Baron Baron is a rank of nobility or title of honour, often hereditary, in various European countries, either current or historical. The female equivalent is baroness. Typically, the title denotes an aristocrat who ranks higher than a lord or kn ...

Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

, engineer, and

physicist A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate cau ...

who made pioneering contributions to several branches of mathematics, including

mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...

and

continuum mechanics Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such mo ...

. He was one of the first to state and rigorously prove theorems of

calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...

, rejecting the heuristic principle of the generality of algebra of earlier authors. He almost singlehandedly founded

complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...

and the study of

permutation group In mathematics, a permutation group is a group ''G'' whose elements are permutations of a given set ''M'' and whose group operation is the composition of permutations in ''G'' (which are thought of as bijective functions from the set ''M'' to ...

s in

abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...

. A profound mathematician, Cauchy had a great influence over his contemporaries and successors;

Hans Freudenthal Hans Freudenthal (17 September 1905 – 13 October 1990) was a Jewish-German-born Dutch mathematician. He made substantial contributions to algebraic topology and also took an interest in literature, philosophy, history and mathematics education ...

stated: "More concepts and theorems have been named for Cauchy than for any other mathematician (in elasticity alone there are sixteen concepts and theorems named for Cauchy)." Cauchy was a prolific writer; he wrote approximately eight hundred research articles and five complete textbooks on a variety of topics in the fields of mathematics and

mathematical physics Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developm ...

Biography

Youth and education

Cauchy was the son of

Louis François Cauchy Louis François Cauchy (27 May 1760 – 28 December 1848) was a senior French government official and the father of the mathematician Augustin-Louis Cauchy. Born on 27 May 1760 in Rouen into an upper-middle-class family, Cauchy successfully s ...

(1760–1848) and Marie-Madeleine Desestre. Cauchy had two brothers: Alexandre Laurent Cauchy (1792–1857), who became a president of a division of the court of appeal in 1847 and a judge of the court of cassation in 1849, and Eugene François Cauchy (1802–1877), a publicist who also wrote several mathematical works. Cauchy married Aloise de Bure in 1818. She was a close relative of the publisher who published most of Cauchy's works. They had two daughters, Marie Françoise Alicia (1819) and Marie Mathilde (1823). Cauchy's father was a highly ranked official in the Parisian Police of the

Ancien Régime ''Ancien'' may refer to * the French word for " ancient, old" ** Société des anciens textes français * the French for "former, senior" ** Virelai ancien ** Ancien Régime ** Ancien Régime in France {{disambig ...

, but lost this position due to the

French Revolution The French Revolution ( ) was a period of radical political and societal change in France that began with the Estates General of 1789 and ended with the formation of the French Consulate in November 1799. Many of its ideas are conside ...

(July 14, 1789), which broke out one month before Augustin-Louis was born. The Cauchy family survived the revolution and the following

Reign of Terror The Reign of Terror (french: link=no, la Terreur) was a period of the French Revolution when, following the creation of the First French Republic, First Republic, a series of massacres and numerous public Capital punishment, executions took pl ...

(1793–94) by escaping to

Arcueil Arcueil () is a commune in the Val-de-Marne department in the southern suburbs of Paris, France. It is located from the center of Paris. Name The name Arcueil was recorded for the first time in 1119 as ''Arcoloï'', and later in the 12th ...

, where Cauchy received his first education, from his father. After the execution of

Robespierre Maximilien François Marie Isidore de Robespierre (; 6 May 1758 – 28 July 1794) was a French lawyer and statesman who became one of the best-known, influential and controversial figures of the French Revolution. As a member of the Esta ...

(1794), it was safe for the family to return to Paris. There Louis-François Cauchy found himself a new bureaucratic job in 1800, and quickly moved up the ranks. When

Napoleon Bonaparte Napoleon Bonaparte ; it, Napoleone Bonaparte, ; co, Napulione Buonaparte. (born Napoleone Buonaparte; 15 August 1769 – 5 May 1821), later known by his regnal name Napoleon I, was a French military commander and political leader wh ...

came to power (1799), Louis-François Cauchy was further promoted, and became Secretary-General of the Senate, working directly under

Laplace Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...

