Recently, we have developed the necessary and sufficient conditions under which a rational function $F\left(hA\right)$ approximates the semigroup of operators $exp\left(tA\right)$ generated by an infinitesimal operator $A$. The present paper extends these results to an inhomogeneous equation $u^{\text{'}}\left(t\right)=Au\left(t\right)+f\left(t\right)$.

Let $r_n*\in {ℛ}_{nn}$ be the best rational approximant to $f\left(x\right)=x^{\alpha }$, 1 > α > 0, on [0,1] in the uniform norm. It is well known that all poles and zeros of $r_n*$ lie on the negative axis ${ℝ}_{<0}$. In the present paper we investigate the asymptotic distribution of these poles and zeros as n → ∞. In addition we determine the asymptotic distribution of the extreme points of the error function $e_n=f-r_n*$ on [0,1], and survey related convergence results.