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75-537: Henri Lebesgue was born on 28 June 1875 in Beauvais , Oise . Lebesgue's father was a typesetter and his mother was a school teacher . His parents assembled at home a library that the young Henri was able to use. His father died of tuberculosis when Lebesgue was still very young and his mother had to support him by herself. As he showed a remarkable talent for mathematics in primary school, one of his instructors arranged for community support to continue his education at

150-556: A closed interval . The Lebesgue integral integrates many of these functions (always reproducing the same answer when it does), but not all of them. For functions on the real line, the Henstock integral is an even more general notion of integral (based on Riemann's theory rather than Lebesgue's) that subsumes both Lebesgue integration and improper Riemann integration. However, the Henstock integral depends on specific ordering features of

225-417: A manifold , no matter how distorted it is. In his dissertation, he established a geometric foundation for complex analysis through Riemann surfaces , through which multi-valued functions like the logarithm (with infinitely many sheets) or the square root (with two sheets) could become one-to-one functions . Complex functions are harmonic functions (that is, they satisfy Laplace's equation and thus

300-415: A " Borel tract" Leçons sur l'intégration et la recherche des fonctions primitives . The problem of integration regarded as the search for a primitive function is the keynote of the book. Lebesgue presents the problem of integration in its historical context, addressing Augustin-Louis Cauchy , Peter Gustav Lejeune Dirichlet , and Bernhard Riemann . Lebesgue presents six conditions which it is desirable that

375-638: A Fourier series representing a continuous, almost nowhere-differentiable function, a case not covered by Dirichlet. He also proved the Riemann–Lebesgue lemma : if a function is representable by a Fourier series, then the Fourier coefficients go to zero for large  n . Riemann's essay was also the starting point for Georg Cantor 's work with Fourier series, which was the impetus for set theory . He also worked with hypergeometric differential equations in 1857 using complex analytical methods and presented

450-529: A competition with Weierstrass since 1857 to solve the Jacobian inverse problems for abelian integrals, a generalization of elliptic integrals . Riemann used theta functions in several variables and reduced the problem to the determination of the zeros of these theta functions. Riemann also investigated period matrices and characterized them through the "Riemannian period relations" (symmetric, real part negative). By Ferdinand Georg Frobenius and Solomon Lefschetz

525-756: A fear of speaking in public. During 1840, Riemann went to Hanover to live with his grandmother and attend lyceum (middle school years), because such a type of school was not accessible from his home village. After the death of his grandmother in 1842, he transferred to the Johanneum Lüneburg , a high school in Lüneburg . There, Riemann studied the Bible intensively, but he was often distracted by mathematics. His teachers were amazed by his ability to perform complicated mathematical operations, in which he often outstripped his instructor's knowledge. In 1846, at

600-404: A finite number of values, and each value is taken on a measurable set). Lebesgue's technique for turning a measure into an integral generalises easily to many other situations, leading to the modern field of measure theory . The Lebesgue integral is deficient in one respect. The Riemann integral generalises to the improper Riemann integral to measure functions whose domain of definition is not

675-460: A hamlet of Verbania on Lake Maggiore ), where he was buried in the cemetery in Biganzolo (Verbania). Riemann was a dedicated Christian, the son of a Protestant minister, and saw his life as a mathematician as another way to serve God. During his life, he held closely to his Christian faith and considered it to be the most important aspect of his life. At the time of his death, he was reciting

750-453: A higher ecclesiastical seminary, a lycée and training colleges. Amongst the major companies operating in the town are Nestle and Agco (Massey Ferguson). Also present since 1986 is RS Components , founded by Jerry Vaughan, and now operating from a purpose-built distribution centre to the east of the town Beauvais also has a small airport, Beauvais Tillé , which is used by several low-cost carriers and charter airlines such as Ryanair as

825-461: A natural, geometric treatment of complex analysis. His 1859 paper on the prime-counting function , containing the original statement of the Riemann hypothesis , is regarded as a foundational paper of analytic number theory . Through his pioneering contributions to differential geometry , Riemann laid the foundations of the mathematics of general relativity . He is considered by many to be one of

