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LLL Find the isolated singularities of the following functions and de ermine where they are...

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LLL Find the isolated singularities of the following functions and de ermine where they are removable essential or poles Determine the order of any pole and find the principal part at each pole a z 2 1 d tan z sin g Log 1 COS Z b e sin h Log z 1 f 22 4 i e 1 e2 1 Solution a J R J Groves with g 2 We have that z 1 is a pole of order 2 since 21 1 and g 2 is analytic near z 1 Similarly we have z 1 is a pole of order 2 since 2 1 with h 2 2 1 and h 2 is analytic near 1 a Double poles at 1 principal parts 1 4 z 1 e A Kumjian 8 2 sim 2 1 2k 1 k0 13 The function is analytic on the punctured plane D C 0 It has an isolated singularity at z 0 For 0 we have k 0 24 1 1 k 1 2k 1 22k 1 2 1 2k 1 22k 1 k 1 Hence 0 is an essential singularity for the function since there are an infi nite number of nonzero terms in the Laurent series with negative exponents e J R J Groves The function 2 sin has an isolated singularity at z 0 Since

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