Talk:Laplace transform/Archive 2

Archive 1

Archive 2

List of Laplace transforms

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)

• Not as List of Laplace transforms is (at time of this post). Maybe if there was a long list of complicated ones like for list of integrals. There is List of Fourier transforms which redirects to Fourier transforms. M∧Ŝc²ħεИ_τlk 17:10, 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?

In the properties section, it is stated that the transform of a periodic function $f(t)$ is:

${1 \over 1-e^{-Ts}}\int _{0}^{\infty }e^{-st}f(t)\,dt$

Shouldn't the integral upper limit be $T$?

${1 \over 1-e^{-Ts}}\int _{0}^{T}e^{-st}f(t)\,dt$

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

Where is a reference about a necessary condition for existence of Laplace transform ?

This article claims that "a necessary condition for existence of the integral is that f must be locally integrable on [0, ∞)."

Where is a reference about it?

— Preceding unsigned comment added by 209.150.37.178 (talk • contribs) 13:21, 21 August 2017 (UTC)

Formal definition section can be confusing to beginners

This sentence can be confusing because of ambiguity: "The Laplace transform is a frequency-domain approach for continuous time signals irrespective of whether the system is stable or unstable" Students reading this will possibly not know what "the system" refers to, even though all experienced readers know what it means. I suggest saying something like "Laplace transform representations of signals at nodes of a linear system including outputs is a frequency domain...." Groovamos (talk) 04:18, 27 April 2018 (UTC)

I just removed that sentence. It is completely unnecessary. Sławomir Biały (talk) 11:34, 27 April 2018 (UTC)

Complex angular frequency

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
Biały 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)

In real variables, the $\omega$ in sin$(\omega t)$ is a frequency in radians/second, commonly called angular frequency. What do you call in in sin$(\omega t)$ when $\omega$ is complex? Gah4 (talk) 07:54, 29 May 2018 (UTC)

analysis

Working with the Fourier transform is called Fourier analysis. Is there a similar name in the case of the Laplace transform? There seems to be no page for Laplace or Laplacian analysis. Gah4 (talk) 07:59, 29 May 2018 (UTC)

As a general subject, "Fourier analysis" also includes analysis of the Laplace transform. (Also, "Harmonic analysis" is sometimes used in connection with the analyticity properties of the Laplace transform, e.g., Paley-Wiener theorem.) In the sense of analysis versus synthesis (e.g., Fourier synthesis/analysis), I suspect "Laplace analysis" and "Laplace synthesis" would be understood in context, but such phrases are rarely used in practice in my own experience. Sławomir Biały (talk) 13:05, 29 May 2018 (UTC)

The reason for the question is that I thought Category:Frequency domain analysis should have Fourier analysis in it, and also the Laplace equivalent. The main use of the Laplace transform that I know of, is in analog filter design, though it is also used for solving some differential equations. I don't usually consider filter design as Fourier analysis, but I suppose it is frequency domain analysis. Gah4 (talk) 00:24, 30 May 2018 (UTC)

What does it mean when we say the s-plane is a frequency domain

It took me 3 decades to sort out all of my confusions regarding Laplace transforms and so I would like to help students as much as possible with this article. I wish I had been told in school that integral transforms are a mapping of a process from one domain to its conjugate domain. So what are conjugate domains? They necessarily are dimensioned in inverse units, e.g. t <-> t^-1 for signals or x <-> x^-1 in graphics or optics.

Referring to s as frequency measure has confused more than the beginners. I saw on quora a physics Ph.D candidate asking for clarity on this. I think a paragraph explaining that with the Fourier transform the inverse dimensioning of t into f literally gives frequency, but with the t and s we need much nuance and so we take advantage of the fact that linear processes and systems have sums of exponentials as eigenfunctions. These eignenfunctions must have dimensionless linear functions as exponents. If the exponents are real then a real exponential does not appear as frequency, but if the exponent is imaginary, then the traditional understanding of 'frequency' can be seen. It can be said also that complex exponents allow for exponentially damped sinusoids. At any rate it can be said that calling s frequency measure is more of a convention than a rigorous reference. Groovamos (talk) 05:08, 27 April 2018 (UTC)

That's definitely worth pointing out somewhere in the article. I don't think it is a good idea, though, to drop the "frequency domain" completely, since that is relatively established terminology. Sławomir Biały (talk) 11:35, 27 April 2018 (UTC)

Agree. Maybe the "nuance" I propose would be to say that the convention of the s-plane being the frequency domain, is just that, a convention, since only one of the dimensions refers to frequency, although the dimensional units are consistent in both dimensions. So e.g. with transformed time functions the real and imaginary components of s are both dimensioned t^-1 and used in the same way in linear exponents in exponentials. The exponential of the real component give a real exponential function, the exponential of the imaginary component gives a rotating phasor in complex amplitude space, thus the correctness of frequency to describe the angular rate of the phasor rotating in time.