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900-478: A number (scalar), with the surfaces of constant positive or negative curvature being models of the non-Euclidean geometries . The Riemann metric is a collection of numbers at every point in space (i.e., a tensor ) which allows measurements of speed in any trajectory, whose integral gives the distance between the trajectory's endpoints. For example, Riemann found that in four spatial dimensions, one needs ten numbers at each point to describe distances and curvatures on

975-730: A population of 56,020 as of 2016 , making it the most populous town in the Oise department, and third most populous in Picardy. Together with its suburbs and satellite towns, the metropolitan area of Beauvais has a population of 128,020. The region around Beauvais is called the Beauvaisis. Beauvais was known to the Romans by the Gallo-Roman name of Caesaromagus ( magos is Common Celtic for "field"). The post-Renaissance Latin rendering

1050-791: A recent graduate of the school. At the same time he started his graduate studies at the Sorbonne , where he learned about Émile Borel 's work on the incipient measure theory and Camille Jordan 's work on the Jordan measure . In 1899 he moved to a teaching position at the Lycée Central in Nancy , while continuing work on his doctorate. In 1902 he earned his PhD from the Sorbonne with the seminal thesis on "Integral, Length, Area", submitted with Borel, four years older, as advisor. Lebesgue married

1125-603: A rigorous foundation. The mathematical notion of limit and the closely related notion of convergence are central to any modern definition of integration. In the 19th century, Karl Weierstrass developed the rigorous epsilon-delta definition of a limit, which is still accepted and used by mathematicians today. He built on previous but non-rigorous work by Augustin Cauchy , who had used the non-standard notion of infinitesimally small numbers , today rejected in standard mathematical analysis . Before Cauchy, Bernard Bolzano had laid

1200-513: A rigorous proof of Gibbs' ergodic hypothesis . Beauvais Beauvais ( US : / b oʊ ˈ v eɪ / boh- VAY , French: [bovɛ] ; Picard : Bieuvais ) is a town and commune in northern France , and prefecture of the Oise département , in the Hauts-de-France region , 75 kilometres (47 miles) north of Paris. The commune of Beauvais had

1275-418: A single number. Thus, they have no Riemann integral. Lebesgue invented a new method of integration to solve this problem. Instead of using the areas of rectangles, which put the focus on the domain of the function, Lebesgue looked at the codomain of the function for his fundamental unit of area. Lebesgue's idea was to first define measure, for both sets and functions on those sets. He then proceeded to build

1350-571: A terminal for nearby Paris, to which frequent shuttle buses run. Beauvais has the following schools: Public schools: Private schools: Beauvais is home to AS Beauvais Oise , a football club playing in the Championnat National (as of 2006 ), which is supported by a fine percussion band. Beauvais is twinned with: Bernhard Riemann Georg Friedrich Bernhard Riemann ( German: [ˈɡeːɔʁk ˈfʁiːdʁɪç ˈbɛʁnhaʁt ˈʁiːman] ; 17 September 1826 – 20 July 1866)

1425-426: A tremendous impact on the shape of the field today and his methods have become an essential part of modern analysis. These have important practical implications for fundamental physics of which Lebesgue would have been completely unaware, as noted below. Integration is a mathematical operation that corresponds to the informal idea of finding the area under the graph of a function . The first theory of integration

1500-478: A trigonometrical series representing a bounded function is a Fourier series, that the n Fourier coefficient tends to zero (the Riemann–Lebesgue lemma ), and that a Fourier series is integrable term by term. In 1904-1905 Lebesgue lectured once again at the Collège de France , this time on trigonometrical series and he went on to publish his lectures in another of the "Borel tracts". In this tract he once again treats

1575-638: A variety of rare and extremely rare Anglo-Norman pennies, English and foreign coins, was reputed to have been found in or near Paris. Beauvais was extensively damaged during World War I, and again in World War II during the German advance on Paris in June 1940. Much of the older part of the city was all but destroyed, and the cathedral was badly damaged before being liberated by British forces on 30 August 1944. Beauvais experienced significant rioting during

