Partial fraction decomposition

Over a field in which the denominator splits, a proper rational function is a sum of terms $c/(x-a{)}^k$. The coefficient at a simple pole $a$ is obtained by multiplying by $x-a$ and evaluating at $a$.

For $A\ne 0$ and $m\ne n$, combining the right-hand fractions gives numerator $A(m-n)$ and proves the identity. Integration on an interval avoiding the poles gives $\ln |(A+mx)/(A+nx)|/[A(m-n)]$ plus a constant. The overall $1/A$ cannot be dropped unless the variable has already been normalized to $A=1$.

Let $P$ be a monic polynomial of positive degree $m\ge 1$ with distinct roots of a polynomial $r_j$. The partial fraction decompositionfollows by matching the simple pole residues; the difference is entire and vanishes at infinity. Comparing its expansion there gives ${\sum }_j{r}_j^{\ell }/P^{\prime }(r_j)=0$ for $0\le \ell <m-1$ and ${\sum }_j{r}_j^{m-1}/P^{\prime }(r_j)=1$. These identities give the continuity and derivative jumps needed for constant-coefficient Green functions.