MA530_JAN99

MA530_JAN99 - that f g must have an essential singularity...

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QUALIFYING EXAMINATION JANUARY 1999 MATH 530 - Bell/Catlin Notation: D r ( a ) denotes the disk, { z C : | z - a | <r } . 1. (15 pts) Evaluate the integral Z 0 cos8 x x 2 +1 dx. 2. (15 pts) Find all entire functions f such that f ((1 + i ) z )= f ( z ) for all z C . 3. (15 pts) Let H denote the half plane { x + iy : y> 2 / 2 } . Explain how to construct an explicit one-to-one analytic map from D 1 (0) H onto D 1 (0). 4. (15 pts) Suppose that f is a non-constant entire function and suppose that g is an analytic function on D 1 (0) \{ 0 } that has an essential singularity at 0. Prove
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Unformatted text preview: that f g must have an essential singularity at 0. 5. (20 pts) Suppose that u is a non-constant real-valued harmonic function on the whole complex plane. Show that the level set { z C : u ( z ) = 0 } must be an unbounded set. 6. (20 pts) Find all analytic functions f on D 2 (-1) D 2 (1) such that f ( D 2 (-1)) D 2 (-1) , f ( D 2 (1)) D 2 (1) , f (-1) =-1 and f (1) = 1 ....
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