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 nonzero 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 MittagLeﬄer Theorem....
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 Spring '09
 Complex Numbers, Complex number, Holomorphic function, Zn

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