MA530_AUG03

# MA530_AUG03 - Qualifying Exam MA 530 August 2003 Professor...

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Qualifying Exam August 2003 MA 530 Professor Donnelly I. The first four questions are each worth ten points. 1. Write (1 - i 3) 85 in the form a + ib . 2. Let Γ be the contour consisting of two circles | z - 1 | = 2, counterclockwise, and | z - 1 | = 1, clockwise. Find the value of the winding number in each component of the complement of Γ. 3. How many zeroes does 2 z 2 - e z/ 2 have in | z | < 1?. 4. Find the first five nonzero terms in the Taylor series of sin z z 2 + 1 , about z = 0. II. The remaining four questions are each worth fifteen points. 5. Give an example of a harmonic function, which is not the real part of a holomorphic function. Explain why your answer is correct. 6. Suppose f is a holomorphic mapping from the unit disc to itself. Show that, for all
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