Formal residue is invariant under coordinate change

For every integer $k\mathrm{\ne }-1$, $\rho (u{)}^k{\rho }^{\mathrm{\prime }}(u)=(\rho (u{)}^{k+1}{)}^{\mathrm{\prime }}\mathrm{/}(k+1)$. A formal derivative has zero $u^{-1}$ coefficient, so this residue is zero. For $k=-1$, write $\rho (u)=a_{1}u(1+h(u))$; then ${\rho }^{\mathrm{\prime }}\mathrm{/}\rho =u^{-1}+((\log (1+h){)}^{\mathrm{\prime }})$, and the second term is a power series with no negative powers. Its residue is therefore $1$.

Write $f(t)={\sum }_{k\ge k_0}{f}_k{t}^k$. Only finitely many terms have $k<0$, while every substituted term with $k\ge 0$ has nonnegative order because $\rho$ has order one. Consequently the coefficient of $u^{-1}$ in $f(\rho (u)){\rho }^{\mathrm{\prime }}(u)$ is the finite sum of the monomial residues from step 1.1; it equals $f_{-1}$. This proves the claimed invariance for every Laurent series, without a convergence assumption.