Gf bernhard riemann biography template

German mathematician (–)

"Riemann" redirects all round. For other people with goodness surname, see Riemann (surname). Operate other topics named after Bernhard Riemann, see List of topics named after Bernhard Riemann.

Not gain be confused with Bernhard Raimann.

Georg Friedrich Bernhard Riemann (German:[ˈɡeːɔʁkˈfʁiːdʁɪçˈbɛʁnhaʁtˈʁiːman]^ⓘ;^[1][2] 17 September – 20 July ) was a German mathematician who made profound contributions to investigation, number theory, and differential geometry. In the field of ideal analysis, he is mostly methodical for the first rigorous disposal of the integral, the Mathematician integral, and his work grandeur Fourier series. His contributions class complex analysis include most remarkably the introduction of Riemann surfaces, breaking new ground in unadorned natural, geometric treatment of unintelligent analysis. His paper on high-mindedness prime-counting function, containing the contemporary statement of the Riemann premiss, is regarded as a foundational paper of analytic number understanding. Through his pioneering contributions finish differential geometry, Riemann laid interpretation foundations of the mathematics take possession of general relativity.^[3] He is held by many to be individual of the greatest mathematicians appreciated all time.^[4][5]

Biography

Early years

Riemann was dropped on 17 September in Breselenz, a village near Dannenberg worry the Kingdom of Hanover. Surmount father, Friedrich Bernhard Riemann, was a poor Lutheran pastor unexciting Breselenz who fought in ethics Napoleonic Wars. His mother, City Ebell, died in Riemann was the second of six descendants. Riemann exhibited exceptional mathematical flair, such as calculation abilities, cause the collapse of an early age but receive from timidity and a grumble of speaking in public.

Education

During , Riemann went to Royalty to live with his grannie and attend lyceum (middle grammar years), because such a sort of school was not susceptible from his home village. Rearguard the death of his grandma in , he transferred deal the Johanneum Lüneburg, a revitalization school in Lüneburg. There, Mathematician studied the Bible intensively, nevertheless he was often distracted provoke mathematics. His teachers were thunderstruck by his ability to meet complicated mathematical operations, in which he often outstripped his instructor's knowledge. In , at representation age of 19, he going on studying philology and Christian system in order to become clever pastor and help with king family's finances.

During the source of , his father, equate gathering enough money, sent Mathematician to the University of Göttingen, where he planned to recite towards a degree in subject. However, once there, he began studying mathematics under Carl Friedrich Gauss (specifically his lectures flaw the method of least squares). Gauss recommended that Riemann compromise up his theological work submit enter the mathematical field; stern getting his father's approval, Mathematician transferred to the University female Berlin in ^[6] During fulfil time of study, Carl Gustav Jacob Jacobi, Peter Gustav Lejeune Dirichlet, Jakob Steiner, and Gotthold Eisenstein were teaching. He stayed in Berlin for two age and returned to Göttingen limit

Academia

Riemann held his first lectures in , which founded class field of Riemannian geometry stomach thereby set the stage cherish Albert Einstein's general theory discover relativity.^[7] In , there was an attempt to promote Mathematician to extraordinary professor status livid the University of Göttingen. Granted this attempt failed, it blunt result in Riemann finally creature granted a regular salary. Check , following the death draw round Dirichlet (who held Gauss's stool at the University of Göttingen), he was promoted to purpose the mathematics department at nobility University of Göttingen. He was also the first to propose using dimensions higher than essentially three or four in uneasiness to describe physical reality.^[8][7]

In grace married Elise Koch; their colleen Ida Schilling was born arranged 22 December ^[9]

Protestant family move death in Italy

Riemann fled Göttingen when the armies of Dynasty and Prussia clashed there interest ^[10] He died of t.b. during his third journey philosopher Italy in Selasca (now trig hamlet of Verbania on Cap Maggiore), where he was covert in the cemetery in Biganzolo (Verbania).

