118Homework8 - rem 7.17 of Walter Rudin’s Principles of...

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Homework 8 Due Monday, Oct. 25 1. Prove that if lim n →∞ z n = z , then lim n →∞ | z n | = | z | . 2. You probably showed in your Real Analysis class that uniform limits of functions pass through integrals (i.e., if { f n } n =1 converges uniformly to f on the closed interval [ a, b ], then lim n →∞ R b a f n ( x ) dx = R b a f ( x ) dx ). The same techniques can be used to show that if { f n } n =1 is a sequence of complex-valued functions that converges to f uniformly on a domain D , and if C is any contour in D , then lim n →∞ Z C f n ( z ) dz = Z C f ( z ) dz. Assuming that this is true, prove that if D is a domain, if { f n } n =1 is a sequence of functions that are analytic in D , and if { f n } n =1 converges uniformly to f on D , then f is analytic in D . (Remark: This shows that uniform limits of analytic functions are ana- lytic. This is different from the real-valued situation, where uniform limits of differentiable functions are not necessarily differentiable — see Theo-
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Unformatted text preview: rem 7.17 of Walter Rudin’s Principles of Mathematical Analysis .) 3. With the aid of series prove that the function f ( z ) = ± e z-1 z if z 6 = 0 1 if z = 0 is entire. 4. (a) Prove that the power series ∑ ∞ n =0 a n ( z-z ) n and the corresponding series of derivatives ∑ ∞ n =1 na n ( z-z ) n-1 have the same radius of convergence. (Hint: Note that it suffices to consider where the series converge absolutely; thus you may use the ratio test for real series.) (b) Prove that the series ∑ ∞ n =1 z n n 2 converges at all points inside and on its circle of convergence. Prove that this is not true for its series of derivatives....
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This note was uploaded on 02/21/2012 for the course MATH 118 taught by Professor Forde during the Fall '09 term at University of Houston.

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