The Unique Inverse of the Laplace Transformation

Abstract

In the previous Chapters, we have consistently considered the L-transformation as the correspondence which relates with each original function its corresponding image function, obviously in a unique manner. This relation may be looked at in the inverse orientation; that is, one may begin with some specific image function and seek the corresponding original functions. This inverse transformation will be designated as L⁻¹-transformation. Clearly, this inverse transformation cannot be unique, for two original functions that differ at a finite number of points, nevertheless have the same image function. Indeed, this conclusion may be carried even further. For this purpose, we introduce the nullfunction n(t), which is characterized by the property that its definite integral vanishes identically for all upper limits:¹

$$ \int\limits_0^\infty {n(\tau )\,d\tau \, = 0} \,\,for\,all\,t\,\,\underline{\underline > } $$

(1)

, For such a nullfunction one concludes, using integration by parts:

$$ \left. {\int\limits_0^\omega {e^{ - st} n\left( t \right)} \;dt = e^{ - st} \int\limits_0^t {n\left( \tau \right)} d\tau } \right|_0^\omega + s\int\limits_0^\omega {e^{ - st} } dt\int\limits_0^t {n\left( \tau \right)} d\tau = 0 $$

hence

$$ \mathop {\lim }\limits_{\omega \to \infty } \int\limits_\infty ^\omega {e^{ - st} n\left( t \right)} \;dt = L\left( n \right) = 0. $$