Evaluate $\underset{z=0}{\text{Res}} \; \frac{(z^6-1)^2}{z^5(2z^4 -5z^2 + 2)}$ . Problem:
Evaluate
$$
\underset{z=0}{\text{Res}} \; \frac{(z^6-1)^2}{z^5(2z^4 -5z^2 + 2)}.
$$
My question:
This is a question from a previous complex analysis qualifying exam that I am trying to work through. I know that the straightforward formula for calculating this residue would be
$$
\frac{1}{4!} \lim_{z\to 0} \frac{d^4}{dz^4} \left(z^5 \cdot \frac{(z^6-1)^2}{z^5(2z^4 -5z^2 + 2)} \right)
$$
However, I'm guessing there is a better method in this case than taking four derivatives or trying to turn this into a Laurent series. Can someone point me in the right direction here?

A variation: It is convenient to use the coefficient of operator $[z^n]$ to denote the coefficient of $z^n$ of a series.

We obtain
\begin{align*}
\color{blue}{\underset{z=0}{\text{Res}}}&\color{blue}{ \; \frac{(z^6-1)^2}{z^5(2z^4 -5z^2 + 2)}}
=[z^{-1}]\frac{\left(z^6-1\right)^2}{z^5\left(2z^4-5z^2+2\right)}\\
&=[z^{4}]\frac{\left(z^6-1\right)^2}{2z^4-5z^2+2}\tag{1}\\
&=\frac{1}{2}[z^{4}]\frac{1}{1+\left(z^4-\frac{5}{2}z^2\right)}\tag{2}\\
&=\frac{1}{2}[z^{4}]\left(1-\left(z^4-\frac{5}{2}z^2\right)+\left(z^4-\frac{5}{2}z^2\right)^2\right)\tag{3}\\
&=\frac{1}{2}\left(-1+\frac{25}{4}\right)\tag{4}\\
&\,\,\color{blue}{=\frac{21}{8}}
\end{align*}
in accordance with other given answers.

Comment:

*

*In (1) we apply the rule $[z^{p-q}]A(z)=[z^p]z^qA(z)$. Here we have $$[z^{-1}]\frac{P(z)}{z^5Q(z)}=[z^{-1}]z^{-5}\frac{P(z)}{Q(z)}=[z^4]\frac{P(z)}{Q(z)}$$


*In (2) we take only the constant term $1$ from the numerator, since other terms do not contribute to $[z^4]$. We also factor out $2$ from the denominator as preparation for the geometric series expansion in the next step.


*In (3) we expand the series up to the third term, since other terms do not contribute to $[z^4]$.


*In (4) we select the coefficient of $z^4$.
Note: The usage of the coefficient of operator $[z^n]$ can be found for instance in section 5.4 in Concrete Mathematics by R.L. Graham, D. Knuth and O. Patashnik or in Generatingfunctionology by H.S. Wilf.
