MA530_AUG98

MA530_AUG98 - (15 pts Let f be a non-constant entire...

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QUALIFYING EXAMINATION August 1998 MATH 530 - Profs. Bell/Catlin Notation: D r ( a ) denotes the disk, { z C : | z - a | <r } . 1. (10 pts) Find all entire functions f such that the real part of f 0 ( z ) is non-negative at every point z C . 2. (15 pts) Evaluate the integral Z 0 x x 2 +1 dx. 3. (15 pts) Suppose that f is a continuous complex valued function on the unit disk that is holomorphic on the sets { Im z> 0 }∩ D 1 (0) and { Im z< 0 }∩ D 1 (0). Prove f is holomorphic on all of D 1 (0). Is the analogue of this problem for harmonic functions true? 4. (15 pts) Find a one-to-one analytic map from { x + iy :2 <y< 3 ,x< 1 } onto { x + iy :5 <y< 8 } . 5.
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Unformatted text preview: (15 pts) Let f be a non-constant entire function such that f ( n ) = 1998 for every n ∈ Z . Can f have at ∞ : a) an essential singularity, b) a pole, c) a removable singularity? 6. (15 pts) Suppose that f is analytic on D 1 (0) and that | f ( z ) | < 1 for all z ∈ D 1 (0). Prove that if f (0) = a 6 = 0, then f has no zeroes in the disk D | a | (0). 7. (15 pts) Show that a one-to-one entire function must be of the form az + b for some complex constants a and b with a 6 = 0....
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