This can get a little messy for the newbies because sooner or later, application may require an understanding that the codomain of the Laplace transform i.e. its evaluation at a point, is a complex density function, with which a differential ds of a line integral at the point, defines the differential amplitude, and phase of the rotating phasor, and the domain point defines the angular frequency and damping time constant. — Preceding unsigned comment added by Groovamos (talk • contribs) 00:21, 29 April 2018 (UTC)

The transform is commonly given using time and (temporal) frequency. In optics and spatial filtering, it would be in position and spatial frequency. More generally, frequency is the reciprocal of a period, which might be a time, distance, or some other quantity. This is also true generally for the complex arguments of circular and hyperbolic trigonometric functions, and complex exponential functions. Gah4 (talk) 22:12, 1 June 2018 (UTC)

I think this clarifies things. It should be in the article. Sławomir Biały (talk) 10:56, 2 June 2018 (UTC)

Differentiation vs Derivative

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)

Addition, division, integration, and differentiation are operations. Sum, quotient, integral, and derivative are the result of applying the operation. Gah4 (talk) 23:08, 19 June 2018 (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)

I think it comes from residue theory, but didn't try to actually show it. It might be worth mentioning, though. Gah4 (talk) 07:44, 29 May 2018 (UTC)

@BenFrantzDale: Yes, I think it is residue theory. I'd point to the Bromwich integral, but it redirects to the Inverse Laplace transform and doesn't say much. It's the limit of a contour integral where the contour is a line with an arc that connects the ends of the line. The arc of contour although infinitely long contributes nothing to the integral, so the contour integral reduces to a line integral. See Residue theorem#An integral along the real axis. Glrx (talk) 22:29, 17 July 2018 (UTC)

Nuclear physics

While it is an example of using L() examples were already given and L() itself has no bearing on quantum mechanics (and chemistry) whatsoever: they are all do-able by other techniques, saying L() is required in Nuclear Physics is simply wrong.

Not nuclear physics but nuclear engineering. It is mostly used for the solutions of some types of linear differential equations, especially ones that come up in nuclear decay series. But yes, QM uses the Fourier transform and not this one. Gah4 (talk) 22:20, 6 February 2019 (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

@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)

You have asked about the inverse transform of a delta function of s which being in two dimensions would properly be :$f(t)={\mathcal {L}}^{-1}\{\delta ^{2}(s+c)\}$ A problem with this is that with the Fourier transform, delta functions in the frequency domain are conceptualized in the limit as the span of the integration in time for the Fourier integral approaches ±infinity limits of integration. With the Laplace transform the integration lower limit is at t=0 (except can be wider with the two-sided transform) and whenever an integral transform integration has a limit to integration that is not infinity, the transform of the original time function cannot be a generalized function of which the delta function is. I think this is a result of the Paley-Wiener theorem if someone would verify. This would preclude your delta function in the s-plane from being generated by any function of one variable in the conjugate domain, and so the illogic of an inverse transform should be apparent. In the bilatera transform the problem is that a function transformed has a characteristic to the left of the origin as decaying faster than exponential order and I think this has the same effect as having a finite lower limit of integration as procluding a generalized function (i.e discontinuous function) as a Laplace transform. Another problem I'm having is that with typical problems in applied science, delta functions in the complex plane to my knowledge do not relate to physical reality. Multidimensional delta functions typically are in spacial domains with uniform dimensional units and dimensional "behavioral" similarities. In the complex s-plane the orthogonal dimensions share dimensional units but do not share the same effect in the time domain, as one generates a decay and the other generates rotating complex phasors. Maybe someone can give an example of a delta function in the complex plane as describing a physical system. Groovamos (talk) 21:38, 21 March 2019 (UTC)

A Question of Linearity

For an article so appearing to be thorough about spaces: linearity v. non-linearity of differential equations isn't mentioned. My books say to use L() ONLY linearly - which hugely limits it's usefulness. Also: L() is infamous for not being best used in many important areas of linear differential equations: such as certain 2nd order linear wave equations - of which L() often makes more of a mess of things than it does to help.

I guess for the first time in 40 years I see the integral transforms described as hugely limited. Especially since I come from the engineering sphere. That they can be a tool to totally bypass the diff eq. stage in a system specification is as spectacular a feature as a ladder is to access a roof. An entire dynamic system can be described by expessing each system element as the 1st order Laplace transform of the impulse response and the equations written as algebraic form ant then a rational function for a node response to system stimulus easily obtained. Then the inverse transform yielding the time response (and summed with the initial conditions) is a scenario that virtually makes possible the entire spectrum of modern technology. Seriously. Groovamos (talk) 00:14, 22 March 2019 (UTC)

Of what use is L for differential equations ?

The use is e^x is the integral and derivative of itself and as such, as a multiple, yields some knowable expressions involving s instead of higher order differentials. However the resulting equation still must be solved and inverses of equations involving s can easily be more complicated than using other methods. It really depends on the class of functions in the initial problem.

