MA530_AUG96

MA530_AUG96 - D, z 6 = 0. Let f ( z ) = a + a 1 z + a 2 z 2...

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MA 530 QUALIFYING EXAMINATION August 1996 Name: 1. Classify the singularities at 0: a )exp ± sin z z ² ,b ) X n =0 n ( z - 1) n , cos ± 1 e z - 1 ² . 2. Evaluate the integrals a ) Z C sin 1 z dz b ) Z C sin 2 1 z dz, where C is the circle | z | =2. 3. Describe the full preimage of the segment [ - 2 , 2] under cos z . Make a picture. 4. Find a conformal map of the upper half-plane, from which the vertical ray [ i, )is removed, onto the upper half-plane. 5. Let f be a meromorphic function in the unit disc D having only one simple pole at z 0
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Unformatted text preview: D, z 6 = 0. Let f ( z ) = a + a 1 z + a 2 z 2 + ... in a neighborhood of 0. Prove the equality z = lim n a n a n +1 . 6. Let f be a holomorphic function in the unit disc D . a) Prove that if f is unjective in D then f ( z ) 6 = 0 for all z D . b) Show that the converse is not true: there is a holomorphic function f in D whose derivative has no zeros in D but f is not injective in D ....
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