitely a point of the complex plane and the integral is over a complex path”.

It was to be published as Cauchy (1826a).

In his letters from Paris Abel wrote that during his stay there in 1826 he bought them and read them “assiduously”. By that time Abel had completed “a long paper on a certain class of transcendental functions”, which is the paper containing Abel’s theorem (see Sect.​ 4.​4). “I showed it to Cauchy, but he scarcely wished to glance at it. And I dare to say without bragging that it is good. I am anxious to know the judgement of the Institut [de France]” Abel wrote confidently in his letters (see Abel 1902, 46)—but he would wait for it in vain.

Apparently, this is the first occurrence in print of the concept of conditionally convergent series.

Dirichlet (1829) gave a counter-example that showed that this “criterion” is false.

The very same year, on September 17, he presented to the Académie a second memoir on the same subject, see Cauchy (1827d).

Abel to Holmboe, in Ore (1957, 147).

They had the explicit titles “Sur la détermination du reste de la série de Lagrange par une intégrale définie” and “Règles de convergence de la série de Lagrange et d’autres séries du même genre”.

One could perhaps recall Buniakowski’s 1825 Dissertation on the application of the calculus of residues to various topics of celestial mechanics, a subject which had likely been suggested by Cauchy.

Lagrange (1797, § 99, 104–105; 1813, § 33, 69–70).

Lagrange series continued to attract the attention of mathematicians for decades, as is shown, for instance, by Heine’s one-page derivation of it (Heine 1857).

Later on, this result was reobtained by Puiseux following Cauchy’s methods of complex analysis, in a note added to the 3rd edition of Lagrange’s Mécanique, see Oeuvres 12, 341–346. Unknown to Cauchy, and possibly to Laplace, in 1817 the Italian astronomer Francesco Carlini had announced a thorough investigation of Kepler’s equation, which included Laplace’s result for the convergence of Lagrange series and a remarkable study of Bessel coefficients for large numbers. The resulting publication (Carlini 1818), however, was flawed by some mistakes that were corrected by Jacobi who published a German translation of it in 1850.

This paper was later reprinted as a separate pamphlet in Rome in 1843.

A well-off nobleman Piola did mathematics as an amateur without looking for any academic position. He collaborated with the Milan Observatory and eventually became the President of the Istituto Lombardo di Scienze e Lettere.

Ten years later, he reprinted it once more in his reborn Exercises, see Cauchy (1841d).

This was later published by Cauchy (with variations that we will point out) in the second volume of his new Exercises. See Cauchy (1841e).

Cauchy tacitly referred here to (2.​35) the formula for the unit disk he found in (1822b), and re-obtained in (1826a).

This explains the name given by Cauchy to his calculus, which is known today as “method of majorants”. (We would denote \(\Lambda f(\tilde{x}) =\sup \vert f(\tilde{x})\vert,\) and \(\vert \tilde{x}\vert = X\)).

The following account of this part of Cauchy’s 1831 Mémoire is based on the printed version (Cauchy 1841e) of it.

However, Cauchy avoided any geometric language.

In the real case one has to assume that the function is of class C ^∞ and, moreover, the following condition is satisfied: \(\frac{\vert {f}^{(n)}(x)\vert } {n!} {k}^{n} <K\), where K and k are positive real numbers.

An Italian translation of this paper was later published at Modena in Memorie della Società Italiana di Scienze 22 (1839), 91–183 followed (on pp. 228–246) by the translation of the paper Calcul des indices des fonctions that had first appeared as a lithograph on June 15, 1833 and later been expanded by Cauchy in a paper published in 1837 in the Journal of the École polytechnique, see Cauchy (1837e).

Cauchy later abandoned this “calculus” in favour of other methods. This perhaps explains why, contrary to his usual habit, Cauchy never re-published this paper.

This mean value equality had also been found by Poisson (1823b, 498) who, however, did not recognise its full scope. As Stäckel remarked, it was “covered by limitless sand dune-like magic formulae” (quoted in Remmert 1991, 205).

As we will see in Sect.​ 7.​10.​4.​1, apparently unaware of Cauchy’s work, Sonya Kovalevskaya rediscovered and generalised this theorem in her thesis (Kovalevskaya 1875).

Cauchy was active for numerous Catholic causes at a time when the Jesuits were believed by many to be interfering with academic freedom, which is the most likely reason why he was passed over in elections at the Collège de France in favour of Libri—a scandal in itself. See Belhoste (1991, 182–189).

See O.C. (1) 4, 369–426. As for the Prague (1835) memoir, it was re-published as such in his new Exercises on December 1840. See O.C. (2) 11, 399–465.

See O.C. (2) 11, 75–133. He added a detailed study of the case in which the characteristic equation is homogeneous, see O.C. (2) 11, 227–264.

According to Kline (1972, 639), Cauchy changed his mind after a correspondence he exchanged with Liouville and Sturm, but we have been unable to find any evidence for this.

