MATH319 Slides 135 The stable rational functions form a differential ring 137 Poles of the transfer function of the damped harmonic oscillator

136 Partial fractions

where here $q\in{\bf C}$ is a constant since $f\in{\cal S}$ is proper, and all the $\lambda_{j}$ have $\Re\lambda_{j}<0$ . So we can take linear combinations of such $f$ , and stay in ${\cal S}$ . Also $f_{1}(s)+f_{2}(s)=(g_{1}(s)h_{2}(s)+h_{1}(s)g_{2}(s))/h_{1}(s)h_{2}(s)\in{\cal S}$ .

(ID2) likewise;

(Diff) Also, we can differentiate

f^{\prime}(s)=\sum_{j=1}^{N}{{-n_{j}a_{j}}\over{(s-\lambda_{j})^{n_{j}+1}}},

and the poles are at $\lambda_{j}$ in open left half plane.

(ii) Whereas $1/(s+1)$ belongs to ${\cal S}$ , the inverse $s+1$ is not proper, hence not in ${\cal S}$ .