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- W2316331505 abstract "1. In the literature on the Laplace transformation much attention has been devoted to inversion formulae, and a considerable number of these is known. In the theory, they often form the basis of representation theorems, that is, statements of necessary and sufficient conditions under which a function is a Laplace transform, or a Laplace transform of a function of a certain class; and in the applications inversion formulae are often used at the last stage of the solution of a problem by means of the Laplace transformation. While many individual inversion formulae have been investigated, it seems that no general principle has developed that will yield all of them. Yet, there is such a principle, and it can be formulated briefly (if somewhat inaccurately) as follows. If there is a singular integral whose kernel N(u, t; k) can be interpreted as the result of a linear operation on e-8u so that N u Lk, t[e5u], then Lk,t is an inversion operator for the Laplace transformation. The extension of this principle to the Laplace-Stieltjes transformation is imnmediate, and it is also clear that, with small changes, the principle applies to other linear functional transformations, notably integral transformations. The principle is not new. In special cases it has been used repeatedly to prove individual inversion formulae. However, it does not appear to have been formulated in a general manner, nor does it seem to have been exploited for discovering inversion operators except possibly for Stieltjes' discovery of the inversion by means of derivatives. 2. The Laplace transform of a function k(t) is defined as" @default.
- W2316331505 created "2016-06-24" @default.
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- W2316331505 date "1950-09-01" @default.
- W2316331505 modified "2023-10-18" @default.
- W2316331505 title "The Inversion of the Laplace Transformation" @default.
- W2316331505 cites W1945843680 @default.
- W2316331505 doi "https://doi.org/10.2307/3029643" @default.
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