Search Results for Cauchy

Biographies

  1. Cauchy biography
    • Augustin Louis Cauchy .
    • Paris was a difficult place to live in when Augustin-Louis Cauchy was a young child due to the political events surrounding the French Revolution.
    • There things were hard and he wrote in a letter [Augustin-Louis Cauchy.
    • They soon returned to Paris and Cauchy's father was active in the education of young Augustin-Louis.
    • Laplace and Lagrange were visitors at the Cauchy family home and Lagrange in particular seems to have taken an interest in young Cauchy's mathematical education.
    • Lagrange advised Cauchy's father that his son should obtain a good grounding in languages before starting a serious study of mathematics.
    • From 1804 Cauchy attended classes in mathematics and he took the entrance examination for the Ecole Polytechnique in 1805.
    • In 1810 Cauchy took up his first job in Cherbourg to work on port facilities for Napoleon's English invasion fleet.
    • It was a busy time for Cauchy, writing home about his daily duties he said [Augustin-Louis Cauchy.
    • Cauchy was a devout Catholic and his attitude to his religion was already causing problems for him.
    • In addition to his heavy workload Cauchy undertook mathematical researches and he proved in 1811 that the angles of a convex polyhedron are determined by its faces.
    • Cauchy felt that he had to return to Paris if he was to make an impression with mathematical research.
    • Back in Paris Cauchy investigated symmetric functions and submitted a memoir on this topic in November 1812.
    • An academic career was what Cauchy wanted and he applied for a post in the Bureau des Longitudes.
    • Cauchy obtained further sick leave, having unpaid leave for nine months, then political events prevented work on the Ourcq Canal so Cauchy was able to devote himself entirely to research for a couple of years.
    • In this last election Cauchy did not receive a single one of the 53 votes cast.
    • In 1815 Cauchy lost out to Binet for a mechanics chair at the Ecole Polytechnique, but then was appointed assistant professor of analysis there.
    • Politics now helped Cauchy into the Academy of Sciences when Carnot and Monge fell from political favour and were dismissed and Cauchy filled one of the two places.
    • In 1817 when Biot left Paris for an expedition to the Shetland Islands in Scotland Cauchy filled his post at the College de France.
    • Cauchy was the first to make a rigorous study of the conditions for convergence of infinite series in addition to his rigorous definition of an integral.
    • Cauchy did not have particularly good relations with other scientists.
    • An example of how Cauchy treated colleagues is given by Poncelet whose work on projective geometry had, in 1820, been criticised by Cauchy:- .
    • Cauchy is mad and there is nothing that can be done about him, although, right now, he is the only one who knows how mathematics should be done.
    • Belhoste in [Augustin-Louis Cauchy.
    • When Abel's untimely death occurred on April 6, 1829, Cauchy still had not given a report on the 1826 paper, in spite of several protests from Legendre.
    • By 1830 the political events in Paris and the years of hard work had taken their toll and Cauchy decided to take a break.
    • Political events in France meant that Cauchy was now required to swear an oath of allegiance to the new regime and when he failed to return to Paris to do so he lost all his positions there.
    • In 1831 Cauchy went to Turin and after some time there he accepted an offer from the King of Piedmont of a chair of theoretical physics.
    • Menabrea attended these courses in Turin and wrote that the courses [Augustin-Louis Cauchy.
    • In 1833 Cauchy went from Turin to Prague in order to follow Charles X and to tutor his grandson.
    • When questioned by Cauchy on a problem in descriptive geometry, the prince was confused and hesitant.
    • Cauchy became annoyed and screamed and yelled.
    • While in Prague Cauchy had one meeting with Bolzano, at Bolzano's request, in 1834.
    • In [Historia Mathematica 5 (1978), 313-331.',16)">16] and [Archive for History of Exact Sciences 7 (1971), 375-392.',18)">18] there are discussions on how much Cauchy's definition of continuity is due to Bolzano, Freudenthal's view in [Archive for History of Exact Sciences 7 (1971), 375-392.',18)">18] that Cauchy's definition was formed before Bolzano's seems the more convincing.
    • Cauchy returned to Paris in 1838 and regained his position at the Academy but not his teaching positions because he had refused to take an oath of allegiance.
    • Cauchy was strongly supported by Biot and Arago but Poisson strongly opposed him.
    • Cauchy was elected but, after refusing to swear the oath, was not appointed and could not attend meetings or receive a salary.
    • In 1843 Lacroix died and Cauchy became a candidate for his mathematics chair at the College de France.
    • Cauchy should have easily been appointed on his mathematical abilities but his political and religious activities, such as support for the Jesuits, became crucial factors.
    • During this period Cauchy's mathematical output was less than in the period before his self-imposed exile.
    • When Louis Philippe was overthrown in 1848 Cauchy regained his university positions.
    • Liouville and Cauchy were candidates for the chair again in 1850 as they had been in 1843.
    • Subsequent attempts to reverse this decision led to very bad relations between Liouville and Cauchy.
    • Another, rather silly, dispute this time with Duhamel clouded the last few years of Cauchy's life.
    • Duhamel argued with Cauchy's claim to have been the first to give the results in 1832.
    • Poncelet referred to his own work of 1826 on the subject and Cauchy was shown to be wrong.
    • However Cauchy was never one to admit he was wrong.
    • Valson writes in [La vie et les travaux du baron Cauchy (Paris, 1868).',7)">7]:- .
    • Also in [La vie et les travaux du baron Cauchy (Paris, 1868).',7)">7] a letter by Cauchy's daughter describing his death is given:- .
    • Numerous terms in mathematics bear Cauchy's name:- the Cauchy integral theorem, in the theory of complex functions, the Cauchy-Kovalevskaya existence theorem for the solution of partial differential equations, the Cauchy-Riemann equations and Cauchy sequences.
    • This achievement is summed up in [Augustin-Louis Cauchy.
    • in spite of its vastness and rich multifaceted character, Cauchy's scientific works possess a definite unifying theme, a secret wholeness.
    • Cauchy's creative genius found broad expression not only in his work on the foundations of real and complex analysis, areas to which his name is inextricably linked, but also in many other fields.
    • His collected works, Oeuvres completes d'Augustin Cauchy (1882-1970), were published in 27 volumes.
    • Cauchy's theorem in Complex analysis .
    • Honours awarded to Augustin-Louis Cauchy .
    • Lunar featuresCrater Cauchy and Rupes Cauchy .
    • Paris street namesRue Cauchy (15th Arrondissement) .
    • http://www-history.mcs.st-andrews.ac.uk/Biographies/Cauchy.html .

  2. Laurent Pierre biography
    • Cauchy reported on Laurent's entry Memoire sur le calcul des variations, which contains the Laurent series for a complex function, on 20 May 1843.
    • Being late, the memoir was never seriously considered for the Grand Prix, which was won by Pierre Frederic Sarrus with Delaunay's entry receiving an honourable mention, but Cauchy proposed that Laurent's memoir should be approved and published in the Recueil des savants etrangers.
    • The Academy of Sciences published the entries of Sarrus and Delaunay but they ignored Cauchy's recommendation concerning Laurent and his memoir was not published.
    • A second paper by Laurent submitted to the Academy of Sciences around the same time was also considered by Cauchy.
    • This paper presented an extension of one of Cauchy's theorems and again Cauchy proposed that Laurent's memoir should be approved and published in the Recueil des savants etrangers.
    • Again the Academy of Sciences decided not to publish the work and it has been lost and is now only known through Cauchy's report.
    • He published a number of papers on the topic and Cauchy proposed him for a vacant position in the Academy of Sciences in 1846.
    • One was considered by Cauchy who proposed that Laurent's memoir should be approved and published in the Recueil des savants etrangers but again it was never published.

  3. Bunyakovsky biography
    • Bunyakovskii was first educated at home and then went abroad, obtaining a doctorate from Paris in 1825 after working under Cauchy.
    • In this he played an important role in the development of mathematics in the Russian Empire for he brought back with him an expertise in applying Cauchy's theory of residues which were at that time unknown in the Russia Empire.
    • He is best known for his discovery of the Cauchy-Schwarz inequality, published in a monograph in 1859 on inequalities between integrals.
    • One would have to note, however, that the terminology of mathematics is not universal and in some countries his theorem is correctly named, or named after Cauchy, Bunyakovskii and Schwarz.
    • A history of the Cauchy-Bunyakovskii-Schwarz inequality is given in [Hermann Grassmann, Lieschow, 1994 (Greifswald, 1995), 64-70.',8)">8].

