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Œuvres Mathématiques de Riemann
Excerpt from ?uvres Mathématiques de RiemannAbout the PublisherForgotten Books publishes hundreds of thousands of rare and classic books.This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works. This text has been digitally restored from a historical edition. Some errors may persist, however we consider it worth publishing due to the work's historical value.The digital edition of all books may be viewed on our website before purchase.
493 pages, Paperback
Published August 24, 2019
About the author
Bernhard Riemann
51books31followersGeorg Friedrich Bernhard Riemann [ˈʁiːman] ( listen) (September 17, 1826 � July 20, 1866) was an influential German mathematician who made lasting contributions to analysis, number theory, and differential geometry, some of them enabling the later development of general relativity.
Riemann was born in Breselenz, a village near Dannenberg in the Kingdom of Hanover in what is the Federal Republic of Germany today. His mother, Charlotte Ebell, died before her children had reached adulthood. Riemann was the second of six children, shy and suffering from numerous nervous breakdowns. Riemann exhibited exceptional mathematical skills, such as calculation abilities, from an early age but suffered from timidity and a fear of speaking in public.
Riemann held his first lectures in 1854, which founded the field of Riemannian geometry and thereby set the stage for Einstein's general theory of relativity. In 1857, there was an attempt to promote Riemann to extraordinary professor status at the University of Göttingen. Although this attempt failed, it did result in Riemann finally being granted a regular salary. In 1859, following Lejeune Dirichlet's death, he was promoted to head the mathematics department at Göttingen. He was also the first to suggest using dimensions higher than merely three or four in order to describe physical reality—an idea that was ultimately vindicated with Albert Einstein's contribution in the early 20th century. In 1862 he married Elise Koch and had a daughter.
Riemann fled Göttingen when the armies of Hanover and Prussia clashed there in 1866. He died of tuberculosis during his third journey to Italy in Selasca (now a hamlet of Verbania on Lake Maggiore) where he was buried in the cemetery in Biganzolo (Verbania). Meanwhile, in Göttingen his housekeeper discarded some of the papers in his office, including much unpublished work. Riemann refused to publish incomplete work and some deep insights may have been lost forever.
Riemann's published works opened up research areas combining analysis with geometry. These would subsequently become major parts of the theories of Riemannian geometry, algebraic geometry, and complex manifold theory. The theory of Riemann surfaces was elaborated by Felix Klein and particularly Adolf Hurwitz. This area of mathematics is part of the foundation of topology, and is still being applied in novel ways to mathematical physics.
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 investigated 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.
Riemann was born in Breselenz, a village near Dannenberg in the Kingdom of Hanover in what is the Federal Republic of Germany today. His mother, Charlotte Ebell, died before her children had reached adulthood. Riemann was the second of six children, shy and suffering from numerous nervous breakdowns. Riemann exhibited exceptional mathematical skills, such as calculation abilities, from an early age but suffered from timidity and a fear of speaking in public.
Riemann held his first lectures in 1854, which founded the field of Riemannian geometry and thereby set the stage for Einstein's general theory of relativity. In 1857, there was an attempt to promote Riemann to extraordinary professor status at the University of Göttingen. Although this attempt failed, it did result in Riemann finally being granted a regular salary. In 1859, following Lejeune Dirichlet's death, he was promoted to head the mathematics department at Göttingen. He was also the first to suggest using dimensions higher than merely three or four in order to describe physical reality—an idea that was ultimately vindicated with Albert Einstein's contribution in the early 20th century. In 1862 he married Elise Koch and had a daughter.
Riemann fled Göttingen when the armies of Hanover and Prussia clashed there in 1866. He died of tuberculosis during his third journey to Italy in Selasca (now a hamlet of Verbania on Lake Maggiore) where he was buried in the cemetery in Biganzolo (Verbania). Meanwhile, in Göttingen his housekeeper discarded some of the papers in his office, including much unpublished work. Riemann refused to publish incomplete work and some deep insights may have been lost forever.
Riemann's published works opened up research areas combining analysis with geometry. These would subsequently become major parts of the theories of Riemannian geometry, algebraic geometry, and complex manifold theory. The theory of Riemann surfaces was elaborated by Felix Klein and particularly Adolf Hurwitz. This area of mathematics is part of the foundation of topology, and is still being applied in novel ways to mathematical physics.
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 investigated 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.
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