Inverse transformation method for linear invariant systems

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:

which is rational function.

If is proper rational function, then . . If is improper rational function, then by means of polynomial divides is possible to obtain

,where already is a proper rational function.

Consequently

Only those cases will be considered in the following chapter, where the inverse transformation may be executed by proper rational functions.