Augustin-Louis Cauchy

Augustin-Louis Cauchy

Augustin-Louis Cauchy [ ogysˈtɛ̃ lwi koˈʃi ] (born August 21, 1789 in Paris , † May 23, 1857 in Sceaux ) was a French mathematician .

As a pioneer of analysis , he further developed the foundations established by Gottfried Wilhelm Leibniz and Sir Isaac Newton , also proving the fundamental statements formally and helping a new understanding of the concept of function to break through. In particular in the group theory and function theory, which he essentially founded , many central statements come from him. His almost 800 publications by and large cover the entire spectrum of mathematics at the time, and in physics he clarified and established in particular the fundamentals of elasticity theory . He occupies a similar position in the development of analysis as Leonhard Euler in the 18th century and in the 19th century he shared his outstanding position as a mathematician in the first half of the century with Carl Friedrich Gauss , but in contrast to him he also published his results without delay and had many students.

Cauchy was a Catholic and a follower of the French rulers of the Bourbons . The latter repeatedly brought him into conflict with the supporters of the republic and the Bonapartists.

Life

Cauchy's father, Louis-François, was a well-read Catholic royalist . At the time of the storming of the Bastille on July 14, 1789, he was the right hand man of Lieutenant General of the Police of Paris, Louis Thiroux de Crosne. He fled to England shortly afterwards, and Louis-François Cauchy lost his post. Augustin-Louis was born a few weeks later, in the middle of the French Revolution . Thiroux returned in April 1794, was arrested and sentenced to death on the same day. Louis-François thereupon took his family with him to their country house in Arcueil , where they lived in poverty, for fear of denunciation . Little Augustin-Louis received basic instruction from his father. The hunger and dangerous situation left a lifelong aversion to revolutions. After the end of the reign of terror , the family returned to Paris, Louis-François made a career again and finally became Secretary General of the Senate after Napoléon's coup . This led to a close acquaintance with the then Interior Minister Pierre-Simon Laplace and Senator Joseph-Louis Lagrange , two important mathematicians. They recognized their son's mathematical talent early on, as Lagrange is said to have said:

“Vous voyez ce petit jeune homme, eh bien! Il nous remplacera tous tant que nous sommes de géomètres "

“Now you see this young man! One day it will surpass all of us simple surveyors. "

- Joseph-Louis Lagrange : to academy colleagues after a conversation with twelve-year-old Cauchy in the Palais du Luxembourg in 1801

and advised his father:

“Ne laissez pas cet enfant toucher un livre de Mathématiques avant l'âge de dix-sept ans. […] Si vous ne vous hâtez de thunder à Augustin une solid éducation littéraire, son goût l'entraînera, il sera un grand mathématicien, mais il ne saura pas même écrire sa langue. "

“Don't let this child touch a math book until they are seventeen. […] If you do not hurry to give Augustine a thorough literary education, he will be carried away by his inclination. He will become a great mathematician, but will hardly be able to write his mother tongue. "

- Joseph-Louis Lagrange

Augustin-Louis Cauchy had two younger brothers: Alexandre Laurent (1792-1857), who like his father became a lawyer and entered the civil service, and Eugène François (1802-1877), a writer.

On the advice of Lagrange, Cauchy first learned classical languages, which should prepare him for further mathematics education. From 1802 he attended the École Centrale du Panthéon for two years , where he particularly shone in Latin . He then decided to embark on a career in engineering, and from 1804 took mathematics lessons, which should prepare him for the entrance examination at the young École polytechnique . In 1805 he passed the entrance examination, which was carried out by the French mathematician and physicist Jean-Baptiste Biot , as the second best . The École Polytechnique was designed to train engineers for France's public service, and students had to choose a particular direction early on. Cauchy chose road and bridge construction. The lessons were very mathematics-heavy. His teachers had well-known names like Lacroix , de Prony , Hachette and Ampère . After two years, Augustin-Louis was top of the class and was allowed to attend the École Nationale des Ponts et Chaussées for further training . Here, too, he was among the best and was allowed to work on the Ourcq Canal during his internship under Pierre Girard . In Paris, the students were anything but apolitical. While most were revolutionary and liberal, Cauchy joined the Congrégation , the secular arm of the Jesuits . He stayed there until it was effectively banned in 1828. After two years of compulsory studies, he left the university in January 1810 as an aspirant engineer .

Napoleon's engineer

In February 1810 Cauchy was commissioned to help build Port Napoléon in Cherbourg , which was then the largest construction site in Europe with around 3,000 workers. The aim was to prepare for the invasion of England. The hours were long and he did math in his little free time. His initial joy and interest in the engineering profession soon waned, and so he decided to embark on a scientific career. However, Cauchy's goal at the time was by no means mathematics. The general scientific view after Euler's death was that the problems of mathematics were almost completely solved. Engineering science and finding new fields of application for mathematics were particularly important.

The research during his time in Cherbourg yielded a small generalization of Euler's polyhedron theorem and a proof of a theorem on the question under which conditions polyhedra with equal faces are identical. Euclid had already formulated the sentence in its elements , but it had never been proven until then. Cauchy made a name for himself in Parisian academic society through this work.

In the summer of 1812 his health deteriorated sharply. Cauchy had not been very healthy since childhood and suffered from occasional depression . The heavy workload in Cherbourg troubled him so that in September he was on sick leave and was given permission to return to his family in Paris. When his health improved, he was not at all anxious to return to work as an engineer and devoted himself to research. Inspired by Lagrange's theorem , he dealt with group theory and found the three axioms that clearly define a determinant .

In the spring of 1813 his sick leave ended. There was no way Cauchy wanted to return to Cherbourg. His former teacher Pierre-Simon Girard gave him the opportunity to continue working on the Ourcq canal project in Paris. His research this year was fruitless: although he developed a method for determining the number of solutions to an algebraic equation of any degree, it was not practical. He applied for over 50 vacancies at the Paris academies, albeit without success - despite the good relationships his father had, who put pressure wherever he could. His scientific colleagues Ampère, Legendre , Louis Poinsot and Emmanuel-François Molard (1772–1829) were appointed, but Cauchy was not. Cauchy took sick leave that summer without pay. The defeat of Napoleon in 1814 benefited him: the Ourcq canal project was interrupted and he was not given a new job. This year also marks the beginning of Cauchy's preoccupation with complex functions.

In December 1815, he won the Paris Academy Grand Prize in Mathematics for his painstaking work on waves in liquids. His solution to Fermat's polygon number problem, submitted in November of the same year, was considered a sensation and made him famous in one fell swoop. Before that, only the cases of the squares (Lagrange) and the cubes (Legendre) had been solved, with new evidence for both cases by Gauss in his Disquisitiones arithmeticae , a work Cauchy had studied. Cauchy had looked at the evidence since 1812. This contributed significantly to his election to the academy and to the fact that he was a professor at the École Polytechnique.

