Regular part at a simple pole ($\frac{N(z)}{(z-z_0)D(z)}=\frac{N(z_0)/D(z_0)}{z-z_0}+\frac{N^{\prime }D-N{D}^{\prime }}{D^2}(z_0)+O(z-z_0)$)

For holomorphic $N,D$ with $N(z_0)D(z_0)\ne 0$, write the quotient as $h(z)/(z-z_0)$ where $h=N/D$. Its Taylor series immediately gives the displayed Laurent series. The residue is $h(z_0)$ and the constant regular term is $h^{\prime }(z_0)$; the latter requires differentiating both numerator and remaining denominator. Higher regular coefficients follow from the higher Taylor coefficients of $h$.