Talk:Laplace transform

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An editor recently added PlanetMath citations to the article (which is a wiki, in violation of our WP:RS guideline). I removed these references, but was reverted by the same editor, with the edit summary "restore PlanetMath citations that provide derivations until they are replace by better refs". If indeed they are to be replaced in the near future by better refs, why can't we just give those better refs? There is no need to have unacceptable refs there at all if they are soon to be replaced by decent ones. However, if as I suspect these "better refs" are merely hypothetical, then we should mark the uncited items as {{citation needed}} in hopes of encouraging people to give better references. This has a much better chance of drawing attention to uncited items than having substandard references in place. In the meantime, I have restored the original consensus revision of the article (without the PlanetMath links). What needs to be discussed (WP:BRD) is why there should be an exception to the rule prohibiting such works as references in this case? I really see no good reason for it. Sławomir Biały (talk) 11:43, 15 April 2012 (UTC)

I have a book or two which can replace the planetmath website. They will be inserted now. F = q(E+v×B) ⇄ ∑ici 13:48, 15 April 2012 (UTC)
The transform table had been populated with many exotic but unsourced entries. I added a column for references[1] and proceeded to add a few simple derivations. Over time, some unreferenced entries were deleted.[2]
This edit made an unsourced change to the unsourced natural logarithm entry. Consequently, I checked the edit and sought a source to make this sourced simplification and correction. The citation leads to a derivation of the transform that may be checked under a sophisticated WP:CALC philosophy. It is not citing for the purpose of supporting some opinion.
Since I had found the PlanetMath transform table, I filled in some other transform table entries -- ones that also lead to derivations.
I do not see any "original consensus revision" of the edits; there has not been any discussion about the references.
The WP:RS is a guideline and editors are cautioned to use common sense. The issue to me is are the derivations reasonable; they are to me. Deleting the references means there is no path for any WP:V.
I haven't studied PlanetMath, it may be a wiki, but it also seems to have an editor model that does not suggest anyone can make a random change. That's a different debate; I don't want to learn the PM editing model and possibly wander over to RSN for this. I don't think it needs to go there.
I've reverted many edits on WP:RS, wiki, and blog grounds. Those reverts have primarily been for statements of opinion -- not only the reference, but also the opinion is removed. I would not simply remove a reference that I thought was poor; I would replace it with something else. Or perhaps request a better citation.
Adding a general citation to the top of the list does not serve the purpose of WP:V; some sort of pinpoint citation should be used. If someone adds a random transform, is it in the general reference or not? Much better are pinpoints such as these.
If there is not a better citation, then the PlanetMath references should be restored. Currently, for example, there is is no verification path for natural logarithm -- the entry that brought in the first PlanetMath citation.
Glrx (talk) 16:56, 16 April 2012 (UTC)
Glrx, for one thing - that website is not durable: given "months of instability and losing information". Then what happens to the table, when information is lost from that website?
Your own statement:
"I haven't studied PlanetMath, it may be a wiki, but it also seems to have an editor model that does not suggest anyone can make a random change. That's a different debate; I don't want to learn the PM editing model and possibly wander over to RSN for this. I don't think it needs to go there."
defeats the point of what you just said. It means you don't really know what you're talking about - you just looked at a page or two and happen to stumble on the Laplace transform page (or any others). Then thought it may be a suitable reference.
However its very relevant (not "a separate issue")... If people can edit PlanetMath, yet NOT cite any sources from their own (any reference for the Laplace transform table???) , how do we know that website is reliable??? Isn't it possible that those editors can get it wrong???
Why PlanetMath anyway - its not even that good. Why not another reliable site like Wolfram Mathworld (though that would be an external link more than a reference)?
About the refs at the top of the table: those citations are for all of the formulae in the table, instead of citing each one individually. The reader can immediately notice the table comes from reliable sources after clicking the linked ref, as they will find academic (degree-level) books, not amateurish websites. In any case those books are far more reliable than PlanetMath, and certainly do "serve the purpose of WP:V" - while your obsession with Planet Math fails that. If you would like inline citations for every single formula in that table, you have Salix alba to say thanks to for doing some of that. I'll try to fill in the other bits if this is what you're really after - but no PlanetMath.
And there is some level of consensus against using sites like PlanetMath for the reasons just said.
Agreed? F = q(E+v×B) ⇄ ∑ici 17:35, 16 April 2012 (UTC)
I couldn't find any reference for the transform of the nth root:
 \sqrt[n]{t} \cdot u(t)\,\rightleftharpoons\, s^{-(n+1)/n} \cdot \Gamma\left(1+\frac{1}{n}\right)
where  \textrm{Re} \{ s \} > 0 \, . However Glrx - you were not able to either, so we're even there. The citation template {{citation needed}} will be added for that function.
About the transform of
{t^q \over \Gamma(q+1)} \cdot u(t)\,\rightleftharpoons\, { 1 \over s^{q+1} }
where  \mathrm{Re}(s) > 0 , \quad \mathrm{Re}(q) > -1\, , wolfram confirms this original statement to be true for complex q, but in the sources I have (one of which is cited) only the real case is given, so I added both.
Happy now? F = q(E+v×B) ⇄ ∑ici 18:56, 16 April 2012 (UTC)
Well no suprise... the square root is of course a special case of the second function, where q = 1/n, since:
t^q \cdot u(t)\,\rightleftharpoons\, { \Gamma(q+1) \over s^{q+1} }
t^{1/n} \cdot u(t)\,\rightleftharpoons\, { \Gamma(\frac{1}{n}+1) \over s^{\frac{1}{n}+1} }
so that row should either be deleted, or state in the ref section how it can be obtained. I'll do it now. F = q(E+v×B) ⇄ ∑ici 19:09, 16 April 2012 (UTC)

