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These are answers submitted by Fay

 Sometime ago there was a number of entries concerning the numerical inversion of Laplace transforms. Has the person concerned tried the direct method of inversion using residues, etc.? Providing that F(s) is a proper fraction in s, then it is strait forward using the distinct roots of the characteristic polynomial and residues, and to obtain a direct and also accurate solution for f(t). Note this method is able to invert F(s), whether it is numeric or symbolic

I am currently compiling a manuscript about the Laplace Transform, and am trying to locate some detailed information about its history. While it has been easy to find information about its development by the two well known players, that is Laplace and Heaviside, I have been unable to obtain very much about the development prior to Laplace, by mathematicians such as Euler? As a result of this I would be grateful of any input as to possible sources of information.

Many thanks all that was very helpful

Can you please give one or more examples of these functions, that are discontinuous but have a continuous antiderivative. As the problem that I am studying, is concerned with the functions that the two Laplace transforms return different values. For example, the Laplace transform of the delta function is either 1 or 0, depending upon which definition of the transform is used, that is int(f(t)*exp(-s*t),t = 0- to infinity) = 1 or int(f(t)*exp(-s*t),t = 0+ to infinity) = 0 Whereas the transform of sin(wt) is always w/(s^2 + w^2). Hence the concern with functions that are continuous or discontinuous in the region [0-,0+]
Further to this discussion, am I correct in thinking that the integral of any continuous function between 0- and 0+ (or more generally a- to a+) is always equal to zero?
Also of course integrating the Delta function from 0- to infinity should yield 1, whereas integrating the Delta function from 0+ to infinity should yield 0.
My appologies, but I did not state the problem very clearly! The difficulty is not with the maths, but with getting Maple to display the lower limit of the integral as 0 subscript + or -
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