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.