Professor at the École polytechnique

The final defeat of Napoleon in 1815 gave Cauchy's career a boost. Louis XVIII became King of France now, and with him restorative forces came to power. As a loyal royalist, Cauchy's father was able to keep his post under the new regime. Scientists with dubious political (i.e. revolutionary) sentiments now had a difficult time. As a strict Catholic, Augustin-Louis did not have these problems, and so he received a position as assistant professor at the École polytechnique in November 1815 and a full professorship in December. In March 1816, the Académie des Sciences was redesigned by the king himself, two liberal members removed and the vacant places occupied by arch-conservative scientists like Cauchy, who took the place of Gaspard Monge .

Cour d'Analyse from 1821

This approach made him no friends. Even if he now had an excellent reputation as a mathematician and his appointments were technically unobjectionable, they still had the stigma of political protection. In addition, Cauchy paid little attention to the opinions of others and was very harsh on the outside, especially against non-Catholics. His supporter Lagrange had died in 1813, and Cauchy managed to make Laplace an enemy as well by calling the methods of Laplace and Poisson as too intuitive and too imprecise. However, he kept a good working relationship with Poisson, who worked in very similar fields, and the two often worked together. He was only close friends with the Catholic Ampère .

As a member of the Académie, one of Cauchy's duties was to review submitted scientific articles. He devoted much of his time to this work, but not necessarily to the delight of the writers. So wrote Niels Henrik Abel : "Cauchy is mad, and you can not do anything about. However, he is currently the only one who knows how to do math. ” Galois and Poncelet had similar bad experiences . It also appeared that Cauchy had partially lost the papers of the young scientists, which he was heavily accused of. Michail Ostrogradski, on the other hand, found only warm words for Cauchy, who even bought the young Russian out of the debt tower several times when he could not pay his rent again.

Cauchy became very eager in class. He thought analysis was a prerequisite for mechanics and other important engineering disciplines. During this time, the textbook Cours d'analyse de l'École Polytechnique was written in 1821 as part of his lectures . He attached great importance to the accuracy of the definitions and introduced much new material, such as his new definition of the derivative , which was based on a limit value and not on the infinitesimal calculus . This met with resistance from the students, for whom Cauchy's lectures were too abstract and too little engineering-oriented; there was also political resentment - once he was even booed. Cauchy had given his 65th lecture of the semester on April 12, 1821. Normally there were 50 lectures per semester, each consisting of 30 minutes of recap and 60 minutes of lecture, and Cauchy had already been lecturing for almost two hours when the students became loud and some left the classroom, whereupon an investigation resulted in both sides being partly to blame . More serious was the resistance on the part of more application-oriented professors like Navier , while on the part of Cauchy only Ampère actively supported him, so that finally a change in the curriculum towards more application-oriented mathematics was implemented.

From 1824 to 1830 he taught part-time at the Collège de France and he also represented Poisson at the Sorbonne.

In April 1818 he married Aloise de Bure (died 1863), the daughter of a respected bookseller and publisher, in whose publishing house Cauchy later published a lot. The two had two daughters, Marie Françoise Alicia (born 1819), later married to Félix d'Escalopier, and Marie Mathilde (born 1823), married to Alfred de Saint-Pol. They had a town house on rue Serpente in Paris (the De Bures house) and a summer residence in Sceaux .

Exile after 1830

In the July Revolution of 1830 , the reactionary King Charles X was overthrown and replaced by the "citizen king" Louis Philippe . The students of the École Polytechnique played a not insignificant role in the street fighting. All of this was too much for Cauchy. He left town in September, leaving his family behind. First he went to Switzerland , to Freiburg , a stronghold of the Jesuits. A return to France now, however, required an oath of loyalty to the new regime, which was out of the question for him. Cauchy stayed away from his family in exile. He lost his post and went to Turin in 1831 , where he was appointed to a chair in theoretical physics . In 1832 Cauchy was elected to the American Academy of Arts and Sciences . As early as 1833 he left the city to join Charles X on the Hradschin in Prague and became tutor of his grandson Henri d'Artois , Duke of Bordeaux.

Charles X had abdicated in August 1830 and declared his grandson heir to the throne. From the age of 14 he raised the right to the title of King of France. Accordingly, his upbringing was a political issue that was also closely followed in France, where some nobles preferred the Bourbons to the throne over Louis-Philippe. Cauchy was chosen because of his scientific merits and his closeness to the Jesuits to teach the prince in mathematics and the natural sciences, especially chemistry and physics. He took this task very seriously, just as he lively supported the prince's claim to the throne. So he prepared conscientiously for the lessons and did next to no research in those years. Here, too, as in Paris and Turin, his lack of talent as a teacher was evident. The prince showed no interest or aptitude for mathematics and understood precious little of what Cauchy was telling him. By the time he was 18 when his education ended, he developed an extensive aversion to mathematics.

In 1834 Augustin-Louis brought his family to join him, whom he had only seen on rare visits to Paris in the previous four years. Two years later, the exiled king's entourage moved on to Gorizia , where the prince celebrated his 18th birthday in 1838. For Cauchy this meant the end of his life as a tutor. Charles X rewarded him for his services with the title of baron , to which Cauchy subsequently attached great importance. Because of the poor health of his mother, who died in 1839, he returned to Paris.

One publication every week

Cauchy was now in the difficult situation that he was no longer a professor because of his refusal to swear the oath of allegiance to the king. Although he was still a member of the Académie des Sciences and was able to participate in scientific life and publish, he could not apply for a new position. An exception was the Bureau des Longitudes , where he expected a loose handling of the oath of allegiance, so that Cauchy applied there. At the end of 1839 he succeeded, but the government insisted on the oath. For the next four years this was ignored at the Bureau; So Cauchy was now a professor again, albeit without a salary .

This began one of his most creative periods. Cauchy had published next to nothing in Prague, although he had thought about a lot, and he was now putting the mature ideas on paper. The Académie had set up a journal, the Comptes Rendus , in which members could publish quickly. Cauchy took advantage of this like no other: between 1839 and February 1848 he published over 300 articles. If you take into account that he did not do research in 1844, almost one article per week remains, an unbelievable speed of creation. He must have so inundated this journal with treatises that in future the number of pages per treatise was limited to four.

Lacroix died in 1843 and a professorship at the Collège de France became vacant. There were three applicants - Liouville , Cauchy and Libri , who had already represented Lacroix and had shown his lack of competence there. He later gained dubious fame as a book thief. During this time, the Jesuits tried to enforce their ideas about teaching at French universities. Cauchy supported this project emphatically and with his own commitment. Libri, on the other hand, was an avowed opponent of the Jesuits, and for this reason Libri was appointed professor. The Ministry took this as an opportunity to remove Cauchy from the Bureau des Longitudes. He then devoted the next year to supporting Jesuit politics.

It was not until the February Revolution of 1848 , which overthrew the citizen king Louis-Philippe, that his situation improved again.

