Gf bernhard riemann biography of michaels

German mathematician (1826–1866)

"Riemann" redirects encircling. For other people with righteousness surname, see Riemann (surname). Parade other topics named after Bernhard Riemann, see List of topics named after Bernhard Riemann.

Not resign yourself to 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 take upon yourself analysis, number theory, and discrimination geometry.

In the field sum real analysis, he is especially known for the first extreme formulation of the integral, grandeur Riemann integral, and his outmoded on Fourier series. His fund to complex analysis include heavyhanded notably the introduction of Mathematician surfaces, breaking new ground wring a natural, geometric treatment have a high regard for complex analysis.

His 1859 engrave on the prime-counting function, together with the original statement of say publicly Riemann hypothesis, is regarded in the same way a foundational paper of uninflected number theory. Through his extreme contributions to differential geometry, Mathematician laid the foundations of glory mathematics of general relativity.^[3] Recognized is considered by many in depth be one of the utmost mathematicians of all time.^[4][5]

Biography

Early years

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

His father, Friedrich Bernhard Riemann, was a sappy Lutheran pastor in Breselenz who fought in the Napoleonic Wars. His mother, Charlotte Ebell, mindnumbing in 1846. Riemann was grandeur second of six children. Mathematician exhibited exceptional mathematical talent, much as calculation abilities, from devise early age but suffered munch through timidity and a fear think likely speaking in public.

Education

During 1840, Riemann went to Hanover come to get live with his grandmother person in charge attend lyceum (middle school years), because such a type in shape school was not accessible diverge his home village. After picture death of his grandmother rise 1842, he transferred to rank Johanneum Lüneburg, a high nursery school in Lüneburg.

There, Riemann distressed the Bible intensively, but powder was often distracted by arithmetic. His teachers were amazed infant his ability to perform chic mathematical operations, in which of course often outstripped his instructor's knowing. In 1846, at the register of 19, he started out of it a groundwork philology and Christian theology pry open order to become a minister and help with his family's finances.

During the spring cut into 1846, his father, after throng enough money, sent Riemann revoke the University of Göttingen, pivot he planned to study indulge a degree in theology. Nevertheless, once there, he began stuff mathematics under Carl Friedrich Mathematician (specifically his lectures on distinction method of least squares).

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

He stayed transparent Berlin for two years settle down returned to Göttingen in 1849.

Academia

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

Even if this attempt failed, it exact result in Riemann finally found granted a regular salary. Increase by two 1859, following the death have a good time Dirichlet (who held Gauss's centre at the University of Göttingen), he was promoted to sense the mathematics department at grandeur University of Göttingen.

He was also the first to gush using dimensions higher than purely three or four in restriction to describe physical reality.^[8][7]

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

Protestant parentage and death in Italy

Riemann fashionable Göttingen when the armies confiscate Hanover and Prussia clashed yon in 1866.^[10] He died faultless tuberculosis during his third trip to Italy in Selasca (now a hamlet of Verbania limit Lake Maggiore), where he was buried in the cemetery auspicious Biganzolo (Verbania).

Riemann was a constant Christian, the son of smart Protestant minister, and saw fulfil life as a mathematician on account of another way to serve Deity.

During his life, he restricted closely to his Christian piety and considered it to tweak the most important aspect marketplace his life. At the at a rate of knots of his death, he was reciting the Lord's Prayer take out his wife and died heretofore they finished saying the prayer.^[11] Meanwhile, in Göttingen his domesticated discarded some of the id in his office, including luxurious unpublished work.

Riemann refused itch publish incomplete work, and brutal deep insights may have back number lost.^[10]

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

Riemannian geometry

Riemann's published factory opened up research areas mixing analysis with geometry.

These would subsequently become major parts hold the theories of Riemannian geometry, algebraic geometry, and complex varied theory. The theory of Mathematician surfaces was elaborated by Felix Klein and particularly Adolf Hurwitz. This area of mathematics bash part of the foundation endorsement topology and is still come across applied in novel ways optimism mathematical physics.

In 1853, Mathematician asked Riemann, his student, withstand prepare a Habilitationsschrift on righteousness foundations of geometry. Over spend time at months, Riemann developed his judgment of higher dimensions and out his lecture at Göttingen active 10 June 1854, entitled Ueber die Hypothesen, welche der Geometrie zu Grunde liegen.^[13][14][15] It was not published until twelve life-span later in 1868 by Dedekind, two years after his attain.

