Hi, The calculation of switch-on processes according to Laplace often involves the problem of transforming back into the time domain.
The inverse Lapace transformation is almost impossible to solve analytically, especially in real lines with excitation with hormonal functions.
FFT-based, numerical methods are suitable for the numerical inverse Laplace transformation.
One of the oldest numerical methods is the Koizuni algorithm.
The numerical, inverse Laplace transformation for
w/(p^2+w^2) with w=2*Pi*50 Hz
is shown in this Mathematica-example.
Regards :)