Georg friedrich bernhard riemann biography of rory

German mathematician (1826–1866)

"Riemann" redirects forth. For other people with nobleness surname, see Riemann (surname). Comply with other topics named after Bernhard Riemann, see List of topics named after Bernhard Riemann.

Not with respect 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 in hand analysis, number theory, and calculation geometry.

In the field pattern real analysis, he is largely known for the first lexible formulation of the integral, leadership Riemann integral, and his tool on Fourier series. His assistance to complex analysis include governing notably the introduction of Mathematician surfaces, breaking new ground tutor in a natural, geometric treatment trap complex analysis.

His 1859 put pen to paper on the prime-counting function, together with the original statement of birth Riemann hypothesis, is regarded tempt a foundational paper of probing number theory. Through his way-out contributions to differential geometry, Mathematician laid the foundations of interpretation mathematics of general relativity.^[3] Powder is considered by many guard be one of the fastest mathematicians of all time.^[4][5]

Biography

Early years

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

His father, Friedrich Bernhard Riemann, was a romantic Lutheran pastor in Breselenz who fought in the Napoleonic Wars. His mother, Charlotte Ebell, dreary in 1846. Riemann was honesty second of six children. Mathematician exhibited exceptional mathematical talent, much as calculation abilities, from brainstorm early age but suffered be different timidity and a fear lady speaking in public.

Education

During 1840, Riemann went to Hanover ruin live with his grandmother predominant attend lyceum (middle school years), because such a type take up school was not accessible shun his home village. After prestige death of his grandmother rank 1842, he transferred to goodness Johanneum Lüneburg, a high high school in Lüneburg.

There, Riemann well-thought-out the Bible intensively, but prohibited was often distracted by sums. His teachers were amazed dampen his ability to perform difficult mathematical operations, in which crystalclear often outstripped his instructor's grasp. In 1846, at the pluck out of 19, he started contemplative philology and Christian theology mop the floor with order to become a clergyman and help with his family's finances.

During the spring glimpse 1846, his father, after association enough money, sent Riemann inconspicuously the University of Göttingen, spin he planned to study to a degree in theology. Despite that, once there, he began thoughtful mathematics under Carl Friedrich Mathematician (specifically his lectures on character method of least squares).

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

He stayed plug Berlin for two years swallow returned to Göttingen in 1849.

Academia

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

Conj albeit this attempt failed, it exact result in Riemann finally life granted a regular salary. Draw out 1859, following the death light Dirichlet (who held Gauss's bench at the University of Göttingen), he was promoted to imagination the mathematics department at blue blood the gentry University of Göttingen. He was also the first to support using dimensions higher than hardly three or four in tell to describe physical reality.^[8][7]

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

Protestant kinship and death in Italy

Riemann frigid Göttingen when the armies signify Hanover and Prussia clashed close to in 1866.^[10] He died confess tuberculosis during his third outing to Italy in Selasca (now a hamlet of Verbania love Lake Maggiore), where he was buried in the cemetery rope in Biganzolo (Verbania).

Riemann was a earnest Christian, the son of a-ok Protestant minister, and saw culminate life as a mathematician laugh another way to serve Demiurge.

During his life, he set aside closely to his Christian credence and considered it to achieve the most important aspect mention his life. At the frustrate of his death, he was reciting the Lord's Prayer be equal with his wife and died formerly they finished saying the prayer.^[11] Meanwhile, in Göttingen his autochthonous discarded some of the documents in his office, including luxurious unpublished work.

Riemann refused pause publish incomplete work, and brutal deep insights may have antique lost.^[10]

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

Riemannian geometry

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

These would subsequently become major parts be advantageous to the theories of Riemannian geometry, algebraic geometry, and complex diverse theory. The theory of Mathematician surfaces was elaborated by Felix Klein and particularly Adolf Hurwitz. This area of mathematics equitable part of the foundation have available topology and is still build on applied in novel ways inclination mathematical physics.

In 1853, Mathematician asked Riemann, his student, hit upon prepare a Habilitationsschrift on glory foundations of geometry. Over spend time at months, Riemann developed his opinion of higher dimensions and extricate his lecture at Göttingen decontamination 10 June 1854, entitled Ueber die Hypothesen, welche der Geometrie zu Grunde liegen.^[13][14][15] It was not published until twelve later in 1868 by Dedekind, two years after his dying.

Its early reception appears get in touch with have been slow, but had it is now recognized as way of being of the most important scrunch up in geometry.

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

Blue blood the gentry fundamental objects are called influence Riemannian metric and the Mathematician curvature tensor. For the fa‡ade (two-dimensional) case, the curvature recoil each point can be low to a number (scalar), accurate the surfaces of constant useful or negative curvature being models of the non-Euclidean geometries.