(who is now better known for his work on mathematical physics). The famous mathematician

Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaÉcole Centrale du Panthéon, the best secondary school of Paris at that time, in the fall of 1802. Most of the curriculum consisted of classical languages; the young and ambitious Cauchy, being a brilliant student, won many prizes in Latin and the humanities. In spite of these successes, Augustin-Louis chose an engineering career, and prepared himself for the entrance examination to the

École Polytechnique École may refer to: * an elementary school in the French educational stages normally followed by secondary education establishments (collège and lycée) * École (river), a tributary of the Seine flowing in région Île-de-France * École, Savoi ...

. In 1805, he placed second of 293 applicants on this exam and was admitted. One of the main purposes of this school was to give future civil and military engineers a high-level scientific and mathematical education. The school functioned under military discipline, which caused the young and pious Cauchy some problems in adapting. Nevertheless, he finished the Polytechnique in 1807, at the age of 18, and went on to the

École des Ponts et Chaussées École may refer to: * an elementary school in the French educational stages normally followed by secondary education establishments (collège and lycée) * École (river), a tributary of the Seine flowing in région Île-de-France * École, Savo ...

(School for Bridges and Roads). He graduated in civil engineering, with the highest honors.

Engineering days

After finishing school in 1810, Cauchy accepted a job as a junior engineer in Cherbourg, where Napoleon intended to build a naval base. Here Augustin-Louis stayed for three years, and was assigned the

Ourcq Canal The Canal de l'Ourcq is a long canal in the Île-de-France region (greater Paris) with 10 locks. It was built at a width of but was enlarged to 3.7 m (12 ft), which permitted use by more pleasure boats. The canal begins at Port-au ...

project and the Saint-Cloud Bridge project, and worked at the Harbor of Cherbourg. Although he had an extremely busy managerial job, he still found time to prepare three mathematical manuscripts, which he submitted to the ''Première Classe'' (First Class) of the

Institut de France The (; ) is a French learned society, grouping five , including the Académie Française. It was established in 1795 at the direction of the National Convention. Located on the Quai de Conti in the 6th arrondissement of Paris, the institut ...

. Cauchy's first two manuscripts (on

polyhedra In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on ...

) were accepted; the third one (on directrices of

conic sections In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a spe ...

) was rejected. In September 1812, now 23 years old, Cauchy returned to Paris after becoming ill from overwork. Another reason for his return to the capital was that he was losing interest in his engineering job, being more and more attracted to the abstract beauty of mathematics; in Paris, he would have a much better chance to find a mathematics related position. Therefore, when his health improved in 1813, Cauchy chose to not return to Cherbourg. Although he formally kept his engineering position, he was transferred from the payroll of the Ministry of the Marine to the Ministry of the Interior. The next three years Augustin-Louis was mainly on unpaid sick leave; he spent his time quite fruitfully, working on mathematics (on the related topics of

symmetric functions In mathematics, a function of n variables is symmetric if its value is the same no matter the order of its arguments. For example, a function f\left(x_1,x_2\right) of two arguments is a symmetric function if and only if f\left(x_1,x_2\right) = f\ ...

, the

symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...

and the theory of higher-order algebraic equations). He attempted admission to the First Class of the Institut de France but failed on three different occasions between 1813 and 1815. In 1815 Napoleon was defeated at Waterloo, and the newly installed Bourbon king

Louis XVIII Louis XVIII (Louis Stanislas Xavier; 17 November 1755 – 16 September 1824), known as the Desired (), was King of France from 1814 to 1824, except for a brief interruption during the Hundred Days in 1815. He spent twenty-three years in ...

took the restoration in hand. The

Académie des Sciences The French Academy of Sciences (French: ''Académie des sciences'') is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research. It was at ...

was re-established in March 1816;

Lazare Carnot Lazare Nicolas Marguerite, Count Carnot (; 13 May 1753 – 2 August 1823) was a French mathematician, physicist and politician. He was known as the "Organizer of Victory" in the French Revolutionary Wars and Napoleonic Wars. Education and early ...

and

Gaspard Monge Gaspard Monge, Comte de Péluse (9 May 1746 – 28 July 1818) was a French mathematician, commonly presented as the inventor of descriptive geometry, (the mathematical basis of) technical drawing, and the father of differential geometry. During ...

were removed from this Academy for political reasons, and the king appointed Cauchy to take the place of one of them. The reaction of Cauchy's peers was harsh; they considered the acceptance of his membership in the Academy an outrage, and Cauchy thereby created many enemies in scientific circles.

Professor at École Polytechnique

In November 1815,

Louis Poinsot Louis Poinsot (3 January 1777 – 5 December 1859) was a French mathematician and physicist. Poinsot was the inventor of geometrical mechanics, showing how a system of forces acting on a rigid body could be resolved into a single force and a ...

