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Math Help - Function - own LT

  1. #1
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    Laplace self transform

    Quote Originally Posted by CaptainBlack
    The caveat in the my last post was supposed to serve as a warning that
    I had my doubts about the last thing that I had written. If you were misled
    I must apologise, sorry.

    RonL
    Here's one that's more along the lines of your expertise.

    What functions of t, if any, f(t) have their Laplace transform F(s) equal to
    F(t) ?
    ?,
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  2. #2
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    Quote Originally Posted by bobbyk
    Here's one that's more along the lines of your expertise.

    What functions of t, if any, f(t) have their Laplace transform F(s) equal to
    F(t) ?
    ?,
    If the question was about the Fourier transform I could tell you the answer off the top of my head, but for the LT I will have to go away and think about it.

    RonL
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  3. #3
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    Laplace self transform

    I didn't ask my question properly!

    What I meant to ask was what function f(t), if any, has as its Laplace Transform f(s)?
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  4. #4
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    I have split this question/sub-thread out of the thread on RC-Networks
    as it is of some interest in itself (that is I wouldn't mind knowing the
    answer myself).

    RonL
    Last edited by CaptainBlack; July 6th 2006 at 07:56 AM.
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  5. #5
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    I am just curious that if there is an answer to this question how do you even begin to solve this problem!
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  6. #6
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    Quote Originally Posted by ThePerfectHacker
    I am just curious that if there is an answer to this question how do you even begin to solve this problem!
    Well there may not be a such a function, but some form of itteration
    might be a viable approach.

    There is (are?) solutions to the equivalent problem for the Fourier Transform.

    RonL
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  7. #7
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    Quote Originally Posted by CaptainBlack
    Well there may not be a such a function, but some form of itteration
    might be a viable approach.
    One thing that always fasincated me was how did Euler ever find the Gamma function? There is no way to do it with diffrencial equations or any other methods. He just was creative and was able to think of such a function.
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  8. #8
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    Laplace self transform

    Quote Originally Posted by ThePerfectHacker
    I am just curious that if there is an answer to this question how do you even begin to solve this problem!
    Well, you solve the integral equation. It may not be all that hard.
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  9. #9
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    Quote Originally Posted by bobbyk
    Well, you solve the integral equation. It may not be all that hard.
    f(s)=\int_0^{\infty}e^{-st}f(t)dt
    What type of equation is this?

    Maybe, you can manipulate it into a diffrencial equation somehow by taking the derivative of both sides. But the problem is that is a definite integral over there. I do not think this is any known type of equation.
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  10. #10
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    Quote Originally Posted by ThePerfectHacker
    f(s)=\int_0^{\infty}e^{-st}f(t)dt
    What type of equation is this?

    Maybe, you can manipulate it into a diffrencial equation somehow by taking the derivative of both sides. But the problem is that is a definite integral over there. I do not think this is any known type of equation.
    You should make your own theorum to solve it.
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  11. #11
    Super Member Rebesques's Avatar
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    What type of equation is this?
    It is a Fredholm integral. But this doesn't mean we can solve it
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  12. #12
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    Quote Originally Posted by Rebesques
    It is a Fredholm integral. But this doesn't mean we can solve it
    Other than for the trivial solution anyway.

    RonL
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  13. #13
    Super Member Rebesques's Avatar
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    For the Fourier transform, there are many though.

    I was reading (fooling around actually ) this book about the heat equation, and there was a formula about its Fourier transform. After some calculations, I ended up with

    \left(\mathcal{F}\rm{e}^{-\frac{x^2}{2}}\right)(\xi)=\rm{e}^{-\frac{\xi^2}{2}},

    and just when I was getting happy with this, there was a formula concerning the Hermite functions H_n(x), namely

    \left(\mathcal{F} H_n(x)\right)(\xi)={\rm i}^n H_n(\xi)

    which means all Hermite functions H_{4n}, \ n\in\mathbb{N} have this property also. As if an infinite number of solutions were not enough, linearity grants us that every linear combination of these functions has the same property; And continuity grants that, every function in the closure (in L^2 say) of these combinations, is again a fixed point...
    Last edited by Rebesques; August 28th 2007 at 11:20 PM.
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  14. #14
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    Laplace self transform

    Thanks for your interest.
    Your Fourier self-transforms are beautiful! And, of course, are on the internet. But I've searched for the Laplace self-transforms and haven't found
    any, although I'm sure there must be some (other than f(t)=0)

    bobbyk
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