Residue of essential singularity

Question

$$f(z)=\sin(z)e^{1/z}$$Find the residue of $f$ at $0$.

I think there is an essential singularity at $z=0$ ?

How do I compute the residue of this... I know how to compute the residue of poles but not essential singularities, is there a trick do to it?

I multiplied out the series for $\sin(z)$ and $e^{1/z}$ and got that the coefficient of $1/z$ is $1/2$ ?

Thanks

Answer 1

Looks like you only took the $z$ term from the sine power series and the $\frac{1}{2z^2}$ term from the series of the exponent.

What you should take is an infinite sum of terms over all powers that sum up to -1.

$\sum (-1)^{n}\frac{1}{(2n+1)!}\frac{1}{(2n+2)!}$