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Complex Variables, Laurent Series and Uniform Convergence

(1) Let G = {z : 0 < abs(z) < R} for some R > 0 and let f be analytic on the punctured

disk G with Laurent Series f(z) = sum a_n*z^n (from n = -oo to oo).

(a) If f_n(z) = sum a_k*z^k (from k =-oo to n), then prove that f_n converges pointwise

f in C(G,C) (all continuous functions from G to C (complex)); i.e., {f_n} converges

uniformly to f on every compact subset of G.

(b) Consider the special case f(z) = 1/(z(1 - z)), 0 < abs(z) < 1. SHow that f_n(z) =

sum z^k (from k = -1 to n) does NOT converge uniformly to f on G.

(2) Consider the sequence of functions f_n(z) = z/n in C(G,C).

(a) Prove that the sequence {f_n} converges uniformly on compact sets in C, and find its

limit function.

(my work): we know compact sets in C (complex) are bounded, so for a compact set K
there exists a constant s_K such that for all z in K, abs(z) <= s_K.
Define s_K = sup abs(z). Then abs(f_n(z)) <= s_K/n for all n. Is this on the right track?

(b) Does {f_n} converge uniformly on C? Explain your answer.

Solution Summary

Complex Variables, Laurent Series and Uniform Convergence are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.