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

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Unformatted text preview: Qualifying Exam MA 530 August 2003 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 2z 2 − ez/2 have in |z | < 1?. 4. Find the first five nonzero terms in the Taylor series of sin z , about z = 0. z2 + 1 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 |a| < 1, |f (a)| 1 ≤ 2 1 − |f (a)| 1 − |a|2 7. If u is harmonic and bounded in 0 < |z | < ρ, show that the origin is a removable singularity, in the sense that u becomes harmonic in |z | < ρ, when u(0) is properly defined. 8. By making the substitution x = iy and noticing that y → ∞ as x → ∞, we formally ∞ ∞ xdx ydy transform the integral into the integral − , which is the 4+4 4+4 x y 0 0 negative of the original integral. Hence, it appears that the value of the integral is zero. But the true value of the integral is found by elementary methods to be π/8. What is the fallacy? Explain using complex analysis. ...
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This document was uploaded on 01/25/2012.

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