How to calculate complex residues

Question

How would one best calculate the residue of

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

At its various poles?

My method is to use L'hopital to calculate $\lim_{z\to root}(z-a)f(z)$ but this is rather slow and laborious. Is there any faster way?

Answer 1

You could use the alternative definition of the residue for a function

$$f(z)=\lim_{z\to c}(z-c) \frac{p(z)}{q(z)}=\frac{p(c)}{q'(c)}$$

Where $c$ is some root of $q(z)$