Paper 3, Section II, F
Part II, 2009
(a) State Runge's theorem on uniform approximation of analytic functions by polynomials.
(b) Let be an unbounded, connected, proper open subset of . For any given compact set and any , show that there exists a sequence of complex polynomials converging uniformly on to the function .
(c) Give an example, with justification, of a connected open subset of , a compact subset of and a point such that there is no sequence of complex polynomials converging uniformly on to the function .