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1650-562: Is Bellovacum from the Belgic tribe the Bellovaci , whose capital it was. In the ninth century, it became a county (comté), which about 1013 passed to the bishops of Beauvais, who became peers of France from the twelfth century. At the coronations of kings, the Bishop of Beauvais wore the royal mantle and went, with the Bishop of Langres , to raise the king from his throne to present him to

1725-456: Is 9.9 °C (1961–1990), and the sunlight annual average of 1669 hours (1991–2010). Hills Bray is provided for the precipitation of Beauvais. The precipitation is 669 mm on average per year (1981–2010), while it is 800 mm on average per year in Bray. However, the frequency of rainfall is high. The average number of days per year above the precipitation of 1 mm is 116 days or every third day. The fog

1800-540: Is often present, it is estimated at 55 days a year. The department is affected by 41 days of average wind year, usually, it comes from the west to the south. The population data in the table and graph below refer to the commune of Beauvais proper in its geography at the given years. The commune of Beauvais absorbed the former communes of Marissel, Saint-Just-des-Marais and Voisinlieu and part of Notre-Dame-du-Thil in 1943. The city's cathedral, dedicated to Saint Peter ( Cathédrale Saint-Pierre de Beauvais ), in some respects,

1875-594: Is the Jacobian variety of the Riemann surface, an example of an abelian manifold. Many mathematicians such as Alfred Clebsch furthered Riemann's work on algebraic curves. These theories depended on the properties of a function defined on Riemann surfaces. For example, the Riemann–Roch theorem (Roch was a student of Riemann) says something about the number of linearly independent differentials (with known conditions on

1950-578: The Cauchy–Riemann equations ) on these surfaces and are described by the location of their singularities and the topology of the surfaces. The topological "genus" of the Riemann surfaces is given by g = w / 2 − n + 1 {\displaystyle g=w/2-n+1} , where the surface has n {\displaystyle n} leaves coming together at w {\displaystyle w} branch points. For g > 1 {\displaystyle g>1}

2025-501: The Collège de Beauvais and then at Lycée Saint-Louis and Lycée Louis-le-Grand in Paris . In 1894, Lebesgue was accepted at the École Normale Supérieure , where he continued to focus his energy on the study of mathematics, graduating in 1897. After graduation he remained at the École Normale Supérieure for two years, working in the library, where he became aware of the research on discontinuity done at that time by René-Louis Baire ,

2100-578: The Dirichlet principle . Karl Weierstrass found a gap in the proof: Riemann had not noticed that his working assumption (that the minimum existed) might not work; the function space might not be complete, and therefore the existence of a minimum was not guaranteed. Through the work of David Hilbert in the Calculus of Variations, the Dirichlet principle was finally established. Otherwise, Weierstrass

2175-565: The Lebesgue–Stieltjes integral generalizes Riemann–Stieltjes and Lebesgue integration, preserving the many advantages of the latter in a more general measure-theoretic framework. During the course of his career, Lebesgue also made forays into the realms of complex analysis and topology . He also had a disagreement with Émile Borel about whose integral was more general. However, these minor forays pale in comparison to his contributions to real analysis ; his contributions to this field had

2250-901: The Lord's Prayer with his wife and died before they finished saying the prayer. Meanwhile, in Göttingen his housekeeper discarded some of the papers in his office, including much unpublished work. Riemann refused to publish incomplete work, and some deep insights may have been lost. Riemann's tombstone in Biganzolo (Italy) refers to Romans 8:28 : Georg Friedrich Bernhard Riemann Professor in Göttingen born in Breselenz, 17 September 1826 died in Selasca, 20 July 1866 Riemann's published works opened up research areas combining analysis with geometry. These would subsequently become major parts of

2325-540: The Musée départemental de l'Oise , was built in the 16th century, partly upon the Gallo-Roman fortifications. The church of Saint-Étienne is a Romanesque-Gothic building (early 12th-late 16th centuries), including, in one of its transept's portals, a sculpture of "Wheel of Life". The railway station, Gare de Beauvais , opened in 1857 is currently served by several TER lines: Beauvais–Tillé Airport , dating from

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2400-694: The Nahel Merzouk protests in 2023. Beauvais lies at the foot of wooded hills on the left bank of the Thérain at its confluence with the Avelon. Its ancient ramparts have been destroyed, and it is now surrounded by boulevards, outside of which run branches of the Thérain. In addition, there are spacious promenades in the north-east of the town. Beauvais experiences an oceanic climate ( Köppen climate classification Cfb ). The average annual temperature

