14.7 Let G be an open set and suppose that {f} is a sequence of functions...

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14.7 Let G be an open set and suppose that {f} is a sequence of functions such that H(G) (k = 1,2,.) and fk f uniformly on G. k (i) Use Morera's theorem to prove that H(G). (ii) Use Cauchy's formula for derivatives to prove that lim f)(a) = f(")(a) (n = 1, 2,...). Deduce corresponding results for uniformly convergent series of holomor- phic functions. 14.7 Let G be an open set and suppose that {f} is a sequence of functions such that H(G) (k = 1,2,.) and fk f uniformly on G. k (i) Use Morera's theorem to prove that H(G). (ii) Use Cauchy's formula for derivatives to prove that lim f)(a) = f(")(a) (n = 1, 2,...). Deduce corresponding results for uniformly convergent series of holomor- phic functions.

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i To prove that f HG using Moreras theorem we need to show that for any simple closed curve C in G t... View the full answer