hw11_2 - n ≥ 1 at z Prove that f/f has a simple pole at z...

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Complex Variables Assignment 11.2 Spring 2011 Due November 16 Exercise 1. Consider the function f ( z ) = e 1 /z - 1. We have seen that it has a sequence of distinct zeros { z n } n =1 with the property that z n 0, and yet it is not the zero function. Why doesn’t this contradict the theorem on the isolation of zeros of analytic functions? Exercise 2. Suppose that f has either a zero or a pole of order
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Unformatted text preview: n ≥ 1 at z . Prove that f /f has a simple pole at z . Exercise 3. Find all the singularities of f ( z ) = ( z-2) 2 ( z + 4) 1-cos( πz ) and identify each as removable, a pole of a certain order, or an essential singularity. Exercise 4. Repeat the preceding exercise for f ( z ) = ( z-2 i ) 4 sin( πi/z )-1 ....
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