MA530_JAN03 - of the domain, then it is identically zero....

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QUALIFYING EXAMINATION JANUARY 2003 MATH 530 - Prof. Bell 1. (10 pts) Suppose P and Q are polynomials and that the degree of P is at least two less than the degree of Q . Prove that the sum of all the residues of P/Q in the complex plane is zero. 2. (20 pts) a) Prove that an uncountable subset of a domain in the complex plane must have a limit point in the domain. b) Prove that an analytic function on a domain can have at most countably many zeroes if it is not identically zero. c) Does there exist a non-constant analytic function on the unit disc with infin- itely many zeroes on the unit disc? Prove or give a counterexample. 3. (20 pts) Prove that if a harmonic function on a domain vanishes on an open subset
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Unformatted text preview: of the domain, then it is identically zero. Can it vanish on a set with a limit point in the domain and not be identically zero? 4. (20 pts) Suppose that f ( z ) has an essential singularity at z = a . Prove that there is a sequence of points z n tending to a such that ( z n-a ) n f ( z n ) tends to ∞ . 5. (10 pts) Does there exist a sequence of polynomials P n ( z ) which converges uni-formly to 1 /z on { z : | z | = 1 } ? Explain. 6. (20 pts) Suppose that f is a continuous complex valued function on a domain Ω. Prove that if f 2 and f 3 are analytic on Ω, then f itself must be analytic....
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