MA530_JAN98

MA530_JAN98 - z 1998 + z + 2001 have in the rst quadrant?...

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MATH 530 Qualifying Exam January 1998 Notation: D r ( a ) denotes the disk, { z C : | z - a | <r } . 1. (15 pts) Evaluate the integral Z 0 -∞ x 2 x 4 + x 2 +1 dx . 2. (15 pts) Find a one-to-one analytic map from D 1 (0) ∩{ x + iy : x,y > 0 } onto D 1 (0). 3. (25 pts) Let F denote the set of analytic functions f on D 1 (0) such that | f ( z ) | < 1 for all z D 1 (0), f (0) = 0, and f 0 (0) = 0. Prove that if f ∈F ,then | f ( z ) |≤| z | 2 for all z D 1 (0). Let M =sup {| f 00 (0) | : f ∈F} . Find all functions, if any, in F such that | f 00 (0) | = M . 4. (15 pts) How many zeroes does the polynomial
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Unformatted text preview: z 1998 + z + 2001 have in the rst quadrant? Explain your answer. 5. (15 pts) Prove that a harmonic function cannot have an isolated zero. 6. (15 pts) Let C 1 (0) denote the unit circle { z C : | z | = 1 } and let f be a function that is analytic on D r (0) for some r &gt; 1. Prove that if f ( C 1 (0)) C 1 (0) \ { 1 } , then f is a constant function....
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