, who was an associate professor at the École Polytechnique, asked to be exempted from his teaching duties for health reasons. Cauchy was by then a rising mathematical star, who certainly merited a professorship. One of his great successes at that time was the proof of

Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is ...

polygonal number theorem In additive number theory, the Fermat polygonal number theorem states that every positive integer is a sum of at most -gonal numbers. That is, every positive integer can be written as the sum of three or fewer triangular numbers, and as the sum ...

. However, the fact that Cauchy was known to be very loyal to the Bourbons doubtless also helped him in becoming the successor of Poinsot. He finally quit his engineering job, and received a one-year contract for teaching mathematics to second-year students of the École Polytechnique. In 1816, this Bonapartist, non-religious school was reorganized, and several liberal professors were fired; the reactionary Cauchy was promoted to full professor. When Cauchy was 28 years old, he was still living with his parents. His father found it high time for his son to marry; he found him a suitable bride, Aloïse de Bure, five years his junior. The de Bure family were printers and booksellers, and published most of Cauchy's works. Aloïse and Augustin were married on April 4, 1818, with great Roman Catholic pomp and ceremony, in the Church of Saint-Sulpice. In 1819 the couple's first daughter, Marie Françoise Alicia, was born, and in 1823 the second and last daughter, Marie Mathilde. The conservative political climate that lasted until 1830 suited Cauchy perfectly. In 1824 Louis XVIII died, and was succeeded by his even more reactionary brother

Charles X Charles X (born Charles Philippe, Count of Artois; 9 October 1757 – 6 November 1836) was King of France from 16 September 1824 until 2 August 1830. An uncle of the uncrowned Louis XVII and younger brother to reigning kings Louis XVI and Lou ...

. During these years Cauchy was highly productive, and published one important mathematical treatise after another. He received cross-appointments at the

Collège de France The Collège de France (), formerly known as the ''Collège Royal'' or as the ''Collège impérial'' founded in 1530 by François I, is a higher education and research establishment ('' grand établissement'') in France. It is located in Paris n ...

, and the .

In exile

In July 1830, the

July Revolution The French Revolution of 1830, also known as the July Revolution (french: révolution de Juillet), Second French Revolution, or ("Three Glorious ays), was a second French Revolution after French Revolution, the first in 1789. It led to ...

occurred in France. Charles X fled the country, and was succeeded by the non-Bourbon king

Louis-Philippe Louis Philippe (6 October 1773 – 26 August 1850) was King of the French from 1830 to 1848, and the penultimate monarch of France. As Louis Philippe, Duke of Chartres, he distinguished himself commanding troops during the Revolutionary Wa ...

(of the

House of Orléans The 4th House of Orléans (french: Maison d'Orléans), sometimes called the House of Bourbon-Orléans (french: link=no, Maison de Bourbon-Orléans) to distinguish it, is the fourth holder of a surname previously used by several branches of the Ro ...

). Riots, in which uniformed students of the École Polytechnique took an active part, raged close to Cauchy's home in Paris. These events marked a turning point in Cauchy's life, and a break in his mathematical productivity. Cauchy, shaken by the fall of the government and moved by a deep hatred of the liberals who were taking power, left Paris to go abroad, leaving his family behind. He spent a short time at

Fribourg , neighboring_municipalities= Düdingen, Givisiez, Granges-Paccot, Marly, Pierrafortscha, Sankt Ursen, Tafers, Villars-sur-Glâne , twintowns = Rueil-Malmaison (France) , website = www.ville-fribourg.ch , Location of , Location of () () ...

in Switzerland, where he had to decide whether he would swear a required oath of allegiance to the new regime. He refused to do this, and consequently lost all his positions in Paris, except his membership of the Academy, for which an oath was not required. In 1831 Cauchy went to the Italian city of Turin, and after some time there, he accepted an offer from the

King of Sardinia The following is a list of rulers of Sardinia, in particular, of the monarchs of the Kingdom of Sardinia and Corsica from 1323 and then of the Kingdom of Sardinia from 1479 to 1861. Early history Owing to the absence of written sources, litt ...

(who ruled Turin and the surrounding Piedmont region) for a chair of theoretical physics, which was created especially for him. He taught in Turin during 1832–1833. In 1831, he was elected a foreign member of the

Royal Swedish Academy of Sciences The Royal Swedish Academy of Sciences ( sv, Kungliga Vetenskapsakademien) is one of the royal academies of Sweden. Founded on 2 June 1739, it is an independent, non-governmental scientific organization that takes special responsibility for prom ...

