Talk:Laplace transform

article states often L is for managing imaginary
No. It was investigated as to if it impacts imaginary and how: but was and is not used with imaginary numbers particularly. I could say D or Dx are for managing imaginary numbers. But neither are they to any extent except in afterthought.

Initial/final value theorems
Can someone explain what it means for a complex number to go to infinity? I could understand |s| -> inf or Re(s) -> but just s? 138.232.68.221 (talk) 08:53, 31 March 2023 (UTC)

Probability Theory
It would be welcome to have $$E$$ defined. Can someone please edit to provide a proper definition for those who are trying to read/understand/use as reference? — Preceding unsigned comment added by DrKC MD (talk • contribs) 18:05, 15 February 2024 (UTC)


 * Done
 * Constant314 (talk) 18:17, 15 February 2024 (UTC)


 * Thanks, but did you mean to say $$E[r]$$ for some random value $$r$$? — Preceding unsigned comment added by DrKC MD (talk • contribs) 22:29, 15 February 2024 (UTC)
 * No. $$r$$ is a random variable. Constant314 (talk) 23:39, 15 February 2024 (UTC)

Fourier transform
Essentially expanding the Laplace transform into real and imaginary exponent parts. The real bit is just $$e^{-\sigma t}$$, imaginary $$e^{-j\omega t}$$. FT of $$f(t)e^{-\sigma t}$$ is $$ \int_0^{\infty} f(t)e^{-\sigma t} e^{-j\omega t} dt$$ when $$f(t) =0, t < 0$$. XabqEfdg (talk) 02:24, 23 May 2024 (UTC)


 * The term "transform" is a bit ambiguous. It can either refer to the operation of transforming a function, or to the result of this operation on a particular function. Using the second meaning, the claim would be false. The term "function" has the same ambiguity, when we say "x is a function of y." It would be more verbose, but maybe "the Laplace transform operator can be defined in terms of the Fourier transform operator" would be clearer? Albie&#39;s relation of misfortune (talk) 03:08, 23 May 2024 (UTC)
 * I don't think that, as put, this should go in the lead. Per MOS:LEAD, the lead should summarize the content in the article. There is a section on the Laplace transform's relationship to the FT, but it does not talk about real arguments. Before putting the interpretation of the Laplace transform in the lead, it should first go in the Fourier transform section. Its placement in the lead may be undue because the Fourier transform relationship is not very important in the rest of the article. I get that there may be some pedagogical value, but see WP:NOTTEXTBOOK. People can also simply scroll down to the Fourier transform section as well. I do think that the Fourier transform section should include a bit about equivalence to the bilateral Laplace transform (which is the context that Signals and Systems and Bracewell put it in), but not the unilateral one which is, after all, just a special case. XabqEfdg (talk) 04:06, 23 May 2024 (UTC)
 * I'm not too worried about where this equation ends up, but there is a reason that the Laplace transform always comes along with the other elements of Fourier analysis, and that needs to be mentioned somewhere in the lead. Albie&#39;s relation of misfortune (talk) 04:19, 23 May 2024 (UTC)
 * Also note that this relationship can be used to explain the region of convergence. I might work it into this article's section on that if I find the time. Albie&#39;s relation of misfortune (talk) 13:51, 23 May 2024 (UTC)
 * Since it is in Signals and Systems, it can go in the body of the article, but it needs to be in the same form that it appears in the reference. "Suitable functions" and any other conditions needs to be clarified. Constant314 (talk) 13:57, 23 May 2024 (UTC)
 * Does that go for the use of "suitable functions" that was already in the lead? Albie&#39;s relation of misfortune (talk) 14:03, 23 May 2024 (UTC)

Demonstration that the Laplace transform can be expressed in terms of the Fourier transform.
$$LHS = \mathcal{L} \{f(t)\}(\sigma\ +\ i \omega)$$ $$= \int_0^\infty f(t)e^{-(\sigma\ +\ i \omega)t} \, dt$$ $$RHS = \mathcal{F}\{f(t)e^{-\sigma t}\}(\omega)$$ $$= \int_{-\infty}^\infty [f(t)e^{-\sigma t}]e^{-i \omega t} \, dt$$ $$= \int_{-\infty}^\infty f(t)e^{-(\sigma\ +\ i \omega)t} \, dt$$

So for suitable functions, $$LHS = RHS.$$ Albie&#39;s relation of misfortune (talk) 02:30, 23 May 2024 (UTC)


 * Except the limits of integration are not the same. Constant314 (talk) 05:22, 23 May 2024 (UTC)
 * Suitable functions are ones such that $$f(\mathbb{R_{<0}}) = \{0\},$$ so the negative part of the integral is zero. The identity appears early on in multiple introductory sources. Albie&#39;s relation of misfortune (talk) 13:59, 23 May 2024 (UTC)

Please explain your objections to the inclusion of the identity, whether in general or in the lead. Albie&#39;s relation of misfortune (talk) 14:02, 23 May 2024 (UTC)


