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Math Help - Fourier Transform of x^2*sin(x)

  1. #16
    A Plied Mathematician
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    ...of course the OP may be an engineering student, in which case the lecturer may just wave their hands over such things and >POOFF< there it is.
    I happen to know that the OP'er is a student at my alma mater (Virginia Tech), and is in the math department. I'm guessing he's studying for his pde Ph.D. prelim, and this question came up. Pde's at Virginia Tech are done fairly rigorously, I believe. So he might have seen the proper integral. Would this be a Lebesgue integral that is needed?
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  2. #17
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    Quote Originally Posted by CaptainBlack View Post
    Post #3. You need a non-standard definition of an integral to do this as an intergral, and I'm pretty sure the OP has never seen such a definition (of course the OP may be an engineering student, in which case the lecturer may just wave their hands over such things and >POOFF< there it is).

    CB
    I wouldn't be so sure of that. I have taking Measure Theory, I have done plenty of PDEs with Distributions and Functionals. I have also taken complex analysis. I am a Math PhD student as it says in my Signature.

    The problem with my definition was that I forgot the 2\pi, but that was just a stupid mistake in typing and then not noticing until now. I like to make stupid mistakes like that.
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  3. #18
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    Quote Originally Posted by Ackbeet View Post
    I happen to know that the OP'er is a student at my alma mater (Virginia Tech), and is in the math department. I'm guessing he's studying for his pde Ph.D. prelim, and this question came up. Pde's at Virginia Tech are done fairly rigorously, I believe. So he might have seen the proper integral. Would this be a Lebesgue integral that is needed?
    No. We need the theory of distributions

    CB
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  4. #19
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    Quote Originally Posted by CaptainBlack View Post
    No. We need the theory of distributions

    CB
    Refer to my previous post.
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  5. #20
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    Quote Originally Posted by lvleph View Post
    I wouldn't be so sure of that. I have taking Measure Theory, I have done plenty of PDEs with Distributions and Functionals. I have also taken complex analysis. I am a Math PhD student as it says in my Signature.
    Yes, all very nice I'm sure, but now go back and read your own posts and tell us where the evidence is that you have the background you now claim. (not that I am disputing what you say, just pointing out that it does not show in this thread)

    CB
    Last edited by CaptainBlack; July 22nd 2010 at 03:10 PM.
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  6. #21
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    Well, why act as if I don't have the knowledge? Why not take me at my world and help me out? Reading the thread over it almost appears as if I cannot integrate by parts, but the only reason I asked about whether I was on the right track with that is because I still couldn't see how this would help. I figured I must have been making a mistake. It is still possible I am missing something, but that is why I am here asking. Either way I still don't see the reluctance to at least send me in the right direction. I am not asking for an answer here and it isn't even a homework problem. I am studying for my PDE qualifying exam and I came across something that I just couldn't solve.
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  7. #22
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    Quote Originally Posted by lvleph View Post
    Well, why act as if I don't have the knowledge? Why not take me at my world and help me out? Reading the thread over it almost appears as if I cannot integrate by parts, but the only reason I asked about whether I was on the right track with that is because I still couldn't see how this would help. I figured I must have been making a mistake. It is still possible I am missing something, but that is why I am here asking. Either way I still don't see the reluctance to at least send me in the right direction. I am not asking for an answer here and it isn't even a homework problem. I am studying for my PDE qualifying exam and I came across something that I just couldn't solve.
    You were sent in the right direction, first to a site with the FT of \sin(x) and the relationship between the FT of $$ x^2f(x) and that of $$ f(x), but then you told us what you said in post #3

    CB
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  8. #23
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    Because on my Qualifying Exam I will not have a table, so I still have to come up with the Fourier Transform of sin(x).

    Now back to the problem.
    \int_{-\infty}^{+\infty}\! \sin x \cdot e^{2\pi i x \lambda}\, dx = \int_{0}^{+\infty}\! \sin x \cdot e^{2\pi i x \lambda}\, dx - \int_{-\infty}^{0}\! \sin x \cdot e^{-2\pi i x \lambda}\, dx
    Which from previous work
    \int_{-\infty}^{+\infty}\! \sin x \cdot e^{2\pi i x \lambda}\, dx = \left.\frac{2\pi i \lambda \sin x - \cos x}{1 - 4\pi^2 \lambda^2}\right|^{+\infty}_0 + \left.\frac{2\pi i \lambda \sin x + \cos x}{1 - 4\pi^2 \lambda^2}\right|^{-\infty}_0
    So from here I am suppose to get some dirac deltas, but I am not seeing it.
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  9. #24
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    Well, it seems to me that if you could prove to yourself that multiplication by x corresponds to operation by i\frac{d}{d\omega}, and if you could prove the identity to yourself that

    F[e^{iax}]=\sqrt{2\pi}\delta(\omega-a),

    then you could finish by using the exponential definition of sin. Instead of referring to the table, be able to derive the table yourself!
    Last edited by Ackbeet; July 22nd 2010 at 04:38 PM. Reason: Pi.
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  10. #25
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    Ahhhh, I forgot about using the exponential definition! I don't know why I miss things like this some times. Thank you.
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  11. #26
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    You're very welcome. Have a good one!
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