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Chapter 2 From Real to Complex Analysis

verfasst von : Umberto Bottazzini, Jeremy Gray

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Abstract

On August 22, 1814 Augustin–Louis Cauchy, a young protegé of Laplace, submitted a long Mémoire to the Institut de France on the calculus of definite integrals that was to mark a turning point in the history of complex analysis, and a first step towards a theory of complex integration. It was in Cauchy’s hands that calculus with complex quantities began to lose the aura of mystery that had accompanied complex numbers since they had first appeared in the work of Italian algebraists Cardano and Bombelli in the Renaissance.

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Vorheriges Kapitel Chapter 1 Elliptic Functions

Nächstes Kapitel Chapter 3 Cauchy’s “Modern Analysis”

This is the case when the three roots of the cubic equation (with real coefficients) are real, but Cardano’s formula gives seemingly complex answers.

“So the elegant and remarkable outcome appears in this miracle of analysis, this marvel of the world of ideas, an almost amphibian object between Being and Non-being that we call imaginary root” (Itaque elegans et mirabile effugium reperit in illo Analyseos miraculo, idealis mundi monstro, pene inter Ens et non-Ens amphibio, quod radicem imaginariam appellamus) (Leibniz 1702, 357).

For d’Alembert’s life and work, see Paty (1998) and Michel and Paty (2002), which includes a complete bibliography of his works by A.-M. Chouillet.

Here we follow the English translation, pp. 42–43.

See Taton and Yushkevich (1980). In this book the variant transliterations Youshkevich, Youshkevitch, and Yushkevich have been standardised to Yushkevich.

Quoted in (Speziali 1983, 428).

In 2005 Ivor Grattan-Guinness challenged the community of historians of mathematics to find it in Euler’s works but the search was unsuccessful.

A child prodigy, Clairaut was elected to the Académie when he was 18, becoming the youngest person ever elected to that Académie. After the success of the expedition to Lapland led by Maupertuis to measure a degree of longitude, in which Clairaut took part, his fame increased quickly and he became the most authoritative French mathematician in the first half of eighteenth century. For a biography of Clairaut, see Brunet (1952) and for Maupertuis, see Terrall (2002).

Clairaut (1739), 427–428. In a subsequent Mémoire Clairaut (1740) used both the terms “complete” and “exact” differential.

Fontaine’s memoir was eventually published in (Fontaine 1764, 24–83). For a detailed analysis of Fontaine’s work, see Greenberg (1995).

See also the rapport written by d’Alembert and De Gua (1742) after the publication of Clairaut’s paper.

For a detailed analysis of Nicolaus I Bernoulli’s and Euler’s relevant papers, see Engelsman (1984).

In a letter to Euler in 1743 Nicolaus I Bernoulli raised doubts about this.

Truesdell (1954, XXI) and Greenberg (1995, 458–459) have pointed out that under certain conditions, such as domains that are not simply connected, Clairaut’s statement is not true.

Nouvelle bibliothèque germanique 4 (1748), 241.

In spite of their importance in the history of fluid dynamics (see Truesdell (1953) and Truesdell (1954)), in our view these papers by d’Alembert, Euler, and Lagrange made no substantial contribution to the development of complex function theory, contrary to a widely-held opinion. See Euler (1983b) and also the Editor’s Introduction to Euler’s Opera Omnia I, 19.

As the editors of d’Alembert (2008) remark, this means that it flows along a closed curve. Quoted in (Truesdell 1954, CXIV).

See d’Alembert (1747a,b). For d’Alembert’s contributions to the theory of partial differential equations, see Demidov (1982).

When this paper appeared in print Lagrange, a protegé of d’Alembert, was about to leave his native Turin to succeed Euler as the Director of the Mathematical Class of the Berlin Académie. He stayed there until 1788 when he moved to Paris, where he spent the rest of his life.

See also their German translation published by Wangerin as (Euler 1898).

The use of the notation ± is ours.

See Euler O.O. (1) 19 and Euler (1983b).

In the notation that Legendre introduced, Δ(n) = Γ(n).

For an account of Laplace’s life and work, see Fox et al. (1978).

For biographies of Cauchy, see Valson (1868) and, better, Belhoste (1991).

On the importance of illegal books and pamphlets in Paris in these years, see Darnton (1995).

Lectures there by Laplace, Lagrange, and Monge have been published in (Dhombres 1992a).

Laplace’s political ability, or, perhaps better, opportunism, enabled him to be named a Marquis in 1817 after Napoleon’s fall and the restoration of the Bourbons.

See Fourcy (1828). For a historical evaluation of the role of the École Polytechnique, see Belhoste et al. (1994), Belhoste et al. (1995), and Belhoste (2003).

In this book Legendre studied integrals related to his Γ-function by means of imaginary substitutions.

The report is reprinted in (Cauchy 1814, 321–327).

Needless to say, Cauchy avoided any reference to a geometrical setting.