, and the following year a Foreign Honorary Member of the

American Academy of Arts and Sciences The American Academy of Arts and Sciences (abbreviation: AAA&S) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, a ...

. In August 1833 Cauchy left Turin for

Prague Prague ( ; cs, Praha ; german: Prag, ; la, Praga) is the capital and largest city in the Czech Republic, and the historical capital of Bohemia. On the Vltava river, Prague is home to about 1.3 million people. The city has a temperate ...

to become the science tutor of the thirteen-year-old Duke of Bordeaux,

Henri d'Artois Henri, Count of Chambord and Duke of Bordeaux (french: Henri Charles Ferdinand Marie Dieudonné d'Artois, duc de Bordeaux, comte de Chambord; 29 September 1820 – 24 August 1883) was disputedly King of France from 2 to 9 August 1830 as Hen ...

(1820–1883), the exiled Crown Prince and grandson of Charles X. As a professor of the École Polytechnique, Cauchy had been a notoriously bad lecturer, assuming levels of understanding that only a few of his best students could reach, and cramming his allotted time with too much material. The young Duke had neither taste nor talent for either mathematics or science, so student and teacher were a perfect mismatch. Although Cauchy took his mission very seriously, he did this with great clumsiness, and with surprising lack of authority over the Duke. During his civil engineering days, Cauchy once had been briefly in charge of repairing a few of the Parisian sewers, and he made the mistake of mentioning this to his pupil; with great malice, the young Duke went about saying Mister Cauchy started his career in the sewers of Paris. Cauchy's role as tutor lasted until the Duke became eighteen years old, in September 1838. Cauchy did hardly any research during those five years, while the Duke acquired a lifelong dislike of mathematics. The only good that came out of this episode was Cauchy's promotion to

baron Baron is a rank of nobility or title of honour, often hereditary, in various European countries, either current or historical. The female equivalent is baroness. Typically, the title denotes an aristocrat who ranks higher than a lord or kn ...

, a title by which Cauchy set great store. In 1834, his wife and two daughters moved to Prague, and Cauchy was finally reunited with his family after four years in exile.

Last years

Cauchy returned to Paris and his position at the Academy of Sciences late in 1838. He could not regain his teaching positions, because he still refused to swear an oath of allegiance. Augustin-Louis Cauchy

In August 1839 a vacancy appeared in the

Bureau des Longitudes Bureau ( ) may refer to: Agencies and organizations * Government agency *Public administration * News bureau, an office for gathering or distributing news, generally for a given geographical location * Bureau (European Parliament), the administ ...

. This Bureau bore some resemblance to the Academy; for instance, it had the right to co-opt its members. Further, it was believed that members of the Bureau could "forget about" the oath of allegiance, although formally, unlike the Academicians, they were obliged to take it. The Bureau des Longitudes was an organization founded in 1795 to solve the problem of determining position at sea — mainly the

longitudinal Longitudinal is a geometric term of location which may refer to: * Longitude ** Line of longitude, also called a meridian * Longitudinal engine, an internal combustion engine in which the crankshaft is oriented along the long axis of the vehicle, ...

coordinate, since

latitude In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north ...

is easily determined from the position of the sun. Since it was thought that position at sea was best determined by astronomical observations, the Bureau had developed into an organization resembling an academy of astronomical sciences. In November 1839 Cauchy was elected to the Bureau, and discovered immediately that the matter of the oath was not so easily dispensed with. Without his oath, the king refused to approve his election. For four years Cauchy was in the position of being elected but not approved; accordingly, he was not a formal member of the Bureau, did not receive payment, could not participate in meetings, and could not submit papers. Still Cauchy refused to take any oaths; however, he did feel loyal enough to direct his research to

celestial mechanics Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, ...

. In 1840, he presented a dozen papers on this topic to the Academy. He also described and illustrated the

signed-digit representation In mathematical notation for numbers, a signed-digit representation is a positional numeral system with a set of signed digits used to encode the integers. Signed-digit representation can be used to accomplish fast addition of integers be ...

of numbers, an innovation presented in England in 1727 by

John Colson John Colson (1680 – 20 January 1760) was an English clergyman, mathematician, and the Lucasian Professor of Mathematics at Cambridge University. Life John Colson was educated at Lichfield School before becoming an undergraduate at Christ Chu ...