The last few years

Photograph of Cauchy from his later years

The February Revolution did not bring his former student Henri von Bourbon to power, as Cauchy had hoped, but rather Charles-Louis-Napoléon Bonaparte (from 1852 Emperor Napoléon III). Initially, however, no new oaths of loyalty were required and Cauchy was able to become professor of mathematical astronomy at the Sorbonne in 1849 after Urbain Le Verrier had switched to a chair in physical astronomy (probably a well-prepared maneuver, according to the biographer of Cauchy Belhost). When Napoleon III. Cauchy did not want to swear allegiance to him when he became emperor in 1852, but an exception was made for him. For his family, however, the February Revolution was a severe blow: his father and his two brothers, who had been high-ranking officials for almost 50 years since the coup of Napoleon Bonaparte and had survived every regime change, lost their posts this time. Louis François Cauchy died shortly afterwards in December 1848.

In 1850, like Liouville, Cauchy again applied for a professorship in mathematics at the Collège de France - Libri had fled. Liouville was elected and an ugly argument broke out between the two of them. Cauchy refused to accept his defeat (the first vote had eleven votes for him, ten for Liouville and two abstentions). The two then got into a scientific dispute: In 1851 Cauchy presented some results to Charles Hermite on double-period functions and proved them using his integral theorem . Liouville believed he could infer the results directly from his Liouville theorem . Cauchy, on the other hand, showed that Liouville's theorem can be proved very easily with the integral theorem.

Cauchy exerted a significant influence on the young mathematicians in France: Even in his last years, when he did little research, he evaluated many submitted articles and criticized them extensively. Cauchy had also tried in recent years to bring his colleagues back to the Catholic faith. He had succeeded in doing this with the mathematician Duhamel . It was with him, of all people, that he had a priority dispute in December 1856, which Ostrogradski was able to resolve to Cauchy's disadvantage. Refusing to admit his mistake, Cauchy became the target of much of the hostility that overshadowed his final months.

He died in Sceaux near Paris in 1857 with his family. After his death, he was honored by adding his name to the list of 72 names on the Eiffel Tower .

Cauchy's estate went into the family of his eldest daughter Alicia (and then into that of their daughter, who married into the Leudeville family). They sent the scientific papers to the Academy of Sciences in 1936 or 1937 because they couldn't do anything with it. Unfortunately, the academy, which had shown great interest in the family at the time of Gaston Darboux, sent the estate back to the family immediately and it was then burned. Only a few notebooks survived. In 1989 some of the private correspondence with his family was rediscovered.

Cauchy was accepted into the Göttingen Academy of Sciences in 1840 at the suggestion of Gauss and was also elected to the Berlin Academy in 1836 (after a first attempt in 1826 failed because there were as many yes and no votes).

plant

Cauchy's work is remarkable: it includes almost 800 articles and various books. It was published in 27 volumes in the Œuvres complètes (Gauthier-Villars, Paris 1882–1974) over the course of almost 100 years .

Cauchy drew inspiration for his research from two sources, mathematics and physics. The great mathematicians before him, such as Euler or Lagrange, had worked without clean mathematical definitions, as they are a matter of course today, and used a lot of intuitive understanding of functions, differentiability or continuity. Cauchy noticed these gaps in preparation for his lectures, and so he was the first to place analysis on a strict methodological basis - one of his great scientific achievements, which is why he is considered one of the first modern mathematicians.

If one had previously argued more intuitively with infinitesimal units, Cauchy introduced limit values ​​for the definition of continuity and differentiability in his lectures Cours d'analysis de l'École Polytechnique (1821) . This enabled an exact definition of the problem and the provability of the theories used.

With the Cours d'Analyse begins the age of rigor and arithmetic analysis . Only the concept of (locally) uniform convergence is missing to give the work the finishing touch. Ignoring this term, Cauchy incorrectly formulated the theorem that convergent series of continuous functions always have continuous limit functions (Cauchy's sum theorem). He wrote about his approach in the Cours d'Analyse : Quant au méthodes, j'ai chercher à leur donner tout la rigueur qu'on exige en géométrie, de manière à ne jamais recourir aux raisons tirées de la généralité de l'algèbre (What As far as methods are concerned, I have tried to give them the rigor that is required in geometry, without always resorting to considerations that result from the general validity of algebra.) The frequently quoted sentence on the one hand represents that of the mathematicians from Euclid in geometry The usual rigor of the methods compared to the flexible methods of the algebraic analysis of the 18th century (Euler, Lagrange), which made the various discoveries in this area possible.

A large part of Cauchy's scientific contributions are listed in his three works Cours d'analysis de l'École Polytechnique (1821), Exercises de mathématique (5 volumes, 1826-30) and Exercises d'analysis et de physique mathématique (4 volumes) written by Cauchy as part of his lectures at the École Polytechnique. The Exercices were more of a kind of private research journal by Cauchy, who was dissatisfied with the fact that the Academy of Sciences was relatively slow to accept his work, which was produced in quick succession, for publication.

From d'Cours analysis of 1821 only appeared a band since the Ecole Polytechnique soon after her under the pressure of more application-oriented de Prony and Navier the curriculum changed with less focus on the basics, what Cauchy responded with new textbooks of the presentation The basics were greatly reduced. His basic work was therefore never used as a textbook at the École Polytechnique.

The following is an example of the structure of some of the lectures from 1821, which already reflect a large part of his research. The most important contributions in his treatises mainly concern sequences and series as well as complex functions.

COURS D'ANALYSE DE L'ECOLE ROYALE POLYTECHNIQUE	Lecture in analysis at the royal polytechnic college
Première part	First part
Analysis algébrique	Algebraic Analysis
1. Des fonctions réelles.	1. Real functions
2. Des quantités infiniment petites ou infiniment grandes, et de la continuité des fonctions. Valeurs singulières des fonctions dans quelques cas particuliers.	2. Infinitely small or infinitely large sizes. Singular function values ​​in certain cases.
3. Des fonctions symétriques et des fonctions alternées. Usage de ces fonctions pour la résolution des equations du premier degré à un nombre quelconque d'inconnues. Des fonctions homogènes.	3. Symmetrical and alternating functions. Use these functions to solve first degree equations with multiple unknowns. Homogeneous functions.
4. Détermination des fonctions entières, d'après un certain nombre de valeurs particulières supposées connues. Applications.	4. Complete determination of entire functions using individual known function values. Applications.
5. Détermination des fonctions continues d'une seule variable propres à vérifier certaines conditions.	5. Determination of continuous functions with one variable under consideration of certain conditions.
6. Des séries (réelles) convergentes et divergentes. Règles sur la convergence des séries. Sommation de quelques séries convergentes.	6. Real divergent and convergent series. Rules of convergence of series. Summation of selected convergent series.
7. Des expressions imaginaires et de leurs modules.	7. Complex expressions and their amounts.
8. Des variables et des fonctions imaginaires.	8. Complex variables and functions.
9. Des séries imaginaires convergentes et divergentes. Sommation de quelques séries imaginaires convergentes. Notations employées pour représenter quelques fonctions imaginaires auxquelles on se trouve conduit par la sommation de ces mêmes séries.	9. Complex convergent and divergent series. Summation of selected convergent complex series. Notation used to represent certain complex functions that occur in series summation.
10. Sur les racines réelles ou imaginaires des equations algébriques dont le premier membre est une fonction rationnelle et entière d'une seule variable. Résolution de quelques equations de cette espèce par l'algèbre ou la trigonometry.	10. Real or complex roots of algebraic equations whose first term is a whole rational function of a variable. Algebraic or trigonometric solution of such equations.
11. Décomposition des fractions rationnelles.	11. Decomposition of rational fractions.
12. Des séries récurrentes.	12. Recursive sequences.