Derivation column[edit]

The "Reference" column was recently changed into "Derivation" in the Table of selected transforms. This should be changed back, and the derivations removed. We don't generally include derivations—especially those that amount to routine calculus exercises, and certainly not in table form. This is far too textbook-ish for an encyclopedia. It serves no encyclopedic purpose whatsoever. Sławomir Biały (talk) 00:52, 21 April 2012 (UTC)

I don't think its that much of a problem in providing alternative explanations, but yes it is text-booky and makes the table too big. We can just state at the beginning of the table that "some Laplace transforms can be obtained from others, using various trigonometric, hyperbolic, and Complex number (etc.) properties and identities". The table will be reverted. F = q(E+v×B) ⇄ ∑ici 08:45, 21 April 2012 (UTC)
Before the two most recent editors (before me just now) come here - I didn't only revert Glrx, but LokiClock also. The edit summary was incomplete. F = q(E+v×B) ⇄ ∑ici 09:40, 21 April 2012 (UTC)

Laplace transform is NOT unitary[edit]

It should be emphasize that the Laplace transform is NOT unitary as opposed to the Fourier transform. Watson1905 (talk) 20:39, 11 February 2014 (UTC)

Laplace Transform of a Random Variable[edit]

Why does it say it is abuse of language to define the Laplace transform of a (nonnegative) random variable? Random variables are defined as measurable functions defined on a probability space (\Omega,\Sigma,P).

The Laplace transform of a random variable is defined in Billingsley's Probability and Measure (which is highly cited and authoritative in probability theory) of a random variable X as \int_{[0,\infty)} e^{st} \mu(dt) where \mu is the probability distribution (which is a measure) of X (F_X instead of \mu is probably a better notation for this setting), which is entirely consistent with formal Lebesgue definition of the Laplace transform above and requires no abuse.

I did originally write part of the section on the Laplace transform in probability before I made an account and it seems to have undergone some revision I don't think is quite correct. It's not the Laplace(-Stieltjes) transform of the probability density function, but rather the Laplace transform of the random variable itself, so it understandably begins to look like an abuse of language when from one side it appears the transform of the PDF and is called the transform of the random variable.

I also worry the statement that says that the Laplace transform with respect to a probability distribution can be written as \int_{0^-}^\infty e^{-st}f(t) dt may be misleading in that it assumes the Lebesgue integral with respect to the probability distribution f reduces to a Riemann integral, which isn't necessarily true (the Lebesgue integrals are defined for discrete and otherwise non-continuous distributions).

Probably a rewrite with references will clear it up, which I'd like to do when I get a chance. — Preceding unsigned comment added by Machi4velli (talkcontribs) 06:25, 24 February 2014 (UTC)

It's called an abuse of language because it's the Laplace transform of the measure associated to the random variable, not of the random variable itself (which is a measurable function in its own right, but this is not the Laplace transform of that measurable function, whatever that might mean.) Sławomir Biały (talk) 18:14, 24 February 2014 (UTC)

Not very clear. Mathematics or physics ?[edit]

I think many articles about mathematics in wikipedia are in general very good.

However I think this article is not very clear and could be improved, in my opinion explained below.

For example, the notions of "s domain" and "time domain" are not very mathematical.

Moreover, hypotheses of results, or theorems, are not always specified. For example, in "Relation to moments", what are the hypotheses required to apply formulas about derivation ? Do we need to know in advance that the derivatives exist ? Or is it sufficient to check that the corresponding integrals are absolutely cnvergent ? This should be clarified, and written in the form of a theorem, with hypotheses and conclusions. And in the last formula it should be said that the derivative is taken at 0.

For me this article seems to be a kind of mixture between a mathematical article and an article of applied science. Maybe it would be better to have 2 articles separately for Laplace transforms, one of mathematical style, and one for applied sciences.

— Preceding unsigned comment added by (talkcontribs) 12:53, 1 August 2014‎

I partially agree. Ideally precise conditions should be stated at some point in the article. I disagree with the proposed separation of the article into two different kinds. It would be much better to do things properly here, including a slightly vague form geared towards applications, followed by a precise form. Generally, it is not our style to present basic facts in the theorem-proof paradigm. Sławomir Biały (talk) 18:42, 1 August 2014 (UTC)
I also agree with Sławomir Biały. The Laplace Transform might be interesting from the standpoint of mathematics (I can't speak for this, but I imagine that it is true), but it is certainly important from the standpoint of "physics" and, more generally, time series analysis. So, some understanding of those important audiences necessarily shapes the content presented here. Of course if the commenter (anonymous, it seems) thinks this Wikiarticle needs a bit of repair, (s domain, time domain, etc.), then please fix it. Sincerely, DoctorTerrella (talk) 11:23, 7 October 2014 (UTC)

Lead paragraphs[edit]

In the lead section it is said that the Laplace transform was introduced by Pierre-Simon Laplace in the context of probability theory. Is this true? Is this so important that it belongs in the lead section? Note that the LT can be used for lots of things, not just probability theory. Sincerely, DoctorTerrella (talk) 16:51, 14 September 2014 (UTC)