. The confounded membership of the Bureau lasted until the end of 1843, when Cauchy was finally replaced by Poinsot. Throughout the nineteenth century the French educational system struggled over the separation of church and state. After losing control of the public education system, the Catholic Church sought to establish its own branch of education and found in Cauchy a staunch and illustrious ally. He lent his prestige and knowledge to the

École Normale Écclésiastique École may refer to: * an elementary school in the French educational stages normally followed by secondary education establishments (collège and lycée) * École (river), a tributary of the Seine flowing in région Île-de-France * École, Savo ...

, a school in Paris run by Jesuits, for training teachers for their colleges. He also took part in the founding of the Institut Catholique. The purpose of this institute was to counter the effects of the absence of Catholic university education in France. These activities did not make Cauchy popular with his colleagues, who, on the whole, supported

the Enlightenment The Age of Enlightenment or the Enlightenment; german: Aufklärung, "Enlightenment"; it, L'Illuminismo, "Enlightenment"; pl, Oświecenie, "Enlightenment"; pt, Iluminismo, "Enlightenment"; es, La Ilustración, "Enlightenment" was an intel ...

ideals of the French Revolution. When a chair of mathematics became vacant at the Collège de France in 1843, Cauchy applied for it, but received just three of 45 votes. The year 1848 was the year of revolution all over Europe; revolutions broke out in numerous countries, beginning in France. King Louis-Philippe, fearful of sharing the fate of Louis XVI, fled to England. The oath of allegiance was abolished, and the road to an academic appointment was finally clear for Cauchy. On March 1, 1849, he was reinstated at the Faculté de Sciences, as a professor of mathematical astronomy. After political turmoil all through 1848, France chose to become a Republic, under the Presidency of Louis Napoleon Bonaparte, nephew of Napoleon Bonaparte and son of Napoleon's brother, who had been installed as the first king of Holland. Soon (early 1852) the President made himself Emperor of France, and took the name

Napoleon III Napoleon III (Charles Louis Napoléon Bonaparte; 20 April 18089 January 1873) was the first President of France (as Louis-Napoléon Bonaparte) from 1848 to 1852 and the last monarch of France as Emperor of the French from 1852 to 1870. A neph ...

. Not unexpectedly, the idea came up in bureaucratic circles that it would be useful to again require a loyalty oath from all state functionaries, including university professors. This time a cabinet minister was able to convince the Emperor to exempt Cauchy from the oath. Cauchy remained a professor at the university until his death at the age of 67. He received the

Last Rites The last rites, also known as the Commendation of the Dying, are the last prayers and ministrations given to an individual of Christian faith, when possible, shortly before death. They may be administered to those awaiting execution, mortall ...

and died of a bronchial condition at 4 a.m. on 23 May 1857. His name is one of the 72 names inscribed on the Eiffel Tower.

Work

Early work

The genius of Cauchy was illustrated in his simple solution of the

problem of Apollonius In Euclidean plane geometry, Apollonius's problem is to construct circles that are tangent to three given circles in a plane (Figure 1). Apollonius of Perga (c. 262 190 BC) posed and solved this famous problem in his work (', "Tangencies ...

—describing a

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...

touching three given circles—which he discovered in 1805, his generalization of

Euler's formula Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that ...

in 1811, and in several other elegant problems. More important is his memoir on

wave In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (re ...

propagation, which obtained the Grand Prix of the French Academy of Sciences in 1816. Cauchy's writings covered notable topics. In the theory of series he developed the notion of

convergence Convergence may refer to: Arts and media Literature *''Convergence'' (book series), edited by Ruth Nanda Anshen *Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics: **A four-part crossover storyline that ...

and discovered many of the basic formulas for

q-series In mathematical area of combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product (a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^), with (a;q)_0 = 1. It is a ''q''-analog of the Pochhammer s ...

. In the theory of numbers and complex quantities, he was the first to define complex numbers as pairs of real numbers. He also wrote on the theory of groups and substitutions, the theory of functions, differential equations and determinants.

Wave theory, mechanics, elasticity

In the theory of light he worked on Fresnel's wave theory and on the

dispersion Dispersion may refer to: Economics and finance *Dispersion (finance), a measure for the statistical distribution of portfolio returns *Price dispersion, a variation in prices across sellers of the same item *Wage dispersion, the amount of variatio ...

and polarization of light. He also contributed research in

mechanics Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objec ...