2475-404: The Riemann–Lebesgue lemma is a best possible result for continuous functions, and gives some treatment to Lebesgue constants . Lebesgue once wrote, "Réduites à des théories générales, les mathématiques seraient une belle forme sans contenu." ("Reduced to general theories, mathematics would be a beautiful form without content.") In measure-theoretic analysis and related branches of mathematics,

2550-605: The method of least squares ). Gauss recommended that Riemann give up his theological work and enter the mathematical field; after getting his father's approval, Riemann transferred to the University of Berlin in 1847. During his time of study, Carl Gustav Jacob Jacobi , Peter Gustav Lejeune Dirichlet , Jakob Steiner , and Gotthold Eisenstein were teaching. He stayed in Berlin for two years and returned to Göttingen in 1849. Riemann held his first lectures in 1854, which founded

2625-504: The real line and so does not generalise to allow integration in more general spaces (say, manifolds ), while the Lebesgue integral extends to such spaces quite naturally. In 1947 Norbert Wiener claimed that the Lebesgue integral had unexpected but important implications in establishing the validity of Willard Gibbs ' work on the foundations of statistical mechanics. The notions of average and measure were urgently needed to provide

2700-401: The zeta function that now bears his name, establishing its importance for understanding the distribution of prime numbers . The Riemann hypothesis was one of a series of conjectures he made about the function's properties. In Riemann's work, there are many more interesting developments. He proved the functional equation for the zeta function (already known to Leonhard Euler ), behind which

2775-507: The 1930s, lies in the north of the city, in Tillé . It is used as a gateway to Paris by several low-cost carriers. Traffic growth is significant: in 1997, 200,000 passengers used it annually, but by 2006, it was more than 1.8 million. Airport usage increased by 40% a year on average between 2001 and 2005. The airport is mainly used for passenger traffic (only 2 to 3 flights involve freight each month) and serves 48 destinations. On 5 October 1930,

2850-583: The British airship R101 crashed just outside Beauvais on its maiden overseas voyage, killing 48 of the 54 people on board. Public transport in Beauvais is provided by Corolis (formerly The Urban Transport network of Beauvaisis French : Transports Urbains du Beauvaisis or TUB ). The transit bus (commuter bus) network consists of 25 regular lines which serve Beauvais and its suburbs, including: To promote cleaner urban transportation and protect

2925-622: The Count of Beauvais was Odo II, Count of Blois . In a charter dated 1056/1060, Eudo of Brittany granted land "in pago Belvacensi" (Beauvais, Picardy) to the Abbey of Angers Saint-Aubin (see Albinus of Angers ). In 1346, the town had to defend itself against the English, who again besieged it in 1433. The siege that it endured in 1472 at the hands of the Duke of Burgundy was rendered famous by

3000-476: The Riemann surface has ( 3 g − 3 ) {\displaystyle (3g-3)} parameters (the " moduli "). His contributions to this area are numerous. The famous Riemann mapping theorem says that a simply connected domain in the complex plane is "biholomorphically equivalent" (i.e. there is a bijection between them that is holomorphic with a holomorphic inverse) to either C {\displaystyle \mathbb {C} } or to

3075-797: The Romanesque and Gothic styles. During the Middle Ages, on 14 January, the Feast of Asses was celebrated in the Beauvais Cathedral, in commemoration of the Flight into Egypt. In the Place de l'Hôtel de Ville and the old streets near the cathedral, several houses are dating from the 12th to the 16th centuries. The Hôtel de ville , close to which stands the statue of Jeanne Hachette , was built in 1752. The episcopal palace, now housing

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3150-730: The Sorbonne to become professor of mathematics at the Collège de France , where he lectured and did research for the rest of his life. In 1922 he was elected a member of the Académie des Sciences . Henri Lebesgue died on 26 July 1941 in Paris . Lebesgue's first paper was published in 1898 and was titled "Sur l'approximation des fonctions". It dealt with Weierstrass 's theorem on approximation to continuous functions by polynomials. Between March 1899 and April 1901 Lebesgue published six notes in Comptes Rendus . The first of these, unrelated to his development of Lebesgue integration, dealt with