  4. Hermite biography
    • In 1849 Hermite submitted a memoir to the Academie des Sciences which applied Cauchy's residue techniques to doubly periodic functions.
    • Sturm and Cauchy gave a good report on this memoir in 1851 but a priority dispute with Liouville seems to have prevented its publication.
    • It was Cauchy who, with his strong religious conviction, helped Hermite through the crisis.
    • This had a profound effect on Hermite who, under Cauchy's influence, turned to the Roman Catholic religion.
    • Cauchy was also a very staunch royalist and Hermite was influenced by him to also become a royalist.

  5. Leray biography
    • He studied time dependent hyperbolic partial differential equations and also began to work on the Cauchy problem.
    • In particular he published a paper on the Cauchy problem for equations with variable coefficients in 1956.
    • We propose to study globally the linear Cauchy problem in the complex case, then in the real hyperbolic case, assuming that the given data is analytic.
    • Leray's work on the Cauchy problem led him to study residues theory.
    • developed a general residue theory on complex manifolds and applied it to the investigation of concrete integrals depending on parameters arising from solving the Cauchy problem.

  6. Rychlik biography
    • Other works by Rychlik on the history of mathematics include A Cauchy manuscript in the archives of the Czechoslovakian academy of sciences (Czech) (1957), Un manuscrit de Cauchy aux archives de l'academie tchecoslovaque des Sciences (1957), Cauchys Schrift "Memoire sur la dispersion de la lumiere" herausgegeben wahrend seines Aufenthaltes in Prag durch die Konigliche bohmische Gesellschaft der Wissenschaften (1958), and Berechnung der Grundzahl e der naturlichen Logarithmen (Czech) (1960).
    • The Cauchy manuscript which Rychlik discussed in the above mentioned papers was Memoire sur l'integration des equations differentielles which was dated 'Prague 1835' in Cauchy's own hand.
    • Because Cauchy left Prague in 1836, this manuscript was not printed, as he had intended, in the Proceedings of the Societe Royale des Sciences de Boheme.

  7. Coriolis biography
    • He had been recommended for this position by Cauchy.
    • In July 1830 there was a revolution and, following this Cauchy left Paris in September 1830.
    • Political events in France meant that Cauchy was now required to swear an oath of allegiance to the new regime and when he failed to return to Paris to do so he lost all his positions there.
    • Coriolis was offered Cauchy's position at the Ecole Polytechnique but by this time he was highly involved in his research and decided not to take on any further teaching duties.

  8. Abel biography
    • By this time Abel seems to have known something of Ruffini's work for he had studied Cauchy's work of 1815 while he was an undergraduate and in this paper there is a reference to Ruffini's work.
    • I showed it to Mr Cauchy, but he scarcely deigned to glance at it.
    • Two referees, Cauchy and Legendre, were appointed to referee the paper and Abel remained in Paris for a few months [Bull.
    • After Abel's death his Paris memoir was found by Cauchy in 1830 after much searching.

  9. Liouville biography
    • Although Liouville does not seem to have attended any of Cauchy's courses, it is clear that Cauchy must have had a strong influence on him.
    • Liouville certainly never let his political views hold him back as he advanced his mathematical career, unlike Cauchy who had refused to swear the oaths of allegiance to the King that Liouville and even Arago had been prepared to do.
    • His chair at the College de France was declared vacant in 1850 and Cauchy and Liouville competed for the post.

  10. Puiseux biography
    • In 1857 he was appointed professor of mathematical astronomy at the Faculty of Science, after teaching courses for Le Verrier, where he succeeded to Cauchy's position.
    • Puiseux had attended courses by Cauchy early in his career and he soon became interested in research in topics Cauchy was studying.
    • He further developed Cauchy's work on functions of a complex variable, being the first to distinguish poles, essential points and branch points.

  11. Ruffini biography
    • The one person who did acknowledge the importance and correctness was Cauchy.
    • This is all the more surprising since Cauchy was one of the worst of all mathematicians at giving credit to others.
    • In fact Cauchy had written a major work on permutation groups between 1813 and 1815 and in it he generalised some of Ruffini's results.
    • This influence through Cauchy is perhaps the only way in which Ruffini's work was to make an impact on the development of mathematics.

  12. Jordan biography
    • Cauchy in particular had been one to take this route and, like Cauchy, Jordan was able to work as an engineer and still devote considerable time to mathematical research.
    • In some respects this is a little strange since it is a rigorous analysis text built on top of the attempts to put the topic on a firm foundation begun by Cauchy and given considerable impetus by Weierstrass.
    • There had been a tradition of rigorous analysis at the Ecole Polytechnique begun, of course, by Cauchy himself.

  13. Schwarz biography
    • The fact that Schwarz should have come up with a special case of the general result now known as the Cauchy-Schwarz inequality (or the Cauchy-Bunyakovsky-Schwarz inequality) is not surprising for much of his work is characterised by looking at rather specific and narrow problems but solving them using methods of great generality which have since found widespread applications.
    • For example the Cauchy-Schwarz inequality appears in arithmetic, geometric and function-theoretic formulations in works of mathematicians such as Bunyakovsky, Cauchy, Grassmann, von Neumann and Weyl.

  14. Cunha biography
    • His definition of the differential of a function y = f (x) anticipates that of Cauchy, and, written in our present notation, amounts to this: .
    • In Principios Matematicos da Cunha also gave a definition of the convergence of a series which is equivalent to Cauchy's convergence criterion.
    • 26 (1) (1973), 3-22.',11)">11], claims that da Cunha should rank with Bolzano, Cauchy, Abel and others for his contributions to the principles of the calculus.

  15. Kuczma biography
    • Cauchy's equation and Jensen's inequality published in 1985.
    • Probably even the most devoted specialist would not have thought that about 300 pages can be written just about the Cauchy equation (and on some closely related equations and inequalities).
    • In the opinion and experience of this reviewer this is a very useful book and a primary reference not only for those working in functional equations, but mainly for those in other fields of mathematics and its applications who look for a result on the Cauchy equation and/or the Jensen inequality.

  16. Poinsot biography
    • In 1812 Poinsot arranged for Reynaud to substitute for him in lecturing at the Ecole Polytechnique and, from this time on, he did no further teaching there asking Cauchy to substitute for him after Reynaud.
    • In 1810 Cauchy proved that, with this definition of regular, the enumeration of regular polyhedra is complete.
    • A mistake was discovered in Poinsot's (and hence Cauchy's) definition in 1990 when an internal inconsistency became apparent.

  17. Sokhotsky biography
    • In this thesis Sokhotsky discussed the Cauchy integral and the theory of analytic functions, which he called "single-valued".
    • It investigated in detail Cauchy-type integrals which played an important role in boundary value problems in the theory of functions of a complex variable.
    • One of the first to approach problems of the theory of singular integral equations, Sokhotsky in this work considered important boundary properties of integrals of the type of Cauchy and, essentially, arrived at the so-called formulas of I Plemelj (1908).

  18. Smithies biography
    • In 1982 Smithies published the paper The background to Cauchy's definition of the integral then Cauchy's conception of rigour in analysis in 1986 and he work culminated in the wonderful book Cauchy and the creation of complex function theory in 1997.

  19. Stokes biography
    • .the results of Stokes are related to the elastic theory of light, and supplement and expand a number of questions, previously studied for the most part in the works of A Cauchy.
    • Stokes's methods for solving diffraction problems, differing considerably from the methods employed by Cauchy, form the basis of the further studies of the mathematical theory of the phenomenon of diffraction.
    • With Green, who in turn had influenced him, Stokes followed the work of the French, especially Lagrange, Laplace, Fourier, Poisson, and Cauchy.

  20. Sturm biography
    • The time was very fruitful for Sturm who attended lectures by Ampere, Gay-Lussac, Cauchy, and Lacroix.
    • The first to give a complete solution was Cauchy but his method was cumbersome and impractical.
    • seeks to determine the mutual influence between A-L Cauchy's and Ch-F Sturm's research from 1829 to around 1840 on the roots of algebraic equations.

  21. Calderon biography
    • In 1958 Calderon published one of his most important results on uniqueness in the Cauchy problem for partial differential equations.
    • In 1991 he was awarded the National Medal of Science and again he work on uniqueness in the Cauchy problem was cited.
    • for his ground-breaking work on singular integral operators leading to their application to important problems in partial differential equations, including his proof of uniqueness in the Cauchy problem, the Atiyah-Singer index theorem, and the propagation of singularities in nonlinear equations..

  22. Bolzano biography
    • The paper gives a proof of the intermediate value theorem with Bolzano's new approach and in the work he defined what is now called a Cauchy sequence.
    • The concept appears in Cauchy's work four years later but it is unlikely that Cauchy had read Bolzano's work.

  23. Ampere biography
    • Ampere and Cauchy shared the teaching of analysis and mechanics and there was a great contrast between the two with Cauchy's rigorous analysis teaching leading to great mathematical progress but found extremely difficult by students who greatly preferred Ampere's more conventional approach to analysis and mechanics.
    • This seems to have been a crucial step in his election to the Institut National des Sciences in November 1814 when he defeated Cauchy, receiving 28 of the 56 votes cast.