The concept of (dis)continuity Cauchy that was implicitly referring to foreshadows the one he was to define in his Cours d’analyse (1821a).

As we will see in Sect. 2.4.1 Cauchy changed his mind in 1823 after introducing the concept of principal value of an integral and his theory of singular integrals.

Cauchy first referred to Argand’s geometrical interpretation and his proof of the fundamental theorem of algebra only in 1849 in his (1849b, 175). In particular, on p. 200 he referred to Argand’s paper in Gergonne’s Annales, adding in a footnote that he presently had under his eyes a copy of the original, anonymous 1806 pamphlet with Argand’s name handwritten on the cover page.

Procès-verbaux des séances de l’Académie 6 (1915), 310.

Quoted in (Del Centina 2005, 63).

For a detailed study of Cauchy’s book, see Bottazzini (1992a).

For a discussion of Cauchy’s concept of continuity and the theorem just mentioned, see Bottazzini (1992a), lxxxi–xcv.

There is a misprint in (1821a, 147). The correct reference is to Theorem 4, Cor. 4.

The proof of the binomial theorem for any complex value μ was to be given by Abel (1826b). It was in this paper that Abel first pointed out that Cauchy’s statement of his theorem about the limit of a series of continuous functions “admits exceptions”.

These are obtained by taking real and imaginary parts of the binomial expansion of \({(2\cos x)}^{m} ={ \left ({e}^{ix} + {e}^{-ix}\right )}^{m}\).

See Burkhardt (1914–1915) and Jahnke (1987).

For a detailed historical account, see Burkhardt (1914–1915, 837–856).

Discussed in (Dhombres 1992b).

For a detailed discussion of Cauchy’s proof of the binomial theorem, see Dhombres (1986), 157–163.

Cauchy’s confusion between minimum and lower bound was shared by everyone at that time, with the possible exception of Bolzano.

Procès-verbaux des séances de l’Académie 7 (1916), 231.

For an account of this paper and related matters, see Burkhardt (1908), esp. pp. 671–680. See also Grattan–Guinness (1990), esp. pp. 690–694.

Procès-verbaux des séances de l’Académie 7 (1916), 271.

See his (1822, 506) and the footnote there by Darboux.

It is his “Sur la résolution analytique des équations de tous les degrés par le moyen des intégrales définies”.

The same statement had already been made by Cauchy in his (1815b, 62), where he referred explicitly to a couple of works by Ruffini (1805) and (1799), which he mistakenly quoted as Théorie des équations numériques.

Further details can be found in a paper he published in 1822 (see below) as well as in the “general remarks and additions” appended to (Cauchy 1823b).

Cauchy supposed x ₀ to be a simple infinity i.e., as Cauchy used to say, a simple root of 1 ∕ f(x), as he had done in his 1814 Mémoire.

In spite of the care Cauchy showed when defining infinitely small quantities as “variables having zero as their limit” in the Cours d’analyse, here and elsewhere in his research papers he kept using the old language of infinitesimals.

Actually, this formula is contained in a footnote Cauchy added in 1825 to his 1814 Mémoire, see Cauchy (1814, 422–423).

A copy of this rather rare pamphlet, placed in a volume with other papers by Cauchy, is kept in the “Fond Mandelbrojt” of the mathematical centre at Luminy.

For a detailed account of this matter, see Bottazzini (1992a), lxvii–lxxii.

Grattan–Guinness (1990, 735–736) points out that Poisson tried to refute Cauchy’s argument in a paper that appeared in the June issue of the Bulletin.

By resorting tacitly to what nowadays is called the uniform continuity of f(x).

This seemingly chronological incongruence can be explained by thinking of the peculiar structure of the Résumé. Each Lesson in it amounts to exactly four printed pages, i.e. a printed double-face folio, and it seems that each Lesson was printed and distributed separately to the students during the academic year 1823 before being collected into a book.

See the modern account of it given by Remmert in (Ebbinghaus et al. 1990, 120–122).

As was already observed by Servois who commented acutely “it seems to me that it is not enough to find values of x that give continually decreasing values to the polynomial, it is also needed that the law of the decrements necessarily leads the polynomial to zero” (quoted in Lebesgue 1937, 152).

After presenting the proof of Argand’s inequality in modern terms, Remmert remarked that it is a special case of the “open mapping theorem” according to which (nonconstant) holomorphic functions map open sets onto open sets (in Ebbinghaus et al. 1990, 120–122).

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Titel: Chapter 2 From Real to Complex Analysis
verfasst von: Umberto Bottazzini
Jeremy Gray
Verlag: Springer New York
Buch: Hidden Harmony—Geometric Fantasies
Print ISBN: 978-1-4614-5724-4

Electronic ISBN: 978-1-4614-5725-1

Copyright-Jahr: 2013
DOI: https://doi.org/10.1007/978-1-4614-5725-1_3

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