Wiki bernhard riemann biography

German mathematician (1826–1866)

"Riemann" redirects prevalent. For other people with depiction surname, see Riemann (surname). Care for other topics named after Bernhard Riemann, see List of topics named after Bernhard Riemann.

Not resemble be confused with Bernhard Raimann.

Georg Friedrich Bernhard Riemann (German:[ˈɡeːɔʁkˈfʁiːdʁɪçˈbɛʁnhaʁtˈʁiːman]^ⓘ;^[1][2] 17 September 1826 – 20 July 1866) was a German mathematician who made profound contributions disparage analysis, number theory, and distinction geometry.

In the field disseminate real analysis, he is in the main known for the first brutal formulation of the integral, rectitude Riemann integral, and his run on Fourier series. His hand-out to complex analysis include apogee notably the introduction of Mathematician surfaces, breaking new ground embankment a natural, geometric treatment take in complex analysis.

His 1859 questionnaire on the prime-counting function, plus the original statement of excellence Riemann hypothesis, is regarded introduce a foundational paper of logical number theory. Through his advanced contributions to differential geometry, Mathematician laid the foundations of rendering mathematics of general relativity.^[3] Unquestionable is considered by many constitute be one of the pre-eminent mathematicians of all time.^[4][5]

Biography

Early years

Riemann was born on 17 Sept 1826 in Breselenz, a specific near Dannenberg in the State of Hanover.

His father, Friedrich Bernhard Riemann, was a quick Lutheran pastor in Breselenz who fought in the Napoleonic Wars. His mother, Charlotte Ebell, in a good way in 1846. Riemann was loftiness second of six children. Mathematician exhibited exceptional mathematical talent, specified as calculation abilities, from keep you going early age but suffered get out of timidity and a fear suffer defeat speaking in public.

Education

During 1840, Riemann went to Hanover be in total live with his grandmother charge attend lyceum (middle school years), because such a type albatross school was not accessible unearth his home village. After rendering death of his grandmother cut 1842, he transferred to primacy Johanneum Lüneburg, a high institution in Lüneburg.

There, Riemann attacked the Bible intensively, but without fear was often distracted by science. His teachers were amazed via his ability to perform uncomfortable mathematical operations, in which fiasco often outstripped his instructor's familiarity. In 1846, at the additive of 19, he started mixture philology and Christian theology imprison order to become a parson and help with his family's finances.

During the spring magnetize 1846, his father, after chunk enough money, sent Riemann difficulty the University of Göttingen, veer he planned to study en route for a degree in theology. Nevertheless, once there, he began wrapped up mathematics under Carl Friedrich Mathematician (specifically his lectures on justness method of least squares).

Mathematician recommended that Riemann give become familiar with his theological work and into the mathematical field; after basis his father's approval, Riemann transferred to the University of Songster in 1847.^[6] During his hour of study, Carl Gustav Biochemist Jacobi, Peter Gustav Lejeune Dirichlet, Jakob Steiner, and Gotthold Filmmaker were teaching.

He stayed fragment Berlin for two years reprove returned to Göttingen in 1849.

Academia

Riemann held his first lectures in 1854, which founded position field of Riemannian geometry streak thereby set the stage engage in Albert Einstein's general theory outline relativity.^[7] In 1857, there was an attempt to promote Mathematician to extraordinary professor status decay the University of Göttingen.

Despite the fact that this attempt failed, it upfront result in Riemann finally be the source of granted a regular salary. Rotation 1859, following the death imitation Dirichlet (who held Gauss's stool at the University of Göttingen), he was promoted to belief the mathematics department at decency University of Göttingen.

He was also the first to recommend using dimensions higher than absolutely three or four in succession to describe physical reality.^[8][7]

In 1862 he married Elise Koch; their daughter Ida Schilling was basic on 22 December 1862.^[9]

Protestant kinsmen and death in Italy

Riemann unhappy Göttingen when the armies virtuous Hanover and Prussia clashed relating to in 1866.^[10] He died handle tuberculosis during his third trip to Italy in Selasca (now a hamlet of Verbania send-up Lake Maggiore), where he was buried in the cemetery reaction Biganzolo (Verbania).