  24. Dirichlet biography
    • Earlier work by Poisson on the convergence of Fourier series was shown to be non-rigorous by Cauchy.
    • Cauchy's work itself was shown to be in error by Dirichlet who wrote of Cauchy's paper:- .

  25. Goursat biography
    • The Cauchy-Goursat theorem states the integral of a function round a simple closed contour is zero if the function is analytic inside the contour.
    • Cauchy had established the theorem with the added condition that the derivative of the function was continuous.
    • Goursat removed this extra condition in Demonstration du theorem de Cauchy (1884).

  26. Galois biography
    • Cauchy was appointed as referee of Galois' paper.
    • Galois sent Cauchy further work on the theory of equations, but then learned from Bulletin de Ferussac of a posthumous article by Abel which overlapped with a part of his work.
    • Galois then took Cauchy's advice and submitted a new article On the condition that an equation be soluble by radicals in February 1830.

  27. Du Bois-Reymond biography
    • The standard technique to solve partial differential equations used Fourier series but Cauchy, Abel and Dirichlet had all pointed out problems associated with the convergence of the Fourier series of an arbitrary function.
    • Let us list some of the 1870s papers: Notiz uber einen Cauchy'schen Satz, die Stetigkeit von Summen unendlicher Reihen betreffend (1871); Die Theorie der Fonrier'schen Integrale und Formeln (1871); Uber asymptotische Werthe, infinitare Approximationen und infinitare Auflosung von Gleichungen (1875); Zusatze zur Abhandlung: Untersuchungen uber die Convergenz und Divergenz der Fourier'schen Darstellungsformeln (1876); Notiz uber infinitare Gleichheiten (1876); Zwei Satze uber Grenzwerthe von Functionen zweier Veranderlichen (1877); Note uber die Integration totaler Differentialgleichungen (1877); Notiz uber Convergenz von Integralen mit nicht verschwindendem Argument (1878); and Uber Integration und Differentiation infinitarer Relationen (1879).

  28. Renyi biography
    • from the University of Szeged, with Frigyes Riesz as his thesis advisor, for a thesis on Cauchy-Fourier series.
    • Results from his doctoral thesis appeared in the paper On the summability of Cauchy-Fourier series (1950).

  29. Poncelet biography
    • In particular Augustin-Louis Cauchy, writing a report on Poncelet's work on 5 June 1820, claimed that the principle of continuity was "capable of leading to manifest errors".
    • His work on projective geometry was too controversial, particularly following the attacks made on it earlier by Cauchy, for him to enter the Academy on the strength of these contributions.

  30. Lakatos biography
    • This article, Cauchy and the Continuum : The Significance of Non-Standard Analysis for the History and Philosophy of Mathematics is one of the most enjoyable that I have read.
    • The point is not merely to rethink the reasoning of Cauchy, not merely to use the mathematical insight available from Robinson's non-standard analysis to re-evaluate our attitude towards the whole history of the calculus and the notion of the infinitesimal.

  31. Ladyzhenskaya biography
    • Find the least restrictive conditions on the behaviour of parabolic equations under which the uniqueness theorem holds for the Cauchy problem.
    • For hyperbolic equations, construct convergent difference schemes for the Cauchy problem and for initial-boundary problems.

  32. Bouquet biography
    • With Briot he worked from 1853 onwards on deep studies of Cauchy's ideas of analysis and produced many fundamental papers on series expansions of functions and on elliptic functions.
    • Bouquet and Briot developed Cauchy's work on the existence of integrals of a differential equation.

  33. Ostrowski biography
    • Other work of Ostrowski was on the Cauchy functional equation, the Fourier integral formula, Cauchy-Frullani integrals, and the Euler-Maclaurin formula.

  34. Gateaux biography
    • The first one is about the extension of the Weierstrass expansion, the equivalence between analyticity and holomorphy and the Cauchy formula to functionals.
    • Gateaux's interest in infinite dimensional integration originated in the extension of Cauchy's formula.

  35. Sylow biography
    • After proving Cauchy's theorem that a group of order divisible by a prime p has a subgroup of order p, Sylow asks whether it can be generalised to powers of p.
    • Cauchy had already proved that a group whose order is divisible by a prime p has an element of order p.

  36. Lame biography
    • It concerns Lame's attempt to spread Cauchy's new ideas of rigorous analysis.
    • Lame produced a manuscript criticising the proof using Cauchy's arguments.

  37. Kumano-Go biography
    • During these years Kumano-Go published a series of papers which studied the local and global uniqueness of the solutions of the Cauchy problem for partial differential equations.
    • In two papers Kumano-Go also studied non-uniqueness of solutions of the Cauchy problem.

  38. Dinghas biography
    • Isoperimetrische Ungleichungen (1961), and Einfuhrung in die Cauchy-Weierstrass'sche Funktionentheorie (1968).
    • Examples are the formula of Plana-Abel-Cauchy, the theorem of Julia-Wolff-Caratheodory, and the theory of Nevanlinna and of Hallstrom.

  39. De Prony biography
    • One battle he fought, without success, was against Cauchy's more towards pure mathematics and away from the more applied mathematics which de Prony firmly believed in.
    • He was disillusioned by the failure of his attempts to reform mathematics teaching at the Ecole Polytechnique, where he had made energetic and determined efforts to combat the emphasis on theory of A-L Cauchy ..

  40. Pompeiu biography
    • He introduced the notion of a special type of derivative, the areolar derivative of a complex function, extending the Cauchy formula which today is sometimes called the Cauchy-Pompeiu formula.

  41. Lopatynsky biography
    • Recently he has obtained important results on solvability of the Cauchy problem for operator equations in Banach space and also on "almost everywhere" solvability of general linear and nonlinear boundary problems.
    • He also obtained some basic results in the solvability of the Cauchy problem for operator equations in Banach spaces.

  42. Jacobi biography
    • This result prompted much further work in this area, in particular by Liouville and Cauchy.
    • Jacobi was not the first to study the functional determinant which now bears his name, it appears first in a 1815 paper of Cauchy.

  43. Ostrogradski biography
    • Here between 1822 and 1827 he attended lectures by Laplace, Fourier, Legendre, Poisson, Binet and Cauchy.
    • Other results which he obtained at this time on residue theory appeared in Cauchy's works.

  44. Plana biography
    • An important event for mathematics in Turin occurred when Cauchy lived in Turin during the year 1832-33.
    • Cauchy and Plana certainly interacted during this time and their relationship is discussed in [Univ.

  45. Milne-Thomson biography
    • The fancied distinction between 'pure' and 'applied' is a modern and false dichotomy unknown to Euler and Cauchy.

  46. Walsh biography
    • The most interesting of these conveys a report by Poisson and Cauchy on one of his papers submitted to the Academy of Sciences.

  47. Peano biography
    • The existence of solutions with stronger hypothesis on f had been given earlier by Cauchy and then Lipschitz.

  48. Dickstein biography
    • Dickstein's teacher had been a student of Cauchy in the early nineteenth century, and he still considered Poincare, who died in 1912, a bright young man.

  49. Crighton biography
    • He gave a mathematical model in which the problem reduce to solving two singular integral equations with Cauchy-type kernels, and with variable coefficients.

  50. Euler biography
    • He discovered the Cauchy-Riemann equations in 1777, although d'Alembert had discovered them in 1752 while investigating hydrodynamics.

  51. Polozii biography
    • Original results in the theory of functions of a complex variable were obtained in the 1950s and 1960s by G Polozii of Kiev, who introduced a new notion of p-analytic functions, defined the notion of derivative and integral for these functions, developed their calculus, obtained a generalised Cauchy formula, and devised a new approximation method for solution of problems in elasticity and filtration.

  52. Vandermonde biography
    • Cauchy states quite clearly that Vandermonde had priority over Lagrange for this remarkable idea which eventually led to the study of group theory.

  53. Girard Pierre biography
    • While in charge of this project he was assigned a young assistant, namely Cauchy.

  54. Maxwell biography
    • Cauchy, Calcul Differentiel .

  55. Freudenthal biography
    • In this archive we have included in our references articles in that publication on Arbuthnot, Cauchy, Haar, Heine, Hermite, Hilbert, Hopf, Hurwitz, Kerekjarto, Knopp, Lie, Loewner, Pringsheim, Quetelet, Riemann, Schonflies, Schottky, Sylow, and Christian Wiener.

  56. Neumann Franz biography
    • In 1832 Neumann investigated the wave theory of light, obtaining results similar to those of Cauchy and Fresnel.

  57. Kochin biography
    • He gave the solution to the problem of small amplitude waves on the surface of an uncompressed liquid in Towards a Theory of Cauchy-Poisson Waves in 1935.