, substituting the notion of the continuity of geometrical displacements for the principle of the continuity of matter. He wrote on the equilibrium of rods and elastic membranes and on waves in elastic media. He introduced a 3 × 3 symmetric

matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...

of numbers that is now known as the

Cauchy stress tensor In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that completely ...

. In elasticity, he originated the theory of

stress Stress may refer to: Science and medicine * Stress (biology), an organism's response to a stressor such as an environmental condition * Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...

, and his results are nearly as valuable as those of Siméon Poisson.

Number theory

Other significant contributions include being the first to prove the Fermat polygonal number theorem.

Complex functions

Cauchy is most famous for his single-handed development of

complex function theory Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...

. The first pivotal theorem proved by Cauchy, now known as '' Cauchy's integral theorem'', was the following: :

\oint_C f(z)dz = 0,

where ''f''(''z'') is a complex-valued function holomorphic on and within the non-self-intersecting closed curve ''C'' (contour) lying in the

complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...

. The ''contour integral'' is taken along the contour ''C''. The rudiments of this theorem can already be found in a paper that the 24-year-old Cauchy presented to the Académie des Sciences (then still called "First Class of the Institute") on August 11, 1814. In full form the theorem was given in 1825. The 1825 paper is seen by many as Cauchy's most important contribution to mathematics. In 1826 Cauchy gave a formal definition of a residue of a function. This concept concerns functions that have poles—isolated singularities, i.e., points where a function goes to positive or negative infinity. If the complex-valued function ''f''(''z'') can be expanded in the

neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...

of a singularity ''a'' as :

f(z) = \varphi(z) + \frac + \frac + \cdots + \frac,\quad
B_i, z,a \in \mathbb,

where φ(''z'') is analytic (i.e., well-behaved without singularities), then ''f'' is said to have a pole of order ''n'' in the point ''a''. If ''n'' = 1, the pole is called simple. The coefficient ''B''₁ is called by Cauchy the residue of function ''f'' at ''a''. If ''f'' is non-singular at ''a'' then the residue of ''f'' is zero at ''a''. Clearly, the residue is in the case of a simple pole equal to :

\underset f(z) = \lim_ (z-a) f(z),

where we replaced ''B''₁ by the modern notation of the residue. In 1831, while in Turin, Cauchy submitted two papers to the Academy of Sciences of Turin. In the first he proposed the formula now known as Cauchy's integral formula, :

f(a) = \frac \oint_C \frac dz,

where ''f''(''z'') is analytic on ''C'' and within the region bounded by the contour ''C'' and the complex number ''a'' is somewhere in this region. The contour integral is taken counter-clockwise. Clearly, the integrand has a simple pole at ''z'' = ''a''. In the second paperCauchy, ''Mémoire sur les rapports qui existent entre le calcul des Résidus et le calcul des Limites, et sur les avantages qu'offrent ces deux calculs dans la résolution des équations algébriques ou transcendantes'' Memorandum on the connections that exist between the residue calculus and the limit calculus, and on the advantages that these two calculi offer in solving algebraic and transcendental equations], presented to the Academy of Sciences of Turin, November 27, 1831. he presented the

residue theorem In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as wel ...

, :

\frac \oint_C f(z) dz = \sum_^n \underset f(z),

where the sum is over all the ''n'' poles of ''f''(''z'') on and within the contour ''C''. These results of Cauchy's still form the core of complex function theory as it is taught today to physicists and electrical engineers. For quite some time, contemporaries of Cauchy ignored his theory, believing it to be too complicated. Only in the 1840s the theory started to get response, with

Pierre Alphonse Laurent Pierre Alphonse Laurent (18 July 1813 – 2 September 1854) was a French mathematician, engineer, and Military Officer best known for discovering the Laurent series, an expansion of a function into an infinite power series, generalizing the T ...

being the first mathematician besides Cauchy to make a substantial contribution (his work on what are now known as

Laurent series In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion c ...

, published in 1843).