 * The lead should summarize the body. Put it in the body first along with a complete description.  At this time, I think that it is not notable enough for the lede, even if properly developed in the body.  If there are multiple reliable sources, then cite some.  My mind can be changed, if the evidence is there. Constant314 (talk) 14:10, 23 May 2024 (UTC)
 * Would you be willing to write it into the body yourself? That would probably be easier for you than correcting me as I make a dozen attempts at getting it right. Here are some supporting sources I was able to find using a search engine: I'm still interested in making this article more accessible, but I might focus on creating more diagrams and illustrations rather than disturbing the text. The fourth source above provides some good ideas. Albie&#39;s relation of misfortune (talk) 14:59, 23 May 2024 (UTC)
 * Thanks for listing those resources. The first one is sufficient.  I could not get the second one to download.  Oppenheim only makes the equivalence for the two-sided Laplace transform.  That is the form that it needs to be shown here.  I have no quibble with then adding a note that if f(t)=0 for t<0, then the result is the same as the singled sided LT.
 * I am not inclined to add the material myself because I am not sure that I fully understand it, and I have limited time also. We need to keep it out of the lede until it is right.  Even then, it doesn't belong in the lede unless it is sufficiently notable.  Let us debate that after getting into the body.
 * I think that there are two good ways to proceed.
 * You can create a subpage under your user page and polish there. Invite editors here to collaborate there.  I will join you.
 * Or you can work on it down in the body.  made some good suggestions.  It is ok to have some work in progress in the body when there are no egregious errors.
 * I would like to encourage you to go ahead and make your mistakes. We all do that.  No one is upset.  It is part of the collaborative process.  It is really the only way to learn the culture and expectations of editors here in the STEM community.
 * Just my opinion: course notes are fine for discussions on the talk page, but often they less desirable in the article. The main problems are that 1. They go away sometimes, and 2. often there is missing context that the lecturer provides verbally.
 * You may find the following essay helpful: WP:OTHER. Constant314 (talk) 15:44, 23 May 2024 (UTC)
 * I'm surprised you think the current state of the article meets the standards you hold changes to it against. For example, the only reference in the current Fourier section is to the mysterious Hungarian journal Magyar Hiradástechnika. Albie&#39;s relation of misfortune (talk) 19:06, 23 May 2024 (UTC)
 * Please make improvements. However, other bad practice does not justify additional bad practice. Constant314 (talk) 19:45, 23 May 2024 (UTC)
 * I think I'll be tearing my hair out if I continue this any further. Albie&#39;s relation of misfortune (talk) 20:12, 23 May 2024 (UTC)

The relationship with the Fourier transform is already discussed in the article, in the section on relationships to other transforms. Tito Omburo (talk) 15:55, 23 May 2024 (UTC)


 * "The" is the wrong article here. Albie&#39;s relation of misfortune (talk) 18:38, 23 May 2024 (UTC)
 * Well, we link to the Paley-Wiener theorem, if this is what you're worried about. Tito Omburo (talk) 20:46, 23 May 2024 (UTC)
 * It isn't anything that complex. It's just a direct but informative algebraic restatement. Albie&#39;s relation of misfortune (talk) 20:49, 23 May 2024 (UTC)
 * It's not really an algebraic statement though, it involves analysis. Tito Omburo (talk) 20:51, 23 May 2024 (UTC)
 * Yes. The Fourier transform section is the obvious place to put it. Constant314 (talk) 21:36, 23 May 2024 (UTC)

Rather than completely remove cross-correlation, why not clarify it comment column?
The edit https://en.wikipedia.org/w/index.php?title=Laplace_transform&curid=18610&diff=1230024344&oldid=1226060581 removed cross-correlation from properties. But wouldn't it be more useful to keep it and say it generally but not always is valid in the comment column. Other properties that don't always hold have a clarifier in the comment column about when the property holds. Or maybe move it out of the table and after the table write a sentence on how cross-correlation isn't a general property. Em3rgent0rdr (talk) 15:51, 20 June 2024 (UTC)


 * Fine if you have a WP:RS. <b style="color: #4400bb;">Constant314</b> (talk) 15:54, 20 June 2024 (UTC)


 * I think it's basically never true as stated. For example, it does not check on basis functions $$exp(-at)$$. Tito Omburo (talk) 16:20, 20 June 2024 (UTC)


 * The convolution property for the unilateral Laplace transform applies only when $$f(t), g(t) = 0$$ for $$t < 0.$$ We have $$[f(t) \star g(t)](t) = [f^*(-t) * g(t)](t),$$ so the unilateral Laplace transformation of the cross-correlation of $$f$$ and $$g$$ is $$F^*(-s^*) \cdot G(s),$$ if $$f(t) = 0$$ for $$t > 0$$ and $$g(t) = 0$$ for $$t < 0.$$ I believe these would be the appropriate conditions if the formula is added again. I may be wrong though and have no RS for the change, so I have not made it. I don't think it is really necessary if it is easily derived from the convolution formula. XabqEfdg (talk) 07:09, 21 June 2024 (UTC)
 * It's true that one can certainly derive *something* from the convolution property, but the domain of integration would not be causal (it would be [-t,0]). A reliable source is required. Tito Omburo (talk) 10:09, 21 June 2024 (UTC)