Sequences and ranks

In the theory of sequences and series , Cauchy developed many important criteria for their convergence.

The Cauchy sequence is fundamental to the theory of sequences and series . In the Cours d'analysis, Cauchy used the Cauchy criterion for series, which can be applied analogously to sequences in order to show their convergence. However, he did not give any real proof that Cauchy sequences converge in R. Bernard Bolzano had already proven in 1817 that the limit value of a Cauchy sequence must be clearly determined, but both Bolzano and Cauchy apparently assumed the existence of this limit value in R as clearly given. Only in the theory of real numbers founded by Eduard Heine and Georg Cantor (cf. construction of R from Q ) was this deficiency eliminated by simply defining R as a set of (equivalence classes of) fundamental sequences. In honor of Cauchy, these have since been called Cauchy episodes. In the early 1970s, there was controversy over Ivor Grattan-Guinness ' claim that Cauchy plagiarized Bolzano.

Cauchy showed the convergence of the geometric series and derived the quotient criterion and the root criterion from it. The latter means that a series of real numbers converges if, starting with an nth summand in the series, the nth root of this summand is smaller than a number smaller than 1. Most of the time, the root criterion can be checked practically with the help of the limit value of the nth root.

Cauchy-Hadamard's formula follows a similar idea, with which one can determine the radius of convergence of a power series . It is calculated as the upper limit of the quotient of two neighboring coefficients of a power series.

The limit value set of Cauchy finally states that the arithmetic mean of the elements of a convergent sequence tends to the limit of this sequence.

The Cauchy compression rate can be a criterion specifies how selected members of a series (hence compacted) as a criterion for a strictly monotonically falling number is used.

In the series product set, he proved for the first time that the so-called Cauchy product series of two convergent series also converges under special conditions. This proof is often used for the convergence analysis of power series.

In addition to the series product theorem, Cauchy also provided further information about the power series. Above all, he proved Taylor's theorem for the first time with formal rigor and in this context developed the Cauchy residual term of a Taylor series .

He was the first to strictly prove the convergence of the sequence already examined by Leonhard Euler , the limit of which is Euler's number . ${\displaystyle \textstyle (1+{\frac {1}{n}})^{n}}$  ${\displaystyle e}$

A special application of convergent sequences can be found in Cauchy's principal value , with the help of which integrals of functions with poles can be determined. One investigates here whether the integral of the function converges in the vicinity of the pole.

Differential and integral calculus

Cauchy's definition of the derivative as a limit value can also be found in the Cours d'Analyse . His contemporaries Lagrange and Laplace had defined the derivative using Taylor series because they assumed that a continuous function could be represented uniquely by an infinite Taylor series, the derivative was then simply the second coefficient of the series. Cauchy refuted this assumption for the first time.

In the integral calculus , Cauchy was also the first (also in the Cours d'Analyse ) to use a definition of a limit value process in which the integration interval is divided into ever smaller sub-intervals and the length of each sub-interval is multiplied by the function value at the beginning of the interval. Cauchy also developed the Cauchy formula for multiple integration .

At the end of the 20th and beginning of the 21st century there was a renaissance of research on Cauchy and a reassessment of his numerous contributions to analysis in the context of conceptualizations of his time (and less from the point of view of later development, for example in the Weierstrass School). One aspect of this is the controversial debate about a possible interpretation of Cauchy in the sense of the later non-standard analysis . Cauchy uses the term infinitely small size explicitly in his Cours d'Analyse. Abraham Robinson and Imre Lakatos already investigated the question of whether some well-known errors (from a later point of view) in Cauchy's work were based on the fact that one should take Cauchy's use of infinitesimals seriously (a form of non-standard analysis). This was also represented by another pioneer of non-standard analysis, Detlef Laugwitz , and for example by Detlef Spalt (who later interpreted Cauchy a little differently with a functional concept that was still radically different from his contemporaries). Among other things, it was about the so-called Cauchy's sum theorem, which, in the usual interpretation of analysis, was the false assertion of Cauchy in his Cours d'analysis of 1821 that a convergent series of continuous functions would be continuous, for which Abel already gave a counterexample in 1826. If one replaces point by point with uniform convergence, the sentence can be saved ( Philipp Ludwig Seidel , George Gabriel Stokes 1847), and the debate centered on whether Cauchy was also correct here, if one assumed that he had interpreted him in the sense of nonstandard analysis (see also the History section in Uniform Convergence ). The majority of Cauchy researchers reject this as an example of retrospective interpretation from a modern point of view, but also developed a much more nuanced picture of Cauchy's understanding of analysis. Examples of recent mathematical historians who have dealt intensively with Cauchy's contributions are Ivor Grattan-Guinness, Hans Freudenthal , Judith Grabiner , Umberto Bottazzini , Frank Smithies (especially function theory) and Amy Dahan-Dalmédico (especially the applications in physics and the group concept) . Spalt, who moved away from Laugwitz's point of view in the 1990s, tried to understand Cauchy from his own conceptual system and pointed out that he used a different functional term than is common today, but which also differs radically from that of the then common algebraic analysis differed (paradigm shift) and which he took over from his teacher Lacroix. He understood (so Spalt) function values ​​as extended quantities, which in turn depended on other extended quantities (the variables), and interpreted Cauchy's proof of the sum theorem in the sense of the concept of continuous convergence, later introduced by Constantin Carathéodory , from which the uniform convergence follows. Cauchy himself came back to the sums theorem in 1853, and this work was seen by Grattan-Guinness and Bottazzini as the beginning of uniform convergence, but this is also controversial.

Grabiner pointed out in particular that epsilontics in the strict justification of analysis goes back to Cauchy, even if this is not always clear, since Cauchy used a wide variety of methods and did not express a lot in formulas, but rather described it with words. There were approaches to this already in the 18th century with the help of inequalities (d'Alembert, Euler, Lagrange et al.) And two precursors of Cauchy (Gauß and Bolzano) came close to him, but it was essentially Cauchy who systematized this and strictly reasoned.

Cauchy gave the first proof of the mean value theorem in differential calculus (1823 in his lectures on calculus).

