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. sin COS Z (a) z/(221)2 (d) tanz= (b) (e) 2sin() COS 2 (g) Log (1) (h) Log (z-1) (c) 2-1 (f) 2/4 Solution a) (J.R.J Groves) (i) e/(x+1) We have that z = 1 is a pole of order 2 since (2-1) 9(2) (2-1)2 with g (2) = +12 and 9 (2) is analytic near z = 1. Similarly we have, z=-1 is a pole of order 2 since h() (z+1)2 = with h (2) and h (2) is analytic near = 1. (a) Double poles at 1, principal parts (1/4) (1). (e) (A. Kumjian) 13 The function is analytic on the punctured plane D=C\{0}. It has an isolated singularity at 20. For 20 we have 2 sin ( ) = k=0 (-1)* (2k+1)! 2k+1 () Hence k0 1 (-1)* (2k+1)! 2k-1==+ k-1 (-1)* (2k+1)!2k-1 1 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 functionsin () has an isolated singularity at 2 = 0. Since