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If the Laplace transformation of the input function can be written in rational function (it is a valid condition in many cases, such as Dirac-delta, Heavyside unit step function, harmonic step functions /like sin, cos/,….etc.), then the output response function in the extended complex frequency domain can be expressed as follow:
{EQUATION(size="80")}$Y(s) = \frac{\sum_{l=1}^{m}b_{l}s^{l}}{\sum_{k=1}^{n}a_{k}s^{k}} = \frac{M(s)}{N(s)}{EQUATION} which is rational function.
If {EQUATION(size="80")}$Y(s){EQUATION} is proper rational function, then {EQUATION(size="80")}$\lim_{s \rightarrow \infty} Y(s) = \lim_{s\rightarrow \infty} \frac{M(s)}{N(s)} = 0 {EQUATION}. . If {EQUATION(size="80")}$Y(s){EQUATION} is improper rational function, then by means of polynomial divides is possible to obtain
{EQUATION(size="80")}$\frac{M(s)}{N(s)} = C_{0} + \frac{P(s)}{N(s)}{EQUATION} ,where {EQUATION(size="80")}$\frac{P(s)}{N(s)}{EQUATION} already is a proper rational function.
Consequently {EQUATION(size="80")}$y(t) = \mathcal{L}^{-1}\left\{ C_{0} + \frac{P(s)}{N(s)}\right\} = C_{0}\delta (t) + \mathcal{L}^{-1}\left\{\frac{P(s)}{N(s)}\right\}{EQUATION}
Only those cases will be considered in the following chapter, where the inverse transformation may be executed by proper rational functions.