MA530_JAN00 - , and then follows a line back to e i , then...

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QUALIFYING EXAMINATION JANUARY 2000 MATH 530 - Prof. Bell 1. (20 pts) Prove that ln | z | cannot have a harmonic conjugate on the domain { z :1 < | z | < 2 } . 2. (20 pts) Suppose that { a n } n =1 is a sequence of complex numbers in the unit disk. What can you say about the radius of convergence of the series n =1 a n z n if | a n |→ 1as n →∞ ? What can you say about the radius of convergence if the set { a n } is dense in the unit disk? 3. (20 pts) Suppose that γ 1 and γ 2 are two continuously differentiable curves that cross at a point z 0 in the complex plane and that their tangent vectors make an angle α at z 0 . If the two curves are contained in the zero set of a harmonic function that is not identically zero, what are the possible values of α ?I f α =0 , what can you say about the two curves near z 0 ? 4. (20 pts) Calculate Z 0 ln x ( x 3 +1) dx. by integrating a meromorphic function around a contour γ described as follows. Let α =2 π/ 3. The contour γ follows the real axis from the ± to R , then follows the circle Re it from t =0to t
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Unformatted text preview: , and then follows a line back to e i , then follows the circle e it back to . 5. (20 pts) Let z n be a sequence of distinct non-zero complex numbers such that z n as n , and let m n be a sequence of positive integers. Let g be a meromorphic function on the plane having simple poles with residue m n at z n and having no other poles. If z 6 = z n for all n , let z be any path from 0 to z which avoids the set { z n } . Dene f ( z ) = exp Z z g ( ) d . Prove that f ( z ) is independent of the choice of z (although the integral itself might not be). Prove that f is analytic on the complement of { z n } , that f has removable singularities at each point z n , and that the extension of f has a zero of order m n at z n . You have shown that the Weierstra Theorem follows from the Mittag-Leer Theorem....
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This document was uploaded on 01/25/2012.

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