3225-558: The University of Göttingen), he was promoted to head the mathematics department at the University of Göttingen. He was also the first to suggest using dimensions higher than merely three or four in order to describe physical reality. In 1862 he married Elise Koch; their daughter Ida Schilling was born on 22 December 1862. Riemann fled Göttingen when the armies of Hanover and Prussia clashed there in 1866. He died of tuberculosis during his third journey to Italy in Selasca (now

3300-424: The age of 19, he started studying philology and Christian theology in order to become a pastor and help with his family's finances. During the spring of 1846, his father, after gathering enough money, sent Riemann to the University of Göttingen , where he planned to study towards a degree in theology . However, once there, he began studying mathematics under Carl Friedrich Gauss (specifically his lectures on

3375-541: The environment, the city began to develop a "Green Plan" ( Plan vert ). Ultimately, the goal is to have a network of 20 km (12 mi) bicycle paths. The mayor of Beauvais is Franck Pia, elected in September 2022. He succeeded Caroline Cayeux , who stepped down to become a deputy minister in the Borne government . The industry of Beauvais comprises, besides the state manufacture of tapestry , which dates from 1664,

3450-510: The extension of Baire's theorem to functions of two variables. The next five dealt with surfaces applicable to a plane, the area of skew polygons , surface integrals of minimum area with a given bound, and the final note gave the definition of Lebesgue integration for some function f(x). Lebesgue's great thesis, Intégrale, longueur, aire , with the full account of this work, appeared in the Annali di Matematica in 1902. The first chapter develops

3525-400: The field of Riemannian geometry and thereby set the stage for Albert Einstein 's general theory of relativity . In 1857, there was an attempt to promote Riemann to extraordinary professor status at the University of Göttingen . Although this attempt failed, it did result in Riemann finally being granted a regular salary. In 1859, following the death of Dirichlet (who held Gauss 's chair at

3600-831: The field of real analysis , he discovered the Riemann integral in his habilitation . Among other things, he showed that every piecewise continuous function is integrable. Similarly, the Stieltjes integral goes back to the Göttinger mathematician, and so they are named together the Riemann–Stieltjes integral . In his habilitation work on Fourier series , where he followed the work of his teacher Dirichlet, he showed that Riemann-integrable functions are "representable" by Fourier series. Dirichlet has shown this for continuous, piecewise-differentiable functions (thus with countably many non-differentiable points). Riemann gave an example of

3675-524: The fifteenth and seventeenth centuries; but its chief artistic treasures are stained glass windows of the thirteenth, fourteenth and sixteenth centuries, the most beautiful of them from the hand of the Renaissance artist, Engrand Le Prince, a native of Beauvais. To him also due to some of the stained glass in St. Etienne, the second church of the town, and an interesting example of the transition stage between

3750-405: The foundations of geometry. Over many months, Riemann developed his theory of higher dimensions and delivered his lecture at Göttingen on 10 June 1854, entitled Ueber die Hypothesen, welche der Geometrie zu Grunde liegen . It was not published until twelve years later in 1868 by Dedekind, two years after his death. Its early reception appears to have been slow, but it is now recognized as one of

3825-439: The fundamental groundwork of the epsilon-delta definition. See here for more. Bernhard Riemann followed up on this by formalizing what is now called the Riemann integral . To define this integral, one fills the area under the graph with smaller and smaller rectangles and takes the limit of the sums of the areas of the rectangles at each stage. For some functions, however, the total area of these rectangles does not approach

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3900-497: The graph. This surprising relationship between two major geometric operations in calculus, differentiation and integration, is now known as the Fundamental Theorem of Calculus . It has allowed mathematicians to calculate a broad class of integrals for the first time. However, unlike Archimedes' method, which was based on Euclidean geometry , mathematicians felt that Newton's and Leibniz's integral calculus did not have

