1. The problem statement, all variables and given/known data Find the residue at each pole of zsin(pi*z)/(4z^2 - 1) 2. Relevant equations An isolated singular point z0 of f is a pole of order m if and only if f(z) can be written in the form: f(z) = phi(z)/(z-z0)^m where phi(z) is analytic and nonzero at z0. Moreover, Res(z=z0) f(z) = phi(z0) if m = 1 and Res(z=z0) f(z) = phi^(m-1)(z0)/(m-1)! if m >= 2 3. The attempt at a solution I don't know what I'm missing here, the problem seems really easy. I factored it to z*sin(pi*z)/[(2z+1)(2z-1)] so f(z) has simple poles at z = 1/2 and z = -1/2 For z = 1/2, we have f(z) = phi(z)/(z-1/2) where phi(z) = z*sin(pi*z)/(z+1/2) Plugging in z = 1/2 in phi(z) I get a residue of 1/4. Similarly, I get a residue of -1/4 at the pole of z = -1/2. But the answer is -1/8 and 1/8 for the residues respectively, and I can't figure out what I'm doing wrong.