Function theory

Leçons sur le calcul différentiel , 1829

Cauchy's achievements in the field of function theory , i.e. the study of complex functions, were groundbreaking. Euler and Laplace had already used the complex plane of numbers in an intuitive way to calculate real integrals, but without being able to justify this procedure with a proof. It was Laplace who piqued Cauchy's interest in this method. Cauchy began to deal systematically with complex functions in 1814. That year he sent an essay ( Mémoire sur les intégrales définies, prises entre des limites imaginaires ) to the French Academy of Sciences, but it was not published until 1825. In the Cours d'Analyse he was the first to formally define a function of complex variables and was in fact the only one who systematically dealt with function theory until around 1840 ( Carl Friedrich Gauß also dealt with it and was familiar with many of the results of Cauchy and beyond, published but nothing until 1831). Its contribution to this area is correspondingly large.

In his famous essay Sur les intégrales définies in 1814, he began to integrate real functions using rectangles in the complex number plane in order to calculate real integrals. This is where the Cauchy-Riemann differential equations appear for the first time , which combine complex differentiability and partial differential equations: A complex-valued function is complexly differentiable if and only if it is totally differentiable and satisfies the above-mentioned system of Cauchy-Riemann equations. What follows is a proof of Cauchy's integral theorem for the rectangle. Finally, the article deals with the case that the function has simple poles in the rectangle and contains the residual theorem for the case of integration over a rectangle. The first published example of an evaluation of an integral by means of an integration path in the complex came from Siméon Denis Poisson (1820), who, however, was familiar with the then unpublished work by Cauchy.

He continued to pursue these approaches over the next ten years and generalized them to any integration paths (assuming that the Jordanian curve theorem applies) and also to multiple poles.

All holomorphic functions can be differentiated as often as desired with the help of Cauchy's integral formula . With these derivatives one can then represent holomorphic functions as power series.

With Cauchy's majorant method (Calcul des Limites, first published by him in 1831 in a work on celestial mechanics), the existence of the solutions of a differential equation with a holomorphic function as the right side can be investigated. The basis for this is the power series expansion of the solution (see also the section on differential equations).

Cauchy saw the complex numbers as purely symbolic expressions. He did not use the geometric interpretation until 1825. Later (in the Compte rendu 1847) he tried to further reduce the use of complex numbers to real sizes by interpreting them as arithmetic modulo in the ring of polynomials influenced by the number theoretical work of Gauss . That was an anticipation of the later work of Leopold Kronecker . ${\displaystyle X^{2}+1}$

Differential equations

The Cauchy problem is named after him, these are initial value problems in which the solutions are sought for the entire space. He may have had the ideas for the initial value problem named after him in his great treatise and academy prize paper on waves in liquids from 1815. The essentially new finding was that the existence of a solution could be proven (even if the solution was not known) and had to and whose uniqueness had to be ensured by special initial and boundary value conditions.

To prove the existence and uniqueness of solutions to differential equations, he used two methods. For the initial value problem he used Euler's polygon method (sometimes also named after Cauchy). He developed this in the 1820s and presented it in the first volume of his Exercices d'Analyse. Cauchy assumed the continuity of the function and its derivation, which was relaxed by Rudolf Lipschitz in 1875 (Lipschitz condition) and the sentence after Cauchy and Lipschitz named, but also after Émile Picard and Lindelöf ( theorem of Picard-Lindelöf ). His second method had a broader range of application and was also used by him in the complex, his calcul des limites , developed by him in several papers in the Comptes Rendus from 1839 to 1842 (later also referred to as the method of the majorant function). In the initial value problem given above (with an analytical function ) this would correspond to a Taylor expansion around the initial value at a point , with the higher derivatives in the coefficients of the Taylor series being obtained by successive derivation of the differential equation, evaluated at the point . The method was simplified by Charles Briot and Jean-Claude Bouquet and its representation later became the standard form. Cauchy probably also knew a third method, which is now named after Picard (the iteration method of the method of successive approximation, first used by Joseph Liouville). Cauchy transferred his method of Calcul des Limites to partial differential equations, which he initially reduced to systems of differential equations. An existence theorem on the Cauchy problem of partial differential equations is named after him and Sofja Kowalewskaja (she found the theorem independently in 1875 and in a somewhat improved form) ( Cauchy-Kowalewskaja theorem ). Cauchy published a series of papers on this in 1842 in the Academy's Comptes Rendu. ${\displaystyle {\dot {y}}=f(t,y(t)),\,y(t_{0})=y_{0}}$ ${\displaystyle f}$ ${\displaystyle P_{0}}$ ${\displaystyle P_{0}}$

In the case of partial differential equations of the first order, he was one of the founders of the method of characteristics in 1819 (independently of Johann Friedrich Pfaff ). In the case of two variables, however, this was already known to Gaspard Monge and also to Ampère.

Cauchy mainly investigated linear partial differential equations with constant coefficients, which he found in applications such as hydrodynamics, elasticity theory or optics, already understood as operator equations and primarily with the method of Fourier transformation (which he first applied to ordinary differential equations) and from 1826 also treated with his residual calculus. Cauchy used the Fourier transform method extensively and used it with greater skill than any of his contemporaries including Fourier and Poisson. He also wrote the first correct formulation of the inverse formula, which he said he found independently of Fourier, but named after him.

Functional equations

In Chapter 5 of his analysis algébrique , Cauchy examined the four functional equations

${\displaystyle {\begin{aligned}\Phi (x+y)&=\Phi (x)+\Phi (y)\\\Phi (x+y)&=\Phi (x)\cdot \Phi (y)\\\Phi (x\cdot y)&=\Phi (x)+\Phi (y)\\\Phi (x\cdot y)&=\Phi (x)\cdot \Phi (y)\end{aligned}}}$

and proved that the continuous solutions have the form , (with positive ), respectively . For the first of these functional equations, the name Cauchy functional equation or Cauchy's functional equation has since become established. ${\displaystyle \Phi (x)=ax\;}$ ${\displaystyle \Phi (x)=a^{x}\;}$ ${\displaystyle a\;}$ ${\displaystyle \Phi (x)=a\log x\;}$ ${\displaystyle \Phi (x)=x^{a}\;}$

Contributions to physics

His research in elasticity theory was fundamental for today's applications. Cauchy developed the stress tensor of a cube, with the 9 parameters of which the stress at one point of an elastic body can be fully described. In contrast, the Cauchy number indicates the ratio of inertial forces to elastic forces when the sound vibrates in a body. According to Cauchy's similarity model, two bodies have the same elasticity behavior if they have the same Cauchy number. The importance of this finding is that it is possible to use models to investigate the stability of real structures. The theoretical knowledge of Cauchy in the theory of elasticity made possible the structural research of Navier at the École Polytechnique and others. His biographer Hans Freudenthal considered this to be his greatest contribution to science. In continuum mechanics, Cauchy-Euler's laws of motion are named after him and Euler, and Cauchy elasticity .