The integral representation theorem and its “elementary” proof were included by Moigno in his Leçons inspired by Cauchy’s work, see Moigno (1840–1844, 1, 152–157).

Not good enough as a mathematician to be a “new Abel”, as Holmböe too precipitously called him, Broch made a career as a mathematics professor at the University of Christiania.

The paper was published in Crelle’s Journal as Broch (1842).

Interestingly enough, Cauchy omitted to mention the paper by his friend Piola (1834). Apparently Cauchy was also unaware of Gregory (1837) which seems to have provided the first exposition of the residue theory in England. The first account of the calculus of residues in America was to be given by Benjamin Peirce in the second volume of his (1846). Curiously, Peirce avoided mentioning Cauchy by name, but used Cauchy’s peculiar \(\mathcal{E}\) notation. The first mention of Cauchy by name was given in Kummell (1879). For a detailed historical survey of the development of the residue theory, see Burkhardt’s Excurs betr. die Entwicklungsgeschichte von Cauchy’s Residuentheorie in Burkhardt (1914–1915, 1001–1032).

Gabriel Oltramare was born in Geneva and studied in Paris. After completing his studies he returned to Geneva where he spent all his academic career teaching at the University there. At the start of his review of Oltramare’s (1899) in the Bulletin of the AMS E. O. Lovett hailed him “the venerable dean of the faculty of sciences of Geneva, who is probably the oldest living pupil of Cauchy” (Lovett 1899, 109).

Recall our earlier remark in the Introduction that Cauchy often used “continuous” to mean something more like “complex analytic”.

For Cauchy’s 1842 contributions to the theory of differential equations, see Cooke (1984, 24–27).

A former pupil of the École Polytechnique, which he entered in 1830, Pierre A. Laurent graduated two years later as one of the best students of his class. After a two years’ training at the École d’Application at Metz he was sent to Algeria as a lieutenant in the army. He returned to France around 1840 and spent six years working as a military engineer on the enlargement of the port of Le Havre. In this period he became interested in mathematics and submitted a paper to the Académie that Cauchy and Liouville were charged to review. They gave a positive report and recommended the paper for publication, but the recommendation was not followed by the Académie. Later on, the same fate befell a paper on the calculus of variations that Laurent had submitted for the Grand Prize in mathematics for 1842. The paper had arrived late and had therefore not been considered for the Prize. Once more Laurent’s paper was positively reviewed by Cauchy and Liouville and unsuccessfully recommended for publication. Disappointed, Laurent turned to applied mathematics and published papers on the theory of light and other subjects. He kept working as a military engineer and doing research in applied mathematics until his untimely death in 1854. See his obituary in Bertrand (1890a).

Laurent’s original memoir was never published. Its main content was included in the (posthumous) article (Laurent 1863).

It is worth remarking that more than 20 years after the publication of this volume, Cauchy still referred to the definition of continuity given there as “new”.

See Cauchy (1844a, 147). Cauchy had implicitly referred to the continuity of an imaginary function of an imaginary variable in the same terms as stated here as early as 1823. See, e.g., Cauchy (1823b, 330).

Later a professor at Genf, Cellérier devoted himself essentially to mechanics. He later gave examples of a continuous, nowhere differentiable function and of an infinitely differentiable function that cannot be expanded in Taylor series, which were published posthumously in Cellérier (1890). See JFM 22.0386.01 for Hurwitz’s review.

The proof was provided by Catalan (1843) in a short note which followed Chebyshev’s paper.

Among his subsequent papers (Chebyshev 1857) is worth mentioning. There he showed that a skilful interpretation of the integration by parts provided an easy method for obtaining Lagrange series. He applied this to the solution of Kepler equation and proved that the expansions of the eccentric anomaly and the vector radius in power series in the eccentricity η are convergent provided that η < 0. 66274, thus re-obtaining Cauchy’s estimate. In addition, he proved that an upper bound for the remainder of the series is given by (η ∕ 0. 66274) ⁿ .

Apparently Cauchy set great store by this result for he re-published this section two years later, on October 19, 1846 as a separate paper in the Exercises as Cauchy (1846n).

Liouville to Fleury, January 16, 1866, quoted in Neuenschwander (1984), Engl. trl. in Lützen (1990, 85).

There is no record of it in the Comptes rendus of the Académie. As Lützen (1990, 125) has conjectured, “it may just have been a private communication”, perhaps a chat they had after the meeting in which Liouville communicated his “general principle”.

For Thompson’s stay in Paris, and his contacts with Liouville and Cauchy, see Lützen (1990, 135–146).

In the concluding pages Cauchy referred once more to his criticism of Lamarle, and in a Post scriptum added to this paper Cauchy was pleased to remark that Björling, in a memoir on infinite series submitted to the Academy of Uppsala in 1846 had seemingly shared his view about the conventions needed when dealing with x ^a and logx. According to Gårding (1998, 14) “one of the recurrent themes and the object of a correspondence [of Björling] with Cauchy was the definitions of the functions x ^y and log _β x when x and y are complex numbers and β is positive”. In 1846 Björling submitted to the Academy of Uppsala another memoir dealing with the convergence of series which, perhaps at Cauchy’s suggestion, was translated into French and eventually appeared in Liouville’s Journal (Björling 1852).

