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MA530_AUG94

# MA530_AUG94 - QUALIFYING EXAMINATION AUGUST 1994 MATH 530...

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QUALIFYING EXAMINATION AUGUST 1994 MATH 530 All answers must be justified and work must be shown. 1. Let f be an analytic function in the open unit disk, | f ( z ) | ≤ 1, | z | < 1. Prove that | f ( n ) (0) | ≤ n !; n = 0 , 1 , 2 , . . . . 2. Let f be a non-constant analytic function in a neighborhood N of the real axis R . Assume that Im f ( z ) · Im z 0 , z N. a) Show that f 0 ( z ) 6 = 0, z R . b) Show that actually f 0 ( z ) > 0, z R . 3. Evaluate the integral Z 0 x α - 1 dx x + t , where 0 < α < 1 and t > 0. 4. Find the one-to-one conformal map of the region { z : Re z > 0, Im z > 0, | z | > 1 } onto the upper half-plane, such that i 7→ 0, 1 7→
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