Riemann was a dedicated Christly, the son of a Dissenter minister, and saw his strive as a mathematician as selection way to serve God. Away his life, he held powerfully to his Christian faith charge considered it to be nobleness most important aspect of empress life. At the time look up to his death, he was reading the Lord's Prayer with rulership wife and died before they finished saying the prayer.^[11] Interval, in Göttingen his housekeeper dumped some of the papers behave his office, including much encrypted work. Riemann refused to publicize incomplete work, and some concave insights may have been lost.^[10]

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

Riemannian geometry

Riemann's published works unbolt up research areas combining study with geometry. These would briefly become major parts of influence theories of Riemannian geometry, algebraical geometry, and complex manifold shyly. The theory of Riemann surfaces was elaborated by Felix Designer and particularly Adolf Hurwitz. That area of mathematics is trace of the foundation of configuration and is still being factual in novel ways to precise physics.

In , Gauss by choice Riemann, his student, to coach a Habilitationsschrift on the framework of geometry. Over many months, Riemann developed his theory suggest higher dimensions and delivered monarch lecture at Göttingen on 10 June , entitled Ueber decease Hypothesen, welche der Geometrie zu Grunde liegen.^[13][14][15] It was wail published until twelve years succeeding in by Dedekind, two era after his death. Its completely reception appears to have antique slow, but it is at the moment recognized as one of character most important works in geometry.

The subject founded by that work is Riemannian geometry. Mathematician found the correct way end up extend into n dimensions significance differential geometry of surfaces, which Gauss himself proved in tiara theorema egregium. The fundamental objects are called the Riemannian measure and the Riemann curvature tensor. For the surface (two-dimensional) crate, the curvature at each standardize can be reduced to uncluttered number (scalar), with the surfaces of constant positive or interdict curvature being models of rendering non-Euclidean geometries.

The Riemann measured is a collection of in profusion at every point in trimming (i.e., a tensor) which allows measurements of speed in plebeian trajectory, whose integral gives prestige distance between the trajectory's endpoints. For example, Riemann found go off in four spatial dimensions, creep needs ten numbers at inculcate point to describe distances near curvatures on a manifold, thumb matter how distorted it wreckage.

Complex analysis

In his dissertation, filth established a geometric foundation pull out complex analysis through Riemann surfaces, through which multi-valued functions alike the logarithm (with infinitely repeat sheets) or the square starting point (with two sheets) could evolve into one-to-one functions. Complex functions lookout harmonic functions^{[citation needed]} (that deterioration, they satisfy Laplace's equation boss thus the Cauchy–Riemann equations) untruthful these surfaces and are designated by the location of their singularities and the topology persuade somebody to buy the surfaces. The topological "genus" of the Riemann surfaces go over given by , where dignity surface has leaves coming confound at branch points. For high-mindedness Riemann surface has parameters (the "moduli").

His contributions to that area are numerous. The famed Riemann mapping theorem says think it over a simply connected domain bother the complex plane is "biholomorphically equivalent" (i.e. there is a-okay bijection between them that task holomorphic with a holomorphic inverse) to either or to prestige interior of the unit clique. The generalization of the supposition to Riemann surfaces is nobleness famous uniformization theorem, which was proved in the 19th hundred by Henri Poincaré and Felix Klein. Here, too, rigorous proofs were first given after interpretation development of richer mathematical reach (in this case, topology). Characterize the proof of the years of functions on Riemann surfaces, he used a minimality example, which he called the Dirichlet principle. Karl Weierstrass found spiffy tidy up gap in the proof: Mathematician had not noticed that crown working assumption (that the existed) might not work; depiction function space might not break down complete, and therefore the stand of a minimum was quite a distance guaranteed. Through the work for David Hilbert in the Stone of Variations, the Dirichlet statute was finally established. Otherwise, Weierstrass was very impressed with Mathematician, especially with his theory look upon abelian functions. When Riemann's bore appeared, Weierstrass withdrew his carve from Crelle's Journal and sincere not publish it. They difficult a good understanding when Mathematician visited him in Berlin rivet Weierstrass encouraged his student Hermann Amandus Schwarz to find alternatives to the Dirichlet principle replace complex analysis, in which powder was successful. An anecdote give birth to Arnold Sommerfeld^[16] shows the encumbrance under obligation which contemporary mathematicians had have a crush on Riemann's new ideas. In , Weierstrass had taken Riemann's essay with him on a timeout to Rigi and complained defer it was hard to lacking clarity. The physicist Hermann von Physicist assisted him in the profession overnight and returned with representation comment that it was "natural" and "very understandable".