A professor at the Military Academy in Turin, Chiò had previously had a lively argument with the Italian followers of Lagrange, and Menabrea in particular, because of his criticism of Lagrange’s work.

Later on, in 1853 Cauchy claimed once more to have found in Saint Venant’s (1845) paper the source of inspiration to his theory of “algebraic keys” which essentially reduced to a method for solving linear systems. This raised an argument with Grassmann, who wrote to the French Académie to claim priority; the commission charged by the Académie to settle the matter (of which Cauchy was a member) never came to a conclusion. See Grassmann (1894–1911, 3.2, 174–203).

Victor Puiseux had graduated from the École Normale Supérieure in Paris in 1841 with a thesis in astronomy and celestial mechanics. In 1849 he took up a teaching position there as maître de conferences. Puiseux attended Cauchy’s courses at the Faculté des sciences, and according to Belhoste (1991, 233) “these two men also quickly became good friends”. In 1857 he was to succeed Cauchy as professor at the Faculté des sciences and subsequently he held a position at the Bureau de Longitudes. He was elected to the Académie des sciences in Paris in 1871; Bertrand (1890b) remarked “The election was due to his merit, but its unanimity, to his character”.

According to Markushevich (1996, 192), Puiseux here established “the identity (under the hypothesis of his study) of continuous extension [of a branch of an algebraic equation] with analytic continuation”.

In modern terms, they correspond to the orbits of the subgroups of the Galois group of f(u, z) = 0, of which he calculated the order.

Perhaps because of this passage Manning (1974–1975, 371) claimed mistakenly that Puiseux possessed the concept of an essential singularity.

At the same meeting of the Académie Puiseux presented a new memoir (1851) Recherches sur les fonctions algébriques (see C. R. Acad. Sci. Paris 32 (1851), 93) which Cauchy, Sturm, and Binet were charged to review, they being the members of the commission already charged to review a memoir on the same subject previously submitted in 1850 by Puiseux.

Apparently he did so in a series of notes, but for most of them only a short summary is recorded in the Comptes rendus of the Académie (see O.C. 1 (11) 350–354; 384–385).

In fact, Cauchy slightly modified to fit it to Puiseux’s context, even adopting the latter’s notation.

We discuss this memoir below, see Sect.​ 4.​2.​4.

Liouville’s theory of elliptic functions and Cauchy’s claims are discussed in detail in Sect.​ 4.​2.​4.

Charles Auguste Briot graduated from the École Normale Supérieure at the head of his class in 1838 and held a succession of teaching positions before becoming a professor at the Sorbonne and the École Normale Supérieure in 1864. There he concentrated mostly on thermodynamics and rational mechanics, but he also teamed up with Jean-Claude Bouquet, an old friend from his school days, who by 1874 was also a professor at the Sorbonne and they wrote a series of successful textbooks on several elementary topics. They were highly regarded in their day as teachers: Briot was awarded the Prix Poncelet of the Académie des sciences in 1882 and Bouquet was particularly praised by Jules Tannery and by Hermite, with whom he shared political and religious views.

Cauchy’s theorem has raised many controversial interpretations among the historians of mathematics. We do not need to enter such a discussion here and limit ourself to remarking that there is hardly any doubt that by 1853 Cauchy had changed his mind about his theorem since 1821. Why else would he have admitted the need of “restrictions” to make the theorem correct?

They were later to be published as Laurent (1863). This paper is interesting in many ways. Not only does it provide a detailed account of Laurent’s achievements in complex function theory, it also shows how Laurent applied complex functions to the study of the equilibrium distribution of heat.

We note here that not only can we not confirm the story that the Comptes rendus adopted a policy of requiring articles to be no longer than four pages in order to keep itself from being swamped by Cauchy’s contributions, but that this article of eleven pages would seem to refute it. The story likely has its origins in Bell (1937, 287). The procès-verbaux of the French Académie on July 13, 1835 do record, however, that a limit of six to eight pages was to be imposed, which evidently Cauchy flouted. This may have been one of the reasons why Biot publicly criticised Cauchy in 1842 in the Journal des savants for flooding the Comptes rendus with badly written, verbose papers; see Biot (1842) and Belhoste (1991, 192). The situation today is than members of the Académie are limited to six pages, and non-members to four, but we cannot determine when this rule was introduced, although it does seem that it is being obeyed.

For a detailed account, see Mitrinović and Kečkić (1993, 64–67).

We follow here the detailed account given in Mitrinović and Kečkić (1993, 93–109).

We quote from Markushevich (1996, 233), whose account of Sokhotskii’s thesis we follow here.