Get On Riemann's Theory of Algebraic Functions and their PDF

By Felix Klein

This quantity introduces Riemann's method of multiple-value services and the geometrical illustration of those services through Riemann surfaces. It concentrates at the kind of services that may be outlined upon those surfaces, proscribing the therapy to rational capabilities and their integrals demonstrating how Riemann's mathematical rules approximately Abelian integrals may be derived through reflecting upon the circulate of electrical present on surfaces. This paintings is without doubt one of the most sensible introductions to the origins of topological difficulties.

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Project from the point ξ = 0, η = 0, ζ = 1, let the plane of projection be the xy plane, and the opposite tangent-plane the ξη plane. Then we have ξ= x2 x , + y2 + 1 η= x2 y , + y2 + 1 ζ= x2 1 . ] introductory remarks. 21 with the fact that the infinitely distant parts of the plane are drawn together to one point of the sphere, the point from which we project, so that it is no longer merely symbolical to speak of the point z = ∞ on the sphere. ∗ The propositions of the theory of surfaces from which this statement follows are now briefly set forth in a form sufficiently general to serve for certain future purposes.

G. be so chosen that each has a non-vanishing modulus of periodicity at one only of the 2p cross-cuts (where, of course, to each cross-cut, one, and only one, potential is assigned). And in ai ui the constants ai can be so chosen that this expression has at each cross-cut the same modulus of periodicity as u. Then u − ai ui is a constant and we have the formula just given. Passing now from the potentials u to the functions u + iv, finite everywhere, suppose, for simplicity, that coordinates x, y, employed on the surface (§ 6), are such that u and v are connected by the equations ∂v ∂u = , ∂x ∂y ∂u ∂v =− .

It would appear that we must go to work in a more roundabout way. Let us first think of thermo-electric currents. Let the xy plane be covered, partly with material I, partly with material II, and let the strength of the films be so arranged that the conductivity shall be everywhere the same. If we now contrive that the two parts of the contour separated by z0 and z1 may be kept at constant and different temperatures, an electric streaming of the kind required will be set up. And the electrostatic potential, by the principles of the theory of thermo-electricity, exhibits discontinuities on both parts of the said contour.

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On Riemann's Theory of Algebraic Functions and their Integrals by Felix Klein

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