Residue formula for a quotient at a simple denominator zero
= Residue formula for a quotient at a simple denominator zero
{title2=$\operatorname{Res}_a(g/h)=g(a)/h'(a)$}
If $g,h$ are <holomorphic> near $a$, $h(a)=0$ and $h'(a)\ne0$, write $h(z)=(z-a)k(z)$ with $k(a)=h'(a)$. Then $g/h=(g/k)/(z-a)$, so its <residue> is $g(a)/h'(a)$. If $g(a)\ne0$ this is a <simple pole>; if $g(a)=0$ the singularity is removable and the formula gives zero.