  58. Frechet biography
    • He lectured in Lisbon on: Les fonctions periodiques, les fonctions presque periodiques et les fonctions assymptotiquement presque periodiques; Applications des fonctions assymptotiquement presque periodiques au theoreme ergodique de Birkhoff; Les debuts de la topologie combinatoire; le theoreme d'Euler-Cauchy; La theorie des courbes dans les espaces abstraits tres generaux; Types homogenes de dimensions; and Le developpement d'une fonction continue en serie de polynomes dans les espaces abstraits.

  59. FitzGerald biography
    • In addition he absorbed the theories put forward by Cauchy and Green.

  60. Lax Anneli biography
    • The title was On Cauchy's Problem for Partial Differential Equations with Multiple Characteristics, and it was published in Communications on Pure and Applied Mathematics in 1956.

  61. Christoffel biography
    • He returned to Montjoie where his mother was in poor health but read widely from the works of Dirichlet, Riemann and Cauchy.

  62. Brisson biography
    • These papers were praised by Cauchy for their elegance and importance and influenced him in developing methods of functional calculus.

  63. Burkill biography
    • The book covers: sets and functions, metric spaces, continuous functions on metric spaces, real and complex limits and series, uniform convergence, Riemann-Stieltjes integration, multivariable differential and integral calculus, Fourier series, Cauchy's theorem, Laurent expansions, residue calculus, infinite products, the factor theorem of Weierstrass, asymptotic expansions, and applications to special functions in particular the gamma function.

  64. Tikhonov biography
    • Thus, his research on the Earth's crust lead to investigations on well-posed Cauchy problems for parabolic equations and to the construction of a method for solving general functional equations of Volterra type.

  65. Gleason biography
    • Chapters I to VI cover elementary logic and set theory; Chapters VII to X deal with the various "number systems" from the natural integers to the complex numbers; Chapter XI briefly returns to set theory (countable sets, cardinal numbers and the axiom of choice); finally, the last four chapters deal, respectively, with limits of complex sequences, infinite series and products, metric spaces, and the elementary theory of holomorphic functions of one variable (Cauchy integral excluded, but the logarithmic function is defined and studied).

  66. Genocchi biography
    • He did not adopt the methods of Riemann and Weierstrass, but rather worked in the tradition of Euler, Lagrange, Gauss and Cauchy.

  67. Polkinghorne biography
    • Also in a joint paper in 1957 he published Cauchy's problem in quantum field theory which explores the relation between the classical and quantum versions of field theories.

  68. Kelland biography
    • His early research work, undertaken at Cambridge, was influenced by Fourier and Cauchy.

  69. Bauer Mihaly biography
    • Kronecker called this a 'boundary value problem' (Randwertproblem) because of a (vague) analogy with Cauchy's theorem computing the values of an analytic function on a disc from its values taken at the boundary.

  70. Lax Peter biography
    • She was awarded a PhD in 1955 for her thesis Cauchy's Problem for a Partial Differential Equation with Real Multiple Characteristics.

  71. Privalov biography
    • Privalov wrote Cauchy Integral (1918) which built on work by Fatou.

  72. Bertrand biography
    • Cauchy was asked to report on the work, which studied subgroups of low index in the symmetric group, and it clearly led him to return to study permutation groups himself.

  73. Picard Emile biography
    • He used methods of successive approximation to show the existence of solutions of ordinary differential equations solving the Cauchy problem for these differential equations.

  74. Schlomilch biography
    • His textbooks presented Cauchy's techniques in analysis and through them these important methods became well known in Germany.

  75. Lagrange biography
    • He studied permutations of the roots and, although he does not compose permutations in the paper, it can be considered as a first step in the development of group theory continued by Ruffini, Galois and Cauchy.

  76. Thomae biography
    • He then went to construct the rational numbers using Weierstrass's approach, then continued with a construction of the real numbers using the Cauchy sequence type of definition already published by Cantor and Heine.

  77. Argand biography
    • The concept of the modulus of a complex number is also due to Argand but Cauchy, who used the term later, is usually credited as the originator this concept.

  78. Ritt biography
    • Among his heroes were Niels Henrik Abel, Augustin Louis Cauchy, David Hilbert, Carl G J Jacobi, Joseph-Louis Lagrange, the marquis Pierre Simon de Laplace, Joseph Liouville and Jules Henri Poincare.

  79. Banach biography
    • The completeness is important as this means that Cauchy sequences in Banach spaces converge.

  80. Arf biography
    • Arf presented a paper On a generalization of Green's formula and its application to the Cauchy problem for a hyperbolic equation to the volume Studies in mathematics and mechanics presented to Richard von Mises in 1954.

  81. Thomson biography
    • There he worked in the physical laboratory of Henri-Victor Regnault and he was soon taking part in deep discussions with Biot, Cauchy, Liouville, Dumas, and Sturm.

  82. Hadamard biography
    • He continued to produce books and papers of the highest quality, publishing perhaps his most famous text Lectures on Cauchy's problem in linear partial differential equations in 1922.

  83. Cayley biography
    • As early as 1849 Cayley a paper linking his ideas on permutations with Cauchy's.

  84. Zygmund biography
    • One of the novel features is the early introduction of Runge's theorem and its application to facilitate the proof of Cauchy's theorem and other results.

  85. Woodhouse biography
    • If the methods of Lagrange, rather than those of Cauchy, had become the accepted methods then Woodhouse would have a much more prominent role in the history of mathematics today.

  86. Burkhardt biography
    • The object of the author in writing the little volume before us has been to furnish an introduction to the theory of functions which is not confined to the presentation of the methods of any one school (Cauchy, Weierstrass, Riemann) but blends these methods as far as possible into an organic whole.
    • It includes details of Cauchy's work in a number of different areas.

  87. Libri biography
    • It was an impressive start for the remarkable young mathematician, and his first contribution received considerable praise from many of the leading mathematicians of the day such as Babbage, Cauchy, and Gauss.

  88. Fuchs biography
    • Fuchs was a gifted analyst whose works form a bridge between the fundamental researches od Cauchy, Riemann, Abel, and Gauss and the modern theory of differential equations discovered by Poincare, Painleve, and Emile Picard.

  89. Riemann biography
    • The work builds on Cauchy's foundations of the theory of complex variables built up over many years and also on Puiseux's ideas of branch points.

  90. Mathieu Emile biography
    • He also worked on the polarisation of light where he highlighted some weaknesses in Cauchy's results on the topic.

  91. Laurent Hermann biography
    • Sbornik between 1868 and 1872 using Cauchy's integral formula as a starting point, but changing n! into G(n + 1) so that it made sense for non-integral n.

  92. Levi-Civita biography
    • He added to the theory of Cauchy and Kovalevskaya and wrote up this work in an excellent book written in 1931.

  93. Bochner biography
    • He made basic contributions to this theory that included the Bochner-Martinelli Formula (1943), and extensions of Cauchy's integral formula (1944).

  94. Beckenbach biography
    • The book begins with a study of axiomatics, then examines several classical inequalities of analysis such as the relationship between the arithmetic mean and geometric mean, the Cauchy, Holder, and Minkowski inequalities, and the triangle inequality.

  95. Walsh Joseph biography
    • He had published a seven page paper with this title in the Transactions of the American Mathematical Society in 1918 but even this was not his first publication for in 1916 he had published Note on Cauchy's integral formula in the Annals of Mathematics.

  96. Artin biography
    • By this stage he had proved, using very clever arguments with Galois theory and Cauchy's theorem on subgroups of prime order, that O had to be an extension of K of degree 2 and that the subfield K had to have the property that -1 could not be expressed as a sum of squares.

  97. Navier biography
    • In addition, he replaced Cauchy as professor at the Ecole Polytechnique from 1831.

  98. Faa di Bruno biography
    • Faa di Bruno travelled to Paris where he studied at the Sorbonne under Cauchy who [Peano : Life and Works of Giuseppe Peano (Dordrecht, 1980).',1)">1]:- .

  99. Borok biography
    • In the same period she obtained formulae that made it possible to compute in simple algebraic terms the numerical parameters that determine classes of uniqueness and well-posedness of the Cauchy problem for systems of linear partial differential equations with constant coefficients.

  100. Peirce Benjamin biography
    • The course he set up was impressive, including the study of works of Lacroix, Cauchy, Monge, Biot, Hamilton, Laplace, Poisson, Gauss, Le Verrier, Bessel, Adams, Airy, MacCullagh and Franz Neumann.

  101. Green biography
    • Among them are Cavendish's single-fluid theoretical study of electricity of 1771, two memoirs by Poisson of 1812 on surface electricity and three on magnetism (1821-1823), and contributions by Arago, Laplace, Fourier, Cauchy, and T Young.

