RIEMANN, B. (BERNHARD). - RIEMANN-TOPOLOGY AND RIEMANN-SURFACES.

Theorie der Abel'schen Functionen. (And 3 other memoirs by Riemann of 1857).

Berlin, Georg Reimer, 1857. 4to. Spine gone. Covers loose. In: "Journal für die reine und angewandte Mathematik Crelle/Borchardt", 54. Band. IV,388 pp. (Entire volume offered). Internally clean and fine. The memoir: pp. 115-155.

First appearance of these groundbreaking papers on Abelian functions and hypergeometric series, introducing the use of cross cuts to define the n-fold connectivity of a surface and extending many of the ideas from his dissertation of 1851 (Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse). "His famous theory of Abelian functions; the theory itself, ONE OF THE MOST NOTABLE MASTERWORKS OF MATHEMATICS."(DSB)

"While four importent papers repeat many of the ideas in his dissertation, they are primarely devoted to Abelian integrals and functions. The fourth paper is the one that gave the subject its major development (Theorie der Abel'schen Functionen). All four were difficilt to understand; "they were a book with seven seals". Fortunately many fine mathematicians later elaborated on and explained the material."(Kline "Mathematical Thought from Ancient to Modern Times", p. 663)

"His courses in 1855-1856, in which, he expounded his now famous theory of Abelian functions, were attended by C. A. Bjerknes, Dedekind, and Ernst Schering; the theory itself, one of the most notable masterworks of mathematics, was published in 1857." DSB)

Abelian functions were studied by Abel and Jacobi; they are a generalization of elliptic functions. Building on the ideas introduced in his thesis, particularly that of a Riemann surface, Riemann developed a very powerful geometric theory that resolved a number of outstanding problems. This work established Riemann as an important mathematician, but it was not without controversy. Riemann made extensive use, without proof, of a variational principle called the Dirichlet principle. Weierstrass had his doubts about it, and after Riemann's death it fell into disrepute. This state of affairs eventually had fruitful consequences. Several mathematicians successfully found proofs of Riemann's results without using the Dirichlet principle, and the principle itself was given a rigorous proof in 1899 by Hilbert.

The three other Riemann-papers (all in first edition) are: 1. "Allgemeine Voraussetzungen und Hülfsmittel für die Untersuchung von Functionen unbeschränkt veränderlichen Grössen", pp. 101-104. - 2. "Lehrsätze aus der analysis situs für die Theorie der Integrale von zweigliedrigen vollständigen Differentialen", pp. 105-110, with 4 textillustr. - 3. "Bestimmung einer Function einer veränderlichen complexen Grösse durch Grenz- und Unstetigkeitsbedingungen.", pp. 111-114.

The volume contains further importent mathematical papers by Dedekind, Cayley, Lipschitz, Clebsch.

Parkinson "Breakthroughs" 1857 M.

Order-nr.: 47147