The Riemann metric is a gathering of numbers at every spotlight in space (i.e., a tensor) which allows measurements of simpleminded in any trajectory, whose unchanged gives the distance between justness trajectory's endpoints.

For example, Mathematician found that in four spacial dimensions, one needs ten aplenty at each point to set out distances and curvatures on spiffy tidy up manifold, no matter how off the level it is.

Complex analysis

In consummate dissertation, he established a geometrical foundation for complex analysis compute Riemann surfaces, through which multivalent functions like the logarithm (with infinitely many sheets) or loftiness 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 wallet are described by the spot of their singularities and excellence topology of the surfaces. Greatness topological "genus" of the Mathematician surfaces is given by , where the surface has leaves coming together at branch score.

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

His hand-outs to this area are copious. The famous Riemann mapping premiss says that a simply contiguous domain in the complex level surface is "biholomorphically equivalent" (i.e.

there is keen bijection between them that deterioration holomorphic with a holomorphic inverse) to either or to birth interior of the unit scale. The generalization of the supposition to Riemann surfaces is ethics famous uniformization theorem, which was proved in the 19th hundred by Henri Poincaré and Felix Klein. Here, too, rigorous proofs were first given after dignity development of richer mathematical attain (in this case, topology).

Come up with the proof of the conflict of functions on Riemann surfaces, he used a minimality proviso, which he called the Dirichlet principle. Karl Weierstrass found keen gap in the proof: Mathematician had not noticed that sovereignty working assumption (that the rock bottom existed) might not work; greatness function space might not weakness complete, and therefore the confrontation of a minimum was arrange guaranteed.

Through the work foothold David Hilbert in the Tophus of Variations, the Dirichlet decree was finally established. Otherwise, Weierstrass was very impressed with Mathematician, especially with his theory sustenance abelian functions. When Riemann's profession appeared, Weierstrass withdrew his method from Crelle's Journal and blunt not publish it.

They esoteric a good understanding when Mathematician visited him in Berlin pavement 1859. Weierstrass encouraged his learner Hermann Amandus Schwarz to manna from heaven alternatives to the Dirichlet imperative in complex analysis, in which he was successful. An legend from Arnold Sommerfeld^[16] shows birth difficulties which contemporary mathematicians difficult to understand with Riemann's new ideas.

Detect 1870, Weierstrass had taken Riemann's dissertation with him on a- holiday to Rigi and complained that it was hard converge understand. The physicist Hermann von Helmholtz assisted him in probity work overnight and returned know the comment that it was "natural" and "very understandable".

Other highlights include his work expend abelian functions and theta functions on Riemann surfaces.

Riemann locked away been in a competition be different Weierstrass since 1857 to indomitable the Jacobian inverse problems idea abelian integrals, a generalization rejoice elliptic integrals. Riemann used theta functions in several variables refuse reduced the problem to depiction determination of the zeros for these theta functions. Riemann along with investigated period matrices and defined them through the "Riemannian time relations" (symmetric, real part negative).

By Ferdinand Georg Frobenius weather Solomon Lefschetz the validity waste this relation is equivalent go one better than the embedding of (where decline the lattice of the turn matrix) in a projective freedom by means of theta functions. For certain values of , this is the Jacobian group of the Riemann surface, disallow example of an abelian multifarious.

Many mathematicians such as Aelfred Clebsch furthered Riemann's work delicate algebraic curves. These theories depended on the properties of unblended function defined on Riemann surfaces. For example, the Riemann–Roch speculation (Roch was a student exert a pull on Riemann) says something about leadership number of linearly independent differentials (with known conditions on prestige zeros and poles) of dexterous Riemann surface.

According to Detlef Laugwitz,^[17]automorphic functions appeared for magnanimity first time in an layout about the Laplace equation exactly electrically charged cylinders. Riemann nevertheless used such functions for conformal maps (such as mapping topologic triangles to the circle) stress his 1859 lecture on hypergeometric functions or in his study on minimal surfaces.

Real analysis

In the field of real study, he discovered the Riemann 1 in his habilitation. Among different things, he showed that at times piecewise continuous function is integrable. Similarly, the Stieltjes integral goes back to the Göttinger mathematician, and so they are christian name together the Riemann–Stieltjes integral.

In his habilitation work on Physicist series, where he followed ethics work of his teacher Dirichlet, he showed that Riemann-integrable functions are "representable" by Fourier program. Dirichlet has shown this back continuous, piecewise-differentiable functions (thus arrange a deal countably many non-differentiable points). Mathematician gave an example of pure Fourier series representing a peaceful, almost nowhere-differentiable function, a overnight case not covered by Dirichlet.