3975-662: The greatest mathematicians of all time. Riemann was born on 17 September 1826 in Breselenz , a village near Dannenberg in the Kingdom of Hanover . His father, Friedrich Bernhard Riemann, was a poor Lutheran pastor in Breselenz who fought in the Napoleonic Wars . His mother, Charlotte Ebell, died in 1846. Riemann was the second of six children. Riemann exhibited exceptional mathematical talent, such as calculation abilities, from an early age but suffered from timidity and

4050-635: The heroism of the town's women, under the leadership of Jeanne Hachette , whose memory is still celebrated by a procession on 27 June (the feast of Sainte Angadrême ), during which women take precedence over men. A significant hoard of coins from the High Middle Ages became known as the Beauvais Hoard because some of the English and European coins found with the lot were from the French abbey located in Beauvais. The hoard, which contained

4125-438: The integral for what he called simple functions ; measurable functions that take only finitely many values. Then he defined it for more complicated functions as the least upper bound of all the integrals of simple functions smaller than the function in question. Lebesgue integration has the property that every function defined over a bounded interval with a Riemann integral also has a Lebesgue integral, and for those functions

4200-475: The integral should satisfy, the last of which is "If the sequence f n (x) increases to the limit f(x), the integral of f n (x) tends to the integral of f(x)." Lebesgue shows that his conditions lead to the theory of measure and measurable functions and the analytical and geometrical definitions of the integral. He turned next to trigonometric functions with his 1903 paper "Sur les séries trigonométriques". He presented three major theorems in this work: that

4275-436: The interior of the unit circle. The generalization of the theorem to Riemann surfaces is the famous uniformization theorem , which was proved in the 19th century by Henri Poincaré and Felix Klein . Here, too, rigorous proofs were first given after the development of richer mathematical tools (in this case, topology). For the proof of the existence of functions on Riemann surfaces, he used a minimality condition, which he called

4350-462: The manufacture of various kinds of cotton and woollen goods, brushes, toys, boots and shoes, and bricks and tiles. Market-gardening flourishes in the vicinity and an extensive trade is carried on in grain and wine. The town is the seat of a bishop, a prefect and a Court of Assizes ; it has Tribunals of First Instance and commerce, together with a Chamber of Commerce , a branch of the Bank of France ,

4425-472: The most daring achievement of Gothic architecture , consists only of a transept and quire with apse and seven apse-chapels. The vaulting in the interior exceeds 46 m or 150 feet in height. The cathedral underwent a major repair and restoration process in 2008. The small Romanesque church of the 10th century known as the Basse Oeuvre occupies the site destined for the nave; much of its east end

4500-495: The most important works in geometry. The subject founded by this work is Riemannian geometry . Riemann found the correct way to extend into n dimensions the differential geometry of surfaces, which Gauss himself proved in his theorema egregium . The fundamental objects are called the Riemannian metric and the Riemann curvature tensor . For the surface (two-dimensional) case, the curvature at each point can be reduced to

4575-452: The people. De Bello Gallico II 13 reports that as Julius Caesar was approaching a fortified town called Bratuspantium in the land of the Bellovaci , its inhabitants surrendered to him when he was about 5 Roman miles away. Its name is Gaulish for "place where judgements are made", from * bratu-spantion . Some say that Bratuspantium is Beauvais. Others theorise that it is Vendeuil-Caply or Bailleul sur Thérain. From 1004 to 1037,

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4650-490: The sister of one of his fellow students, and he and his wife had two children, Suzanne and Jacques. After publishing his thesis, Lebesgue was offered in 1902 a position at the University of Rennes , lecturing there until 1906, when he moved to the Faculty of Sciences of the University of Poitiers . In 1910 Lebesgue moved to the Sorbonne as a maître de conférences , being promoted to professor starting in 1919. In 1921 he left

4725-473: The solutions through the behaviour of closed paths about singularities (described by the monodromy matrix ). The proof of the existence of such differential equations by previously known monodromy matrices is one of the Hilbert problems. Riemann made some famous contributions to modern analytic number theory . In a single short paper , the only one he published on the subject of number theory, he investigated

4800-483: The subject in its historical context. He expounds on Fourier series, Cantor-Riemann theory, the Poisson integral and the Dirichlet problem . In a 1910 paper, "Représentation trigonométrique approchée des fonctions satisfaisant a une condition de Lipschitz" deals with the Fourier series of functions satisfying a Lipschitz condition , with an evaluation of the order of magnitude of the remainder term. He also proves that

