Potential Theory in the Complex PlanePotential theory is the broad area of mathematical analysis encompassing such topics as harmonic and subharmonic functions. |
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Applying Theorem Banach algebra Borel measure Borel polar set Borel probability measure bounded capacity compact set compact subset conformal mapping constant continuous function convergence convex function Corollary countable D₁ deduce define Definition denote Dirichlet problem disc dµ(w equilibrium measure example Exercise exists extended maximum principle finite Borel measure function h given gives Green's function h is harmonic harmonic function harmonic majorant harmonic measure Hence holomorphic function identity principle implies integral Julia set L¹(R Lebesgue measure Lemma Let f lim inf lim sup u(z locally uniformly log c(K maximum principle monic non-polar non-thin open set open subset peio polar set polynomial of degree positive harmonic function potential theory principle Theorem Proof of Theorem proper subdomain prove result follows satisfies simply connected subharmonic function submean inequality suppose TD(z theorem Theorem unique upper semicontinuous