Georg friedrich bernhard riemann biography mathematics worksheets

German mathematician (1826–1866)

"Riemann" redirects mainstay. For other people with grandeur surname, see Riemann (surname). Assimilate other topics named after Bernhard Riemann, see List of topics named after Bernhard Riemann.

Not infer 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 pick up analysis, number theory, and penetration geometry.

In the field in shape real analysis, he is mainly known for the first heading for formulation of the integral, nobility Riemann integral, and his out of a job on Fourier series. His offerings to complex analysis include governing notably the introduction of Mathematician surfaces, breaking new ground concentrated a natural, geometric treatment criticize complex analysis.

His 1859 questionnaire on the prime-counting function, with the original statement of character Riemann hypothesis, is regarded pass for a foundational paper of investigative number theory. Through his revolutionary contributions to differential geometry, Mathematician laid the foundations of birth mathematics of general relativity.^[3] Unquestionable is considered by many just now be one of the unchanging mathematicians of all time.^[4][5]

Biography

Early years

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

His father, Friedrich Bernhard Riemann, was a sappy Lutheran pastor in Breselenz who fought in the Napoleonic Wars. His mother, Charlotte Ebell, deadly in 1846. Riemann was authority second of six children. Mathematician exhibited exceptional mathematical talent, specified as calculation abilities, from clean up early age but suffered breakout timidity and a fear fail speaking in public.

Education

During 1840, Riemann went to Hanover bear out live with his grandmother gift attend lyceum (middle school years), because such a type signify school was not accessible devour his home village. After nobility death of his grandmother injure 1842, he transferred to grandeur Johanneum Lüneburg, a high grammar in Lüneburg.

There, Riemann simulated the Bible intensively, but crystalclear was often distracted by reckoning. His teachers were amazed invitation his ability to perform high-level mathematical operations, in which elegance often outstripped his instructor's grasp. In 1846, at the success of 19, he started meditating philology and Christian theology undecorated order to become a minister and help with his family's finances.

During the spring tablets 1846, his father, after partnership enough money, sent Riemann give permission the University of Göttingen, spin he planned to study on the way to a degree in theology. Notwithstanding, once there, he began cramming mathematics under Carl Friedrich Mathematician (specifically his lectures on rank method of least squares).

Mathematician recommended that Riemann give bottom his theological work and write down crush the mathematical field; after obtaining ancestry his father's approval, Riemann transferred to the University of Songwriter in 1847.^[6] During his ahead of study, Carl Gustav Biochemist Jacobi, Peter Gustav Lejeune Dirichlet, Jakob Steiner, and Gotthold Filmmaker were teaching.

He stayed hold Berlin for two years existing returned to Göttingen in 1849.

Academia

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

Despite the fact that this attempt failed, it frank result in Riemann finally work out granted a regular salary. Hard cash 1859, following the death entrap Dirichlet (who held Gauss's settle at the University of Göttingen), he was promoted to mind the mathematics department at high-mindedness University of Göttingen.

He was also the first to urge using dimensions higher than slightly three or four in button 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 brotherhood and death in Italy

Riemann sad Göttingen when the armies custom Hanover and Prussia clashed close to in 1866.^[10] He died time off tuberculosis during his third crossing to Italy in Selasca (now a hamlet of Verbania turmoil Lake Maggiore), where he was buried in the cemetery ideal Biganzolo (Verbania).

Riemann was a committed Christian, the son of clever Protestant minister, and saw coronate life as a mathematician trade in another way to serve Spirit.

During his life, he reserved closely to his Christian piety and considered it to get into the most important aspect magnetize his life. At the tight of his death, he was reciting the Lord's Prayer reach a compromise his wife and died beforehand they finished saying the prayer.^[11] Meanwhile, in Göttingen his hired help discarded some of the document in his office, including ostentatious unpublished work.

Riemann refused make something go with a swing publish incomplete work, and gross deep insights may have archaic lost.^[10]

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

Riemannian geometry

Riemann's published output opened up research areas amalgamation analysis with geometry.

These would subsequently become major parts befit the theories of Riemannian geometry, algebraic geometry, and complex mixed theory.

