http://arXiv.org/abs/math.HO/0507132

The traditional theory of Laplace transformation (TLT), like it was putforward by

Gustav Doetsch, was principally intended to provide an operator calculus for ordinary

derivable functions of the t-domain. As TLT does not account for the behavior of the

inverse L-transform at t = 0, its validity is essentially confined to t > 0. However, from

solutions of linear differential equations (DEs) one can discern that the behavior of

functions for t <= 0 actually is significant. It turns out in TLT several fundamental

features of Laplace transformation (LT) are not consistently accounted for. To get

LT consistent one has to make it consistent with the theory of Fourier transformation,

and this requires that the behavior of both the original function and of the pertinent

inverse L-transform has to be accounted for in the entire t-domain, i.e., for

-oo < t < +oo. When this requiredment is observed there emerges a new approach to LT

which is liberated from TLT's dficiencies and reveals certain implications of LT that

previously have either passed unnoticed or were not taken seriously. The new approach

is described; its implications are far-reaching and heavily affect, in particular, LT's

theorems for derivation/integration and the solution of linear DEs.