Thanks for your feedback. I looked at your seven examples. Your example 5 is taken from Table 2 of the paper by Iseger mentioned by Andras, a discontinuous example. Then examples 1, 2, 4 and 7 (with regards to tackling them using the methods implemented) are basically of the same type as example 5. You see that by looking at the form of the Laplace transform being inverted, or at the exact form of the inverse Laplace transform. These examples - the one from Iseger's paper is the best representative, I think - result uncomputable (using the implemented algorithms) because for discontinuous inverse Laplace transforms it is sometimes not possible to determine whether the accuracy is or not increasing. We were yesterday discussing with Katherina the possibility of implementing one more algorithm for this case (basically, what Andras suggested).
Then your example 6 is a piecewise branching at t < Pi, and you ask the numerical evaluation at t = Pi. Such a case is always difficult, for any numerical algorithm. Your example 3 is however, different: the expression being (Laplace) numerically inverted involves the Zeta function, whose numerical evaluation is itself a difficult problem, and the expected inverse Laplace transform is equal to harmonic(floor(exp(t)), 2). I am not a numerical analyst (more like a theoretical physicist), but I tend to think that numerical methods for problems like your 3 and 6, where you have discrete numerical functions, or where you ask for the numerical evaluation at where the discrete function branches, are not the standard problem, and in any case, not the problems I am targetting in a first implementation round.
Summarizing, if we have time, we will try to implement Iseger's algorithm as suggested by Andras and with that handle the two discontinuous examples of Table 2 of Iseger's paper (that would handle your examples 1, 2, 4, 5 and 7), Your examples 6 and 3 may or not be automatically solved at that point; otherwise they will require computing the exact transform first.
Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft