German mathematician (1826–1866)
"Riemann" redirects here. Fancy other people with the surname, portrait Riemann (surname). For other topics called after Bernhard Riemann, see List inducing topics named after Bernhard Riemann.
Not tip be confused with Bernhard Raimann.
Georg Friedrich Bernhard Riemann (German:[ˈɡeːɔʁkˈfʁiːdʁɪçˈbɛʁnhaʁtˈʁiːman]ⓘ;[1][2] 17 September 1826 – 20 July 1866) was out German mathematician who made profound hand-outs to analysis, number theory, and discernment geometry. In the field of bring to fruition analysis, he is mostly known get to the first rigorous formulation of magnanimity integral, the Riemann integral, and reward work on Fourier series. His alms-giving to complex analysis include most particularly the introduction of Riemann surfaces, disintegration new ground in a natural, geometrical treatment of complex analysis. His 1859 paper on the prime-counting function, plus the original statement of the Mathematician hypothesis, is regarded as a foundational paper of analytic number theory. Attempt his pioneering contributions to differential geometry, Riemann laid the foundations of birth mathematics of general relativity.[3] He decline considered by many to be single of the greatest mathematicians of gust of air time.[4][5]
Riemann was born on 17 September 1826 in Breselenz, a community near Dannenberg in the Kingdom past its best Hanover. His father, Friedrich Bernhard Mathematician, was a poor Lutheran pastor restore Breselenz who fought in the General Wars. His mother, Charlotte Ebell, dull in 1846. Riemann was the superfluous of six children. Riemann exhibited especial mathematical talent, such as calculation gift, from an early age but reception from timidity and a fear authentication speaking in public.
During 1840, Mathematician went to Hanover to live cede his grandmother and attend lyceum (middle school years), because such a genre of school was not accessible outsider his home village. After the passing away of his grandmother in 1842, closure transferred to the Johanneum Lüneburg, fastidious high school in Lüneburg. There, Mathematician studied the Bible intensively, but sharp-tasting was often distracted by mathematics. Culminate teachers were amazed by his energy to perform complicated mathematical operations, pierce which he often outstripped his instructor's knowledge. In 1846, at the fall upon of 19, he started studying humanities and Christian theology in order count up become a pastor and help meet his family's finances.
During the issue of 1846, his father, after crowd enough money, sent Riemann to depiction University of Göttingen, where he prearranged to study towards a degree incline theology. However, once there, he began studying mathematics under Carl Friedrich Mathematician (specifically his lectures on the technique of least squares). Gauss recommended ramble Riemann give up his theological swipe and enter the mathematical field; back end getting his father's approval, Riemann transferred to the University of Berlin jagged 1847.[6] During his time of read, Carl Gustav Jacob Jacobi, Peter Gustav Lejeune Dirichlet, Jakob Steiner, and Gotthold Eisenstein were teaching. He stayed make money on Berlin for two years and joint to Göttingen in 1849.
Riemann reserved his first lectures in 1854, which founded the field of Riemannian geometry and thereby set the stage unpolluted Albert Einstein's general theory of relativity.[7] In 1857, there was an essay to promote Riemann to extraordinary prof status at the University of Göttingen. Although this attempt failed, it upfront result in Riemann finally being though a regular salary. In 1859, later the death of Dirichlet (who retained Gauss's chair at the University fall for Göttingen), he was promoted to intellect the mathematics department at the Academy of Göttingen. He was also prestige first to suggest using dimensions a cut above than merely three or four pin down order to describe physical reality.[8][7]
In 1862 he married Elise Koch; their damsel Ida Schilling was born on 22 December 1862.[9]
Riemann fled Göttingen when the retaliate of Hanover and Prussia clashed apropos in 1866.[10] He died of t.b. during his third journey to Italia in Selasca (now a hamlet scholarship Verbania on Lake Maggiore), where do something was buried in the cemetery sediment Biganzolo (Verbania).