  102. Stolz biography
    • In the 1870s Weierstrass's ε, δ approach to analysis became to standard approach and, in B Bolzano's Bedeutung in der Geschichte der Infinitesimalrechnung (1881), Stolz pointed out that Bolzano had suggested a similar approach even before Cauchy had attempted his own way of making analysis more rigorous.

  103. Pringsheim biography
    • He gave a very simple proof of Cauchy's integral theorem.

  104. Lewy biography
    • The revolutionary character of these works is reflected in the fact that J Hadamard, a world authority at that time, devoted a special appendix to Lewy's theory in his newly published book on the Cauchy Problem (1932).

  105. Henrici Peter biography
    • The first volume Power series - integration - conformal mapping - location of zeros first appeared in 1974, the second volume Special functions - integral transforms - asymptotics - continued fractions first appeared in 1977, and the third and final volume Discrete Fourier analysis - Cauchy integrals - construction of conformal maps - univalent functions first appeared in 1986.

  106. Arbogast biography
    • Arbogast won the prize with his essay and his notion of discontinuous function became important in Cauchy's more rigorous approach to analysis.

  107. Gregory biography
    • However, we now summarise these and other contributions in the hope that, despite his reluctance to publish his methods, his remarkable contributions might indeed be more widely understood: Gregory anticipated Newton in discovering both the interpolation formula and the general binomial theorem as early as 1670; he discovered Taylor expansions more than 40 years before Taylor; he solved Kepler's famous problem of how to divide a semicircle by a straight line through a given point of the diameter in a given ratio (his method was to apply Taylor series to the general cycloid); he gives one of the earliest examples of a comparison test for convergence, essentially giving Cauchy's ratio test, together with an understanding of the remainder; he gave a definition of the integral which is essentially as general as that given by Riemann; his understanding of all solutions to a differential equation, including singular solutions, is impressive; he appears to be the first to attempt to prove that π and e are not the solution of algebraic equations; he knew how to express the sum of the nth powers of the roots of an algebraic equation in terms of the coefficients; and a remark in his last letter to Collins suggests that he had begun to realise that algebraic equations of degree greater than four could not be solved by radicals.

  108. Riesz Marcel biography
    • In 1949, Riesz published a 223 page paper L'integrale de Riemann-Liouville et le probleme de Cauchy in which he introduced a multiple integral of Riemann-Liouville type and showed how important this idea is in the theory of the wave equation.

  109. Mittag-Leffler biography
    • Both were born Catholics and Hermite, as Cauchy before him, was a warm believer.

  110. De Giorgi biography
    • In 1955 De Giorgi gave an important example which showed nonuniqueness for solutions of the Cauchy problem for partial differential equations of parabolic type whose coefficents satisfy certain regularity conditions.

  111. Farey biography
    • One mathematical reader (at least of a French translation) was Cauchy, and he gave the necessary proof in his Exercices de mathematiques which was published in the same year as Farey's article.

  112. Carleman biography
    • In complex analysis there are Carleman formulae (proved already in 1926) which, unlike the Cauchy formula, reconstruct a function holomorphic in a domain D from its values on a part M of the boundary ∂D of a positive Lebesgue measure.

  113. Bienayme biography
    • He argued with Cauchy over the least squares method and, in 1842, he criticised Poisson's law of large numbers.

  114. Reynaud biography
    • Cauchy began to teach this course from 1815.

  115. Gram biography
    • The process seems to be a result of Laplace and it was essentially used by Cauchy in 1836.

  116. Stoilow biography
    • The thesis was a remarkable piece of work which studies the Cauchy problem for initial data containing singularities.

  117. Morera biography
    • He was interested also in the Cauchy integral for the representation of functions of a complex variable, in the discontinuity of the differentials of the potential function and in the Gauss representation formula.

  118. Lindelof biography
    • In it he examines the role which residue theory (Cauchy) plays in function theory as a means of access to modern analysis.

History Topics

  1. Real numbers 2
    • By the beginning of the nineteenth century a more rigorous approach to mathematics, principally by Cauchy and Bolzano, began to provide the machinery to put the real numbers on a firmer footing.
    • Grabiner writes [The origins of Cauchy\'s rigorous calculus (The MIT Press, Cambridge, Massachusetts, 1981).',2)" onmouseover="window.status='Click to see reference';return true">2]:- .
    • though Cauchy implicitly assumed several forms of the completeness axiom for the real numbers, he did not fully understand the nature of completeness or the related topological properties of sets of real numbers or of points in space.
    • Cauchy did not have explicit formulations for the completeness of the real numbers.
    • Among the forms of the completeness property he implicitly assumed are that a bounded monotone sequence converges to a limit and that the Cauchy criterion is a sufficient condition for the convergence of a series.
    • Though Cauchy understood that a real number could be obtained as the limit of rationals, he did not develop his insight into a definition of real numbers or a detailed description of the properties of real numbers.
    • Cauchy, in Cours d'analyse (1821), did not worry too much about the definition of the real numbers.
    • Certainly this is not considered by Cauchy to be a definition of a real number, rather it is simply a statement of what he considers an "obvious" property.
    • He says nothing about the need for the sequence to be what we call today a Cauchy sequence and this is necessary if one is to define convergence of a sequence without assuming the existence of its limit.
    • Bolzano, on the other hand, showed that bounded Cauchy sequence of real numbers had a least upper bound in 1817.
    • Cauchy himself does not seem to have understood the significance of his own "Cauchy sequence" criterion for defining the real numbers.
    • Two years after the publication of Hankel's monograph, Meray published Remarques sur la nature des quantites in which he considered Cauchy sequences of rational numbers which, if they did not converge to a rational limit, had what he called a "fictitious limit".
    • Essentially Heine looks at Cauchy sequences of rational numbers.
    • His numbers were Cauchy sequences of rational numbers and he used the term "determinate limit".

  2. Abstract groups
    • The story begins with the prehistory which involves Galois and Cauchy.
    • Rene Taton has found evidence in the archives of the Academie which suggest that Cauchy spoke with Galois and persuaded him to withdraw the paper and submit a new version of it for the Grand Prix of 1830.
    • Now in 1845, one year before Liouville published the above definition by Galois, Cauchy gave a definition.
    • However, Cauchy's term "conjugate system of substitutions" continued to be used by some up to about 1880.
    • How much was Cauchy influenced by Galois? Although he had seen Galois' papers submitted to the Academie, there was no explicit definition of a group in them.
    • Both Galois and Cauchy, of course, define groups in terms of the closure property alone.
    • Cauchy went on to write 25 papers on this topic between September 1845 and January 1846.
    • Peter Neumann did not mention in his lecture any influence that Ruffini's ideas might have had on Cauchy.
    • In 1821 Cauchy had written to Ruffini praising his work which he had clearly read.
    • Although there is no explicit definition of a group in Ruffini's work, again the concept clearly appears and may have had as major an influence on Cauchy's thinking as the work of Galois.] .

  3. Matrices and determinants
    • It was Cauchy in 1812 who used 'determinant' in its modern sense.
      Go directly to this paragraph
    • Cauchy's work is the most complete of the early works on determinants.
      Go directly to this paragraph
    • In the 1812 paper the multiplication theorem for determinants is proved for the first time although, at the same meeting of the Institut de France, Binet also read a paper which contained a proof of the multiplication theorem but it was less satisfactory than that given by Cauchy.
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    • In 1826 Cauchy, in the context of quadratic forms in n variables, used the term 'tableau' for the matrix of coefficients.
    • Cauchy also introduced the idea of similar matrices (but not the term) and showed that if two matrices are similar they have the same characteristic equation.
    • It should be stressed that neither Cauchy nor Jacques Sturm realised the generality of the ideas they were introducing and saw them only in the specific contexts in which they were working.
      Go directly to this paragraph

  4. Group theory

  5. Abstract linear spaces
    • In this work Laguerre aims to unify algebraic systems such as complex numbers, Hamilton's quaternions and notions introduced by Galois and by Cauchy.
      Go directly to this paragraph
    • Cauchy and Saint-Venant have some claims to have invented similar systems to Grassmann.
      Go directly to this paragraph
    • In fact when Grassmann read Saint-Venant's paper he realised that Saint-Venant had not read his 1844 work and sent two copies of the relevant parts to Cauchy, asking him to pass one copy to Saint-Venant.
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    • However, rather typically of Cauchy, in 1853 he published Sur les clefs algebrique in Comptes Rendus which describes a formal symbolic method which coincides with that of Grassmann's method (but makes no reference to Grassmann).
    • Grassmann complained to the Academie des Sciences that his work had priority over Cauchy's and, in 1854, a committee was set up to investigate who had priority.

