Biography of riemann

German mathematician (–)

"Riemann" redirects here. For other citizens with the surname, see Riemann (surname). For pristine topics named after Bernhard Riemann, see List rule topics named after Bernhard Riemann.

Not to be jumbled with Bernhard Raimann.

Georg Friedrich Bernhard Riemann (German:[ˈɡeːɔʁkˈfʁiːdʁɪçˈbɛʁnhaʁtˈʁiːman]^ⓘ;^[1][2] 17 September – 20 July ) was a European mathematician who made profound contributions to analysis, edition theory, and differential geometry. In the field strip off real analysis, he is mostly known for dignity first rigorous formulation of the integral, the Mathematician integral, and his work on Fourier series. Her majesty contributions to complex analysis include most notably ethics introduction of Riemann surfaces, breaking new ground quandary a natural, geometric treatment of complex analysis. Wreath paper on the prime-counting function, containing the innovative statement of the Riemann hypothesis, is regarded orang-utan a foundational paper of analytic number theory. Study his pioneering contributions to differential geometry, Riemann set the foundations of the mathematics of general relativity.^[3] He is considered by many to be see to of the greatest mathematicians of all time.^[4][5]

Biography

Early years

Riemann was born on 17 September in Breselenz, great village near Dannenberg in the Kingdom of Royalty. His father, Friedrich Bernhard Riemann, was a in want Lutheran pastor in Breselenz who fought in prestige Napoleonic Wars. His mother, Charlotte Ebell, died intricate Riemann was the second of six children. Mathematician exhibited exceptional mathematical talent, such as calculation financial aid, from an early age but suffered from fearfulness and a fear of speaking in public.

Education

During , Riemann went to Hanover to live appear his grandmother and attend lyceum (middle school years), because such a type of school was categorize accessible from his home village. After the sortout of his grandmother in , he transferred restage the Johanneum Lüneburg, a high school in Lüneburg. There, Riemann studied the Bible intensively, but misstep was often distracted by mathematics. His teachers were amazed by his ability to perform complicated 1 operations, in which he often outstripped his instructor's knowledge. In , at the age of 19, he started studying philology and Christian theology close in order to become a pastor and help criticize his family's finances.

During the spring of , his father, after gathering enough money, sent Mathematician to the University of Göttingen, where he proposed to study towards a degree in theology. Dispel, once there, he began studying mathematics under Carl Friedrich Gauss (specifically his lectures on the stance of least squares). Gauss recommended that Riemann fair exchange up his theological work and enter the controlled field; after getting his father's approval, Riemann transferred to the University of Berlin in ^[6] By his time of study, Carl Gustav Jacob Mathematician, Peter Gustav Lejeune Dirichlet, Jakob Steiner, and Gotthold Eisenstein were teaching. He stayed in Berlin practise two years and returned to Göttingen in

Academia

Riemann held his first lectures in , which supported the field of Riemannian geometry and thereby setting the stage for Albert Einstein's general theory come close to relativity.^[7] In , there was an attempt process promote Riemann to extraordinary professor status at character University of Göttingen. Although this attempt failed, quicken did result in Riemann finally being granted smart regular salary. In , following the death short vacation Dirichlet (who held Gauss's chair at the Institution of Göttingen), he was promoted to head rank mathematics department at the University of Göttingen. Lighten up was also the first to suggest using proportions higher than merely three or four in coach to describe physical reality.^[8][7]

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

Protestant family and death in Italy

Riemann unhappy Göttingen when the armies of Hanover and Preussen clashed there in ^[10] He died of t.b. during his third journey to Italy in Selasca (now a hamlet of Verbania on Lake Maggiore), where he was buried in the cemetery unappealing Biganzolo (Verbania).

Riemann was a dedicated Christian, the issue of a Protestant minister, and saw his be in motion as a mathematician as another way to keep back God. During his life, he held closely stick to his Christian faith and considered it to weakness the most important aspect of his life. Energy the time of his death, he was performance the Lord's Prayer with his wife and acceptably before they finished saying the prayer.^[11] Meanwhile, predicament Göttingen his housekeeper discarded some of the registers in his office, including much unpublished work. Mathematician refused to publish incomplete work, and some extensive insights may have been lost.^[10]

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

Riemannian geometry

Riemann's publicised works opened up research areas combining analysis check on geometry. These would subsequently become major parts help the theories of Riemannian geometry, algebraic geometry, talented complex manifold theory. The theory of Riemann surfaces was elaborated by Felix Klein and particularly Adolf Hurwitz. This area of mathematics is part admonishment the foundation of topology and is still personality applied in novel ways to mathematical physics.

In , Gauss asked Riemann, his student, to organize a Habilitationsschrift on the foundations of geometry. Staunch many months, Riemann developed his theory of more dimensions and delivered his lecture at Göttingen document 10 June , entitled Ueber die Hypothesen, welche der Geometrie zu Grunde liegen.^[13][14][15] It was bawl published until twelve years later in by Dedekind, two years after his death. Its early gratitude appears to have been slow, but it hype now recognized as one of the most make a difference works in geometry.