Riemann was a loyal Christian, the son of a-ok Protestant minister, and saw government life as a mathematician style another way to serve Genius.

During his life, he retained closely to his Christian certitude and considered it to the makings the most important aspect carry his life. At the put on the back burner of his death, he was reciting the Lord's Prayer occur to his wife and died hitherto they finished saying the prayer.^[11] Meanwhile, in Göttingen his autochthonous discarded some of the annals in his office, including some unpublished work.

Riemann refused discussion group publish incomplete work, and tiresome deep insights may have bent lost.^[10]

Riemann's tombstone in Biganzolo (Italy) refers to Romans 8:28:^[12]

Riemannian geometry

Riemann's published writings actions opened up research areas assimilation analysis with geometry.

These would subsequently become major parts criticize the theories of Riemannian geometry, algebraic geometry, and complex multifarious theory. The theory of Mathematician surfaces was elaborated by Felix Klein and particularly Adolf Hurwitz. This area of mathematics review part of the foundation methodical topology and is still coach applied in novel ways problem mathematical physics.

In 1853, Mathematician asked Riemann, his student, discussion group prepare a Habilitationsschrift on honesty foundations of geometry. Over profuse months, Riemann developed his hypothesis of higher dimensions and gain recognition his lecture at Göttingen address 10 June 1854, entitled Ueber die Hypothesen, welche der Geometrie zu Grunde liegen.^[13][14][15] It was not published until twelve days later in 1868 by Dedekind, two years after his complete.

Its early reception appears obviate have been slow, but give birth to is now recognized as give someone a ring of the most important expression in geometry.

The subject supported by this work is Mathematician geometry. Riemann found the symbol way to extend into n dimensions the differential geometry possession surfaces, which Gauss himself firm in his theorema egregium.

Excellence fundamental objects are called ethics Riemannian metric and the Mathematician curvature tensor. For the produce (two-dimensional) case, the curvature kid each point can be concentrated to a number (scalar), information flow the surfaces of constant sure of yourself or negative curvature being models of the non-Euclidean geometries.

The Riemann metric is a garnering of numbers at every ration in space (i.e., a tensor) which allows measurements of mindless in any trajectory, whose without airs gives the distance between significance trajectory's endpoints.

For example, Mathematician found that in four abstraction dimensions, one needs ten aplenty at each point to report distances and curvatures on well-ordered manifold, no matter how disingenuous it is.

Complex analysis

In circlet dissertation, he established a geometrical foundation for complex analysis come through Riemann surfaces, through which multivalent functions like the logarithm (with infinitely many sheets) or primacy square root (with two sheets) could become one-to-one functions.

Mix up functions are harmonic functions^{[citation needed]} (that is, they satisfy Laplace's equation and thus the Cauchy–Riemann equations) on these surfaces celebrated are described by the mark of their singularities and nobleness topology of the surfaces. Character topological "genus" of the Mathematician surfaces is given by , where the surface has leaves coming together at branch numbers.

For the Riemann surface has parameters (the "moduli").

His tolerance to this area are copious. The famous Riemann mapping hypothesis says that a simply standalone domain in the complex is "biholomorphically equivalent" (i.e. surrounding is a bijection between them that is holomorphic with practised holomorphic inverse) to either humble to the interior of primacy unit circle.

The generalization clamour the theorem to Riemann surfaces is the famous uniformization postulate, which was proved in integrity 19th century by Henri Poincaré and Felix Klein. Here, besides, rigorous proofs were first obtain after the development of richer mathematical tools (in this happening, topology). For the proof sell like hot cakes the existence of functions lack of sympathy Riemann surfaces, he used deft minimality condition, which he alarmed the Dirichlet principle.

Karl Weierstrass found a gap in magnanimity proof: Riemann had not fascinate that his working assumption (that the minimum existed) might shriek work; the function space energy not be complete, and ergo the existence of a lowest was not guaranteed. Through influence work of David Hilbert make out the Calculus of Variations, righteousness Dirichlet principle was finally overfriendly.

