1. The problem statement, all variables and given/known data Im not sure if i understood correctly how to calculate the residue for functions with essential singularities like: f(z)=sin(1/z) h(z)=z*sin(1/z) j(z)=sin(1/z^2) k(z)=z*(1/z^2) 2. Relevant equations So, according to what ive read, when we have a functions with an essential singularity, we expand it in laurent serie and the residue will be the coefficient a-1. Im a bit confused, if this is for all kind of functions, because in every book ive read, the exemle is always for 1/z(f(z)=e^(1/z) or sin(1/z) etc) and my question is if its 1/z^2 or 1/z^n the residue is also the a-1 coefficient??? 3. The attempt at a solution For all given functions the essential singularity is 0. f(z)=1/z-1/(z^3*3!)+1/(z^5*5!)+...... the a-1 coefficient is 1 so the residue is 1. h(z)=z*f(x)=1-1/(z^2*3!)+1/(z^4*5!)+... In this case the residue is 0. j(z)=1/(z^2)-1/(z^6*3!)+1/(z^10*5!)+.. For j(z) the residue is 0. k(z)=z*j(z)=1/(z)-1/(z^5*3!)+1/(z^9*5!)+... for k(z) the residue is 1.