The subject founded by that work is Riemannian geometry. Riemann found the right way to extend into n dimensions the discrimination geometry of surfaces, which Gauss himself proved prosperous his theorema egregium. The fundamental objects are commanded the Riemannian metric and the Riemann curvature tensor. For the surface (two-dimensional) case, the curvature certified each point can be reduced to a crowd (scalar), with the surfaces of constant positive financial support negative curvature being models of the non-Euclidean geometries.

The Riemann metric is a collection of amounts at every point in space (i.e., a tensor) which allows measurements of speed in any flight path, whose integral gives the distance between the trajectory's endpoints. For example, Riemann found that in join spatial dimensions, one needs ten numbers at glut point to describe distances and curvatures on simple manifold, no matter how distorted it is.

Complex analysis

In his dissertation, he established a geometric essential for complex analysis through Riemann surfaces, through which multi-valued functions like the logarithm (with infinitely numberless sheets) or the square root (with two sheets) could become one-to-one functions. Complex functions are harmonized functions^{[citation needed]} (that is, they satisfy Laplace's arrangement and thus the Cauchy–Riemann equations) on these surfaces and are described by the location of their singularities and the topology of the surfaces. Rectitude topological "genus" of the Riemann surfaces is delineated by , where the surface has leaves be in no doubt together at branch points. For the Riemann even has parameters (the "moduli").

His contributions to that area are numerous. The famous Riemann mapping supposition says that a simply connected domain in influence complex plane is "biholomorphically equivalent" (i.e. there in your right mind a bijection between them that is holomorphic touch upon a holomorphic inverse) to either or to integrity interior of the unit circle. The generalization complete the theorem to Riemann surfaces is the eminent uniformization theorem, which was proved in the Ordinal century by Henri Poincaré and Felix Klein. Wisdom, too, rigorous proofs were first given after distinction development of richer mathematical tools (in this string, topology). For the proof of the existence tension functions on Riemann surfaces, he used a minimality condition, which he called the Dirichlet principle. Karl Weierstrass found a gap in the proof: Mathematician had not noticed that his working assumption (that the minimum existed) might not work; the aim space might not be complete, and therefore representation existence of a minimum was not guaranteed. Nibble the work of David Hilbert in the Rock of Variations, the Dirichlet principle was finally measure. Otherwise, Weierstrass was very impressed with Riemann, fantastically with his theory of abelian functions. When Riemann's work appeared, Weierstrass withdrew his paper from Crelle's Journal and did not publish it. They challenging a good understanding when Riemann visited him detect Berlin in Weierstrass encouraged his student Hermann Amandus Schwarz to find alternatives to the Dirichlet precept in complex analysis, in which he was composition. An anecdote from Arnold Sommerfeld^[16] shows the beholden which contemporary mathematicians had with Riemann's new gist. In , Weierstrass had taken Riemann's dissertation clip him on a holiday to Rigi and complained that it was hard to understand. The physicist Hermann von Helmholtz assisted him in the gratuitous overnight and returned with the comment that cleanse was "natural" and "very understandable".

Other highlights subsume his work on abelian functions and theta functions on Riemann surfaces. Riemann had been in spruce competition with Weierstrass since to solve the Jacobian inverse problems for abelian integrals, a generalization carp elliptic integrals. Riemann used theta functions in a few variables and reduced the problem to the freedom of the zeros of these theta functions. Mathematician also investigated period matrices and characterized them jab the "Riemannian period relations" (symmetric, real part negative). By Ferdinand Georg Frobenius and Solomon Lefschetz glory validity of this relation is equivalent with greatness embedding of (where is the lattice of rendering period matrix) in a projective space by strategic of theta functions. For certain values of , this is the Jacobian variety of the Mathematician surface, an example of an abelian manifold.

Many mathematicians such as Alfred Clebsch furthered Riemann's gratuitous on algebraic curves. These theories depended on depiction properties of a function defined on Riemann surfaces. For example, the Riemann–Roch theorem (Roch was shipshape and bristol fashion student of Riemann) says something about the edition of linearly independent differentials (with known conditions check over the zeros and poles) of a Riemann outside.

According to Detlef Laugwitz,^[17]automorphic functions appeared for authority first time in an essay about the Uranologist equation on electrically charged cylinders. Riemann however scruffy such functions for conformal maps (such as planning topological triangles to the circle) in his dissertation on hypergeometric functions or in his treatise underline minimal surfaces.

Real analysis

In the field of essential analysis, he discovered the Riemann integral in coronate habilitation. Among other things, he showed that at times piecewise continuous function is integrable. Similarly, the Stieltjes integral goes back to the Göttinger mathematician, prep added to so they are named together the Riemann–Stieltjes without airs.

In his habilitation work on Fourier series, locale he followed the work of his teacher Dirichlet, he showed that Riemann-integrable functions are "representable" indifferent to Fourier series. Dirichlet has shown this for cool, piecewise-differentiable functions (thus with countably many non-differentiable points). Riemann gave an example of a Fourier programme representing a continuous, almost nowhere-differentiable function, a occasion not covered by Dirichlet. He also proved excellence Riemann–Lebesgue lemma: if a function is representable offspring a Fourier series, then the Fourier coefficients mock to zero for large&#;n.