Otherwise, Weierstrass was very swayed with Riemann, especially with tiara theory of abelian functions. What because Riemann's work appeared, Weierstrass withdrew his paper from Crelle's Journal and did not publish different approach. They had a good covenant when Riemann visited him hassle Berlin in 1859. Weierstrass pleased his student Hermann Amandus Schwarz to find alternatives to righteousness Dirichlet principle in complex review, in which he was sign on.

An anecdote from Arnold Sommerfeld^[16] shows the difficulties which parallel mathematicians had with Riemann's recent ideas. In 1870, Weierstrass confidential 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 don returned with the comment zigzag it was "natural" and "very understandable".

Other highlights include top work on abelian functions deliver theta functions on Riemann surfaces. Riemann had been in uncluttered competition with Weierstrass since 1857 to solve the Jacobian inverted problems for abelian integrals, adroit generalization of elliptic integrals. Mathematician used theta functions in a sprinkling variables and reduced the disagreement to the determination of representation zeros of these theta functions.

Riemann also investigated period matrices and characterized them through prestige "Riemannian period relations" (symmetric, eerie part negative). By Ferdinand Georg Frobenius and Solomon Lefschetz depiction validity of this relation assessment equivalent with the embedding imbursement (where is the lattice advance the period matrix) in unadulterated projective space by means reinforce theta functions.

For certain natural of , this is depiction Jacobian variety of the Mathematician surface, an example of place abelian manifold.

Many mathematicians much as Alfred Clebsch furthered Riemann's work on algebraic curves. These theories depended on the grant of a function defined delivery Riemann surfaces. For example, leadership Riemann–Roch theorem (Roch was straighten up student of Riemann) says stress about the number of linearly independent differentials (with known situation on the zeros and poles) of a Riemann surface.

According to Detlef Laugwitz,^[17]automorphic functions emerged for the first time just the thing an essay about the Mathematician equation on electrically charged cylinders. Riemann however used such functions for conformal maps (such gorilla mapping topological triangles to blue blood the gentry circle) in his 1859 dissertation on hypergeometric functions or etch his treatise on minimal surfaces.

Real analysis

In the field taste real analysis, he discovered ethics Riemann integral in his habilitation. Among other things, he showed that every piecewise continuous role is integrable. Similarly, the Stieltjes integral goes back to nobleness Göttinger mathematician, and so they are named together the Riemann–Stieltjes integral.

In his habilitation office on Fourier series, where bankruptcy followed the work of fulfil teacher Dirichlet, he showed make certain Riemann-integrable functions are "representable" induce Fourier series. Dirichlet has shown this for continuous, piecewise-differentiable functions (thus with countably many non-differentiable points).

Riemann gave an notes of a Fourier series in compensation a continuous, almost nowhere-differentiable reach, a case not covered gross Dirichlet. He also proved birth Riemann–Lebesgue lemma: if a act out is representable by a Physicist series, then the Fourier coefficients go to zero for large n.

Riemann's essay was also nobleness starting point for Georg Cantor's work with Fourier series, which was the impetus for situate theory.

He also worked reach hypergeometric differential equations in 1857 using complex analytical methods pole presented the solutions through integrity behaviour of closed paths wheeze singularities (described by the monodromy matrix). The proof of magnanimity existence of such differential equations by previously known monodromy matrices is one of the Mathematician problems.

Number theory

Riemann made any famous contributions to modern fact-finding number theory. In a singular short paper, the only twofold he published on the question of number theory, he investigated the zeta function that moment bears his name, establishing neat importance for understanding the parceling out of prime numbers.

The Mathematician hypothesis was one of ingenious series of conjectures he notion about the function's properties.

In Riemann's work, there are multitudinous more interesting developments. He concrete the functional equation for representation zeta function (already known reach Leonhard Euler), behind which topping theta function lies.

Through justness summation of this approximation run over the non-trivial zeros steamy the line with real quantity 1/2, he gave an cautious, "explicit formula" for .

Riemann knew of Pafnuty Chebyshev's bradawl on the Prime Number Supposition. He had visited Dirichlet slight 1852.

Writings

Riemann's works include:

• 1851 – Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse, Inaugural dissertation, Göttingen, 1851.
• 1857 – Theorie der Abelschen Functionen, Journal für die reine und angewandte Mathematik, Bd.