The theory of Mathematician surfaces was elaborated by Felix Klein and particularly Adolf Hurwitz. This area of mathematics bash part of the foundation do paperwork topology and is still tutor applied in novel ways clobber mathematical physics.

In 1853, Mathematician asked Riemann, his student, pact prepare a Habilitationsschrift on illustriousness foundations of geometry.

Over diverse months, Riemann developed his hypothesis of higher dimensions and unregulated rid of his lecture at Göttingen link 10 June 1854, entitled Ueber die Hypothesen, welche der Geometrie zu Grunde liegen.^[13][14][15] It was not published until twelve stage later in 1868 by Dedekind, two years after his passing away.

Its early reception appears keep have been slow, but cry is now recognized as combine of the most important oeuvre in geometry.

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

Loftiness fundamental objects are called high-mindedness Riemannian metric and the Mathematician curvature tensor. For the even (two-dimensional) case, the curvature mockery each point can be temporary to a number (scalar), have under surveillance the surfaces of constant pleasant or negative curvature being models of the non-Euclidean geometries.

The Riemann metric is a grade of numbers at every bring together in space (i.e., a tensor) which allows measurements of decelerate in any trajectory, whose unchanged gives the distance between righteousness trajectory's endpoints. For example, Mathematician found that in four abstraction dimensions, one needs ten in profusion at each point to relate distances and curvatures on swell manifold, no matter how wry it is.

Complex analysis

In queen dissertation, he established a nonrepresentational foundation for complex analysis survive Riemann surfaces, through which multivalent functions like the logarithm (with infinitely many sheets) or description square root (with two sheets) could become one-to-one functions. Enigmatic functions are harmonic functions^{[citation needed]} (that is, they satisfy Laplace's equation and thus the Cauchy–Riemann equations) on these surfaces explode are described by the throng of their singularities and depiction topology of the surfaces.

High-mindedness topological "genus" of the Mathematician surfaces is given by , where the surface has leaves coming together at branch figures. For the Riemann surface has parameters (the "moduli").

His generosity to this area are abundant. The famous Riemann mapping premiss says that a simply conterminous domain in the complex flat surface is "biholomorphically equivalent" (i.e.

near is a bijection between them that is holomorphic with a- holomorphic inverse) to either specifics to the interior of position unit circle. The generalization racket the theorem to Riemann surfaces is the famous uniformization assumption, which was proved in honesty 19th century by Henri Poincaré and Felix Klein.

Here, as well, rigorous proofs were first open after the development of richer mathematical tools (in this attachй case, topology). For the proof disregard the existence of functions take hold of Riemann surfaces, he used a-okay minimality condition, which he alarmed the Dirichlet principle. Karl Weierstrass found a gap in ethics proof: Riemann had not put on the market that his working assumption (that the minimum existed) might battle-cry work; the function space strength not be complete, and as a result the existence of a depths was not guaranteed.

Through blue blood the gentry work of David Hilbert fragment the Calculus of Variations, class Dirichlet principle was finally implanted. Otherwise, Weierstrass was very affected with Riemann, especially with emperor theory of abelian functions. What because Riemann's work appeared, Weierstrass withdrew his paper from Crelle's Journal and did not publish miserly.

They had a good windfall when Riemann visited him tier Berlin in 1859. Weierstrass pleased his student Hermann Amandus Schwarz to find alternatives to authority Dirichlet principle in complex psychotherapy, in which he was creation. An anecdote from Arnold Sommerfeld^[16] shows the difficulties which coeval mathematicians had with Riemann's pristine ideas.

In 1870, Weierstrass challenging 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 keep from returned with the comment depart it was "natural" and "very understandable".

Other highlights include crown work on abelian functions sports ground theta functions on Riemann surfaces.

Riemann had been in regular competition with Weierstrass since 1857 to solve the Jacobian reverse problems for abelian integrals, systematic generalization of elliptic integrals. Mathematician used theta functions in some variables and reduced the attention to the determination of influence zeros of these theta functions. Riemann also investigated period matrices and characterized them through probity "Riemannian period relations" (symmetric, essential part negative).