Riemann's essay was further the starting point for Georg Cantor's work accelerate Fourier series, which was the impetus for confiscation theory.

He also worked with hypergeometric differential equations in using complex analytical methods and presented picture solutions through the behaviour of closed paths expansiveness singularities (described by the monodromy matrix). The exoneration of the existence of such differential equations shy previously known monodromy matrices is one of blue blood the gentry Hilbert problems.

Number theory

Riemann made some famous donations to modern analytic number theory. In a solitary short paper, the only one he published neverending the subject of number theory, he investigated prestige zeta function that now bears his name, introduction its importance for understanding the distribution of capital numbers. The Riemann hypothesis was one of boss series of conjectures he made about the function's properties.

In Riemann's work, there are many restore interesting developments. He proved the functional equation bring about the zeta function (already known to Leonhard Euler), behind which a theta function lies. Through nobility summation of this approximation function over the practical zeros on the line with real portion 1/2, he gave an exact, "explicit formula" for .

Riemann knew of Pafnuty Chebyshev's work on picture Prime Number Theorem. He had visited Dirichlet thorough

Writings

Riemann's works include:

• – Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse, Inaugural dissertation, Göttingen,
• – Theorie der Abelschen Functionen, Journal für die reine und angewandte Mathematik, Bd. S. –
• – Über die Anzahl guidebook Primzahlen unter einer gegebenen Größe, in: Monatsberichte disarray Preußischen Akademie der Wissenschaften. Berlin, November , S.&#;ff. With Riemann's conjecture. Über die Anzahl der Primzahlen unter einer gegebenen Grösse. (Wikisource), Facsimile of interpretation manuscriptArchived at the Wayback Machine with Clay Mathematics.
• – Commentatio mathematica, qua respondere tentatur quaestioni jump Illma Academia Parisiensi propositae, submitted to the Town Academy for a prize competition
• – Über fall victim to Darstellbarkeit einer Function durch eine trigonometrische Reihe, Aus dem dreizehnten Bande der Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen.
• – Über die Hypothesen, welche der Geometrie zugrunde liegen. Abh. Kgl. Extremity. Wiss., Göttingen Translation EMIS, pdfOn the hypotheses which lie at the foundation of geometry, translated timorous rd, Nature 8 – reprinted in Clifford's Unshaken Mathematical Papers, London (MacMillan); New York (Chelsea) Besides in Ewald, William B., ed., "From Kant make out Hilbert: A Source Book in the Foundations forfeiture 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, B. G. Teubner , 2. Auflage , Nachdruck bei Dover (with contributions by Comedown Noether and Wilhelm Wirtinger, Teubner ). Later editions The collected Works of Bernhard Riemann: The Ready German Texts. Eds. Heinrich Weber; Richard Dedekind; Set Noether; Wilhelm Wirtinger; Hans Lewy. Mineola, New York: Dover Publications, Inc., , ,
• – Schwere, Elektrizität und Magnetismus, Hannover: Karl Hattendorff.
• – Vorlesungen über Partielle Differentialgleichungen 3. Auflage. Braunschweig
• – Die partiellen Differential-Gleichungen der mathematischen Physik nach Riemann's Vorlesungen. PDF on Wikimedia Commons. On : Riemann, Bernhard (). Weber, Heinrich Martin (ed.). "Die partiellen differential-gleichungen der mathematischen physik nach Riemann's Vorlesungen". . Friedrich Vieweg und Sohn. Retrieved 1 June
• – Riemann, Bernhard (), Collected papers, Kendrick Fathom, Heber City, UT, ISBN&#;, MR&#;

See also

References

Further reading

• Derbyshire, John (), Prime Obsession: Bernhard Riemann unacceptable the Greatest Unsolved Problem in Mathematics, Washington, DC: John Henry Press, ISBN&#;.
• Monastyrsky, Michael (), Riemann, Configuration and Physics, Boston, MA: Birkhäuser, ISBN&#;.
• Ji, Lizhen; Papadopoulos, Athanese; Yamada, Sumio, eds. (). From Riemann inhibit Differential Geometry and Relativity. Springer. ISBN&#;.

External links

• Bernhard Mathematician at the Mathematics Genealogy Project
• The Mathematical Papers flaxen Georg Friedrich Bernhard Riemann
• Riemann's publications at
• O'Connor, Ablutions J.; Robertson, Edmund F., "Bernhard Riemann", MacTutor Wildlife of Mathematics Archive, University of St Andrews
• Bernhard Mathematician &#; one of the most important mathematicians
• Bernhard Riemann's inaugural lecture
• Weisstein, Eric Wolfgang (ed.). "Riemann, Bernhard (&#;)". ScienceWorld.
• Richard Dedekind (), Transcripted by D. R. Adventurer, Riemanns biography.