 Solution

A simple zero means $h(a)=0$ and $h^{\prime }(a)\ne 0$. The local factorization is $h(z)=(z-a)h_1(z)$, where $h_1$ is holomorphic and $h_1(a)=h^{\prime }(a)\ne 0$. ThusThe numerator is holomorphic near $a$, so its constant Taylor coefficient is the coefficient of $(z-a{)}^{-1}$ in the quotient. ThereforeIf $g(a)=0$, the possible pole cancels and the residue is zero; the formula does not require $g/h$ actually to have a pole. This is the residue formula for a quotient at a simple denominator zero.