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Laplace Transforms

  1. May 3, 2013 #1
    Recently in my DiffEq class, we learned how to use, and come up with, Laplace transforms. After doing my homework, I realized that Laplace Transforms are my new favorite concept in math(just beating out double/triple integrals and their applications)! The transforms just look so elegant on a white board!

    The theorem we were taught said this: ##L##{##F(t)##}##=∫^∞_0e^{-st}F(t)dt##. My professor mentioned that there are more transforms out there, but we only had time for this one. Are the other transforms out there of this same form? Where can I find more information on them?
     
  2. jcsd
  3. May 3, 2013 #2

    MathematicalPhysicist

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    It's not a theorem, it's its definition.

    As for others, yes, there are plenty and as much as there are applications of it.

    Radon transform, Mellin transform, etc...

    In Functional Analysis we look on a kernel $$K(x,y)$$ and look on the next operator:

    $$Kf(x) := \int f(y)K(x,y)dy$$

    Obviously we can do this for a function f that has as many variables we wish, and then we look for possible convergence conditions of the integral, and other features as the theory of these kernels has progressed.
     
  4. May 3, 2013 #3

    SteamKing

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  5. May 3, 2013 #4

    AlephZero

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    Here's a list to get you started :smile: http://en.wikipedia.org/wiki/List_of_transforms

    The "integral transforms" have similar forms to the Laplace transform. The most similar is the Fourier transform.

    You might also look at the Z transform, which is analogous to the Laplace transform but for a series instead of a continuous function.
     
  6. May 3, 2013 #5
  7. May 3, 2013 #6

    I like Serena

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    Fourier Transform and Fourier Series, which is a whole family of transforms.
    In particular, the Laplace Transform is the same as the Fourier Transform, except for a factor ##i##.
    The z-transform is also a (discrete) variant of these same transforms.
     
  8. May 7, 2013 #7
    Thank you all for your input, and I'm sorry I've not gotten a chance to reply. (Finals >.<)

    Can't wait to finish up with finals so I can spare some time to look into those that you all have mentioned!
     
  9. Dec 21, 2013 #8
    How to solve D.E by Laplace transform ??
     
  10. Dec 21, 2013 #9

    SteamKing

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    It's not cool to hijack an older thread at PF. Set up your own thread to ask a new question.

    You can find out how to solve D.E.s using Laplace transforms by searching the net. You can google 'Laplace transform' to get you started.
     
  11. Dec 29, 2013 #10
    The biggest problem with these transforms is that not exist the rule of composition of function (chain rule, in derivative; integration for substituion, in integration). Thus, you need to calculate a mountain of possible cases.
     
  12. Dec 30, 2013 #11
    Fourier Transform is useful for solve linear system too?
     
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