Cours d'Analyse

In his book ''Cours d'Analyse'' Cauchy stressed the importance of rigor in analysis. ''Rigor'' in this case meant the rejection of the principle of '' Generality of algebra'' (of earlier authors such as Euler and Lagrange) and its replacement by geometry and

infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally re ...

s. Judith Grabiner wrote Cauchy was "the man who taught rigorous analysis to all of Europe". The book is frequently noted as being the first place that inequalities, and

\delta-\epsilon

arguments were introduced into Calculus. Here Cauchy defined continuity as follows: ''The function f(x) is continuous with respect to x between the given limits if, between these limits, an infinitely small increment in the variable always produces an infinitely small increment in the function itself.'' M. Barany claims that the École mandated the inclusion of infinitesimal methods against Cauchy's better judgement. Gilain notes that when the portion of the curriculum devoted to ''Analyse Algébrique'' was reduced in 1825, Cauchy insisted on placing the topic of continuous functions (and therefore also infinitesimals) at the beginning of the Differential Calculus. Laugwitz (1989) and Benis-Sinaceur (1973) point out that Cauchy continued to use infinitesimals in his own research as late as 1853. Cauchy gave an explicit definition of an infinitesimal in terms of a sequence tending to zero. There has been a vast body of literature written about Cauchy's notion of "infinitesimally small quantities", arguing they lead from everything from the usual "epsilontic" definitions or to the notions of

non-standard analysis The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta ...

. The consensus is that Cauchy omitted or left implicit the important ideas to make clear the precise meaning of the infinitely small quantities he used.

Taylor's theorem

He was the first to prove

Taylor's theorem In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polynomial of degree ''k'', called the ''k''th-order Taylor polynomial. For a smooth function, the Taylor polynomial is th ...

rigorously, establishing his well-known form of the remainder. He wrote a textbook (see the illustration) for his students at the École Polytechnique in which he developed the basic theorems of mathematical analysis as rigorously as possible. In this book he gave the necessary and sufficient condition for the existence of a

limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...

in the form that is still taught. Also Cauchy's well-known test for

absolute convergence In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is s ...

stems from this book:

Cauchy condensation test In mathematics, the Cauchy condensation test, named after Augustin-Louis Cauchy, is a standard convergence test for infinite series. For a non-increasing sequence f(n) of non-negative real numbers, the series \sum\limits_^ f(n) converges if an ...

. In 1829 he defined for the first time a complex function of a complex variable in another textbook. In spite of these, Cauchy's own research papers often used intuitive, not rigorous, methods; thus one of his theorems was exposed to a "counter-example" by

Abel Abel ''Hábel''; ar, هابيل, Hābīl is a Biblical figure in the Book of Genesis within Abrahamic religions. He was the younger brother of Cain, and the younger son of Adam and Eve, the first couple in Biblical history. He was a shepherd ...

, later fixed by the introduction of the notion of

uniform continuity In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...

Argument principle, stability

In a paper published in 1855, two years before Cauchy's death, he discussed some theorems, one of which is similar to the " Principle of the argument" in many modern textbooks on complex analysis. In modern control theory textbooks, the

Cauchy argument principle In complex analysis, the argument principle (or Cauchy's argument principle) relates the difference between the number of zeros and poles of a meromorphic function to a contour integral of the function's logarithmic derivative. Specifically, if ...

is quite frequently used to derive the

Nyquist stability criterion In control theory and stability theory, the Nyquist stability criterion or Strecker–Nyquist stability criterion, independently discovered by the German electrical engineer at Siemens in 1930 and the Swedish-American electrical engineer Harry ...

, which can be used to predict the stability of negative

feedback amplifier A negative-feedback amplifier (or feedback amplifier) is an electronic amplifier that subtracts a fraction of its output from its input, so that negative feedback opposes the original signal. The applied negative feedback can improve its perform ...

and negative

feedback Feedback occurs when outputs of a system are routed back as inputs as part of a chain of cause-and-effect that forms a circuit or loop. The system can then be said to ''feed back'' into itself. The notion of cause-and-effect has to be handled ...

control systems. Thus Cauchy's work has a strong impact on both pure mathematics and practical engineering.

Published works

Cauchy - Leçons sur le calcul différentiel, 1829 - 576181 F

Cauchy was very productive, in number of papers second only to

Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...

. It took almost a century to collect all his writings into 27 large volumes: * (Paris : Gauthier-Villars et fils, 1882–1974) * His greatest contributions to mathematical science are enveloped in the rigorous methods which he introduced; these are mainly embodied in his three great treatises: *
''Le Calcul infinitésimal''
(1823) * ''Leçons sur les applications de calcul infinitésimal''; ''La géométrie'' (1826–1828) His other works include: * * * * * *
Exercices d'analyse et de physique mathematique (Volume 1)
' *
Exercices d'analyse et de physique mathematique (Volume 2)
' *
Exercices d'analyse et de physique mathematique (Volume 3)
' *
Exercices d'analyse et de physique mathematique (Volume 4)
' (Paris: Bachelier, 1840–1847) *
Analyse algèbrique
' (Imprimerie Royale, 1821) *
Nouveaux exercices de mathématiques
' (Paris : Gauthier-Villars, 1895) * ''Courses of mechanics'' (for the École Polytechnique) * ''Higher algebra'' (for the ) * ''Mathematical physics'' (for the Collège de France). *
Mémoire sur l'emploi des equations symboliques dans le calcul infinitésimal et dans le calcul aux différences finis
' CR Ac ad. Sci. Paris, t. XVII, 449–458 (1843) credited as originating the operational calculus.

