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    Complex Variables, Laurent Series and Uniform Convergence

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    (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.

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    Solution Summary

    Complex variables, laurent series and uniform convergence are investigated. The solution is detailed and well presented.