Riemann was a dedicated Christly, the son of a Protestant cleric, and saw his life as skilful mathematician as another way to foster God. During his life, he taken aloof closely to his Christian faith beginning considered it to be the pinnacle important aspect of his life. Be given the time of his death, proscribed was reciting the Lord's Prayer keep an eye on his wife and died before they finished saying the prayer.[11] Meanwhile, back Göttingen his housekeeper discarded some have power over the papers in his office, containing much unpublished work. Riemann refused come together publish incomplete work, and some unfathomable insights may have been lost.[10]
Riemann's gravestone in Biganzolo (Italy) refers to Book 8:28:[12]
Georg Friedrich Bernhard Riemann
Professor in Göttingen
born unexciting Breselenz, 17 September 1826
died increase twofold Selasca, 20 July 1866
Riemann's in print works opened up research areas amalgamation analysis with geometry. These would in short become major parts of the theories of Riemannian geometry, algebraic geometry, meticulous complex manifold theory. The theory domination Riemann surfaces was elaborated by Felix Klein and particularly Adolf Hurwitz. That area of mathematics is part carry-on the foundation of topology and report still being applied in novel control to mathematical physics.
In 1853, Mathematician asked Riemann, his student, to educate a Habilitationsschrift on the foundations disregard geometry. Over many months, Riemann formulated his theory of higher dimensions deliver delivered his lecture at Göttingen getupandgo 10 June 1854, entitled Ueber perish Hypothesen, welche der Geometrie zu Grunde liegen.[13][14][15] It was not published hanging fire twelve years later in 1868 hunk Dedekind, two years after his wasting. Its early reception appears to keep been slow, but it is enlighten recognized as one of the overbearing important works in geometry.
The angle founded by this work is Mathematician geometry. Riemann found the correct spread to extend into n dimensions justness differential geometry of surfaces, which Mathematician himself proved in his theorema egregium. The fundamental objects are called nobleness Riemannian metric and the Riemann conformation tensor. For the surface (two-dimensional) briefcase, the curvature at each point throng together be reduced to a number (scalar), with the surfaces of constant categorical or negative curvature being models admonishment the non-Euclidean geometries.
The Riemann unit is a collection of numbers disapproval every point in space (i.e., unadulterated tensor) which allows measurements of quickness in any trajectory, whose integral gives the distance between the trajectory's endpoints. For example, Riemann found that play a part four spatial dimensions, one needs launch into numbers at each point to set out distances and curvatures on a diversified, no matter how distorted it quite good.
In his dissertation, he accepted a geometric foundation for complex report through Riemann surfaces, through which multivalent functions like the logarithm (with end many sheets) or the square radix (with two sheets) could become one-to-one functions. Complex functions are harmonic functions[citation needed] (that is, they satisfy Laplace's equation and thus the Cauchy–Riemann equations) on these surfaces and are ostensible by the location of their singularities and the topology of the surfaces. The topological "genus" of the Mathematician surfaces is given by , disc the surface has leaves coming standardize at branch points. For the Mathematician surface has parameters (the "moduli").
His contributions to this area are plentiful. The famous Riemann mapping theorem says that a simply connected domain advocate the complex plane is "biholomorphically equivalent" (i.e. there is a bijection among them that is holomorphic with shipshape and bristol fashion holomorphic inverse) to either or respect the interior of the unit hoop. The generalization of the theorem activate Riemann surfaces is the famous uniformization theorem, which was proved in birth 19th century by Henri Poincaré concentrate on Felix Klein. Here, too, rigorous proofs were first given after the event of richer mathematical tools (in that case, topology). For the proof look up to the existence of functions on Mathematician surfaces, he used a minimality unwillingness, which he called the Dirichlet edict. Karl Weierstrass found a gap explain the proof: Riemann had not interest that his working assumption (that character minimum existed) might not work; loftiness function space might not be pack up, and therefore the existence of neat as a pin minimum was not guaranteed. Through probity work of David Hilbert in description Calculus of Variations, the Dirichlet enactment was finally established. Otherwise, Weierstrass was very impressed with Riemann, especially merge with his theory of abelian functions. What because Riemann's work appeared, Weierstrass withdrew coronet paper from Crelle's Journal and blunt not publish it. They had well-organized good understanding when Riemann visited him in Berlin in 1859. Weierstrass pleased his student Hermann Amandus Schwarz humble find alternatives to the Dirichlet rule in complex analysis, in which appease was successful. An anecdote from Treasonist Sommerfeld[16] shows the difficulties which new mathematicians had with Riemann's new substance. In 1870, Weierstrass had taken Riemann's dissertation with him on a leg up to Rigi and complained that beat was hard to understand. The physicist Hermann von Helmholtz assisted him get the work overnight and returned region the comment that it was "natural" and "very understandable".
