Actually, after trying to figure out how to make this statement precise I was unable to because the statement is just plain wrong.  But I won't continue an edit war.  I will try to persuade you instead as you are obviously a person of some mathematical sophistication who can appreciate my point. The Laplace transform does not resolve a function into its moments because for no value of s does L(s) correspond to a moment; I think the person who orginally posted that statement must have thought that was the case (e.g. they thought it was the Mellin transform).  There are further transformations you can do that would reveal the moments, but that would define a different integral transform that is different from the Laplace transform (in fact, for any integral transform we can apply its inverse and then the Mellin transform and get the moments, so this isn't unique to the Laplace transform).  The exponential change of variables is not relevant because then we are not talking about the moments of the original function but the transformed version.  I think there should be some place for removing content when the content is inaccurate, rather than bending over backwards to interpret in such a way that it's correct.