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Georg Friedrich Bernhard Riemann (pronounced REE mahn or in 'ri:man; September 17, 1826 – July 20, 1866) was a German mathematician who made important contributions to analysis and differential geometry, some of them paving the way for the later development of general relativity.

In 1847, his father (Friedrich Riemann), after gathering enough money to send Riemann to university, allowed him to stop studying theology and start studying mathematics. He was sent to the renowned University of Göttingen, where he first met Carl Friedrich Gauss, and attended his lectures on the method of least squares.

In 1847, Riemann moved to Berlin, where Jacobi, Dirichlet, and Steiner were teaching. He stayed in Berlin for two years and returned to Göttingen in 1849.

Riemann made major contributions to real analysis. He defined the Riemann integral by means of Riemann sums, developed a theory of trigonometric series that are not Fourier series—a first step in generalized function theory—and studied the Riemann-Liouville differintegral.

He made some famous contributions to modern analytic number theory. In a single short paper (the only one he published on the subject of number theory), he introduced the Riemann zeta function and established its importance for understanding the distribution of prime numbers. He made a series of conjectures about properties of the zeta function, one of which is the well-known Riemann hypothesis.

He applied the Dirichlet principle from variational calculus to great effect; this was later seen to be a powerful heuristic rather than a rigorous method. Its justification took at least a generation. His work on monodromy and the hypergeometric function in the complex domain made a great impression, and established a basic way of working with functions by consideration only of their singularities.

In 1853, Gauss asked his student Riemann to prepare a Habilitationsschrift on the foundations of geometry. Over many months, Riemann developed his theory of higher dimensions. When he finally delivered his lecture at Göttingen in 1854, the mathematical public received it with enthusiasm, and it is one of the most important works in geometry. It was titled Über die Hypothesen welche der Geometrie zu Grunde liegen (loosely: "On the foundations of geometry"; more precisely, "On the hypotheses which underlie geometry"), and was published in 1868.

The subject founded by this work is Riemannian geometry. Riemann found the correct way to extend into n dimensions the differential geometry of surfaces, which Gauss himself proved in his theorema egregium. The fundamental object is called the Riemann curvature tensor. For the surface case, this can be reduced to a number (scalar), positive, negative or zero; the non-zero and constant cases being models of the known non-Euclidean geometries.

- 1868.“On the hypotheses which lie at the foundation of geometry” in Ewald, William B., ed., 1996. “From Kant to Hilbert: A Source Book in the Foundations of Mathematics” , 2 vols. Oxford Uni. Press: 652-61.

- Riemann hypothesis
- Riemann zeta function
- Riemann integral
- Riemann sum
- Riemann lemma
- Riemannian manifold
- Riemann mapping theorem
- Riemann-Hilbert problem
- Riemann-Hurwitz formula
- Riemann-von Mangoldt formula
- Riemann surface
- Riemann-Roch theorem
- Riemann theta function
- Riemann-Siegel theta function
- Riemann's differential equation
- Riemann matrix
- Riemann sphere
- Riemannian metric tensor
- Riemann curvature tensor
- Cauchy-Riemann equations
- Hirzebruch-Riemann-Roch theorem
- Riemann-Lebesgue lemma
- Riemann-Stieltjes integral
- Riemann-Liouville differintegral
- Riemann series theorem
- Riemann's 1859 paper introducing the complex zeta function
- Prime Obsession
- The Music of the Primes

- John Derbyshire, "Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics" (John Henry Press, 2003) ISBN 0-309-08549-7

- The Mathematical Papers of Georg Friedrich Bernhard Riemann
- All publications of Riemann can be found at: http://www.emis.de/classics/Riemann/
- Bernhard Riemann - one of the most important mathematicians
- Bernhard Riemann's inaugural lecture

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Last updated on Monday October 06, 2008 at 17:20:12 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Monday October 06, 2008 at 17:20:12 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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