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THE DISC THE PLANE AND THE EXTENDED PLANE
THE MAJ0R RESULTS
0bserve analytic continuation analytic function assume atlas atlases Aut(A bijection boundary centre chart choice closed path compact subset complete analytic function complex conformal structure conformally equivalent construct containing deduce defined Definition denote differential end-point example Exercise exists f and g f is constant f is holomorphic f is meromorphic f is univalent finite function f germ h is continuous harmonic function Hausdorff space hence holomorphic function homeomorphism homotopic hyperbolic initial point integral Let f Let G lift metric space Mobius transformations neighbourhood number of zeros obtain open cover open disc open sets open subset parametric disc Perron family plane domain proof of Theorem properties prove quotient space quotient topology Re[f regular covering surface result Riemann surface satisfies sequence Show simply connected single valued smooth covering surface subdomain subharmonic functions sup u(z Suppose that f supremum throughout topological space torus union unique valued function