Towards an Effectivisation of the Riemann Theorem.pdf

4 0 0 2 TOWARDS AN EFFECTIVISATION OF THE RIEMANN THEOREM y a M S.Natanzon 6 ] Moscow State University, Independent University of V Moscow, Institute for Theoretical and Experimental Physics C . h t a m Abstract [ Let Q be a connected and simply connected domain on the Riemann sphere, not coinciding 2 with the Riemann sphere and with the whole complex plane C. Then, according to the v Riemann Theorem, there exists a conformal bijection between Q and the exterior of the 8 unit disk. In this paper we find an explicit form of this map for a broad class of domains 2 2 with analytic boundaries. 9 2000 Math. Sabj. Class. 30C, 37K. 0 3 1. Introduction. 0 / Let Q be a connected and simply connected domain on the Riemann sphere, not coinciding h t with the Riemann sphere and with the whole complex plane C. Then, according to Riemann a m Theorem, there exists a conformal bijection between Q and the exterior of the unit disk ¯ : U = {u ∈ C||u | 1}. In this paper we find an explicit form of this map for a broad class of v i domains with analytic boundaries. X Without loss of generally it is possible to assume that ∞ ∈ Q − ∂Q and 0 ∈ Q◦ , where r ¯ a Q◦ = C − (Q ∪ ∂Q) and ∂Q is the boundary of Q. Let H be the set of all such domains. The harmonic moments t0 = 1 dxdy, tk = − 1 z−k dxdy (k = 1, 2...) (1.1) π πk Q◦ Q form a coordinate system t = (t , t , t