hw3_2 - hood of ∞ and that lim z →∞ 1 p z = 0...

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Complex Variables Assignment 3.2 Spring 2011 Due September 19 Exercise 1. Recall that a function f ( z ) is defined on a neighborhood of if there exists an R > 0 so that the domain of f contains { z : | z | > R } . a. Show that if f ( z ) is defined on a neighborhood of if and only if g ( w ) = f (1 /w ) is defined on a deleted neighborhood of 0. b. Show that lim z →∞ f ( z ) = a if and only if lim w 0 f (1 /w ) = a . Exercise 2. If p ( z ) is a non-constant polynomial, prove that 1 p ( z ) is defined on a neighbor-
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Unformatted text preview: hood of ∞ and that lim z →∞ 1 p ( z ) = 0. [ Suggestion: You can avoid an argument involving ± by using the preceding exercise and the limit laws.] Exercise 3. Let log w denote the branch of the logarithm with arg w ∈ (-π,π ]. a. Where is log( z 2 ) continuous? What about log( z 3 )? b. Let w 1 / 2 = e 1 2 log w . Where is (1 + z ) 1 / 2 continuous? What about (1-z ) 1 / 2 ?...
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This note was uploaded on 02/29/2012 for the course MATH 4364 taught by Professor Ryandaileda during the Fall '11 term at Trinity University.

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