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Home_11 - Homework 10 Stephen Taylor June 1 2005 Page 110...

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Homework 10 Stephen Taylor June 1, 2005 Page 110 - 111 1. Each of the following functions f has an isolated singularity at z = 0. Determine its nature; if it is a removable singularity defined f (0) so that f is analytic at z = 0; if it is a pole find the singular part; if it is an essential singularity determine f ( { z : 0 < | z | < δ } ) for arbitrarily small values of δ . (a) f ( z ) = sin z z This function has a removable singularity at z = 0 and f (0) = 1. (b) f ( z ) = cos z z This function has a pole at z = 0 with singular part 1 /z . (c) f ( z ) = cos z - 1 z This function has a removable singularity at z = 0 and f (0) = 0. (d) f ( z ) = exp( z - 1 ) This function has an essential singularity at z = 0. f ( { z : 0 < | z | < δ } ) assumes every value in the set C - { 0 } an infinite number of times by Picard’s Great Theorem. (e) f ( z ) = log( z +1) z 2 This function has a pole at z = 0 with singular part 1 /z . (f) f ( z ) = cos( z - 1 ) z - 1 This function has an essential singularity at z = 0. f ( { z : 0 < | z | < δ } ) assumes every value in the set C an infinite number of times by Picard’s Great Theorem.
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