Eheh, as far as I know they're currectly used mostly in finance, for theorethical (i.e. easier representation of characteristic functions) and practical (i.e. spectral analysis of time series, adjusting for seasonality) reasons. I'm still a bit skeptical though, since the results aren't much better compared to what's obtained by the usage of simpler formulae. In economics, they're used to control for seasonality effects aswell, so perhaps they might be useful for managing interest rates, but I haven't been able to keep up with the lastest research in that realm.
ENTj 7-3-8 sx/sp
Around here, they're introduced at about the senior level of undergraduate math, and they're used in mid- and high-level electrical engineering classes in order to get equations of systems--such as control systems and electronic filters--into a form that can be manipulated and studied in various ways.
I suppose that they could be used in all sorts of systems, whenever one has to translate from one domain to another. My Fourier teacher's pet peeve was that engineers tended to call the two domains "the time domain" and "the frequency domain," when the two domains to be mapped need not be frequency and time
Out of curiosity, I searched and came across a forum link that seemed to try to tie prospect theory to the Fourier transform, but I'm not sure how the transform is actually used..
I don't know if this will be good or bad news for most of you, but higher math is rarely used once you graduate.
At my school at least Laplace and maybe fourier transforms were introduced in our ODE class right after Calc 3. Myself and several others took that 1st semester of sophomore year. Most people [who take it] probably take it 2nd semester of sophomore year.
Laplace and Fourier transforms are both examples of "integral transforms" which is a major method of solving PDE's [partial differential equations]. My EE friends said they used them a LOT in their various device classes. I forgot where I first saw it [though I still think it was ODE], but Fourier transform was hit hard in our 400 level elective PDE class. As a physics person, we got them in quantum mech, and maybe E&M as well. You could also see them in various "applied math" or "math for physics peeps" classes. In physics we used laplace transforms, well never I think. It was always Fourier. Until later there was some "greens functions solutions" for PDE's, but virtually always it was "separation of variables." I think we only really did Fourier due to its connections/applications in quantum mechanics.
Fwiw, I don't think Fourier or Laplace transforms are particularly "special" or "difficult" math, and as a math/physics person we went far beyond them, but for a typical or average engineer yeah you might max our around there.
This is so, so true.. it's not used in the real world nearly as often as it's used in classes. I guess its benefit, though, is that (a) it enables you to learn the context of the problems that you solve in the real world, which helps to reinforce why thinks work the way they do in reality, and (b) it demonstrates to employers that you can actually think at that level.
I do wish that coursework actually did a better job of explaining how the stuff you learn is actually used, or if they were honest about the fact that it is only primarily used to establish context, though. This is why I'd totally recommend a co-op program for any aspiring engineer if they can get their hands on one.
However, some of the statistics concepts that I end up using in my career are pretty insane and far beyond what I ever learned in the advanced stats classes that I ever took for my degrees
I was once asked to 'check' the use of complex conjugates from a Shell knowledge expert. My head exploded. I still can't remember how to do complex conjugates or what it means. Most of my math at work is fairly trivial, only made complex by the scale I work with (often enough to touch 500 megs in a spreadsheet).
Example lazily off the top of my head, here.. on one project, we did use some neat little time window correlation stuff as inspired by this paper. Not the most elegant example, but it does show that some higher math/stats is actually used outside of the classroom environment.
Oh, and that project? It had to do with behavioral economics.