NUMERICAL INTEGRATION OF DIFFERENTIAL EQUATIONS - Computer Methods for Analysis of Mixed-Mode Switching Circuits - page 61

cuits to circuits, it is a natural choice to apply Pad é approximation to
rather than Let where we arrive at

Next, we approximate using Padé polynomial given earlier

It was shown in [5] that under the condition all the poles
of are simple and are located in the right half of the complex
plane. As a result,

where and are the residues and poles of respectively.
The integral in Laplace inverse transform can thus be evaluated using
the residue theorem by closing the path of integration along an infinite
arc around the poles in the right half plane.

where is the displacement from the time origin. If is a real function,

then

where and Let us examine the properties of the
preceding numerical Laplace inversion in detail:

Because and where and are Dirac
impulse function and unit step function, respectively, the Laplace