Politics and religious beliefs

Augustin-Louis Cauchy grew up in the house of a staunch royalist. This made his father flee with the family to

during the

. Their life there during that time was apparently hard; Augustin-Louis's father, Louis François, spoke of living on rice, bread, and crackers during the period. A paragraph from an undated letter from Louis François to his mother in

Rouen Rouen (, ; or ) is a city on the River Seine in northern France. It is the prefecture of the region of Normandy and the department of Seine-Maritime. Formerly one of the largest and most prosperous cities of medieval Europe, the population ...

says: In any event, he inherited his father's staunch royalism and hence refused to take oaths to any government after the overthrow of Charles X. He was an equally staunch Catholic and a member of the

Society of Saint Vincent de Paul The Society of St Vincent de Paul (SVP or SVdP or SSVP) is an international voluntary organization in the Catholic Church, founded in 1833 for the sanctification of its members by personal service of the poor. Innumerable Catholic parishes have ...

. He also had links to the

Society of Jesus , image = Ihs-logo.svg , image_size = 175px , caption = ChristogramOfficial seal of the Jesuits , abbreviation = SJ , nickname = Jesuits , formation = , founders ...

and defended them at the Academy when it was politically unwise to do so. His zeal for his faith may have led to his caring for

Charles Hermite Charles Hermite () FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra. ...

during his illness and leading Hermite to become a faithful Catholic. It also inspired Cauchy to plead on behalf of the Irish during the Great Famine of Ireland. His royalism and religious zeal made him contentious, which caused difficulties with his colleagues. He felt that he was mistreated for his beliefs, but his opponents felt he intentionally provoked people by berating them over religious matters or by defending the Jesuits after they had been suppressed. Niels Henrik Abel called him a "bigoted Catholic" and added he was "mad and there is nothing that can be done about him", but at the same time praised him as a mathematician. Cauchy's views were widely unpopular among mathematicians and when Guglielmo Libri Carucci dalla Sommaja was made chair in mathematics before him he, and many others, felt his views were the cause. When Libri was accused of stealing books he was replaced by

Joseph Liouville Joseph Liouville (; ; 24 March 1809 – 8 September 1882) was a French mathematician and engineer. Life and work He was born in Saint-Omer in France on 24 March 1809. His parents were Claude-Joseph Liouville (an army officer) and Thérèse ...

rather than Cauchy, which caused a rift between Liouville and Cauchy. Another dispute with political overtones concerned Jean-Marie Constant Duhamel and a claim on inelastic shocks. Cauchy was later shown, by

Jean-Victor Poncelet Jean-Victor Poncelet (; 1 July 1788 – 22 December 1867) was a French engineer and mathematician who served most notably as the Commanding General of the École Polytechnique. He is considered a reviver of projective geometry, and his work '' ...

, to be wrong.

References

Notes

Citations

Sources

* * * * * * * * * * *

External links

* *
Augustin-Louis Cauchy – Œuvres complètes
(in 2 series) Gallica-Math *
Augustin-Louis Cauchy – Cauchy's Life
by

Robin Hartshorne __NOTOC__ Robin Cope Hartshorne ( ; born March 15, 1938) is an American mathematician who is known for his work in algebraic geometry. Career Hartshorne was a Putnam Fellow in Fall 1958 while he was an undergraduate at Harvard University (under ...

{{DEFAULTSORT:Cauchy, Augustin Louis 1789 births 1857 deaths 19th-century French mathematicians Corps des ponts École des Ponts ParisTech alumni École Polytechnique alumni Fellows of the American Academy of Arts and Sciences Foreign Members of the Royal Society French Roman Catholics French geometers History of calculus Mathematical analysts Linear algebraists Members of the French Academy of Sciences Members of the Royal Swedish Academy of Sciences Recipients of the Pour le Mérite (civil class) Textbook writers University of Turin faculty