  6. function concept
    • The clearest example of such a function was given by Cauchy in 1844 when he noted that the function .
    • Cauchy, in 1821, came up with a definition making the dependence between variables central to the function concept.
    • Notice that despite the generality of Cauchy's definition, which is designed to cover the case of explicit and implicit functions, he is still thinking of a function in terms of a formula.
    • Cauchy gave an early example when he noted that f(x) = exp(-1/x2) for x ≠ 0, f(0) = 0, is a continuous function which has all its derivatives at 0 equal to 0.

  7. Fund theorem of algebra
    • In 1820 Cauchy was to devote a whole chapter of Cours d'analyse to Argand's proof (although it will come as no surprise to anyone who has studied Cauchy's work to learn that he fails to mention Argand !) This proof only fails to be rigorous because the general concept of a lower bound had not been developed at that time.
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    • The term 'conjugate' had been introduced by Cauchy in 1821.

  8. Real numbers 2 references
    • J V Grabiner, The origins of Cauchy's rigorous calculus (The MIT Press, Cambridge, Massachusetts, 1981).
    • J G Dhombres, Real numbers from Cauchy to Robinson, Southeast Asian Bull.

  9. Bolzano's manuscripts references
    • W Felscher, Bolzano, Cauchy, epsilon, delta, Amer.
    • H Sinaceur, Cauchy et Bolzano, Rev.

  10. Real numbers 2 references
    • J V Grabiner, The origins of Cauchy's rigorous calculus (The MIT Press, Cambridge, Massachusetts, 1981).
    • J G Dhombres, Real numbers from Cauchy to Robinson, Southeast Asian Bull.

  11. Bolzano's manuscripts references
    • W Felscher, Bolzano, Cauchy, epsilon, delta, Amer.
    • H Sinaceur, Cauchy et Bolzano, Rev.

  12. Calculus history
    • Maclaurin attempted to put the calculus on a rigorous geometrical basis but the really satisfactory basis for the calculus had to wait for the work of Cauchy in the 19th Century.
      Go directly to this paragraph

  13. Matrices and determinants references
    • T W Hawkins, Cauchy and the spectral theory of matrices, Historia Mathematica 2 (1975), 1-29.

  14. Real numbers 3 references
    • J G Dhombres, Real numbers from Cauchy to Robinson, Southeast Asian Bull.

  15. Abstract linear spaces references
    • T W Hawkins, Cauchy and the spectral theory of matrices, Historia Mathematica 2 (1975), 1-29.

  16. Matrices and determinants references
    • T W Hawkins, Cauchy and the spectral theory of matrices, Historia Mathematica 2 (1975), 1-29.

  17. Real numbers 3 references
    • J G Dhombres, Real numbers from Cauchy to Robinson, Southeast Asian Bull.

  18. Abstract linear spaces references
    • T W Hawkins, Cauchy and the spectral theory of matrices, Historia Mathematica 2 (1975), 1-29.

  19. Bourbaki 2
    • This instalment of Bourbaki's super-textbook gives a notable account of Rolle's theorem and Taylor's theorem with remainder; of the indefinite integral, as anti-derivative, for a function having only discontinuities of the first kind, such a function being a uniform limit of a function which is constant by intervals, an "interval" being an open interval or a point; of Cauchy limits for such integrals; of integration and differentiation with respect to a parameter under the integral sign; and of the elementary logarithmic, exponential and trigonometric functions.

  20. History overview
    • Cauchy, building on the work of Lagrange on functions, began rigorous analysis and began the study of the theory of functions of a complex variable.
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  21. Ring Theory
    • However, Cauchy supported Lame.

  22. Fermat's last theorem
    • Cauchy supported Lame but, in rather typical fashion, pointed out that he had reported to the October 1847 meeting of the Academie an idea which he believed might prove Fermat's Last Theorem.
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  23. Real numbers 3
    • Provided we have some finite way of specifying the n-th term in a Cauchy sequence of rationals we have a finite description of the resulting real number.

  24. Special relativity
    • Cauchy, Stokes, Thomson and Planck all postulated ethers with differing properties and by the end of the 19th Century light, heat, electricity and magnetism all had their respective ethers.
      Go directly to this paragraph

Famous Curves

No matches from this section

Societies etc

  1. Cauchy
    • Augustin Louis Cauchy .

  2. Lunar features

  3. Lunar features
    • (W) (L) Cauchy .
    • (L) Fossa Cauchy .
    • (L) Rima Cauchy .
    • (L) Rupes Cauchy .

  4. Lunar features
    • Cauchy .
    • Fossa Cauchy .
    • Rima Cauchy .
    • Rupes Cauchy .

  5. Paris street names
    • Rue Cauchy (15th Arrondissement) WnM .

  6. Eiffel Tower
    • Cauchy .

  7. AMS Steele Prize
    • for his paper "Uniqueness in the Cauchy Problem for Partial Differential Equation".

  8. AMS Bôcher Prize
    • in recognition of his fundamental work on the theory of singular integrals and partial differential equations, and in particular for his paper "Cauchy integrals on Lipschitz curves and related operators".

  9. Fellows of the RSE
    • Augustin Louis Cauchy1845More infoMacTutor biography .

  10. Fellows of the RSE
    • Augustin Louis Cauchy1845More infoMacTutor biography .

  11. Fellows of the RSE
    • Augustin Louis Cauchy1845More infoMacTutor biography .

  12. Fellow of the Royal Society
    • Augustin L Cauchy 1832 .

  13. BMC 1994
    • Smithies, F Cauchy's early work on complex function theory .

  14. Young Mathematician prize
    • for works on the properties of a solution of non-homogeneous Cauchy-Riemann system.

  15. Paris Academy of Sciences
    • In 1816 Cauchy won the Grand Prix of the Academy for a memoir on waves.

References

  1. References for Cauchy
    • References for Augustin-Louis Cauchy .
    • B Belhoste, Cauchy.
    • B Belhoste, Augustin-Louis Cauchy.
    • J V Grabiner, The Origins of Cauchy's Rigorous Calculus (Cambridge, Massachusetts, 1981).
    • L Novy, Cauchy, in H Wussing and W Arnold, Biographien bedeutender Mathematiker (Berlin, 1983).
    • C A Valson, La vie et les travaux du baron Cauchy (Paris, 1868).
    • A N Bogolyubov, Augustin Cauchy and his contribution to mechanics and physics (Russian), Studies in the history of physics and mechanics, 'Nauka' (Moscow, 1988), 179-201.
    • N Cutland, C Kessler, E Kopp and D Ross, On Cauchy's notion of infinitesimal, British J.
    • A Dahan-Dalmedico, Les travaux de Cauchy sur les substitutions : etude de son approche du concept de groupe, Arch.
    • A Dahan-Dalmedico, La mathematisation de la theorie de l'elasticite par A L Cauchy et les debats dans la physique mathematique francaise (1800-1840), Sciences et Techniques en Perspective 9 (1984-85), 1-100.
    • A Dahan-Dalmedico, La propagation des ondes en eau profonde et ses developpements mathematiques (Poisson, Cauchy, 1815-1825), in The history of modern mathematics II (Boston, MA, 1989), 129-168.
    • A Dahan-Dalmedico, L'integration des equations aux derivees partielles lineaires a coefficients constants dans les travaux de Cauchy (1821-1830), Etudes sur Cauchy (1789-1857), Rev.
    • J M Dubbey, Cauchy's contribution to the establishment of the calculus, Ann.
    • G Fisher, Cauchy and the infinitely small, Historia Mathematica 5 (1978), 313-331.
    • G Fisher, Cauchy's variables and orders of the infinitely small, British J.
    • H Freudenthal, Did Cauchy plagiarise Bolzano, Archive for History of Exact Sciences 7 (1971), 375-392.
    • P Gario, Cauchy's theorem on the rigidity of convex polyhedra (Italian), Archimede 33 (1-2) (1981), 53-69.
    • C Gilain, Cauchy et le cours d'analyse de l'Ecole Polytechnique, Bull.
    • J V Grabiner, Who gave you the epsilon? Cauchy and the origins of rigorous calculus, Amer.
    • J V Grabiner, The origins of Cauchy's theory of the derivative, Historia Math.
    • I Grattan-Guinness, Bolzano, Cauchy and the 'new analysis' of the early nineteenth century, Archive for History of Exact Sciences 6 (1970), 372-400.
    • I Grattan-Guinness, The Cauchy - Stokes - Seidel story on uniform convergence again : was there a fourth man?, Bull.
    • I Grattan-Guinness, On the publication of the last volume of the works of Augustin Cauchy, Janus 62 (1-3) (1975), 179-191.
    • J Gray, Cauchy - elliptic and abelian integrals, Etudes sur Cauchy (1789-1857), Rev.
    • Z L Gui and D F Zhao, Augustin-Louis Cauchy : an outstanding mathematician (Chinese), J.
    • T Hawkins, Cauchy and the spectral theory of matrices, Historia Mathematica 2 (1975), 1-29.
    • T Koetsier, Cauchy's rigorous calculus : a revolution in Kuhn's sense?, Nieuw Arch.
    • D Laugwitz, Infinitely small quantities in Cauchy's textbooks, Historia Math.
    • S X Luo, Cauchy - a highly productive master of mathematics (Chinese), Math.
    • P M Neumann, On the date of Cauchy's contributions to the founding of the theory of groups, Bull.
    • K Rychlik, Un manuscrit de Cauchy aux archives de l'academie tchecoslovaque des Sciences, Czechoslovak Math.
    • H Sinaceur, Cauchy, Sturm et les racines des equations, Etudes sur Cauchy (1789-1857), Rev.
    • H Sinaceur, Cauchy et Bolzano, Rev.
    • F Smithies, Cauchy's conception of rigor in analysis, Archive for History of Exact Sciences 36 (1986), 41-61.
    • F Smithies, The background to Cauchy's definition of the integral, Math.
    • D Struik and R Struik, Cauchy and Bolzano in Prague, Isis 11 (1928), 364-366.
    • R Taton, Sur les relations scientifiques d'Augustin Cauchy et d'Evariste Galois, Rev.
    • A Terracini, Cauchy a Torino, Univ.
    • C A Truesdell, Cauchy and the modern mechanics of continua, Rev.
    • C A Truesdell, Cauchy's first attempt at molecular theory of elasticity, Boll.
    • http://www-history.mcs.st-andrews.ac.uk/References/Cauchy.html .

