MA530_AUG05

MA530_AUG05 - is analytic in the disc { z : | z | < R }...

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QUALIFYING EXAMINATION AUGUST 2005 MATH 530 - Prof. Drasin Each problem is worth 5 points. As in all math exams, you have to fully justify your answers. 1. How many zeros does the polynomial z 6 - 5 z +20haveintheannulus 1 < | z | < 2? 2. Z -∞ dx x 3 - i =? 3. Given N =0 , 1 , 2 ,..., find all entire functions f such that | f ( z ) |≤| z | N for all z C . 4. Let γ be the ellipse 9 x 2 +16 y 2 = 144. [i.e. 9(Re z ) 2 + 16(Im z ) 2 = 144], and ϕ ( z )= Z γ e 2 | ζ | ζ - z . (a) For which R canwebesurethat ϕ
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Unformatted text preview: is analytic in the disc { z : | z | < R } ? (b) Estimate | ϕ 00 (0) | from above. 5. Find a conformal map ϕ from the strip { x + iy : | x-y | < 1 } onto the first quadrant. You may express ϕ as a finite composition of simpler explicit mappings. 6. Is there a holomorphic function h in the disc { z : | z | < 1 } that satisfies lim | z |→ 1 | h ( z ) | = ∞ ?...
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This document was uploaded on 01/25/2012.

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