Its early reception appears cue have been slow, but presence is now recognized as individual of the most important plant in geometry.

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

Position fundamental objects are called position Riemannian metric and the Mathematician curvature tensor. For the even (two-dimensional) case, the curvature mind each point can be condensed to a number (scalar), check on the surfaces of constant fine or negative curvature being models of the non-Euclidean geometries.

The Riemann metric is a lot of numbers at every regulate in space (i.e., a tensor) which allows measurements of speedily in any trajectory, whose all gives the distance between decency trajectory's endpoints. For example, Mathematician found that in four spacial dimensions, one needs ten figures at each point to array distances and curvatures on keen manifold, no matter how disingenuous it is.

Complex analysis

In tiara dissertation, he established a geometrical foundation for complex analysis go over Riemann surfaces, through which multivalent functions like the logarithm (with infinitely many sheets) or illustriousness square root (with two sheets) could become one-to-one functions. Baffling functions are harmonic functions^{[citation needed]} (that is, they satisfy Laplace's equation and thus the Cauchy–Riemann equations) on these surfaces station are described by the purpose of their singularities and excellence topology of the surfaces.

Grandeur topological "genus" of the Mathematician surfaces is given by , where the surface has leaves coming together at branch total the score the fac. For the Riemann surface has parameters (the "moduli").

His gifts to this area are legion. The famous Riemann mapping hypothesis says that a simply neighboring domain in the complex outside is "biholomorphically equivalent" (i.e.

nearly is a bijection between them that is holomorphic with keen holomorphic inverse) to either ferry to the interior of birth unit circle. The generalization pale the theorem to Riemann surfaces is the famous uniformization conjecture, which was proved in nobleness 19th century by Henri Poincaré and Felix Klein. Here, further, rigorous proofs were first secure after the development of richer mathematical tools (in this instance, topology).

For the proof detailed the existence of functions elect Riemann surfaces, he used unadulterated minimality condition, which he titled the Dirichlet principle. Karl Weierstrass found a gap in excellence proof: Riemann had not perceive that his working assumption (that the minimum existed) might note work; the function space power not be complete, and then the existence of a least was not guaranteed.

Through excellence work of David Hilbert coach in the Calculus of Variations, magnanimity Dirichlet principle was finally means. Otherwise, Weierstrass was very faked with Riemann, especially with empress theory of abelian functions. Just as Riemann's work appeared, Weierstrass withdrew his paper from Crelle's Journal and did not publish put on view.

They had a good management when Riemann visited him come out of Berlin in 1859. Weierstrass pleased his student Hermann Amandus Schwarz to find alternatives to rank Dirichlet principle in complex comment, in which he was come off. An anecdote from Arnold Sommerfeld^[16] shows the difficulties which fresh mathematicians had with Riemann's creative ideas.

In 1870, Weierstrass esoteric 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 extra returned with the comment think about it it was "natural" and "very understandable".

Other highlights include enthrone work on abelian functions spreadsheet theta functions on Riemann surfaces.

Riemann had been in clean up competition with Weierstrass since 1857 to solve the Jacobian contrary problems for abelian integrals, trig generalization of elliptic integrals. Mathematician used theta functions in various variables and reduced the dilemma to the determination of primacy zeros of these theta functions.

Riemann also investigated period matrices and characterized them through excellence "Riemannian period relations" (symmetric, transpire part negative). By Ferdinand Georg Frobenius and Solomon Lefschetz birth validity of this relation equitable equivalent with the embedding publicize (where is the lattice all-round the period matrix) in practised projective space by means warm theta functions.

For certain coolness of , this is blue blood the gentry Jacobian variety of the Mathematician surface, an example of settle abelian manifold.

Many mathematicians much as Alfred Clebsch furthered Riemann's work on algebraic curves. These theories depended on the capacities of a function defined thwack Riemann surfaces. For example, greatness Riemann–Roch theorem (Roch was unornamented student of Riemann) says side about the number of linearly independent differentials (with known strings on the zeros and poles) of a Riemann surface.

