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analytic curves analytic functions argument principle boundaries of A(t boundary continuum bounded component bounded doubly connected circle circular slits class of mappings compact subset complement conformal deformation classes conformal mapping conformally equivalent conformally isotopic contained continuous family corresponds define dense distinguished inner boundary Doctor of Philosophy domain bounded domain convergence doubly connected domain equations extremal domain f and g F fl f omits f(fl f(il family of domains fixed disk formal moduli function F Garabedian and Schiffer Green's function harmonic measure Ik(f implicit function theorem interior variation isotopic to g Lemma level curve limit function mapping of XL mappings f mappings of type minus the slit Morse and Heins multiply connected domain normalized mapping function outer boundary positively oriented Proof rank less rank N*l real axis Refsk regular and schlicht sequence shows slit circular ring Theorem U.3 triply connected domain Vitali sense winding numbers