Question

(c) Locate and classify all the singularities of the function Then compute the residues at each of these singularities. (d) Give an example of a function with a pole of order 5 at 1 -2i, an essential singularity at 1, and a removable singularity at 0. Justify all assertions.

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Answer 1

f(z) = sin z/z^2 - pi^2 the singularity of f(z) is, z^2 - pi^2 = 0 z = plusminus pi here z = plusminus pi are simple pole residence of f(z) at z = pi is, res(f(z), z = pi) = lim_z rightarrow pi g(z)/z - pi, g(z) = sin z/z + pi = sin pi/(pi + pi) = 0 residence of f(z) at z = -pi is, Res(f(z), z = -pi) = lim_z rightarrow -pi g(z)/z + pi, g(z) = sinz/z - pi = sin(-pi)/-2pi = 0 therefore Res(f(z), z = -pi) = Res(f(z), z = -pi) = Res (f(z), z = pi) = 0 d) Let f(z) = 1/(z - 1 + 2i)^5 has a pole of order 5 at z = infinity z rightarrow 1 - 2i hence it is a pole of order 5 at z = 1 - 2i Let f(z) = {sin z/z, 5 z notequalto 0 z = 0 Here lim_z rightarrow 0 f(z) exist but notequalto f(0) so z = 0 is a removable singularity