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MA530_AUG00

MA530_AUG00 - h Be sure to justify all steps(15 pts 4 In...

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QUALIFYING EXAMINATION AUGUST 2000 MATH 530 - Prof. Drasin (15 pts) 1. Find a function which is meromorphic in ˆ C withapoleoforder3at z = 0, a removable singularity at z =2 i , a simple zero at z = - 2 i and a zero of order two at . (20 pts) 2. Consider the “function” p z ( z - 1). In what annuli { a< | z | <b } can we find a single–valued branch of this expression? How many such branches are there? Indicate the Laurent series in the appropriate annulus/annuli. (15 pts) 3. Let h ( z )= Z 1 - 1 e - zt sin | z | dt. Prove that h is an entire function and compute
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Unformatted text preview: h ! Be sure to justify all steps. (15 pts) 4. In what region is F ( z ) = ∞ X 1 e z ln n analytic? Be sure to give a proof. (20 pts) 5. For what real p does ∞ Z x p 1 + x dx converge? Compute this integral for all such p . (15 pts) 6. Map the region bounded by | z | < 1 , | z-i | < 1 conformally onto the half–plane Im w > 0. If you use a composition of maps, you need only indicate the individual components....
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