Other highlights include his work on abelian functions and theta functions rebellion Riemann surfaces. Riemann had anachronistic in a competition with Weierstrass since to solve the Jacobian inverse problems for abelian integrals, a generalization of elliptic integrals. Riemann used theta functions hillock several variables and reduced interpretation problem to the determination dear the zeros of these theta functions. Riemann also investigated age matrices and characterized them from one side to the ot the "Riemannian period relations" (symmetric, real part negative). By Ferdinand Georg Frobenius and Solomon Lefschetz the validity of this coherence is equivalent with the embedding of (where is the opening of the period matrix) bind a projective space by course of action of theta functions. For think values of , this not bad the Jacobian variety of birth Riemann surface, an example give evidence an abelian manifold.

Many mathematicians such as Alfred Clebsch furthered Riemann's work on algebraic stroll. These theories depended on righteousness properties of a function careful on Riemann surfaces. For draw, the Riemann–Roch theorem (Roch was a student of Riemann) says something about the number boss linearly independent differentials (with pronounce conditions on the zeros deed poles) of a Riemann sector.

According to Detlef Laugwitz,^[17]automorphic functions appeared for the first hold your fire in an essay about representation Laplace equation on electrically crammed cylinders. Riemann however used specified functions for conformal maps (such as mapping topological triangles pass on the circle) in his disquisition on hypergeometric functions or infant his treatise on minimal surfaces.

Real analysis

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

In his habilitation ditch on Fourier series, where loosen up followed the work of enthrone teacher Dirichlet, he showed cruise Riemann-integrable functions are "representable" newborn Fourier series. Dirichlet has shown this for continuous, piecewise-differentiable functions (thus with countably many non-differentiable points). Riemann gave an sample of a Fourier series allowing for regarding a continuous, almost nowhere-differentiable work, a case not covered near Dirichlet. He also proved nobleness Riemann–Lebesgue lemma: if a overhaul is representable by a Mathematician series, then the Fourier coefficients go to zero for large&#;n.

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

He also worked mess up hypergeometric differential equations in demand complex analytical methods and throb the solutions through the ethics of closed paths about singularities (described by the monodromy matrix). The proof of the continuance of such differential equations near previously known monodromy matrices assessment one of the Hilbert pressure.

Number theory

Riemann made some well-known contributions to modern analytic hand out theory. In a single small paper, the only one oversight published on the subject refer to number theory, he investigated primacy zeta function that now bears his name, establishing its desirability for understanding the distribution distinctive prime numbers. The Riemann composition was one of a lean-to of conjectures he made contest the function's properties.

In Riemann's work, there are many finer interesting developments. He proved interpretation functional equation for the zeta function (already known to Leonhard Euler), behind which a theta function lies. Through the addition of this approximation function assign the non-trivial zeros on illustriousness line with real portion 1/2, he gave an exact, "explicit formula" for .

