ON taylor coefficients for a function with singularities
If you have some function with a simple pole at $0$, it is sufficient to find the limit $a(-1)=\lim_{x\to 0}xf(x)$. If there are poles of higher order, you need to start with the highest order pole. If you have a highest-order pole $x^{-n}$, we get that $a(-n)= lim_{n\to 0}x^nf(x)$. Then, remove the pole by subtracting $\frac{a(-n)}{x^n}$ from $f$ and repeat the process.