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:
where . 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
where , 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:
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 .

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 as where is the probability distribution (which is a measure) of ( instead of 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 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)

Clone articles[edit]

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. (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]


  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)
The Laplace-Carson transform is related to Heaviside's operator calculus on which Carson wrote a book which was well known in its time, earlier in the 20th century. There is a short WP article about the transform which should be kept for the history of the subject. As the article says, the transform is still used in railway engineering. JFB80 (talk) 20:27, 19 January 2016 (UTC)

Redundant information in the transform?[edit]

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 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[edit]

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 :

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 means that the topology has changed since is essentially .

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

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

@BenFrantzDale: 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[edit]

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


which is to say that


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


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: so we have


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 we have


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 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 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)

partial differential equation[edit]

What does the Laplace transformation do ? Faisale1994 (talk) 03:01, 17 November 2015 (UTC)

Differentiation vs Derivative[edit]

Recent edits to the tables changed "differentiation" to "derivative". That conflicts with other names used in the tables. The previously used words in the tables are operations: e.g., "differentiation", "integration", and "multiplication". The result of those respective operations are "derivative", "integral", and "product". (I'm not sure what the results of "convolution" and "correlation" are, but the words denote operations.) Replacing "differentiation" with "derivative" breaks the pattern. Glrx (talk) 21:29, 15 December 2015 (UTC)

Complex angular frequency[edit]

This term has just been introduced into the definition of the Laplace transform. But it is not in general use. It is also inaccurate since s is a complex number and takes the form a + iω where ω is the angular frequency. In a stable system a gives the damping factor which has nothing to do with frequency. JFB80 (talk) 06:54, 21 March 2016 (UTC)

There is potential confusion here. Generally, s has units of frequency, and one does refer to the s-domain in some applications as the "frequency domain" or "complex frequency domain". I feel that calling it a "complex angular frequency" might be extrapolating a little too much (surely that term would be reserved for the imaginary part of s only?) For the moment, I have left in the first mention of complex angular frequency as a compromise, but I would welcome a more widely supported term. Sławomir
12:00, 21 March 2016 (UTC)
I too would support a more widely used term (though this one gets a significant number of hits for this use and closely related meanings, including in texts). I also feel that it is not perfect. I think we should make some effort to avoid confusion with the standard term "frequency", however, because it is clearly different, but similar enough to be confusing. At the very least, we should draw attention to the fact that the two uses are not quite the same. It does already occur in other articles that use the concept (Pole–zero plot, Two-port network, Sallen–Key topology) and I did locate the definition in the IEC link that I provided, so it should not be treated as "not in general use", at least not until a more suitable term or other approach is found. Perhaps "frequency" is used as a short term when the context is understood, but never used without context? And on a related note, the unit is of angular frequency (rad⋅s−1) and not of frequency (Hz or events per second) that is meant here; people are generally at pains to distinguish these when it is not abundantly clear from context. —Quondum 15:50, 21 March 2016 (UTC)
I would not call it complex angular frequency. "Angular frequency" seems similar to the cringe-worthy "rate of speed". A frequency is already an angular speed (cycles/second or radians/second). Effectively saying "angular radians per second" is just odd. Looking at the angular frequency article, it starts out just talking about ω, so it suggests the confusion of a complex ω rather than ω being just the imaginary component of a complex frequency.
I would use just "complex frequency'. Glrx (talk) 18:42, 24 March 2016 (UTC)
Unfortunately this is a classic problem of terminology. "Complex frequency" does seem to occur a lot and is less clumsy, but it is also a misnomer. (It is incorrect to say that frequency can be measured in radians per second. Angular speed aka the awkward "angular frequency" has units rad/s, but frequency is not angular speed at all: it measures events per second as defined by SI, and nothing sinusoidal or rotational is implied at all.) A more accurate name would be "complex damping constant"; it is equivalent to what one might refer to as "complex angular speed" multiplied by i. What we need to do is find the most-used terms for this notable concept, and choose one of them. —Quondum 19:13, 24 March 2016 (UTC)

I have tried to alleviate the problem by removing direct use of any awkward terms for s, at one point simply listing some of the terms that seem to be frequently used. That no term has become "standard" shows that this awkwardness of terminology is more general, but in the encyclopaedic context we would do well to avoid emphasizing any one of the terms used. I hope that this finds more general approval. —Quondum 05:27, 27 March 2016 (UTC)

List of Laplace transforms[edit]

The article List of Laplace transforms was recently forked off of this article. I reverted the removal of the table. Does having a separate article actually benefit likely readers? Sławomir Biały (talk) 12:50, 17 May 2016 (UTC)

  • I do not think it is in readers' interests to have a separate List of Laplace transforms. I myself have used this article as a quick reference. Having the Laplace transforms listed in a separate, hard-to-find table is not helpful. Furthermore, there are more eyes watching this page, which serves as a safeguard against inaccuracy. Because of different conventions for special functions, one needs to be very careful in adding identities to articles like this. They need to be checked and re-checked against different sources. So having a separate list is not likely to lead to greater accuracy. Unless the list were to contain many more entries that cannot be easily accommodated in the main article, I do not think there should be such a list. Sławomir Biały (talk) 12:50, 17 May 2016 (UTC)
  • Merge back into Laplace transforms. If the list was unwieldy and unbalanced the parent article, I could see a case for breaking out the list. But in this case the list is fairly short and manageable and is more valuable in the parent article for the reasons Sławomir gives above. --Mark viking (talk) 20:14, 17 May 2016 (UTC)
  • I was the one who forked the list off of this article. I believe it is better to have it separate as the list gets buried inside the main article. Having its own article allows to expand the list without making the main article too long. We don't have to remove the list from the main article, just make it shorter and link to the full list with the template that I added. Also, there is lists of other mathematical things such as List of second moments of area or List of centroids, so I believe we could do the same with the Laplace transform. --IngenieroLoco (talk) 15:26, 18 May 2016 (UTC)
  • Oppose fork at this time. The article should have a list of transforms, and the current list is not long enough that it needs to be shortened and split to a separate article. Maintaining two lists is more work and the fork repeats some properties. Delete the {{main}} link from this aticle and have List of Laplace transforms redirect to Laplace transform#Table of transforms. Glrx (talk) 18:13, 21 May 2016 (UTC)

Periodic transform property typo?[edit]

In the properties section, it is stated that the transform of a periodic function is:

Shouldn't the integral upper limit be ?

--Jmendeth (talk) 21:10, 14 January 2017 (UTC)