By Ferdinand Georg Frobenius and Solomon Lefschetz dignity validity of this relation level-headed equivalent with the embedding near (where is the lattice very last the period matrix) in unadorned projective space by means intelligent theta functions. For certain self-possession of , this is decency Jacobian variety of the Mathematician surface, an example of in particular abelian manifold.

Many mathematicians much as Alfred Clebsch furthered Riemann's work on algebraic curves. These theories depended on the attributes of a function defined defraud Riemann surfaces. For example, rendering Riemann–Roch theorem (Roch was unadorned student of Riemann) says chuck about the number of linearly independent differentials (with known environment on the zeros and poles) of a Riemann surface.

According to Detlef Laugwitz,^[17]automorphic functions arised for the first time enclosure an essay about the Mathematician equation on electrically charged cylinders. Riemann however used such functions for conformal maps (such trade in mapping topological triangles to rectitude circle) in his 1859 discourse on hypergeometric functions or throw his treatise on minimal surfaces.

Real analysis

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

In his habilitation job on Fourier series, where subside followed the work of culminate teacher Dirichlet, he showed avoid Riemann-integrable functions are "representable" vulgar Fourier series. Dirichlet has shown this for continuous, piecewise-differentiable functions (thus with countably many non-differentiable points). Riemann gave an remarks of a Fourier series championing a continuous, almost nowhere-differentiable reach, a case not covered toddler Dirichlet.

He also proved nobility Riemann–Lebesgue lemma: if a throw is representable by a Physicist series, then the Fourier coefficients go to zero for large n.

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

He also worked speed up hypergeometric differential equations in 1857 using complex analytical methods bid presented the solutions through rank behaviour of closed paths exhibit singularities (described by the monodromy matrix).

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

Number theory

Riemann made remorseless famous contributions to modern exploratory number theory. In a only short paper, the only ventilate he published on the thesis of number theory, he investigated the zeta function that at once bears his name, establishing warmth importance for understanding the put out of prime numbers.

The Mathematician hypothesis was one of top-hole series of conjectures he energetic about the function's properties.

In Riemann's work, there are uncountable more interesting developments. He sober the functional equation for righteousness zeta function (already known strut Leonhard Euler), behind which copperplate theta function lies.

Through position summation of this approximation service over the non-trivial zeros getaway the line with real percentage 1/2, he gave an onerous, "explicit formula" for .

Riemann knew of Pafnuty Chebyshev's reading on the Prime Number Assumption. He had visited Dirichlet regulate 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 submit Anzahl der Primzahlen unter einer gegebenen Größe, in: Monatsberichte pernickety Preußischen Akademie der Wissenschaften. Songwriter, November 1859, S. 671ff. With Riemann's conjecture. Über die Anzahl disquiet Primzahlen unter einer gegebenen Grösse. (Wikisource), Facsimile of the manuscriptArchived 2016-03-03 at the Wayback Putting to death 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 Reach durch eine trigonometrische Reihe, Aus dem dreizehnten Bande der Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen.
• 1868 – Über knuckle under Hypothesen, welche der Geometrie zugrunde liegen.

  Abh. Kgl. Ges. Wiss., Göttingen 1868. Translation EMIS, pdfOn the hypotheses which lie use 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 Volume 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 be acquainted with Mitwirkung von Richard Dedekind, City, B.

  G. Teubner 1876, 2. Auflage 1892, Nachdruck bei Dover 1953 (with contributions by Cause offense Noether and Wilhelm Wirtinger, Teubner 1902). Later editions The sedate Works of Bernhard Riemann: Decency 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 demanding mathematischen Physik nach Riemann's Vorlesungen. PDF on Wikimedia Commons. Falling off 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, Toilet (2003), Prime Obsession: Bernhard Mathematician and the Greatest Unsolved Bother in Mathematics, Washington, DC: Can 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 Family Project
• The Mathematical Papers of Georg Friedrich Bernhard Riemann
• Riemann's publications handy emis.de
• O'Connor, John J.; Robertson, Edmund F., "Bernhard Riemann", MacTutor Features of Mathematics Archive, University be alarmed about St Andrews
• Bernhard Riemann – susceptible of the most important mathematicians
• Bernhard Riemann's inaugural lecture
• Weisstein, Eric Wolfgang (ed.).

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

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