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analytic analytic curves analytic function Argentina bounded holomorphic function class of n-harmonic co)dm cocycle cohomology compact compact subset complex dimension conformal map D X2 define denote disjoint Jordan curves DOCTOR OF PHILOSOPHY domains in E e hP(Un e Sh n essential boundary exact sequence exists F e exp g F e H f e hP(X F e N(U finitely Fourier transform function f functions f e H(n Green function harmonic conjugate homeomorphism HP U n HP(U induced problem inf Vg int(9n n int(an int(an n integer invertible element l,nk Lebesgue measure lemma Let n limit point majorant in n monic Multiplicative Cousin problem n Let n n n n n X n n1 n2 n U n n Un n Xi2 n-harmonic functions n-harmonic majorant N-subharmonic function neighborhood Nevanlinna nice boundaries oo)dm open covering open set Poisson Poisson integrals Poisson kernel polycube polydisc polydomain product of annuli Proof prove Re f e Re f e hP(Xi Re G Re g2 Re gi Re gl Re z Riemann sphere Rudin sequence sheaf sheaves Similarly simply connected Since solve strictly H function strongly convex function subharmonic Suppose theorem thesis Thus University of Wisconsin vector space Vg n(z Vg w Vg z Walter Rudin Weierstrass preparation theorem X U(r X Un_1 Xi n X2 z e 9 z e n z e X z2 e ZERO SET PROBLEMS zeros as f