Riemann knew of Pafnuty Chebyshev's work prickliness the Prime Number Theorem. Inaccuracy had visited Dirichlet in

Writings

Riemann's works include:

• – Grundlagen für eine allgemeine Theorie instability Functionen einer veränderlichen complexen Grösse, Inaugural dissertation, Göttingen,
• – Theorie der Abelschen Functionen, Newspaper für die reine und angewandte Mathematik, Bd. S. –
• – Über die Anzahl der Primzahlen unter einer gegebenen Größe, in: Monatsberichte der Preußischen Akademie slow down Wissenschaften. Berlin, November , S.&#;ff. With Riemann's conjecture. Über fall victim to Anzahl der Primzahlen unter einer gegebenen Grösse. (Wikisource), Facsimile countless the manuscriptArchived at the Wayback Machine with Clay Mathematics.
• – Commentatio mathematica, qua respondere tentatur quaestioni ab Illma Academia Parisiensi propositae, submitted to the Town Academy for a prize competition
• – Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe, Aus dem dreizehnten Bande calm Abhandlungen der Königlichen Gesellschaft conflict Wissenschaften zu Göttingen.
• – Über die Hypothesen, welche der Geometrie zugrunde liegen. Abh. Kgl. Side. Wiss., Göttingen Translation EMIS, pdfOn the hypotheses which lie claim the foundation of geometry, translated by rd, Nature 8 – reprinted in Clifford's Collected Accurate Papers, London (MacMillan); New Royalty (Chelsea) Also in Ewald, William B., ed., "From Kant essay Hilbert: A Source Book be thankful for the Foundations of Mathematics", 2 vols. Oxford Uni. Press: &#;
• – Bernhard Riemann's Gesammelte Mathematische Werke und wissenschaftlicher Nachlass. herausgegeben von Heinrich Weber unter Mitwirkung von Richard Dedekind, Leipzig, All thumbs. G. Teubner , 2. Auflage , Nachdruck bei Dover (with contributions by Max Noether ray Wilhelm Wirtinger, Teubner ). Following editions The collected Works signal Bernhard Riemann: The Complete Germanic Texts. Eds. Heinrich Weber; Richard Dedekind; M Noether; Wilhelm Wirtinger; Hans Lewy. Mineola, New York: Dover Publications, Inc., , ,
• – Schwere, Elektrizität top secret Magnetismus, Hannover: Karl Hattendorff.
• – Vorlesungen über Partielle Differentialgleichungen 3. Auflage. Braunschweig
• – Die partiellen Differential-Gleichungen der mathematischen Physik nach Riemann's Vorlesungen. PDF method Wikimedia Commons. On : Riemann, Bernhard (). Weber, Heinrich Actor (ed.). "Die partiellen differential-gleichungen set out mathematischen physik nach Riemann's Vorlesungen". . Friedrich Vieweg und Sohn. Retrieved 1 June
• – Riemann, Bernhard (), Collected papers, Kendrick Press, Heber City, Say-so, ISBN&#;, MR&#;

See also

References

Further reading

• Derbyshire, John (), Prime Obsession: Bernhard Riemann and the Greatest Up in the air Problem in Mathematics, Washington, DC: John Henry Press, ISBN&#;.
• Monastyrsky, Archangel (), Riemann, Topology and Physics, Boston, MA: Birkhäuser, ISBN&#;.
• Ji, Lizhen; Papadopoulos, Athanese; Yamada, Sumio, system. (). From Riemann to Discernment Geometry and Relativity. Springer. ISBN&#;.

External links

• Bernhard Riemann at the Science Genealogy Project
• The Mathematical Papers entrap Georg Friedrich Bernhard Riemann
• Riemann's publications at
• O'Connor, John J.; Guard, Edmund F., "Bernhard Riemann", MacTutor History of Mathematics Archive, Founding of St Andrews
• Bernhard Riemann &#; one of the most leading mathematicians
• Bernhard Riemann's inaugural lecture
• Weisstein, Eric Wolfgang (ed.). "Riemann, Bernhard (&#;)". ScienceWorld.
• Richard Dedekind (), Transcripted hunk D. R. Wilkins, Riemanns biography.