4875-411: The theories of Riemannian geometry , algebraic geometry , and complex manifold theory. The theory of Riemann surfaces was elaborated by Felix Klein and particularly Adolf Hurwitz . This area of mathematics is part of the foundation of topology and is still being applied in novel ways to mathematical physics . In 1853, Gauss asked Riemann, his student, to prepare a Habilitationsschrift on

4950-492: The theory of measure (see Borel measure ). In the second chapter he defines the integral both geometrically and analytically. The next chapters expand the Comptes Rendus notes dealing with length, area and applicable surfaces. The final chapter deals mainly with Plateau's problem . This dissertation is considered to be one of the finest ever written by a mathematician. His lectures from 1902 to 1903 were collected into

5025-464: The two integrals agree. Furthermore, every bounded function on a closed bounded interval has a Lebesgue integral and there are many functions with a Lebesgue integral that have no Riemann integral. As part of the development of Lebesgue integration, Lebesgue invented the concept of measure , which extends the idea of length from intervals to a very large class of sets, called measurable sets (so, more precisely, simple functions are functions that take

5100-402: The validity of this relation is equivalent with the embedding of C n / Ω {\displaystyle \mathbb {C} ^{n}/\Omega } (where Ω {\displaystyle \Omega } is the lattice of the period matrix) in a projective space by means of theta functions. For certain values of n {\displaystyle n} , this

5175-462: The zeros and poles) of a Riemann surface. According to Detlef Laugwitz , automorphic functions appeared for the first time in an essay about the Laplace equation on electrically charged cylinders. Riemann however used such functions for conformal maps (such as mapping topological triangles to the circle) in his 1859 lecture on hypergeometric functions or in his treatise on minimal surfaces . In

5250-407: Was a German mathematician who made profound contributions to analysis , number theory , and differential geometry . In the field of real analysis , he is mostly known for the first rigorous formulation of the integral, the Riemann integral , and his work on Fourier series . His contributions to complex analysis include most notably the introduction of Riemann surfaces , breaking new ground in

5325-450: Was built from 1500 to 1548. In 1573 the fall of a too-ambitious central tower stopped work again, after which little addition was made. Its façades, especially that on the south, exhibit all the richness of the late Gothic style. The carved wooden doors of both the north and the south portals are masterpieces respectively of Gothic and Renaissance workmanship. The church possesses an elaborate astronomical clock (1866) and tapestries of

5400-528: Was demolished to make room for the new cathedral. Begun in 1247, under Bishop William of Grès (Guillaume de Grès, Guillaume de Grez), an extra 5 metres (16 feet) were added to the height, to make it the tallest cathedral in Europe: the work was interrupted in 1284 by the collapse of the vaulting of the choir, a disaster that produced a temporary failure of nerve among the masons working in Gothic style. The transept

5475-424: Was developed by Archimedes in the 3rd century BC with his method of quadratures , but this could be applied only in limited circumstances with a high degree of geometric symmetry. In the 17th century, Isaac Newton and Gottfried Wilhelm Leibniz discovered the idea that integration was intrinsically linked to differentiation , the latter being a way of measuring how quickly a function changed at any given point on

5550-543: Was successful. An anecdote from Arnold Sommerfeld shows the difficulties which contemporary mathematicians had with Riemann's new ideas. In 1870, Weierstrass had taken Riemann's dissertation with him on a holiday to Rigi and complained that it was hard to understand. The physicist Hermann von Helmholtz assisted him in the work overnight and returned with the comment that it was "natural" and "very understandable". Other highlights include his work on abelian functions and theta functions on Riemann surfaces. Riemann had been in

5625-410: Was very impressed with Riemann, especially with his theory of abelian functions . When Riemann's work appeared, Weierstrass withdrew his paper from Crelle's Journal and did not publish it. They had a good understanding when Riemann visited him in Berlin in 1859. Weierstrass encouraged his student Hermann Amandus Schwarz to find alternatives to the Dirichlet principle in complex analysis, in which he

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