How to compute residue of $f(z)=z^3e^{\frac{1}{z}}$?

Question

How to compute residue of $$f(z)=z^3e^{\frac{1}{z}}$$

I find there is an essential singularity in $z=0$, and I have

$$ f(z) = \sum_0^\infty \frac{z^{3-k}}{k!}$$

but how to compute residue in 0 ?

Answer 1

Write out the series as

$$f(z) = z^3+ \frac{z^2}{1}+\frac{z}{2}+\frac{1}{6}+\frac{1}{24z}+\cdots$$

the residue is the coefficent of $\frac{1}{z}$ and therefore the residue is $\frac{1}{24}$