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First edition, very rare separately-paginated offprint (journal pagination 161-172), of this sequel to Riemann's epoch-making papers on geometric complex function theory (and Riemann surfaces), 'Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse' (1851) (his dissertation) and 'Theorie der Abel'schen Functionen' (1857). The Dissertation is "one of the most important achievements of 19th century mathematics" (Laugwitz, Bernhard Riemann, 1826-1866), "which marked a new era in the development of the theory of analytic functions" (Kolmogorov & Yushkevich, Mathematics in the 19th century, Vol. II, p. 199), introducing geometric and topological methods, notably the idea of a 'Riemann surface.' "Riemann's doctoral thesis is, in short, a masterpiece" (Derbyshire, Prime Obsession, p. 121). Riemann's 1857 paper on Abelian functions, "one of the most notable masterworks of mathematics" (DSB), develops much further the methods of the Dissertation, using them to give the general solution of the Jacobian inversion problem and to establish an early version of the famous Riemann-Roch theorem. Stillwell (Mathematics and its History, 3rd edition, p. 310) considers Riemann's 1857 paper perhaps his greatest work, which did for algebraic geometry what his Habilitationsvortrag [Ueber die Hypothesen, welche die Geometrie zu Grunde liegen, 1854] did for differential geometry. The present paper corrects some of the proofs in 'Theorie der Abel'schen Functionen' involving zeros of theta functions, and provides some further developments. 4to, pp. [ii], 12. Modern boards. Seller Inventory # ABE-1505828399526
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