Cauchy's research on light also has a certain connection with the theory of elasticity. At that time the aim was to investigate the nature of light waves with the aid of dispersion , i.e. the wavelength-dependent propagation speed of light when passing through a prism . Cauchy had already investigated wave equations in 1815 and was primarily concerned with linear partial differential equations in his studies of elasticity, which he was able to use for the investigation of light waves. It was assumed that the space had to be filled with a medium comparable to a liquid, the so-called ether , because the waves needed a carrier for their propagation. From this research Cauchy empirically derived a simple connection between the refractive index of the prism and the wavelength of light.

Cauchy also dealt with celestial mechanics, where he also made detailed perturbation calculations. In 1845 he checked the complex orbit calculation of the asteroid Pallas by Urbain Le Verrier with a simpler method.

Other services

The Cauchy distribution or t-distribution with one degree of freedom is characterized by the fact that it has no moments. The integral of the expectation values ​​does not converge here.

The Cauchy-Schwarz inequality states that the absolute value of the scalar product of two vectors is never greater than the product of the respective vector norms. This knowledge is used, for example, as the basis for the correlation coefficient in statistics.

A valuable contribution to stochastics is the principle of convergence with probability 1, with which a sequence of random variables almost certainly converges to a random variable.

In geometry he proved around 1812 that convex polyhedra are rigid (Cauchy's theorem of rigidity for polyhedra). At that time he also gave one of the first strict proofs of Euler's polyhedron substitution. The set of Cauchy is the area of a convex body as means over the areas of the parallel projections and is an early result of integral geometry .

In linear algebra he published a treatise on determinants (1812), thereby popularizing the term and proving fundamental properties (such as the determination of the matrix inverse with its help and the determinant product theorem at the same time as Binet: Binet-Cauchy theorem ). In 1829, at the same time as Carl Gustav Jacobi, he published the general theory of the principal axis transformation of a square shape through orthogonal transformations, which generalized and unified earlier studies by Euler and Lagrange. Cauchy also proved that the eigenvalues ​​of a symmetric n × n matrix are real. Cauchy's work was related to second-order n-dimensional surfaces and was also one of the first works on n-dimensional geometry. In 1815 he also founded the theory of permutations (which he initially referred to as substitution and only later, as it is today, as permutation) and introduced the terms common today, including cycle representation. He came back to this in his exercises in the 1840s. He already considered special substitutions, conjugated substitutions and the interchangeability properties, but did not yet break through to the group concept, which only developed with Arthur Cayley (who in turn built on Cauchy). In these investigations, he followed up on Lagrange. A contribution to Cauchy's group theory is Cauchy's theorem of 1845.

In 1815 he published a proof of Fermat's polygonal number theorem , which helped to cement his reputation. He also tried the Fermat conjecture and, like Ernst Eduard Kummer, found in the discussion that followed Gabriel Lamé's attempt to prove in 1847 that the unambiguous prime factorization is not always given in the algebraic number fields under consideration. First, however, in March 1847 he competed with Lamé to prove the conjecture, assuming the uniqueness of the prime factorization of into complex factors ( Pierre Wantzel had meanwhile claimed to have found proof of this assumption). On May 17, the always skeptical Joseph Liouville read a letter from Ernst Eduard Kummer to the academy, who announced that he had already proven the ambiguity of the prime factorization three years ago. Lamé recognized this quickly and stopped the publications, Cauchy continued to publish up to August on polynomials in circular fields, but increasingly independent of the question of the Fermat conjecture and Kummer's ideas. ${\displaystyle x^{n}+y^{n}}$ ${\displaystyle n}$

Reception in Germany

In Germany, Cauchy found high recognition both through Gauss, although Gauss did not comment on him in the correspondence he received or otherwise and left the reviews of Cauchy's writings in the Göttinger Gelehrten Anzeiger to others, but prompted his acceptance into the Göttingen Academy, as well as, for example by Carl Gustav Jacobi (who counted him with Gauss to the leading geometers). According to Karin Reich , his textbooks initially met with a mixed, sometimes quite negative reception ( Martin Ohm, for example, in his review of the Cours d'Analyse of 1829, found it bad if one only had to limit oneself to convergent series and the usual algebraic analysis as formal Manipulation of sequences and series in the succession of Euler would sacrifice), and it was not until the 1840s that this changed with the textbooks of Oskar Schlömilch (Handbuch der Algebraischen Analysis 1845) and Johann August Grunert . Although Schlömilch first made Cauchy's innovations generally known in Germany in his textbook, he too missed the beauty of the architectural structure and the life of the invention . As late as 1860, Moritz Abraham Stern admitted in his analysis textbook that Cauchy had ushered in a new era in analysis, but also criticized artificiality, opacity compared to Euler and known errors (Cauchy's sum theorem).

Bernhard Riemann was familiar with the contributions of the French school of Cauchy when he was building up the theory of functions. As a student he read the Cours d'Analyse, used a short note in the Comptes Rendus of the Cauchy Academy of 1851 with the Cauchy-Riemann differential equation (the note was based on many earlier works by Cauchy) and knew and used in later lectures the textbooks by Briot and Bouquet and the work of Puiseux, which expanded Cauchy's teaching. Even if he did not always explicitly refer to it as a source in his publications, but assumed this as general knowledge. In his lectures he also integrated the power series approach to function theory, which is historically linked to Cauchy and Weierstrass, with his own geometrical-potential-theoretical approach (conformal maps, Cauchy-Riemann differential equation as a basis) and was flexible in his choice of means, such as for example Erwin Neuenschwander showed when examining the lecture notes. Conversely, many of the findings of geometric function theory ascribed to Riemann can already be found in Cauchy, even if, as Laugwitz noted, Cauchy made it difficult for himself by deliberately avoiding the geometric interpretation of the complex numbers. There is an anecdote from the memories of James Joseph Sylvester , who referred to a conversation with a former fellow student of Riemann, that Riemann in his Berlin time (1847 to 1849) after reading intensely the recently published works by Cauchy in the Comptes Rendus said that there was a new mathematics in front of you.

Cauchy also valued Weierstrass. In treatises that were first published in his collected works in 1894 and that he wrote as a student in 1841/42, he anticipated essential parts of the theory of functions. He later claimed that he had not read Cauchy's works at the time and was mainly influenced by Abel, but Cauchy's influence was so great that it could have happened indirectly. After that he lived largely in isolation until he was called to Berlin in 1856. In his lectures he adhered primarily to his own system and his own research and adapted the research of others to it, so that his student Leo Koenigsberger once complained about the many Cauchy's discoveries not having learned anything. It was the Weierstrasse School that had the greatest international influence in the second half of the 19th century.