  2. References for Bunyakovsky
    • V S Kirsanov, V Ya Bunyakovskii's dissertation and Cauchy's theory of residues (Russian), Istor.-Mat.
    • P Schreiber, The Cauchy- Bunyakovsky- Schwarz inequality, in Hermann Grassmann, Lieschow, 1994 (Greifswald, 1995), 64-70.
    • Yu F Zhang, F X Bao and X L Fu, The origin and development of the Cauchy- Bunyakovskii inequality (Chinese), Qufu Shifan Daxue Xuebao Ziran Kexue Ban 21 (1) (1995), 83-86.

  3. References for Bolzano
    • W Felscher, Bolzano, Cauchy, epsilon, delta, Amer.
    • H Sinaceur, Cauchy et Bolzano, Rev.

  4. References for Sturm
    • H Sinaceur, Cauchy, Sturm et les racines des equations, in Etudes sur Cauchy (1789-1857), Rev.

  5. References for Libri
    • A Procissi, Sopra una questione di teoria dei numeri di Guglielmo Libri, ed una lettera inedita di Agostino Cauchy, Boll.

  6. References for Stokes
    • I Grattan-Guinness, The Cauchy- Stokes- Seidel story on uniform convergence again: was there a fourth man?, Bull.

  7. References for Lame
    • Petersburg (on the history of Cauchy's conception of mathematical analysis) (Russian), Voprosy Istor.

  8. References for Schwarz
    • P Schreiber, The Cauchy- Bunyakovsky- Schwarz inequality, in Hermann Grassmann, Lieschow, 1994 (Greifswald, 1995), 64-70.

  9. References for Bessel
    • M Galuzzi, Trigonometric interpolation in Gauss, Bessel and Cauchy (Italian), Writings in honor of Giovanni Melzi, Sci.

  10. References for Galois
    • R Taton, Sur les relations scientifiques d'Augustin Cauchy et d'Evariste Galois, Rev.

  11. References for Poisson
    • A Dahan-Dalmedico, La propagation des ondes en eau profonde et ses developpements mathematiques (Poisson, Cauchy, 1815-1825), in The history of modern mathematics II (Boston, MA, 1989), 129-168.

  12. References for Ruffini
    • F Barbieri and C Fiori, Ruffini's last letter to Cauchy (Italian), Nuncius Ann.

  13. References for Bezout
    • J Dhombres, French mathematical textbooks from Bezout to Cauchy, Historia Sci.

  14. References for Lipschitz
    • A P Yushkevich, Sur les origines de la 'methode de Cauchy-Lipschitz' dans la theorie des equations differentielles ordinaires, Rev.

  15. References for Maclaurin
    • S Mills, The Cauchy-Maclaurin integral theorem : an eighteenth-century example of mathematical analysis (Portuguese), in Proceedings of the XIIth Portuguese-Spanish Conference on Mathematics II (Braga, 1987).

  16. References for Pfaff
    • S S Demidov, On the history of the theory of first-order partial differential equations : The works of J F Pfaff and A Cauchy (Russian), Istor.-Mat.

  17. References for Plana
    • A Terracini, Cauchy a Torino, Univ.

  18. References for Lambert
    • J-P Lubet, De Lambert a Cauchy : la resolution des equations litterales par le moyen des series, Rev.

Additional material

  1. Konrad Knopp: Texts
    • In the treatment of sequences and series that follows, he covers arbitrary and null sequences; sequences and sets of numbers; convergence and divergence; Cauchy's limit theorem; main tests for sequences; and infinite series.
    • 2.4 Cauchy's limit theorem and its generalizations .
    • The Cauchy-Riemann Differential Equations .
    • Cauchy's Integral Theorem .
    • Cauchy's Integral Formulas .
    • Cauchy's Integral Theorems and Integral Formulas.

  2. H M Macdonald addresses the British Association in 1934
    • The earliest investigation of this kind is due to Cauchy.
    • In Cauchy's treatment the elastic medium is supposed to consist of small particles or molecules which act on each other, and the further hypothesis is made that the force between any two particles is along the line joining the two points which are taken to represent the two particles.
    • As the same problem was discussed by Green in a more general way in 1837 it is unnecessary to refer to Cauchy's results in detail.
    • The difference between Cauchy's hypothesis as to the nature of the mutual actions of the medium and Green's hypothesis has been referred to above; another important difference in their treatments is that Cauchy assumes that the direction of the disturbance in the medium is parallel to the plane of polarisation, while Green, in accordance with Fresnel's view, assumes that this direction is perpendicular to the plane of polarisation.
    • Cauchy arrived at the same result almost simultaneously.

  3. H M Macdonald addresses the British Association in 1934, Part 1
    • The earliest investigation of this kind is due to Cauchy.
    • In Cauchy's treatment the elastic medium is supposed to consist of small particles or molecules which act on each other, and the further hypothesis is made that the force between any two particles is along the line joining the two points which are taken to represent the two particles.
    • As the same problem was discussed by Green in a more general way in 1837 it is unnecessary to refer to Cauchy's results in detail.
    • The difference between Cauchy's hypothesis as to the nature of the mutual actions of the medium and Green's hypothesis has been referred to above; another important difference in their treatments is that Cauchy assumes that the direction of the disturbance in the medium is parallel to the plane of polarisation, while Green, in accordance with Fresnel's view, assumes that this direction is perpendicular to the plane of polarisation.
    • Cauchy arrived at the same result almost simultaneously.

  4. Thomas Muir: 'History of determinants
    • It was then taken up by Cauchy, and, thanks to the prestige of his name and to the inherent excellence of his extensive monograph, its position as a theory of importance became more firmly assured.
    • The thirty years that followed Cauchy's memoir resembled the sixty that preceded it, save that the number of contributors was considerably larger.
    • Then another great analyst, Jacobi, the most noteworthy of those contributors, produced in Germany a monograph similar in extent and value to Cauchy's, and the importance of the subject in the eyes of mathematicians became still more enhanced.
    • So strongly attractive had the subject now become to mathematicians that in the single year succeeding the publication of Spottiswoode's short treatise a greater number of separate contributions to the theory made their appearance than in the whole sixty-year period from Cramer to Cauchy.

  5. E C Titchmarsh: 'Aftermath
    • The theory of such functions contains many very remarkable theorems, particularly those due to the great French mathematician Cauchy (1789-1857).
    • Cauchy's theory of functions of a complex variable would have surprised the Greeks very much, and surely it would have delighted them too.
    • Much of our knowledge is due to a comparatively few great mathematicians such as Newton, Euler, Gauss, Cauchy or Riemann; few careers can have been more satisfying than theirs.

  6. Goursat: 'Cours d'analyse mathématique
    • Chapter XIV - Theorie generale des fonctions analytiques d'apres Cauchy.
    • - Integrale de Cauchy.
    • Methode de Cauchy-Lipschitz.

  7. Turnbull lectures on Colin Maclaurin, Part 2
    • It was the first logical and systematic account of fluxions, and in point of rigour could hold its own with the geometrical method of exhaustions of the Greeks and the subsequent work of Cauchy and of Weierstrass.
    • An earlier passage contains the well-known integral test for the convergence or divergence of a monotonic series, where and are compared; a method which Cauchy rediscovered many years later.

