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Math Help - Laplace transform and Fourier transform what is the different?

  1. #1
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    Laplace transform and Fourier transform what is the different?

    Hello,
    I want to know what is the different between Laplace transform and Fourier transform?
    Can anyone show me? and help me about it?
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    Well, for one, they don't have the same definition. What else do you want to know? You have to be specific.
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    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Hamed View Post
    Hello,
    I want to know what is the different between Laplace transform and Fourier transform?
    Can anyone show me? and help me about it?
    They have a different kernel. In Laplace, the kernel is e^{-st}. In Fourier, the kernel is e^{-2\pi i x t}

    Also, they are defined over different domains. Laplace is calculated for t>0, where as in Fourier, t\in(-\infty,\infty).

    As Bruno said, please be more specific in what you're asking. These are just a couple of the differences between these two transforms.
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    My master told me they are same frequency domain and I said no they are different and Fourier is frequency domain!
    What is Laplace domain?
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    Grand Panjandrum
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    Quote Originally Posted by Hamed View Post
    My master told me they are same frequency domain and I said no they are different and Fourier is frequency domain!
    What is Laplace domain?
    For an absolutely integrable real function f(x) which is zero for all x<0, then (Ff)(\omega)=(Lf)(i \omega) (assuming I have done the algebra right and give or take a multiplicative constant due to how the FT is defined)

    CB
    Last edited by CaptainBlack; December 29th 2010 at 01:52 AM.
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    Your are right!
    But I need show diffrent
    In wiki I read this :
    The Laplace transform is related to the Fourier transform, but whereas the Fourier transform resolves a function or signal into its modes of vibration, the Laplace transform resolves a function into its moments. Like the Fourier transform, the Laplace transform is used for solving differential and integral equations
    Can anyone explain it more for me?
    Last edited by Hamed; December 29th 2010 at 03:57 AM.
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  7. #7
    Grand Panjandrum
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    Quote Originally Posted by Hamed View Post
    Your are right!
    But I need show diffrent
    In wiki I read this :

    Can anyone explain it more for me?
    Exactly what do you not understand in the phrase "the Fourier transform resolves a function or signal into its modes of vibration"?

    And the phrase "the Laplace transform resolves a function into its moments" (you can Google for the definition of what the moments of a function are if need be)

    Both are of use in differential equations because of the relationship between the FT and LT of a derivative and the function itself, and certain other properties which the two transforms have in common such as the convolution theorems, ...

    CB
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  8. #8
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    vibration and moments I can not find exact defenation!
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  9. #9
    Grand Panjandrum
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    Quote Originally Posted by Hamed View Post
    vibration and moments I can not find exact defenation!
    The Fourier transform gives a representation of a function in terms of a linear supposition of sinusoids, that it analyses the function/signal into vibrational components.

    The Laplace transform is related to the moment generating function, and allows the generation of the moments when they exist (in fact they both do).

    CB
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