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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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