What is the residue of $f(z) = z\sin(z + {1 \over z})$ at $z_0 = 0$?

Question

I want to determine the residue of $f(z) = z\sin(z + {1 \over z})$ at $z_0 = 0$. I think that this an essential singularity and I want to work with the Laurent series which should be given by:

$$z \sin\left(z + {1 \over z}\right) = z \sum_{n=0}^{\infty} {(-1)^n \over (2n + 1)!} \left(z + {1 \over z}\right)^{2n+1}.$$

I don't know how to continue

Answer 1

Hint: $f(z)$ is an even function of $z$.