Finding the residue of $e^{\frac {-3} {z^2}} $

Question

I'm going through past exam papers and came across the following question

find the residue of $f(z)=e^{\frac {-3} {z^2}} $ at $z=0$

I know how to find the residue and the residue theorem but I'm unsure how to find it for this question. Any help would be greatly appreciated. Thanks

Answer 1

What about the usual Laurent (Taylor) series for the exponential?:

$$e^{-\frac3{z^2}}=\sum_{n=0}^\infty\frac{(-1)^n3^n}{z^{2n}n!}\implies\text{ the residue is zero...as expected, since all the powers}$$

of $\;z\;$ are even.