Laplace transformation updated

The traditional theory of Laplace transformation (TLT) as it was put forward 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. In TLT several fundamental features of Laplace transformation (LT) evidently are not under control. To get LT consistent one has to make it consistent with the theory of Fourier transformation, and this requires that the behavior for t ≤ 0 of both the original function and of the pertinent inverse L-transform has to be accounted for. When this requirement is observed there emerges a new kind of description of LT which is liberated from TLT's deficiencies and which reveals certain implications of LT that previously have either passed unnoticed or were not taken seriously. The new concept is described; its implications are far-reaching and principally concern LT's derivation theorem and the solution of linear DEs.