Honors

The lunar crater Cauchy , the asteroid (16249) Cauchy and the Rupes Cauchy are named after him.

literature

• Bruno Belhoste : Augustin-Louis Cauchy. A biography. Springer, New York 1985, 1991, ISBN 3-540-97220-X
• Umberto Bottazzini : Geometrical Rigor and 'modern' analysis. ” An introduction to Cauchy's Cours d'analysis, foreword to the facsimile edition of Cauchy's Cours d'Analyse, Bologna 1990
• Amy Dahan-Dalmédico : Mathematisations: Augustin-Louis Cauchy et l'École Française. Ed. du choix, Argenteuil 1992 & Albert Blanchard, Paris 1992
  • Review by Craig Fraser , in Isis. A Journal of the History of Science , 86, 1995, pp. 501f. doi : 10.1086 / 357285
• Giovanni Ferraro: The rise and development of the theories of series up to the early 1820s, Springer 2008
• Hans Freudenthal : Cauchy, Augustin-Louis . In: Charles Coulston Gillispie (Ed.): Dictionary of Scientific Biography . tape 3 : Pierre Cabanis - Heinrich von Dechen . Charles Scribner's Sons, New York 1971, p. 131-148 .
• Craig Fraser : Cauchy. In: Dictionary of Scientific Biography , Volume 2, Scribners 2008, pp. 75-79
• Judith Grabiner : The Origins of Cauchy's Rigorous Calculus , MIT Press 1981, Dover 2005
• Judith Grabiner: Who gave you the epsilon? Cauchy and the origins of rigorous calculus , Amer. Math. Monthly, Vol. 90, 1983, pp. 185-194. On-line
• Ivor Grattan-Guinness : The development of the foundations of mathematical analysis from Euler to Riemann, MIT Press, Cambridge, 1970
• Ivor Grattan-Guinness, Ivor Cooke (eds.), Landmark writings in the history of mathematics, Elsevier 2005 (therein by Grattan-Guinness: Cours d'analysis and Resumé of the calculus (1821, 1823), by F. Smithies: Two memoirs on complex function theory (1825, 1827)).
• Hans Niels Jahnke (Ed.): A history of analysis, American Mathematical Society 2003 (therein Jesper Lützen : The foundation of analysis in the 19th century, Umberto Bottazzini: Complex function theory 1780-1900)
• Frank Smithies : Cauchy and the creation of complex function theory , Cambridge UP 1997
• Thomas Sonar : 3000 Years of Analysis , Springer 2011 (biography p. 503ff)
• Detlef D. Spalt : Analysis in change and in conflict. A history of formation of their basic concepts , Verlag Karl Alber 2015
• Klaus Viertel: History of Uniform Convergence , Springer 2014

Fonts (selection)

• Oeuvres complètes , 1st series, Paris: Gauthier-Villars, 12 volumes, 1882 to 1900, series 2, 15 volumes (published until 1974), series 1, digitized, ETH , Gallica-Math
  • In 1981, previously unpublished lectures by Cauchy at the École Polytechnique on differential equations from the beginning of the 1820s were also published in the Gesammelte Werken: Equations différentielles ordinaires. Cours inédits. Fragment , Paris: Études Vivantes, New York: Johnson Reprint, 1981 (foreword Christian Gilain)
• Mémoire sur les intégrales définies, prices entre des limites imaginaires , Paris: De Bure 1825, Archives
  • It appeared in 500 copies and had 69 pages. A reprint was made in Bulletin des sciences mathématiques 1874, Volume 7, pp. 265-304, Volume 8, pp. 43-55, 148-159 and in the Oeuvres, Series 2, Volume 15, 1974, pp. 41-89
  • German edition: Treatise on certain integrals between imaginary borders , Ostwald's classic, Ed. Paul Stäckel , Leipzig 1900, archive
• Mémoire sur les intégrales définies , Mémoires présentés par divers savants à l'Académie des Sciences, Ser. 2, Volume 1, 1827, pp. 601–799 (reprinted in the Oeuvres, Series 1, Volume 1, 1882, pp. 319–506, it essentially dates from 1814)
• Cours d'analysis de l'École royale polytechnique , Volume 1, Paris: Imprimerie Royale 1821, Archives
  • German translation: Textbook of Algebraic Analysis , Königsberg 1828 (translator CLB Huzler), digitized version , Berlin edition: Springer 1885 (Ed. Carl Itzigsohn) SUB Göttingen
  • English translation with commentary: Robert Bradley, Edward Sandifer: Cauchy's Cours d'Analyse: An annotated translation , Springer 2009
• Résumée des lecons données a l'école royal polytechnique sur le calcul infinitésimal , Paris: De Bure 1823
• Leçons sur les applications du calcul infinitésimal à la géométrie , Paris: Imprimerie Royale 1826, Archives
  • German translation by Heinrich Christian Schnuse: Lectures on the Application of Differential Enrichment in Geometry , 1840
• Exercices de mathématiques , 5 volumes, Paris, De Bure fréres 1826 to 1830, Archives, Volume 1
• Leçons sur le calcul différentiel , Paris 1829
  • A German translation by Heinrich Christian Schnuse came out in Braunschweig in 1836: Lecture on differential calculus combined with Fourier's methods of solving certain equations .
• Leçons de calcul différentiel et de calcul intégral , Paris 1844, Archives
  • German translation by Schnuse: Lecture on integral calculus , Braunschweig 1846
• Exercices d'analysis et de physique mathématique , 4 volumes, Paris: Bachelier, 1840 to 1847, Archives, Volume 1

Web links

• Literature by and about Augustin-Louis Cauchy in the catalog of the German National Library
• Works by and about Augustin-Louis Cauchy  in the German Digital Library
• John J. O'Connor, Edmund F. Robertson :  Augustin-Louis Cauchy. In: MacTutor History of Mathematics archive .
• Augustin-Louis Cauchy , entry in Bibm @ th –La bibliothèque des mathématiques
• Cauchy, Augustin (1789-1857) on the MathCS.org websiteby Bert G. Wachsmuth
• Entry in the catalog of the Bibliothèque nationale de France

