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MA530_JAN00 - and then follows a line back to ±e iα 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 | → 1 as 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 ? If 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 = 0 to
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Unformatted text preview: , and then follows a line back to ±e iα , 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 γ z be any path from 0 to z which avoids the set { z n } . Define f ( z ) = exp ±Z γ z g ( ζ ) dζ ² . Prove that f ( z ) is independent of the choice of γ 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 Weierstraß Theorem follows from the Mittag-Leffler Theorem....
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