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Fourier Transform of 1/t

  1. Feb 20, 2008 #1
    1. The problem statement, all variables and given/known data
    Find the Fourier Transform of [tex] \frac {1}{t} [/tex]



    2. Relevant equations
    Euler's equations I think...


    3. The attempt at a solution
    I tried splitting up the integral into two. One from [tex] -\inf [/tex] to 0 and the other from 0 to [tex] \inf [/tex]. Not much help there. I tried using [tex] e^{ix} = cos(x) + isin(x) [/tex]. Im pretty sure that is the way to go, but I cant seem to make it work. I think the answer is plus or minus i (from google searches), but I cant make the steps to get there. Could someone give me some tips, or out line the steps? Thank you
     
  2. jcsd
  3. Feb 20, 2008 #2

    quasar987

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    It would help to know that the integral from 0 to infinity of sinx/x is pi/2 !
     
  4. Feb 20, 2008 #3
    Thank you, that does help. My teach. said dont use a table though... But this is better than nothing.

    What is the integral from 0 to inf for cosx/x ?
     
  5. Feb 20, 2008 #4

    quasar987

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    cos(x) = sin(x+pi/2)
     
  6. Feb 20, 2008 #5

    Dick

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    The integral of cos(x)/x from 0 to infinity just plain does not exist. As far as I know you can't do things like the fourier transform of 1/t by changing them into real integrals. You have to express them as contour integrals in the complex plane and pick a convergent contour or pull a residue theorem argument. Or do you know some trick I don't??
     
  7. Feb 20, 2008 #6

    quasar987

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    No, I suppose you'Re right!
     
  8. Feb 20, 2008 #7
    bah, thats not what I want to hear!

    We did some complex integration with poles in a different class. I didnt get it at all. I dont think that is required for this class. Im gonna stick with the sinx/x = pi/2 unless somebody has a better idea.
     
  9. Feb 20, 2008 #8

    Dick

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    If you've looked up the results then you should know that the integral of (1/t)*exp(i*t*x) depends on a discrete function of the value of x. That's a pretty sure sign that a contour choice is involved. Neglect this at your own risk.
     
  10. Sep 27, 2009 #9
    Hello. I am new to fourier transforms. Also I have not studied contour integration. In entry 309 in the table on wikipedia the answer to the fourier transform of 1/t = − i*pi*sgn(w).

    The answer I get is i*pi*sgn(t). I'm not sure where the (-) comes from. I get, skipping a few steps: the integral with limits from -inf to inf of isin(wt)/t dt.

    From my notes the integral from -inf to inf of sin(wt)/t would be = pi*sgn(w). I would assume when an imaginary number is in there you just treat it as a constant?

    What am I missing here? Is my assumption wrong?

    Thanks.
     
  11. Nov 25, 2010 #10
    Hey,
    Using Euler's formula, I'v found the FT of 1/(Pi*t) as -j. integration of cos(x)/x from -inf to inf is zero, as odd function. And using integration of sin(x)/x from -inf to inf = Pi. Using these two we easily can get FT of 1/(Pi.t) is equal to -j.
    Using a known FT of rectangular(t/Tau) and X(0) or x(0) formulas of FT and IFT we can get the integration of sin(x)/x.
     
    Last edited: Nov 25, 2010
  12. Sep 8, 2011 #11
    fourier sine transform of 1/sqrt x

    can u plz help me out with fourier sine transform of 1/ sqrt x
     
  13. Sep 8, 2011 #12
    i need the solution asap...
     
  14. Sep 8, 2011 #13
    are u here?????quasar987
     
  15. Nov 14, 2012 #14
    ok, 1/t is like 1/w, if you times the numerator and denominator by j its like the Duality property (j* 1/(jt) ), so its like j2*pi*x(-w) = 2j*pi(-0.5+u(-w))
    as you can see from the 1/jw transformation on the table.
     
  16. Nov 14, 2012 #15

    Zondrina

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    This thread is like 4 years old. Why did you grave dig it?
     
  17. Nov 14, 2012 #16
    Because I wanted to know the answer and I didn't think it had been adoquately addressed, because It hadn't, so I solved it for the next person to find it on google.
     
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