Individual evidence

1. ↑ ^{a b} Claude Alphonse Valson: La vie et les travaux du baron Cauchy: membre de la̕cadémie des sciences , Gauthier-Villars, Paris 1868, p.18 Google Digitaisat
2. ^ Gerhard Kowalewski : Great mathematicians. A walk through the history of mathematics from ancient times to modern times. 2nd Edition. JF Lehmanns Verlag, Munich / Berlin 1939
3. ↑ Belhost, Cauchy, p. 46
4. ↑ There were two semesters a year and the semester started in November and actually ended in March, so it was overdone. For instruction at the École Polytechnique, see Belhost, La formation d'une technocratie, Belin, 2003, p. 372
5. ↑ Belhost, Cauchy, 1991, p. 224
6. ↑ Belhost, Cauchy, 1991, p. 363
7. ↑ after Reinhold Remmert: Funktionentheorie I. Springer, Berlin 1984, ISBN 3-540-12782-8
8. ^ Cauchy, Cours d'Analyse, 1821, Introduction, p. Ii
9. ↑ Robert Bradley, Edward Sandifer, Preface to the new edition of the Cours d'Analyze, Springer 2009, p XII
10. ↑ Robert Bradley, Edward Sandifer: Cauchy's Cours d'Analyze: An annotated translation, Springer 2009, p VIII
11. ↑ Grattan-Guinness: Bolzano, Cauchy and the "new analysis" of the early nineteenth century, Archive for History of Exact Sciences, Volume 6, 1970, pp. 372-400.
12. ↑ Hans Freudenthal: Did Cauchy plagiarize Bolzano? Archive for History of Exact Sciences, Volume 7, 1971, pp. 375-392.
13. ↑ For example, the addendum by Craig Fraser to the more recent Cauchy research in the new Dictionary of Scientific Biography 2008 as a supplement to the older essay by Hans Freudenthal, who already listed numerous contributions by Cauchy in the 1970s and was an outspoken admirer of his work.
14. ^ Definition of infinitement petit in the 1821 edition, p. 4
15. ↑ Lakatos, Cauchy and the Continuum, Mathematical Intelligencer, 1978, No. 9. The essay was originally from 1966 and is based on discussions with Abraham Robinson.
16. ↑ Laugwitz, Infinitely small quantities in Cauchy's textbooks, Historia Mathematica, Volume 14, 1987, pp. 258-274
17. ^ Laugwitz, Definite Values ​​of Infinite Sums: Aspects of the Foundations of Infinitesimal Analysis around 1820, Archive for History of Exact Sciences, Volume 39, 1989, pp. 195–245
18. ↑ Spalt, Cauchy's Continuum, Arch. Hist. Exact Sciences, Vol. 56, 2002, pp. 285-338.
19. ↑ Klaus Viertel, History of Uniform Convergence, 2015, p. 32f, with Viertel's own analysis of Cauchy's work from 1853
20. ^ Grabiner, Who gave you the Epsilon? Cauchy and the origins of rigorous calculus, American Mathematical Monthly, March 1983, p. 185
21. ↑ Jean-Luc Verley, Analytical Functions, in: Geschichte der Mathematik 1700-1900, Vieweg 1985, p. 145. There is particular reference to a letter to Bessel in 1811.
22. ^ Nahin, An imaginary tale, Princeton UP 1998, p. 190
23. ^ Nahin, An imaginary tale, 1998, p. 196
24. ↑ Verley, Analytical Functions, in Dieudonne, Gesch. der Math., Vieweg 1985, p. 145
25. ↑ Verley in Dieudonné, Gesch. der Math., 1985, p. 146
26. ↑ Freudenthal, Dict. Sci. Biogr.
27. ^ Morris Kline , Mathematical thought from ancient to modern times, Oxford UP 1972, pp. 717ff
28. ^ Cauchy-Lipschitz theorem , Encyclopedia of Mathematics, Springer
29. ^ Kline, Mathematical thought ..., pp. 718f
30. ^ Freudenthal, Dictionary of Scientific Biography
31. ^ Morris Kline, Mathematical thought ..., 1972, p. 703
32. ^ Freudenthal, Article Cauchy, Dictionary of Scientific Biography. In the era of Weierstrass, however, this method for solving differential equations according to Freudenthal took a back seat in favor of other methods.
33. ↑ The treatment of the Fourier transformation including the inverse formula can be found in his price publication from 1815 on water waves, printed in series 1, volume 1 of the works
34. ^ Freudenthal, Cauchy, Dictionary of Scientific Biography. After Freudenthal his most famous achievement in astronomy.
35. ↑ See Aigner, Ziegler: The Book of Proofs , in which Cauchy's proof is presented
36. ↑ Cauchy IIe mémoire sur les polygones et les polyèdres , J. Fac. Polytechnique, Vol. 9, 1813, pp. 87-98.
37. ↑ H.-W. Alten, H. Wussing a. a., 4000 years of algebra, Springer 2003, p. 401
38. ↑ Alten u. a., 4000 years of algebra, Springer, 2003, p. 400
39. ↑ Belhoste, Cauchy, p. 212
40. ↑ Jacobi is often quoted as saying that Dirichlet alone would know what rigor would be in a mathematical proof; if Gauss were to call a proof strict, he (Jacobi) would probably be both pro and con with Cauchy Dirichlet sure.
41. ↑ Reich, Cauchy and Gauss. Cauchy's reception in the vicinity of Gauss. Archive for History of Exact Sciences, Volume 57, 2003, pp. 433-463. To the appreciation of Jacobi p. 453.
42. ↑ Hans Niels Jahnke , Algebraische Analysis, in: D. Spalt, Rechnen mit dem Unendlichen, Springer 1990, pp. 103-122. In Germany there was a firm establishment of algebraic analysis in school lessons and also the special form of combinatorial analysis by Carl Friedrich Hindenburg . The tradition of algebraic analysis in Germany also ended in school lessons with the reforms of Felix Klein .
43. ↑ Jahnke, Algebraische Analysis, in: Spalt, Rechnen mit dem Unendlichen, 1990, p. 118
44. ↑ Laugwitz, Riemann, 1996, p. 91. He borrowed it from Göttingen in 1847.
45. ↑ Laugwitz, Riemann, 1996, p. 86. He even knew the Cauchy-Hadamard formula, but apparently had forgotten the origin of his knowledge in Cauchy's Cours d'Analyse, which he borrowed as a student, and derived it in a more laborious manner. Laugwitz, Riemann, p. 96.
46. ↑ Neuenschwander, Riemann and the “Weierstrasse” principle of analytical continuation through power series, Annual Report DMV, Volume 82, 1980, pp. 1–11. Riemann was also familiar with works by Weierstrass from 1856/57.
47. ^ Neuenschwander Studies in the history of complex function theory II: Interactions among the French school, Riemann, and Weierstrass. , Bulletin of the American Mathematical Society, 1981, pp. 87-105 ( online )
48. ^ Neuenschwander: About the interactions between the French school, Riemann and Weierstrasse. An overview with two source studies, Archive for History of Exact Sciences, Volume 24, 1981, pp. 221-255.
49. ↑ Laugwitz, Riemann, Birkhäuser 1996, p. 83
50. ↑ Laugwitz, Riemann, 1996, p. 115. Laugwitz deals with p. 114ff, the influence of Cauchy on Riemann.
51. ^ Neuenschwander, On the Interactions of the French School, Riemann and Weierstrass, Archive for History of Exact Sciences, Volume 24, 1981, p. 229
52. ^ Neuenschwander, Riemann, Weierstrass and the French, 1981, p. 232
53. ↑ The edition from 1821 has 568 pages and differs in several points from the edition that appeared in the Oeuvres, Series 2, Volume 3 in 1897 and is more of a second edition.
54. ↑ Vice-Rector at the Höhere Stadtschule von Löbenicht in Königsberg
55. ↑ Schnuse was born in Braunschweig in 1808, attended the Collegium Carolinum there and studied mathematics in Göttingen until 1834. In 1835 he received his doctorate from Christian Gerling in Marburg. He was a full-time translator of mathematical works and reviewed them and lived in Heidelberg and Munich, among others. He died around 1878.

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This article was added to the list of excellent articles on June 15, 2005 in this version .