= Regular part at a simple pole
{title2=$\frac{N(z)}{(z-z_0)D(z)}=\frac{N(z_0)/D(z_0)}{z-z_0}+\frac{N'D-ND'}{D^2}(z_0)+O(z-z_0)$}
For holomorphic $N,D$ with $N(z_0)D(z_0)\ne0$, 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 $h'(z_0)$; the latter requires differentiating both numerator and remaining denominator. Higher regular coefficients follow from the higher Taylor coefficients of $h$.
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