Paper 3, Section II, F

Topics in Analysis
Part II, 2009

(a) State Runge's theorem on uniform approximation of analytic functions by polynomials.

(b) Let Ω\Omega be an unbounded, connected, proper open subset of C\mathbb{C}. For any given compact set KC\ΩK \subset \mathbb{C} \backslash \Omega and any ζΩ\zeta \in \Omega, show that there exists a sequence of complex polynomials converging uniformly on KK to the function f(z)=(zζ)1f(z)=(z-\zeta)^{-1}.

(c) Give an example, with justification, of a connected open subset Ω\Omega of C\mathbb{C}, a compact subset KK of C\Ω\mathbb{C} \backslash \Omega and a point ζΩ\zeta \in \Omega such that there is no sequence of complex polynomials converging uniformly on KK to the function f(z)=(zζ)1f(z)=(z-\zeta)^{-1}.