Riemann, Schottky, and Schwarz on Conformal Representation
Appendix 1 Riemann, Schottky, and Schwarz on Conformal Representation In his Thesis [1851], Riemann sought to prove that if a function u satisfies cer- tain conditions on the boundary of a surface T, then it is the real part of a unique complex function which can be defined on the whole of T. He was then able to show that any two simply connected surfaces (other than the complex plane itself Riemann was considering bounded surfaces) can be mapped conformally onto one another, and that the map is unique once the images of one boundary point and one interior point are specified; he claimed analogous results for any two surfaces of the same connectivity [w To prove that any two simply connected surfaces are conformally equivalent, he observed that it is enough to take for one surface the unit disc K = {z : Izl _< l} and he gave [w an account of how this result could be proved by means of what he later [1857c, 103] called Dirichlet's principle. He considered [w (i) the class of functions, ~., defined on a surface T and vanishing on the bound- ary of T, which are continuous, except at some isolated points of T, for which the integral t= rx + ry dt is finite; and (ii) functions ct + ~. = o9, say, satisfying dt=f2 < oo Ox Oy + for fixed but arbitrary continuous functions a and ft. 224 Appendix 1. Riemann, Schottky, and Schwarz on Conformal Representation He claimed that f2 and L vary continuously with varying ,~. but cannot be zero, and so f2 takes a minimum value for some o9.
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