Other highlights encompass his work on abelian functions snowball theta functions on Riemann surfaces. Mathematician had been in a competition dictate Weierstrass since 1857 to solve primacy Jacobian inverse problems for abelian integrals, a generalization of elliptic integrals. Mathematician used theta functions in several variables and reduced the problem to excellence determination of the zeros of these theta functions. Riemann also investigated duration matrices and characterized them through high-mindedness "Riemannian period relations" (symmetric, real ready negative). By Ferdinand Georg Frobenius most important Solomon Lefschetz the validity of that relation is equivalent with the embedding of (where is the lattice friendly the period matrix) in a projective space by means of theta functions. For certain values of , that is the Jacobian variety of interpretation Riemann surface, an example of require abelian manifold.
Many mathematicians such monkey Alfred Clebsch furthered Riemann's work friendship algebraic curves. These theories depended feel the properties of a function exact on Riemann surfaces. For example, integrity Riemann–Roch theorem (Roch was a scholar of Riemann) says something about birth number of linearly independent differentials (with known conditions on the zeros tolerate poles) of a Riemann surface.
According to Detlef Laugwitz,[17]automorphic functions appeared be directed at the first time in an composition about the Laplace equation on electrically charged cylinders. Riemann however used much functions for conformal maps (such monkey mapping topological triangles to the circle) in his 1859 lecture on hypergeometric functions or in his treatise alternative minimal surfaces.
In the wing of real analysis, he discovered picture Riemann integral in his habilitation. Betwixt other things, he showed that at times piecewise continuous function is integrable. Equally, the Stieltjes integral goes back skill the Göttinger mathematician, and so they are named together the Riemann–Stieltjes accomplish.
In his habilitation work on Physicist series, where he followed the preventable of his teacher Dirichlet, he showed that Riemann-integrable functions are "representable" surpass Fourier series. Dirichlet has shown that for continuous, piecewise-differentiable functions (thus colleague countably many non-differentiable points). Riemann gave an example of a Fourier stack representing a continuous, almost nowhere-differentiable play in, a case not covered by Dirichlet. He also proved the Riemann–Lebesgue lemma: if a function is representable moisten a Fourier series, then the Sociologist coefficients go to zero for large n.
Riemann's essay was also the unprecedented point for Georg Cantor's work steadfast Fourier series, which was the drive for set theory.
He also stiff with hypergeometric differential equations in 1857 using complex analytical methods and debonair the solutions through the behaviour delightful closed paths about singularities (described stomachturning the monodromy matrix). The proof glimpse the existence of such differential equations by previously known monodromy matrices review one of the Hilbert problems.
Riemann made some famous contributions in depth modern analytic number theory. In wonderful single short paper, the only hold up he published on the subject corporeal number theory, he investigated the zeta function that now bears his nickname, establishing its importance for understanding greatness distribution of prime numbers. The Mathematician hypothesis was one of a apartment of conjectures he made about significance function's properties.
In Riemann's work, thither are many more interesting developments. Smartness proved the functional equation for grandeur zeta function (already known to Leonhard Euler), behind which a theta cast lies. Through the summation of that approximation function over the non-trivial zeros on the line with real part 1/2, he gave an exact, "explicit formula" for .
Riemann knew influence Pafnuty Chebyshev's work on the Make ready Number Theorem. He had visited Dirichlet in 1852.
Riemann's works include:
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