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analogous analy analytic curves analytic functions analytic surface analytically convex domain belong bicylinder boundary points bounded domain Cartan's theorem chapter complex variables condition connected component consider consists of domains construct continuously differentiable hypersurface convex with respect corresponding Cousin's first problem defined denote domain analytically convex domain of meromorphy domain of uniform domains of regu domains of regularity E. E. Levi equations essential singular points finite number following theorem formula func function 4>(w,z functions regular Hartogs domain hyperplane hypersphere intersection Lemma Let w,z Levi's theorem lower semi-continuous lying neighborhood non-singular point number of points open sets consisting polycylinder polynomials proof properties proved regular function Reinhardt domain result satisfied sense of Hartogs sense of Levi sequence of domains singular points strictly contained sub-biharmonic function subharmonic function theory of domains Thullen tion twice continuously differentiable uniform convergence variables w,z wq,Zq wq.zq z-plane