# Talk:Laplace transform

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## PlanetMath

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. 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?
"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? 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.
${t^q \over \Gamma(q+1)} \cdot u(t)\,\rightleftharpoons\, { 1 \over s^{q+1} }$
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? 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. 19:09, 16 April 2012 (UTC)

## Derivation column

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. 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. 09:40, 21 April 2012 (UTC)

## Laplace transform is NOT unitary

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

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 ?

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

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 31.39.233.46 (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)

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)

## Clone articles

Not an issue with this article, but it should be noted that there are several articles on Wikipedia now about things which are really just the Laplace transform in different notation: N-transform, Sumudu_transform, Laplace–Carson_transform. The first two, at least, appear to be attempted self-promotion of some scholarship of questionable merit.

85.69.207.227 (talk) 16:03, 12 January 2015 (UTC)

The first two should be deleted. I agree with the analysis at Talk:N-transform and the PROD reason at Sumudu transform. The third just seems to be a seldom used terminology for the ordinary Laplace transform, and so probably should just redirect here. Sławomir Biały (talk) 01:31, 13 January 2015 (UTC)
• The first two don't have secondary sources. the Sumudu transform article claims that the Laplace–Carson transform is used in Eastern Europe rather than the Laplace transform:
Equation (2) is employed in Western countries,[1] and the Laplace–Carson form remains the standard in Eastern Europe.[2]

References

1. ^ Oberhettinger, F. and Badii, L., Tables of Laplace transforms (Berlin: Springer, 1973).
2. ^ Ditkin, V. A. and Prudnikov, A. P., Integral Transforms and Operational Calculus (Oxford: Pergamon, 1965).
Consequently, the article should stay a separate article or redirect to a Laplace–Carson section here. Glrx (talk) 19:00, 17 January 2015 (UTC)
I really doubt "Laplace-Carson transform" is standard anywhere. Google certainly doesn't seem to provide much support for that notion. What exactly does the source say about this? Sławomir Biały (talk) 19:46, 17 January 2015 (UTC)
I don't have the source, so I AGFd. A search for LCt has many hits, so just directing to Lt is too little. Glrx (talk) 20:38, 17 January 2015 (UTC)
There are a few hits, but not enough to make one think that the term is in very wide use, and certainly nothing that would suggest some distinction based on geography. On Google Scholar (which excludes Wikipedia mirrors), there are just 440 hits for Laplace-Carson transform, compared with over 100,000 for Laplace transform. Sławomir Biały (talk) 23:17, 17 January 2015 (UTC)

## Redundant information in the transform?

I understand Fourier transforms and how they relate to the discrete cosine transform and linear transformations of vector spaces. But the Laplace transform is throwing me for a loop. In particular, the inverse Laplace transform doesn't integrate over the entire domain—the complex plane. Obviously the forward Laplace transform maps $R^1\to R^2$ so there's extra information, but it surprises me that in general the inverse Laplace transform doesn't require integrating across the whole function, just along one haphazardly-placed line. What's up with that? —Ben FrantzDale (talk) 03:59, 20 March 2015 (UTC)

## Intuition and inverse transforming the Dirac delta

I understand the Fourier transform and linear integral transforms of functions spaces. But the Laplace transform throws me for a loop. One simple case I'm trying to understand is what the inverse Laplace transform is of a shifted delta function. That is, find $f(t)$:

$f(t) = \mathcal{L}^{-1}\{\delta(s+c)\}$

That is, what are the basis functions of the Laplace transform? This is easy with the Fourier transform: a spike in the Fourier domain at a particular Fourier-domain frequency corresponds to a sine wave in the time domain with an amplitude and phase corresponding to the value in the Fourier domain.

I'm beginning to think that there's something fundamentally odd about the Laplace domain not true of typical linear transforms: First, the fact that the inverse Laplace transform doesn't integrate across the whole s domain tells me that my initial question about a delta function is probably misguided—that the s domain must have redundant information in it. Second, and related, is that $\mathcal{L}:\mathbb{R}\to\mathbb{C}$ means that the topology has changed since $\mathbb{C}$ is essentially $\mathbb{R}^2$.

