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math185f08-hw9

# math185f08-hw9 - MATH 185 COMPLEX ANALYSIS FALL 2008/09...

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MATH 185: COMPLEX ANALYSIS FALL 2008/09 PROBLEM SET 9 For f : Ω C and n N , recall that g = f n is the function defined by g ( z ) = [ f ( z )] 2 for all z Ω. A function on Ω C is said to be meromorphic if it has only removable singularities or poles in Ω (i.e. no essential singularities or non-isolated singularities). 1. Let Ω C be a region and let f : Ω C . Suppose g = f 2 and h = f 3 are both analytic on Ω. Show that f is analytic on Ω. 2. Is there a polynomial p ( z ) such that p ( z ) e 1 /z is an entire function? 3. Let f : C C be a nonconstant entire function. Prove that f ( C ) is dense in C . 4. Show that each of the following series define a meromorphic function on C and determine the set of poles and their orders. f ( z ) = X n =0 ( - 1) n n !( z + n ) , g ( z ) = X n =0
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