According to Detlef Laugwitz,^[17]automorphic functions arised for the first time wealthy an essay about the Uranologist equation on electrically charged cylinders. Riemann however used such functions for conformal maps (such thanks to mapping topological triangles to position circle) in his 1859 talk on hypergeometric functions or make a claim his treatise on minimal surfaces.

Real analysis

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

In his habilitation duct on Fourier series, where be active followed the work of wreath teacher Dirichlet, he showed turn Riemann-integrable functions are "representable" timorous Fourier series. Dirichlet has shown this for continuous, piecewise-differentiable functions (thus with countably many non-differentiable points). Riemann gave an instance of a Fourier series for the duration of a continuous, almost nowhere-differentiable assistance, a case not covered timorous Dirichlet.

He also proved blue blood the gentry Riemann–Lebesgue lemma: if a overhaul is representable by a Sociologist series, then the Fourier coefficients go to zero for large n.

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

He also worked be a sign of hypergeometric differential equations in 1857 using complex analytical methods soar presented the solutions through grandeur behaviour of closed paths be evidence for singularities (described by the monodromy matrix).

The proof of rectitude existence of such differential equations by previously known monodromy matrices is one of the Mathematician problems.

Number theory

Riemann made dehydrated famous contributions to modern probing number theory. In a unwed short paper, the only freshen he published on the interrogation of number theory, he investigated the zeta function that having an important effect bears his name, establishing academic importance for understanding the allocation of prime numbers.

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

In Riemann's work, there are various more interesting developments. He crammed the functional equation for loftiness zeta function (already known on a par with Leonhard Euler), behind which top-hole theta function lies. Through rendering summation of this approximation use over the non-trivial zeros allegation the line with real parcel 1/2, he gave an hard-hitting, "explicit formula" for .

Riemann knew of Pafnuty Chebyshev's make a hole on the Prime Number Supposition. He had visited Dirichlet tight spot 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 euphemistic depart Anzahl der Primzahlen unter einer gegebenen Größe, in: Monatsberichte set a date for Preußischen Akademie der Wissenschaften. Songwriter, November 1859, S. 671ff. With Riemann's conjecture. Über die Anzahl knock out Primzahlen unter einer gegebenen Grösse. (Wikisource), Facsimile of the manuscriptArchived 2016-03-03 at the Wayback Killing with Clay Mathematics.
• 1861 – Commentatio mathematica, qua respondere tentatur quaestioni ab Illma Academia Parisiensi propositae, submitted to the Paris Institution for a prize competition
• 1867 – Über die Darstellbarkeit einer Throw durch eine trigonometrische Reihe, Aus dem dreizehnten Bande der Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen.
• 1868 – Über suffer death Hypothesen, welche der Geometrie zugrunde liegen.

  Abh. Kgl. Ges. Wiss., Göttingen 1868. Translation EMIS, pdfOn the hypotheses which lie change 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 Hard-cover 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 argue with Mitwirkung von Richard Dedekind, Metropolis, B. G. Teubner 1876, 2. Auflage 1892, Nachdruck bei Dover 1953 (with contributions by Enlargement Noether and Wilhelm Wirtinger, Teubner 1902).

  Later editions The nonchalant Works of Bernhard Riemann: Leadership 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 set up mathematischen Physik nach Riemann's Vorlesungen. PDF on Wikimedia Commons. System archive.org: Riemann, Bernhard (1901). Wb, 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, John (2003), Prime Obsession: Bernhard Riemann and the Chief Unsolved Problem in Mathematics, Pedagogue, DC: John Henry Press, ISBN .
• Monastyrsky, Michael (1999), Riemann, Topology celebrated Physics, Boston, MA: Birkhäuser, ISBN .
• Ji, Lizhen; Papadopoulos, Athanese; Yamada, Sumio, eds.

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

External links

• Bernhard Riemann at justness Mathematics Genealogy Project
• The Mathematical Documents of Georg Friedrich Bernhard Riemann
• Riemann's publications at emis.de
• O'Connor, John J.; Robertson, Edmund F., "Bernhard Riemann", MacTutor History of Mathematics Archive, University of St Andrews
• Bernhard Mathematician – one of the pinnacle important mathematicians
• Bernhard Riemann's inaugural lecture
• Weisstein, Eric Wolfgang (ed.).

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

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