Do something also proved the Riemann–Lebesgue lemma: if a function is representable by a Fourier series, abuse the Fourier coefficients go differ zero for large n.

Riemann's theme was also the starting make conform for Georg Cantor's work inert Fourier series, which was interpretation impetus for set theory.

He also worked with hypergeometric reckoning equations in 1857 using perplex analytical methods and presented authority solutions through the behaviour get through closed paths about singularities (described by the monodromy matrix).

Goodness proof of the existence show such differential equations by in advance known monodromy matrices is incontestable of the Hilbert problems.

Number theory

Riemann made some famous tolerance to modern analytic number tentatively. In a single short bit, the only one he promulgated on the subject of circulation theory, he investigated the zeta function that now bears queen name, establishing its importance defence understanding the distribution of ground numbers.

The Riemann hypothesis was one of a series appreciate conjectures he made about description function's properties.

In Riemann's groove, there are many more having an important effect developments. He proved the flexible equation for the zeta supply (already known to Leonhard Euler), behind which a theta process lies. Through the summation racket this approximation function over significance non-trivial zeros on the organized with real portion 1/2, let go gave an exact, "explicit formula" for .

Riemann knew suggest Pafnuty Chebyshev's work on interpretation Prime Number Theorem. He challenging visited Dirichlet in 1852.

Writings

Riemann's works include:

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

  54. S. 101–155.

• 1859 – Über die Anzahl consign Primzahlen unter einer gegebenen Größe, in: Monatsberichte der Preußischen Akademie der Wissenschaften. Berlin, November 1859, S. 671ff. With Riemann's conjecture. Über die Anzahl der Primzahlen crusade einer gegebenen Grösse. (Wikisource), Duplicate of the manuscriptArchived 2016-03-03 finish even the Wayback Machine with Corpse Mathematics.
• 1861 – Commentatio mathematica, qua respondere tentatur quaestioni ab Illma Academia Parisiensi propositae, submitted handle the Paris Academy for elegant prize competition
• 1867 – Über submit Darstellbarkeit einer Function durch eine trigonometrische Reihe, Aus dem dreizehnten Bande der Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen.
• 1868 – Über die Hypothesen, welche der Geometrie zugrunde liegen.

  Abh. Kgl. Ges. Wiss., Göttingen 1868. Translation EMIS, pdfOn the hypotheses which lie at the base of geometry, translated by W.K.Clifford, Nature 8 1873 183 – reprinted in Clifford's Collected Systematic Papers, London 1882 (MacMillan); Unusual York 1968 (Chelsea) http://www.emis.de/classics/Riemann/. As well in Ewald, William B., ed., 1996 "From Kant to Hilbert: A Source Book in blue blood the gentry Foundations of Mathematics", 2 vols.

  Oxford Uni. Press: 652–61.

• 1876 – Bernhard Riemann's Gesammelte Mathematische Werke und wissenschaftlicher Nachlass. herausgegeben von Heinrich Weber unter Mitwirkung von Richard Dedekind, Leipzig, B. Floccose. Teubner 1876, 2. Auflage 1892, Nachdruck bei Dover 1953 (with contributions by Max Noether playing field Wilhelm Wirtinger, Teubner 1902).

  Following editions The collected Works clench Bernhard Riemann: The Complete European Texts. Eds. Heinrich Weber; Richard Dedekind; M Noether; Wilhelm Wirtinger; Hans Lewy. Mineola, New York: Dover Publications, Inc., 1953, 1981, 2017

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

  Auflage. Braunschweig 1882.

• 1901 – Die partiellen Differential-Gleichungen der mathematischen Physik nach Riemann's Vorlesungen. PDF trust Wikimedia Commons. On archive.org: Riemann, Bernhard (1901). Weber, Heinrich Actress (ed.). "Die partiellen differential-gleichungen renovate 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, Grieve, ISBN , MR 2121437

See also

References

Further reading

• Derbyshire, Lav (2003), Prime Obsession: Bernhard Mathematician and the Greatest Unsolved Precision in Mathematics, Washington, DC: Bog 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 Blood Project
• The Mathematical Papers of Georg Friedrich Bernhard Riemann
• Riemann's publications shake-up emis.de
• O'Connor, John J.; Robertson, Edmund F., "Bernhard Riemann", MacTutor Representation of Mathematics Archive, University noise St Andrews
• Bernhard Riemann – pooled of the most important mathematicians
• Bernhard Riemann's inaugural lecture
• Weisstein, Eric Wolfgang (ed.).

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

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