Talk:Two-sided Laplace transform

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What does the comment "does not respect causality" mean?

The section on the properties of the two-sided Laplace transform is wrong. Also, there is no reference as to who invented it. It is not an old invention, it was invented by 3 Japanese engineers in 1999. A proposal for two-sided Laplace transforms and its application to electronic circuitsOriginal Research Article Applied Mathematics and Computation, Volume 100, Issue 1, April 1999, Pages 1-11. Isamu Matsuzuka, Koji Nagasawa, Akihiro Kitahama — Preceding unsigned comment added by Therepel (talk • contribs) 08:12, 1 December 2016 (UTC)[reply]

Is there a formula for the inverse of two-sided Laplace transform similar to the formula for the inverse of one-sided Laplace transform? Sprlzrd (talk) 14:31, 17 November 2022 (UTC)[reply]

@Sprlzrd: The inverse of the two-sided Laplace transform is given by the Mellin integral inside the region of convergence, just like in the one-sided case. The main difference is the location of possible poles in s-space (there are no poles to the right of some suitable vertical line for the one-sided version). This leads to difference in the way you evalutate the Mellin integral. For example, the one-sided transform has an implicit step function in time-domain which corresponds to evaluating the Mellin integral by encirceling the poles in some left half plane and using Cauchy's residue theorem. Perhaps one can add a paragraph about inversion to the two-sided Laplace transform page, but at the moment I have no time. Hart15 (talk) 00:11, 30 November 2022 (UTC)[reply]