By Detlef Laugwitz

The identify of Bernard Riemann is celebrated to mathematicians and physicists worldwide. His identify is indelibly stamped at the literature of arithmetic and physics. This extraordinary paintings, wealthy in perception and scholarship, is addressed to mathematicians, physicists, and philosophers attracted to arithmetic. It seeks to attract these readers in the direction of the underlying principles of Riemann’s paintings and to the improvement of them of their ancient context. This illuminating English-language model of the unique German version can be a huge contribution to the literature of the historical past of mathematics.

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1 1 zero. 1 Bernhard R i e m a n n in his time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . zero. 1. 1 His existence and the advance of his character . . . . . . . . . . . . . . . . . . . . . 1 zero. 1. 2 The political and e c o n o m i c scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . three zero. 1. three Upbringing and schooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . five zero. 1. four R i e m a n n ' s place of origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . nine zero. 1. five G6ttingen and Berlin as facilities of research . . . . . . . . . . . . . . . . . . . . . . . . . 17 zero. 1. 6 R i e m a n n as complete professor: 1 eight five nine - 1 eight 6 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 zero. 2 The golden 1850s in G6ttingen: From Gauss and Dirichlet to R i e m a n n and Dedekind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 zero. 2. 1 R i e m a n n and Dedekind: own conditions . . . . . . . . . . . . . . . . . . 24 zero. 2. 2 in the direction of switch in arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 zero. 2. three Snapshots through an English observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 zero. three impacts within the ultimate years" Riemann among G e r m a n y and Italy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 zero. four zero. four. 1 C o m p e t i n g conceptions of research sooner than Riemann . . . . . . . . . . . . . . forty three R i e m a n n within the historic evolution of study" an outline . . . . . . forty three zero. four. 2 Algebraic research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . forty six zero. four. three Infinitesimal research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . fifty two zero. four. four G e o m e t r i c deliberations: Fourier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . fifty five zero. four. five The restrict notion: Newton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . fifty six zero. four. 6 in the direction of epsilontics: Cauchy and Dirichlet . . . . . . . . . . . . . . . . . . . . . . . fifty seven 1. complicated research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . sixty four 1. 1 The genesis of complicated research as much as R i e m a n n ' s time . . . . . . . . . . . . sixty four 1. 1. 1 initial feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . sixty four 1. 1. 2 The advanced numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . sixty five 1. 1. three complicated capabilities and their derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . sixty nine 1. 1. four Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . seventy three vi Contents 1. 1. five energy sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . seventy seven 1. 1. 6 additional functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . eighty four 1. 1. 7 Multivalued features and R i e m a n n surfaces . . . . . . . . . . . . . . . . . . . . . 88 1. 1. eight D o u b l y periodic capabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ninety two 1. 2 The dissertation of 1851 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ninety six 1. 2. 1 R i e m a n n ' s view of the factors for the paper: Article 20 1. 2. 2 a quick account of the contents of the dissertation . . . . . . . . . . . . . . . . 1. 2. three R i e m a n n ' s s u m m a r y of the dissertation. this system- of the dissertation, half I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ninety six ninety nine Article 20, and half II, Article 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a hundred and one 1. 2. four at the prehistory of the dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 1. 2. five The influence of the dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 1. three The gildings .

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