Terhardt, E. (2005-2006). Laplace transformation updated. In: arXiv.

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.