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MA530_JAN11

# MA530_JAN11 - MATH 530 Qualifying Exam January 2011 G...

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MATH 530 Qualifying Exam January 2011, G. Buzzard, S. Bell 1. (a) (10 points) Define f ( z ) := π - π cos 2 t z - sin t dt. Use the difference quotient definition of complex derivative to show that f is holomorphic on C - [ - 1 , 1] and to write f ( z ) as an integral. (b) (10 points) Can f ( z ) be extended to [ - 1 , 1] to make f ( z ) an entire function? If so, describe how to extend f . If not, prove that no such extension exists. 2. (20 points) Let P n ( z ) = n k =0 z k k ! . Prove that for a given R > 0, there exists a positive integer N such that if n N , then P n ( z ) has exactly n zeroes in { z : | z | > R } . 3. (20 points) Suppose that f has a pole at z 0 and that g is holomorphic on C - D (so that g has an isolated singularity at ). Give necessary and sufficient conditions on the
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