What's going on here? —Ben FrantzDale (talk) 02:47, 26 March 2015 (UTC)

In particular, it seems that the inverse Laplace transform formula says that I integrate along a line parallel to the imaginary axis. If we take the delta function to be a singularity, then that integral is zero. If we hit the delta, then we have
$f(t) = \mathcal{L}^{-1}\{\delta(s+c)\} = \frac{1}{2\pi i}\lim_{T\to\infty}\int_{\gamma-iT}^{\gamma+iT}e^{st}\delta(s+c)\,ds$
$f(t) = \mathcal{L}^{-1}\{\delta(s+c)\} = \frac{e^{-ct}}{2\pi i}$

So what's right? Or is that Laplace-domain function not well-formed for some reason? —Ben

Your question seems more appropriate for WP:Reference desk/Mathematics since it seems more like a request for understanding than a specific comment on this article. --Izno (talk) 16:13, 26 March 2015 (UTC)
I was thinking of it as a shortcoming of the article that there are very odd-seeming things about the Laplace domain that aren't made clear. I may CC WP:Reference desk/Mathematics though. —Ben FrantzDale (talk) 16:40, 26 March 2015 (UTC)

## Simple examples

I'm trying to dope this out. Here's one example that doesn't make sense to me:

$f(t) = \sin(t)$ has $F(s) = \frac{1}{s^2+1}$

which is to say that

$F(s) =\frac{1}{s^2+1} = \int_0^{\infty} e^{-st} \sin(t)\, dt$.

Now, I worked that integral by hand and found that yes, this holds. (Although I may have assumed that

$\lim_{t\to\infty} e^{-(s-i)t} =1$,

which is dubious.)

But noting that the imaginary axis of the s plane is basically the Fourier transform of f, I was expecting to see delta functions, which I don't see (I just see poles). So I plug in a particular value for s: $s:=i/2$ so we have

$\frac{1}{(\frac{i}{2})^2+1} = \int_0^{\infty} e^{-it/2} \sin(t)\, dt$
$\frac{4}{3} = \int_0^{\infty}(\cos(-t/2) + i \sin(-t/2)) \sin(t)\, dt$.

Now, clearly this integrand is a periodic function about zero that never decays. It's an odd function, so the integral from zero to infinity will never go negative. Similarly, if we pick $s:=0$ we have

$F(0) = \frac{1}{1} = 1 = \int_0^\infty \sin(t)\,dt$.

It looks like this relates to Improper_integral#Summability, which mentions the above integral of sine explicitly. Also Cesàro_summation#Ces.C3.A0ro_summability_of_an_integral. This article mentions the Lebesgue integral, which I think relates to this. What's going on here? —Ben FrantzDale (talk) 17:58, 27 March 2015 (UTC)

The Laplace transform is given by an integral only if s is within the region of convergence. Neither s=0 nor s=i/2 is within the region of convergence, which is the half-plane re(s)>0. Although it is still possible to make sense of the Laplace transform, at least formally, on re (s) = 0, it is no longer given by an integral there. The improper Lebesgue integral isnt strong enough to produce convergence, and one needs to regularize the integral, e.g. by a Cesaro method. But in fact the Laplace transform already gives a way to regularize: take $s=\sigma+i \tau$ and take the limit as sigma tends to zero through positive values. (This is called the Abel summation of the integral.) Sławomir Biały (talk) 15:29, 28 March 2015 (UTC)
That makes a lot of sense. I suppose the same applies to the Fourier transform on an infinite domain? That is, that you can use a summation approach to regularized things like the Fourier transform of sin(t). I'm totally familiar with Fourier transforms over finite domains, but I'd never stopped to think about the case of integrating acrossall real numbers. Interestingly, whereas I am used to the Fourier transform of sin producing a delta function, it appears that the $a/(s^2+a^2)$ behavior (with poles rather than deltas) comes from the one-sided infinite integral (that if it is two-sided, then you get deltas). I think this is because direct multiplication in te time domain is convolution in the s domain and so the transform of sin(t) for t >= 0 is the convolution of the transform of sin(t) and the transform of h(t).
Overall I am interested in making sense of the s domain to the left of the poles since for engineering applications, people seem to play fast and loose with the ROC, drawing various diagrams as though the Laplace transform is well-defined everywhere, yet when I try to make physical sense of some points on the Laplace transform where the integral does not converge. I think that's the gap between my present understanding and those people who speak fluently of adding poles to filters to change stability and damping. Again, thanks. That was a huge help! —Ben FrantzDale (talk) 12:55, 30 March 2015 (UTC)