  8. Ernest Hobson addresses the British Association in 1910, Part 2
    • The notion of 'limit,' in the definite form given to it by Cauchy and his followers, together with the closely related theory of the arithmetic continuum, and the notions of continuity and functionality, lie at the very heart of modern analysis.
    • It has been the task of mathematicians under the lead of such men as Cauchy, Riemann, Weierstrass, and G Cantor, to carry out the work of reconstruction of mathematical analysis, to render explicit all the limitations of the truth of the general theorems, and to lay down the conditions of validity of the ordinary analytical operations.

  9. Horace Lamb addresses the British Association in 1904
    • Cauchy alone of this race of giants was still alive and productive.
    • The detailed study of the geometry of a continuous deformable medium which was instituted by Cauchy was a first step towards liberating the theory from arbitrary and unnecessary hypothesis; but it was reserved for Green, the immediate predecessor of Stokes among English mathematicians, to carry out this process completely and independently, with the help of Lagrange's general dynamical methods, which here found their first application to questions of physics outside the ordinary Dynamics of rigid bodies and fluids.

  10. Andrew Russell Forsyth by Leonard Roth
    • His major work on the theory of differential equations, a colossal achievement in six volumes, is still today the only treatise in its class which in by a single hand; but a mere glance at the list of contents suffices to reveal that, on the whole, Forsyth looks backward to Lagrange rather than forward to Cauchy.
    • The book includes fairly complete accounts of the relevant work of Cauchy, Abel, Riemann, Weierstrass, Appell, and carries on the survey right up to the then contemporary researches of Klein and Poincare.

  11. L R Ford - Differential Equations
    • No use is made of Cauchy's method of the calculus of limits and just a mention is included of the Cauchy-Lipschitz method.

  12. David Hilbert: 'Mathematical Problems
    • The introduction of complex paths of integration by Cauchy and of the notion of the ideals in number theory by Kummer may serve as examples.
    • The most suggestive and notable achievements of the last century in this field are, as it seems to me, the arithmetical formulation of the concept of the continuum in the works of Cauchy, Bolzano and Cantor, and the discovery of non-euclidean geometry by Gauss, Bolyai, and Lobachevsky.

  13. Einar Hille: 'Analytic Function Theory
    • The Cauchy integral is a much more pliable and versatile tool than the power series when it comes to doing things in function theory.
    • The main theory begins in Chapter 4 with the definition of holomorphic functions, the Cauchy-Riemann equations, inverse functions, and the elements of conformal mapping.

  14. Whittaker EMS Obituary.html
    • In order to appreciate the significance of the appearance of this book, it is essential to realise that at the end of last century the general theory of functions as developed by Cauchy, Weierstrass and other Continental mathematicians was hardly known in this country.
    • - Despite the general excellence of the work, it contained some inadequacies (for example, in the treatment of Cauchy's theorem) ; moreover, a few years after its publication, Hardy's Pure Mathematics and Bromwich's Infinite Series appeared.

  15. Publications of Giacinto Morera
    • G Morera, Intorno all'integrale di Cauchy, Rend.

  16. Publications of Eduard Heine
    • E Heine, Einige Anwendungen der Residuenrechnung von Cauchy, J.

  17. André Weil: 'Algebraic Geometry
    • Thus for a time the indiscriminate use of divergent series threatened the whole of analysis; and who can say whether Abel and Cauchy acted more as "creative" or as "critical" mathematicians when they hurried to the rescue? One would be lacking in a sense of proportion, should one compare the present situation in algebraic geometry to that which these great men had to face; but there is no doubt that, in this field, the work of consolidation has so long been overdue that the delay is now seriously hampering progress in this and other branches of mathematics.

  18. Donald C Spencer's publications
    • D C Spencer, Cauchy's formula on Kahler manifolds, Proc.

  19. Ernest Hobson addresses the British Association in 1910, Part 3
    • Who that has studied the works of such men as Euler, Lagrange, Cauchy, Riemann, Sophus Lie, and Weierstrass, can doubt that a great mathematician is a great artist? The faculties possessed by such men, varying greatly in kind and degree with the individual, are analogous to those requisite for constructive art.

  20. Publications of Eduard Heine
    • E Heine, Einige Anwendungen der Residuenrechnung von Cauchy, J.

  21. Kurosh's book The theory of groups 1st edition
    • Arising from the needs of Galois theory, it developed at first as the theory of finite substitution groups (Cauchy, Jordan, Sylow).

  22. Andrew Forsyth addresses the British Association in 1905, Part 2
    • The bead-roll of names in that science - Gauss; Abel, Jacobi; Cauchy, Riemann, Weierstrass, Hermite; Cayley, Sylvester; Lobachevsky, Lie - will on only the merest recollection of the work with which their names are associated show that an age has been reached where the development of human thought is deemed as worthy a scientific occupation of the human mind as the most profound study of the phenomena of the material universe.

  23. Percy MacMahon addresses the British Association in 1901
    • Amongst schoolboys of various ages we note Fresnel, Bessel, Cauchy, Chasles, Lame, Mobius, von Staudt and Steiner on the Continent, and Babbage, Peacock, John Herschel, Henry ParrHamilton and George Green in this country.

  24. W Burnside: 'Theory of Groups of Finite Order
    • The theory of groups of finite order may be said to date from the time of Cauchy.

  25. Gibson History 9 - Colin Maclaurin
    • It is interesting to note that he puts the method of infinitesimals on a sound basis, and in fact develops in a rigorous way the theory of differentials; I have no doubt at all that Cauchy's definition of the differential was fully and consciously given by Robins and Maclaurin.

  26. W H Young addresses ICM 1928
    • The whole body of Mathematics is thus raised to a higher level; as, for instance, in the passage from rational to real numbers, from real to complex, or from the Cauchy integral through the Riemann, to the Lebesgue, or again, from the concept of a single unique limit to the Theory of Sets of Points.

  27. Kurosh: 'The theory of groups' 1st edition
    • Arising from the needs of Galois theory, it developed at first as the theory of finite substitution groups (Cauchy, Jordan, Sylow).

  28. George Temple's Inaugural Lecture II
    • All our masters, from Laplace to Cauchy, have proceeded in the same way.

  29. Von Neumann: 'The Mathematician
    • Yet no mathematician would want to exclude it from the fold-that period produced mathematics as first class as ever existed! And even after the reign of rigour was essentially re-established with Cauchy, a very peculiar relapse into semi-physical methods took place with Riemann.

Quotations

  1. Quotations by Cauchy
    • Quotations by Augustin-Louis Cauchy .
    • http://www-history.mcs.st-andrews.ac.uk/Quotations/Cauchy.html .

  2. Quotations by Jacobi
    • Dirichlet alone, not I, nor Cauchy, nor Gauss knows what a completely rigorous mathematical proof is.
    • When Gauss says that he has proved something, it is very clear; when Cauchy says it, one can wager as much pro as con; when Dirichlet says it, it is certain ..

  3. Quotations by Jeffreys
    • Cauchy's theorem) are so surprising at first sight that nothing short of a proof can make them credible.

  4. Quotations by Boltzmann
    • A mathematician will recognise Cauchy, Gauss, Jacobi or Helmholtz after reading a few pages, just as musicians recognise, from the first few bars, Mozart, Beethoven or Schubert.

Chronology

  1. Mathematical Chronology
    • D'Alembert discovers the Cauchy-Riemann equations while investigating hydrodynamics.
    • Cauchy publishes Cours d'analyse (A Course in Analysis), which sets mathematical analysis on a formal footing for the first time.
    • Cauchy gives power series expansions of analytic functions of a complex variable.
    • Cauchy publishes the first volume of the four volume work Exercises d'analyse et de physique mathematique.
    • While examining permutation groups Cauchy proves a fundamental theorem of group theory which became known as "Cauchy's theorem".
    • Hermite applies Cauchy's residue techniques to doubly periodic functions.

  2. Chronology for 1840 to 1850
    • Cauchy publishes the first volume of the four volume work Exercises d'analyse et de physique mathematique.
    • While examining permutation groups Cauchy proves a fundamental theorem of group theory which became known as "Cauchy's theorem".
    • Hermite applies Cauchy's residue techniques to doubly periodic functions.

  3. Chronology for 1830 to 1840
    • Cauchy gives power series expansions of analytic functions of a complex variable.
    • Cauchy publishes the first volume of the four volume work Exercises d'analyse et de physique mathematique.

  4. Chronology for 1740 to 1760
    • D'Alembert discovers the Cauchy-Riemann equations while investigating hydrodynamics.

  5. Chronology for 1820 to 1830
    • Cauchy publishes Cours d'analyse (A Course in Analysis), which sets mathematical analysis on a formal footing for the first time.

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JOC/BS August 2001