By Michael Monastyrsky (auth.), R. O. Wells Jr. (eds.)
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Extra info for Riemann, Topology, and Physics
Grassmann (1809-1877), and A. Cayley (1821-1895). He constructed an example ofa noncommutative but associative multi-dimensional algebra (Clifford algebra). For many years, his results remained important for the physics of elementary particles with half-integer spin (particles liable to Fermi-Dirac statistics). In geometry he developed the work of C. von Staudt (1798-1867), Cayley, and F. Klein (1849-1925) on the foundations of geometry (the relationship between non-Euclidean and projective geometries).
Are polynomials in zd is topologically equivalent to an arbitrary orient able closed two-dimensional surface was clarified later, basically thanks to Felix Klein's paper of 1881/1882 "Uber Riemanns Theorie der algebraischen Funktionen und ihrer Integrale"(On Riemann's theory of algebraic functions and their integrals). Riemann obtained an important formula linking the number of sheets n, the number of branch points b (with corresponding multiplicities), and the genus of the Riemann surface of an algebraic function b g=:2- n +1.
R/J(z) (13) where r/J(z) is a function without singularities. Singularities of the flow defined by the function J(z) are made up of the singularities of the flows defined by the separate terms of (13). Let us consider the behavior of the logarithmic term. To begin with, let us suppose that A is a real number. We choose a disk ofradius r about the point zo: z = Zo + rei¢> and set Alog(z - zo) = u + iv; I. Bernhard Riemann 44 " (b) (a) Figure 3 separating out the real and imaginary parts, we obtain u = A log r, v = At/J.
Riemann, Topology, and Physics by Michael Monastyrsky (auth.), R. O. Wells Jr. (eds.)