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(Im sorry if the question is out of the scope of the forum)

Hi, Im currently a Physics student. I have studied most of the Calculus. Now, according to the book Im using, there is chapter on "Fourier Transform" and then there's a chapter on "Laplace Transform". Now, Iam for some reason inclined to learn"Laplace Transform" (because a Great youtuber has its full series)

So, is the knowledge of Fourier transform necessary for learning Laplace Transform, or I will be fine without it?

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    $\begingroup$ I never studied Laplace but I've taught it in diff. eqs. Like most integral transforms, it's a neat algebraic trick, often applied in textbooks only to DEs that are easy to solve in other ways (the textbooks' fault). For Laplace xforms in DEs, that seems to be its main interest. Most of my students have had no trouble teaching it to themselves. And they like it as a method. (They have also been good at integration.) Laplace seems simpler than Fourier but not as interesting or important. My estimate of its importance may be due to my (lack of) experience (3 yrs as phys major, then math Ph.D.) $\endgroup$
    – user1815
    Commented May 19, 2023 at 15:24
  • $\begingroup$ @raciquel the Laplace transform is a holomorphic function, while the Fourier transform is again a function of one real variable. You may be interested to read about Paley Wiener theorems. $\endgroup$
    – Steven Gubkin
    Commented May 20, 2023 at 17:13
  • $\begingroup$ Related: Should I teach Laplace Transforms? How much? $\endgroup$
    – Dave L Renfro
    Commented May 21, 2023 at 17:59

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I think you should be able to easily understand Laplace transform though it is little tricky at first. I didn't need any knowledge of Fourier Transform.

I don't what book you are reading you might be interested: A first course of Differential Equations with Modelling Application By Dennis G Zill. It covers Laplace transform as well application of Laplace Transform to differential equations. I found it easy to read.

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    $\begingroup$ Either of these two topics can be done first. Then, later, you may learn that they are "the same", in some sense, if you use complex numbers to change variables. $\endgroup$
    – Gerald Edgar
    Commented May 19, 2023 at 16:00
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Laplace Transforms are a fairly common, if not required, topic in the first undergraduate course on differential equations. Essentially the Laplace Transform is used to treat problems with discontinuous forcing functions. It also has the advantage of naturally folding initial conditions into your solution. Typical problems allow for us to trade a differential equation in the time domain for an associated algebraic problem in the so-called frequency domain. The full mathematics is really analysis of distributions and you will not find much of that in the usual textbooks.

Fourier Transforms are not usually taught in the introductory DEQns course, at least in my experience at typical US universities. You might find Fourier analysis in a second course in DEqns where you focused on boundary value problems. However, I had such a course without the transform. The Fourier Transform would seem to enjoy greater popularity in its application to higher mathematics. Generalized Fourier analysis is a thing. Also, the application to quantum mechanics is important, the Fourier transform takes you from position to momentum representation.

Both transforms have a similar application, they're used to convert differential equations to algebraic equations. The Laplace transform treats initial data nicely. The Fourier transform implement boundary conditions naturally. As other comments have already pointed out, both of these transforms are best when implemented in tandem with the full suite of complex analysis tools. Last semester I taught an independent study course where the text (Introduction to Hilbert Spaces with applications by Debnath and Mikusinski) had an example which simultaneously used both Laplace and Fourier transforms in conjunction to solve a partial differential equation.

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