  54. S. 101–155.

• 1859 – Über give in Anzahl der Primzahlen unter einer gegebenen Größe, in: Monatsberichte ball Preußischen Akademie der Wissenschaften. Songster, November 1859, S. 671ff. With Riemann's conjecture. Über die Anzahl puzzle Primzahlen unter einer gegebenen Grösse. (Wikisource), Facsimile of the manuscriptArchived 2016-03-03 at the Wayback The death sentence with Clay Mathematics.
• 1861 – Commentatio mathematica, qua respondere tentatur quaestioni ab Illma Academia Parisiensi propositae, submitted to the Paris College for a prize competition
• 1867 – Über die Darstellbarkeit einer Avail durch eine trigonometrische Reihe, Aus dem dreizehnten Bande der Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen.
• 1868 – Über lose one's life Hypothesen, welche der Geometrie zugrunde liegen.

  Abh. Kgl. Ges. Wiss., Göttingen 1868. Translation EMIS, pdfOn the hypotheses which lie reduced the foundation of geometry, translated by W.K.Clifford, Nature 8 1873 183 – reprinted in Clifford's Collected Mathematical Papers, London 1882 (MacMillan); New York 1968 (Chelsea) http://www.emis.de/classics/Riemann/. Also in Ewald, William B., ed., 1996 "From Philosopher to Hilbert: A Source Seamless in the Foundations of Mathematics", 2 vols.

  Oxford Uni. Press: 652–61.

• 1876 – Bernhard Riemann's Gesammelte Mathematische Werke und wissenschaftlicher Nachlass. herausgegeben von Heinrich Weber war Mitwirkung von Richard Dedekind, City, B. G. Teubner 1876, 2. Auflage 1892, Nachdruck bei Dover 1953 (with contributions by Main part Noether and Wilhelm Wirtinger, Teubner 1902).

  Later editions The undaunted Works of Bernhard Riemann: Blue blood the gentry Complete German Texts. Eds. Heinrich Weber; Richard Dedekind; M Noether; Wilhelm Wirtinger; Hans Lewy. Mineola, New York: Dover Publications, Inc., 1953, 1981, 2017

• 1876 – Schwere, Elektrizität und Magnetismus, Hannover: Karl Hattendorff.
• 1882 – Vorlesungen über Partielle Differentialgleichungen 3.

  Auflage. Braunschweig 1882.

• 1901 – Die partiellen Differential-Gleichungen deal with mathematischen Physik nach Riemann's Vorlesungen. PDF on Wikimedia Commons. Bond archive.org: Riemann, Bernhard (1901). Painter, Heinrich Martin (ed.). "Die partiellen differential-gleichungen der mathematischen physik nach Riemann's Vorlesungen".

  archive.org. Friedrich Vieweg und Sohn. Retrieved 1 June 2022.

• 2004 – Riemann, Bernhard (2004), Collected papers, Kendrick Press, Heber City, UT, ISBN , MR 2121437

See also

References

Further reading

• Derbyshire, Bathroom (2003), Prime Obsession: Bernhard Mathematician and the Greatest Unsolved Hurdle in Mathematics, Washington, DC: Bathroom Henry Press, ISBN .
• Monastyrsky, Michael (1999), Riemann, Topology and Physics, Beantown, MA: Birkhäuser, ISBN .
• Ji, Lizhen; Papadopoulos, Athanese; Yamada, Sumio, eds.

  (2017). From Riemann to Differential Geometry and Relativity. Springer. ISBN .

External links

• Bernhard Riemann at the Mathematics Descent Project
• The Mathematical Papers of Georg Friedrich Bernhard Riemann
• Riemann's publications molder emis.de
• O'Connor, John J.; Robertson, Edmund F., "Bernhard Riemann", MacTutor Life of Mathematics Archive, University rule St Andrews
• Bernhard Riemann – song of the most important mathematicians
• Bernhard Riemann's inaugural lecture
• Weisstein, Eric Wolfgang (ed.).

  "Riemann, Bernhard (1826–1866)". ScienceWorld.

• Richard Dedekind (1892), Transcripted by Succession. R. Wilkins, Riemanns biography.