As it says in the article, you can transform a differential equation into a polynomial equation. If you solve that, the inverse transform will get the solution to the original equation. Gah4 (talk) 22:06, 29 March 2019 (UTC)

time

A recent edit removed mention of time in the lede, and I just removed another unnecessary one later on. But it is used all through the article. Can we just use $t$ instead? Certainly in the case of differential equations there is no need for the variable to be time, though it often enough is. Physics often enough uses position and spatial frequency, and calling it just frequency usually isn't too confusing. Gah4 (talk) 22:10, 29 March 2019 (UTC)

Mistake in explanation of Fourier transform of sine

"For example, the function f(t) = cos(ω0t) has a Laplace transform F(s) = s/(s2 + ω02) whose ROC is Re(s) > 0. As s = iω is a pole of F(s), substituting s = iω in F(s) does not yield the Fourier transform of f(t)u(t), which is proportional to the Dirac delta-function δ(ω − ω0)."

This is not true; the Fourier transform (in the sense of distributions) of cos(ω0t) u(t) is not a Dirac delta function, but has an additional Cauchy principal value term. The substitution from the Laplace transform, properly done (ie from the right side) actually gives the correct result. See https://wiki.riteme.site/wiki/Heaviside_step_function#Fourier_transform Smeuuh (talk) 20:39, 26 June 2019 (UTC)

Not very clear. Mathematics or physics ?

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 31.39.233.46 (talk • contribs) 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)

As well as I know, the Laplace transform is rarely used in physics, where the Fourier transform is more popular. Signal processing, done in engineering and especially in EE, mostly uses the Laplace transform, and less often Fourier. There are complications in signal processing with the Fourier transform integral from negative infinity that go away using Laplace. (Not to mention the double sided Laplace transform.). Gah4 (talk) 22:15, 6 February 2019 (UTC)

As for time domain, the Fourier transform is commonly described as time vs. frequency (inverse time), but is also often use in computing spatial frequencies from position data. Both transforms should be able to transform any quantity into a corresponding frequency space. The need for a causal transform mostly comes up in the time domain, though. Gah4 (talk) 22:15, 6 February 2019 (UTC)

Also, as well as I know, the Laplace transform can be used to compute some integrals, independent of the engineering or scientific use for those integrals. Otherwise, it is easiest to describe in terms of time and frequency. Not so easy in terms of (unnamed unit) and (inverse of unnamed unit). Gah4 (talk) 09:12, 25 October 2019 (UTC)

Direction of contour

Under "Inverse Laplace transform", should the direction of the contour be explicitly indicated? —DIV (1.144.105.98 (talk) 03:03, 25 October 2019 (UTC))

Having not thought about this for years, I think it doesn't matter, as long as the straight part goes up. As long as there are no poles at infinity, that is. Otherwise, you choose the one that is easier. Gah4 (talk) 09:58, 25 October 2019 (UTC)

Is the difference between the Region of Convergence and the Region of Absolute Convergence a set of measure zero?

An application of the Root test seems to suggest that if $\int _{0}^{\infty }f(t)e^{-st}\,dt$ converges at $s_{0}$ then it must converge absolutely at all $s$ for which $\Re (s)>\Re (s_{0})$. Therefore the difference between the Region of Convergence and the Region of Absolute Convergence is a set of measure zero. I think this may be worth adding to the article.

To be explicit, the Root test suggests to consider the value of $C(s)=\limsup _{t\to \infty }|f(t)|^{1/t}e^{-s}$. Let's say that for $s=s_{0}$ we have that the integral above converges. Then the Root Test dictates that $C(s_{0})\leq 1$. Consider $s>s_{0}$. We have that $C(s)=C(s_{0})e^{-(s-s_{0})}<C(s_{0})\leq 1$ which implies absolute converence. --Svennik (talk) 08:29, 4 June 2020 (UTC)

@Svennik: Find a reliable source that provides this and it can be considered. I also do not see it straightforward that the root test, a result about infinite series, would apply to an improper integral (counterexample: summing up $e^{2\sin(\pi x)x-x}$ is an absolutely convergent geometric series when discretized to integer values of x, but as it is unbounded, its improper integral over the positive real line diverges). With that said, I think the result itself is sound, at least for well-behaved functions, but no more special than how a point directly on the radius of convergence may also be a convergent point for a power series.--Jasper Deng (talk) 09:22, 4 June 2020 (UTC)

Frequency domain?

If I'm not mistaken, then "frequency domain" in the table of transformations is not correct terminology for Laplace. It should be "s-domain". — Preceding unsigned comment added by 2001:980:93a5:1:2bd:56d1:1e92:3c2a (talk • contribs) 07:04, 23 September 2021 (UTC)

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References

Readability

The occurrence of many Heaviside stepfunctions $u(t)$ in the table "selected Laplace transforms" is redundant and only makes sense if one considers the two-sided Laplace transform. — Preceding unsigned comment added by Hart15 (talk • contribs) 09:46, 30 January 2022 (UTC)

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Integral Function is Potentially Confusing

The transform of the integral function is shown as:

$\int _{0}^{t}f(\tau )d\tau =(u*f)(t)$

While correct mathematically, why have the convolution on the right side? I don't think it adds much and potentially confuses new readers.

https://proofwiki.org/wiki/Laplace_Transform_of_Integral

Tables of Laplace transforms don't generally write it like this. Rhodydog (talk) 20:27, 24 February 2023 (UTC)