= Reciprocal-polynomial root identities
{title2=$\sum_jr_j^{m-1}/P^{\prime}(r_j)=1$}
Let $P$ be a monic <polynomial> of positive degree $m\ge1$ with distinct <roots of a polynomial> $r_j$. The <partial fraction decomposition>
$$
\frac1{P(z)}=\sum_{j=1}^m\frac1{P'(r_j)(z-r_j)}
$$
follows by matching the simple pole residues; the difference is entire and vanishes at infinity. Comparing its expansion there gives $\sum_jr_j^\ell/P'(r_j)=0$ for $0\le\ell<m-1$ and $\sum_jr_j^{m-1}/P'(r_j)=1$. These identities give the continuity and derivative jumps needed for constant-coefficient <Green functions>.
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